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(problems.) Tj
100 Tz
1 0 0 1 148 523 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 169 523 Tm
98 Tz
(emphasis) Tj
100 Tz
1 0 0 1 213 523 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 223 523 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 238 523 Tm
98 Tz
(motivating) Tj
100 Tz
1 0 0 1 288 523 Tm
(and) Tj
1 0 0 1 308 523 Tm
94 Tz
(making) Tj
100 Tz
1 0 0 1 343 523 Tm
90 Tz
(clear) Tj
100 Tz
1 0 0 1 367 523 Tm
95 Tz
(what) Tj
100 Tz
1 0 0 1 391 523 Tm
98 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 440 523 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 477 523 Tm
80 Tz
(is,) Tj
100 Tz
1 0 0 1 491 523 Tm
95 Tz
(what) Tj
100 Tz
1 0 0 1 515 523 Tm
83 Tz
(it) Tj
100 Tz
1 0 0 1 98 510 Tm
(does,) Tj
1 0 0 1 126 510 Tm
95 Tz
(what) Tj
100 Tz
1 0 0 1 151 510 Tm
83 Tz
(it) Tj
100 Tz
1 0 0 1 162 510 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 173 510 Tm
96 Tz
(useful) Tj
100 Tz
1 0 0 1 204 510 Tm
(for) Tj
1 0 0 1 222 510 Tm
(and) Tj
1 0 0 1 242 510 Tm
95 Tz
(what) Tj
100 Tz
1 0 0 1 267 510 Tm
95 Tz
(must) Tj
100 Tz
1 0 0 1 293 510 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 308 510 Tm
95 Tz
(done) Tj
100 Tz
1 0 0 1 334 510 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 347 510 Tm
94 Tz
(produce) Tj
100 Tz
1 0 0 1 386 510 Tm
(a) Tj
1 0 0 1 395 510 Tm
92 Tz
(practical) Tj
100 Tz
1 0 0 1 435 510 Tm
95 Tz
(program) Tj
100 Tz
1 0 0 1 476 510 Tm
(for) Tj
1 0 0 1 493 510 Tm
107 Tz
(imple) Tj
100 Tz
(-) Tj
1 0 0 1 98 498 Tm
97 Tz
(menting) Tj
100 Tz
1 0 0 1 139 498 Tm
88 Tz
(it.) Tj
100 Tz
1 0 0 1 155 498 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 177 498 Tm
97 Tz
(generalization) Tj
100 Tz
1 0 0 1 242 498 Tm
(of) Tj
1 0 0 1 256 498 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 307 498 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 344 498 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 357 498 Tm
(other) Tj
1 0 0 1 385 498 Tm
98 Tz
(equations,) Tj
100 Tz
1 0 0 1 435 498 Tm
95 Tz
(including) Tj
100 Tz
1 0 0 1 480 498 Tm
98 Tz
(nonlinear) Tj
100 Tz
1 0 0 1 98 487 Tm
(and) Tj
1 0 0 1 118 487 Tm
97 Tz
(non-self-adjoint) Tj
100 Tz
1 0 0 1 190 487 Tm
98 Tz
(equations,) Tj
100 Tz
1 0 0 1 238 487 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 249 487 Tm
95 Tz
(discussed) Tj
100 Tz
1 0 0 1 293 487 Tm
(and) Tj
1 0 0 1 312 487 Tm
96 Tz
(motivated.) Tj
100 Tz
1 0 0 1 365 487 Tm
86 Tz
(A) Tj
100 Tz
1 0 0 1 376 487 Tm
97 Tz
(detailed) Tj
100 Tz
1 0 0 1 414 487 Tm
97 Tz
(mathematical) Tj
100 Tz
1 0 0 1 475 487 Tm
91 Tz
(analysis) Tj
100 Tz
1 0 0 1 512 487 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 98 475 Tm
98 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 149 475 Tm
(method,) Tj
1 0 0 1 190 475 Tm
96 Tz
(which) Tj
100 Tz
1 0 0 1 221 475 Tm
(this) Tj
1 0 0 1 241 475 Tm
92 Tz
(paper) Tj
100 Tz
1 0 0 1 269 475 Tm
(tries) Tj
1 0 0 1 293 475 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 306 475 Tm
96 Tz
(avoid) Tj
100 Tz
1 0 0 1 334 475 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 348 475 Tm
96 Tz
(much) Tj
100 Tz
1 0 0 1 377 475 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 390 475 Tm
95 Tz
(possible,) Tj
100 Tz
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71 Tz
(is) Tj
100 Tz
1 0 0 1 444 475 Tm
95 Tz
(available) Tj
100 Tz
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(in) Tj
100 Tz
1 0 0 1 501 475 Tm
82 Tz
(123\].) Tj
100 Tz
1 0 0 1 98 463 Tm
97 Tz
(Anyone) Tj
100 Tz
1 0 0 1 138 463 Tm
(interested) Tj
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88 Tz
(in) Tj
100 Tz
1 0 0 1 198 463 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 217 463 Tm
96 Tz
(theoretical) Tj
100 Tz
1 0 0 1 267 463 Tm
98 Tz
(underpinnings) Tj
100 Tz
1 0 0 1 334 463 Tm
(of) Tj
1 0 0 1 348 463 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 398 463 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 436 463 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 448 463 Tm
92 Tz
(urged) Tj
100 Tz
1 0 0 1 477 463 Tm
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(to) Tj
100 Tz
1 0 0 1 490 463 Tm
94 Tz
(browse) Tj
100 Tz
1 0 0 1 98 451 Tm
97 Tz
(through) Tj
100 Tz
1 0 0 1 136 451 Tm
(that) Tj
1 0 0 1 156 451 Tm
97 Tz
(excellent) Tj
100 Tz
1 0 0 1 198 451 Tm
95 Tz
(text.) Tj
100 Tz
1 0 0 1 122 436 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 143 436 Tm
(equation) Tj
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(studied) Tj
1 0 0 1 219 436 Tm
71 Tz
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100 Tz
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130 Tz
(\(a\(x\)y\202\(x\)\)\202+b\(x\)y\(x\)+c\(x\)0) Tj
100 Tz
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1 0 0 1 311 418 Tm
91 Tz
(on) Tj
100 Tz
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1 0 0 1 326 418 Tm
108 Tz
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100 Tz
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1 0 0 1 499 418 Tm
90 Tz
(\(1.1\)) Tj
100 Tz
1 0 0 1 98 400 Tm
94 Tz
(subject) Tj
100 Tz
1 0 0 1 132 400 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 144 400 Tm
95 Tz
(boundary) Tj
100 Tz
1 0 0 1 188 400 Tm
98 Tz
(conditions) Tj
100 Tz
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120 Tz
(aLy\(L\)) Tj
100 Tz
1 0 0 1 186 382 Tm
129 Tz
(+t3Ly\202\(L\)vL) Tj
100 Tz
/F9 11 Tf
1 0 0 1 499 364 Tm
90 Tz
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100 Tz
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114 Tz
(aRy\(R\)) Tj
100 Tz
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96 Tz
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5 Ts

(3) Tj
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100 Tz
1 0 0 1 221 347 Tm
113 Tz
(\(R\)) Tj
100 Tz
1 0 0 1 248 347 Tm
77 Tz
(YR) Tj
100 Tz
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1 0 0 1 98 329 Tm
96 Tz
(where) Tj
100 Tz
1 0 0 1 129 329 Tm
115 Tz
(a\(x\)) Tj
100 Tz
/F5 9 Tf
1 0 0 1 155 329 Tm
117 Tz
(\>) Tj
100 Tz
/F9 11 Tf
1 0 0 1 166 329 Tm
88 Tz
(0,) Tj
100 Tz
/F7 10 Tf
1 0 0 1 180 329 Tm
115 Tz
(b\(x\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 207 329 Tm
200 Tz
(=) Tj
100 Tz
1 0 0 1 218 329 Tm
(0,) Tj
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1 0 0 1 230 329 Tm
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126 Tz
(c\(x\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 273 329 Tm
92 Tz
(are) Tj
100 Tz
1 0 0 1 290 329 Tm
96 Tz
(given) Tj
100 Tz
1 0 0 1 318 329 Tm
98 Tz
(functions) Tj
100 Tz
1 0 0 1 361 329 Tm
91 Tz
(on) Tj
100 Tz
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1 0 0 1 377 329 Tm
118 Tz
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100 Tz
/F8 8 Tf
1 0 0 1 410 329 Tm
110 Tz
(aL) Tj
100 Tz
/F5 4 Tf
1 0 0 1 425 329 Tm
(~) Tj
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93 Tz
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5 Ts

(13) Tj
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(L) Tj
100 Tz
1 0 0 1 462 329 Tm
114 Tz
(\201PR) Tj
100 Tz
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1 0 0 1 480 329 Tm
(~VL~) Tj
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93 Tz
(and) Tj
100 Tz
/F8 8 Tf
1 0 0 1 99 315 Tm
91 Tz
(YR) Tj
100 Tz
/F9 11 Tf
1 0 0 1 114 315 Tm
92 Tz
(are) Tj
100 Tz
1 0 0 1 130 315 Tm
96 Tz
(given) Tj
100 Tz
1 0 0 1 157 315 Tm
98 Tz
(constants) Tj
100 Tz
1 0 0 1 201 315 Tm
94 Tz
(obeying) Tj
100 Tz
/F11 10 Tf
1 0 0 1 147 300 Tm
108 Tz
(aLPL) Tj
100 Tz
/F1 11 Tf
1 0 0 1 176 300 Tm
157 Tz
(=0) Tj
100 Tz
1 0 0 1 194 300 Tm
79 Tz
(and) Tj
100 Tz
/F12 9 Tf
1 0 0 1 214 300 Tm
129 Tz
(\(aL,) Tj
100 Tz
/F8 8 Tf
1 0 0 1 239 300 Tm
(13L\)) Tj
/F1 11 Tf
1 0 0 1 259 300 Tm
200 Tz
(!=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 271 300 Tm
(\(0,0\)) Tj
1 0 0 1 499 281 Tm
90 Tz
(\(1.3\)) Tj
100 Tz
/F7 10 Tf
1 0 0 1 147 265 Tm
96 Tz
(aRPR) Tj
100 Tz
/F9 11 Tf
1 0 0 1 177 265 Tm
121 Tz
(\>0) Tj
100 Tz
1 0 0 1 196 265 Tm
(and) Tj
/F8 8 Tf
1 0 0 1 215 265 Tm
129 Tz
(\(aR,) Tj
100 Tz
1 0 0 1 241 265 Tm
124 Tz
(PR\)) Tj
100 Tz
/F7 10 Tf
1 0 0 1 262 265 Tm
200 Tz
(!=) Tj
100 Tz
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1 0 0 1 274 265 Tm
104 Tz
(\(0,0\).) Tj
100 Tz
1 0 0 1 99 246 Tm
(Under) Tj
1 0 0 1 130 246 Tm
94 Tz
(reasonable) Tj
100 Tz
1 0 0 1 179 246 Tm
98 Tz
(assumptions) Tj
100 Tz
1 0 0 1 236 246 Tm
96 Tz
(about) Tj
100 Tz
1 0 0 1 264 246 Tm
(the) Tj
1 0 0 1 281 246 Tm
98 Tz
(smoothness) Tj
100 Tz
1 0 0 1 335 246 Tm
(of) Tj
1 0 0 1 348 246 Tm
(the) Tj
1 0 0 1 365 246 Tm
94 Tz
(coefficients) Tj
100 Tz
1 0 0 1 417 246 Tm
(a,) Tj
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80 Tz
(b) Tj
100 Tz
/F9 11 Tf
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(and) Tj
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1 0 0 1 458 246 Tm
83 Tz
(c,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 468 246 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 485 246 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 98 235 Tm
(of) Tj
1 0 0 1 113 235 Tm
90 Tz
(\(1.1\)) Tj
100 Tz
1 0 0 1 138 235 Tm
94 Tz
(subject) Tj
100 Tz
1 0 0 1 173 235 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 186 235 Tm
90 Tz
(\(1.2\)) Tj
100 Tz
1 0 0 1 212 235 Tm
(and) Tj
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(\(1.3\)) Tj
100 Tz
1 0 0 1 259 235 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 270 235 Tm
97 Tz
(known) Tj
100 Tz
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(exist) Tj
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90 Tz
(be) Tj
100 Tz
1 0 0 1 376 235 Tm
97 Tz
(unique.) Tj
100 Tz
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90 Tz
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100 Tz
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86 Tz
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100 Tz
1 0 0 1 465 235 Tm
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(violated,) Tj
100 Tz
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92 Tz
(the) Tj
100 Tz
1 0 0 1 99 223 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 137 223 Tm
94 Tz
(may) Tj
100 Tz
1 0 0 1 158 223 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 176 223 Tm
96 Tz
(exist,) Tj
100 Tz
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(even) Tj
1 0 0 1 246 223 Tm
(if) Tj
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83 Tz
(it) Tj
100 Tz
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(does,) Tj
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83 Tz
(it) Tj
100 Tz
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94 Tz
(may) Tj
100 Tz
1 0 0 1 322 223 Tm
93 Tz
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100 Tz
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100 Tz
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(unique.) Tj
100 Tz
1 0 0 1 123 208 Tm
98 Tz
(Problems) Tj
100 Tz
1 0 0 1 170 208 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 184 208 Tm
(this) Tj
1 0 0 1 205 208 Tm
94 Tz
(type) Tj
100 Tz
1 0 0 1 229 208 Tm
95 Tz
(arise) Tj
100 Tz
1 0 0 1 254 208 Tm
(in) Tj
1 0 0 1 269 208 Tm
96 Tz
(many) Tj
100 Tz
1 0 0 1 298 208 Tm
98 Tz
(situations) Tj
100 Tz
1 0 0 1 346 208 Tm
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1 0 0 1 395 208 Tm
(and) Tj
1 0 0 1 417 208 Tm
95 Tz
(their) Tj
100 Tz
1 0 0 1 442 208 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 483 208 Tm
(must) Tj
1 0 0 1 510 208 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 98 196 Tm
97 Tz
(accomplished) Tj
100 Tz
1 0 0 1 160 196 Tm
94 Tz
(cheaply,) Tj
100 Tz
1 0 0 1 199 196 Tm
93 Tz
(accurately) Tj
100 Tz
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1 0 0 1 265 196 Tm
92 Tz
(reliably.) Tj
100 Tz
1 0 0 1 306 196 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 327 196 Tm
98 Tz
(numerical) Tj
100 Tz
1 0 0 1 374 196 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 412 196 Tm
(of) Tj
1 0 0 1 426 196 Tm
90 Tz
(\(1.1\)) Tj
100 Tz
1 0 0 1 450 196 Tm
93 Tz
(has) Tj
100 Tz
1 0 0 1 468 196 Tm
96 Tz
(traditionally) Tj
100 Tz
1 0 0 1 99 184 Tm
(been) Tj
1 0 0 1 124 184 Tm
95 Tz
(accomplished) Tj
100 Tz
1 0 0 1 188 184 Tm
96 Tz
(using) Tj
100 Tz
1 0 0 1 215 184 Tm
92 Tz
(classical) Tj
100 Tz
1 0 0 1 254 184 Tm
91 Tz
(finite) Tj
100 Tz
1 0 0 1 281 184 Tm
96 Tz
(differences) Tj
100 Tz
1 0 0 1 335 184 Tm
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1 0 0 1 362 184 Tm
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1 0 0 1 374 184 Tm
(the) Tj
1 0 0 1 393 184 Tm
96 Tz
(mesh) Tj
100 Tz
1 0 0 1 421 184 Tm
94 Tz
(spacing) Tj
100 Tz
1 0 0 1 457 184 Tm
86 Tz
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100 Tz
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1 0 0 1 469 184 Tm
(h,) Tj
/F9 11 Tf
1 0 0 1 482 184 Tm
(then) Tj
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92 Tz
(the) Tj
100 Tz
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(error) Tj
1 0 0 1 125 173 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 138 173 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 156 173 Tm
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100 Tz
1 0 0 1 182 173 Tm
95 Tz
(difference) Tj
100 Tz
1 0 0 1 230 173 Tm
95 Tz
(approximation) Tj
100 Tz
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(solution) Tj
100 Tz
1 0 0 1 367 173 Tm
86 Tz
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100 Tz
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1 0 0 1 378 173 Tm
114 Tz
(0\(h) Tj
5 Ts

(2) Tj
5 Ts

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100 Tz
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1 0 0 1 415 173 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 437 173 Tm
93 Tz
(popularity) Tj
100 Tz
1 0 0 1 485 173 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 498 173 Tm
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(finite) Tj
100 Tz
1 0 0 1 98 161 Tm
95 Tz
(difference) Tj
100 Tz
1 0 0 1 146 161 Tm
97 Tz
(methods) Tj
100 Tz
1 0 0 1 186 161 Tm
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(resulted) Tj
100 Tz
1 0 0 1 224 161 Tm
93 Tz
(mainly) Tj
100 Tz
1 0 0 1 257 161 Tm
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(from) Tj
100 Tz
1 0 0 1 282 161 Tm
95 Tz
(their) Tj
100 Tz
1 0 0 1 306 161 Tm
91 Tz
(flexibility) Tj
100 Tz
1 0 0 1 350 161 Tm
(and) Tj
1 0 0 1 369 161 Tm
(ease) Tj
1 0 0 1 391 161 Tm
(of) Tj
1 0 0 1 404 161 Tm
99 Tz
(implementation,) Tj
100 Tz
1 0 0 1 478 161 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 490 161 Tm
89 Tz
(well) Tj
100 Tz
1 0 0 1 511 161 Tm
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(as) Tj
100 Tz
1 0 0 1 99 149 Tm
(the) Tj
1 0 0 1 116 149 Tm
94 Tz
(fact) Tj
100 Tz
1 0 0 1 135 149 Tm
(that) Tj
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(there) Tj
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88 Tz
(was) Tj
100 Tz
1 0 0 1 199 149 Tm
91 Tz
(no) Tj
100 Tz
1 0 0 1 214 149 Tm
97 Tz
(known) Tj
100 Tz
1 0 0 1 246 149 Tm
94 Tz
(real) Tj
100 Tz
1 0 0 1 265 149 Tm
96 Tz
(alternative) Tj
100 Tz
1 0 0 1 314 149 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 350 149 Tm
(for) Tj
1 0 0 1 367 149 Tm
92 Tz
(practical) Tj
100 Tz
1 0 0 1 405 149 Tm
95 Tz
(problems.) Tj
100 Tz
1 0 0 1 124 134 Tm
95 Tz
(However,) Tj
100 Tz
1 0 0 1 171 134 Tm
(there) Tj
1 0 0 1 198 134 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 209 134 Tm
94 Tz
(now) Tj
100 Tz
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(an) Tj
100 Tz
1 0 0 1 247 134 Tm
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(alternative) Tj
100 Tz
1 0 0 1 298 134 Tm
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(method) Tj
100 Tz
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(to) Tj
100 Tz
1 0 0 1 350 134 Tm
91 Tz
(finite) Tj
100 Tz
1 0 0 1 376 134 Tm
96 Tz
(differences) Tj
100 Tz
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1 0 0 1 429 134 Tm
150 Tz
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100 Tz
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1 0 0 1 437 134 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 487 134 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 99 122 Tm
96 Tz
(using) Tj
100 Tz
1 0 0 1 126 122 Tm
93 Tz
(B-splines.) Tj
100 Tz
1 0 0 1 175 122 Tm
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86 Tz
(is) Tj
100 Tz
1 0 0 1 197 122 Tm
95 Tz
(easy) Tj
100 Tz
1 0 0 1 220 122 Tm
(to) Tj
1 0 0 1 232 122 Tm
98 Tz
(implement) Tj
100 Tz
/F9 10 Tf
1 0 0 1 284 122 Tm
(\() Tj
/F9 11 Tf
1 0 0 1 291 122 Tm
97 Tz
(although) Tj
100 Tz
1 0 0 1 333 122 Tm
(not) Tj
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89 Tz
(as) Tj
100 Tz
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100 Tz
1 0 0 1 168 337 Tm
(to) Tj
1 0 0 1 180 337 Tm
(the) Tj
1 0 0 1 197 337 Tm
103 Tz
(solution) Tj
100 Tz
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111 Tz
(of) Tj
100 Tz
1 0 0 1 249 337 Tm
105 Tz
(\(1.1\).) Tj
100 Tz
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106 Tz
(This) Tj
100 Tz
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(space) Tj
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107 Tz
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100 Tz
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105 Tz
(have) Tj
100 Tz
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103 Tz
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100 Tz
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(nice) Tj
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107 Tz
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100 Tz
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111 Tz
(includ) Tj
100 Tz
(-) Tj
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(ing) Tj
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104 Tz
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100 Tz
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106 Tz
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100 Tz
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(\(2\)) Tj
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103 Tz
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100 Tz
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(of) Tj
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105 Tz
(approximating) Tj
100 Tz
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(the) Tj
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103 Tz
(solution) Tj
100 Tz
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102 Tz
(accurately.) Tj
100 Tz
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1 0 0 1 122 311 Tm
91 Tz
(Such) Tj
100 Tz
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(a) Tj
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97 Tz
(representation) Tj
100 Tz
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92 Tz
(exists) Tj
100 Tz
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150 Tz
(-) Tj
100 Tz
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105 Tz
(expansion) Tj
100 Tz
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(in) Tj
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103 Tz
(B-splines) Tj
100 Tz
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(of) Tj
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105 Tz
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(\[6,7,9\].) Tj
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(This) Tj
100 Tz
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75 Tz
(a) Tj
100 Tz
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103 Tz
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108 Tz
(for) Tj
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108 Tz
(representing) Tj
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105 Tz
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100 Tz
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80 Tz
(by) Tj
100 Tz
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102 Tz
(piecewise) Tj
100 Tz
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106 Tz
(polynomials,) Tj
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(that) Tj
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89 Tz
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104 Tz
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111 Tz
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106 Tz
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(each) Tj
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(a) Tj
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105 Tz
(mesh) Tj
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(or) Tj
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(grid.) Tj
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105 Tz
(Here) Tj
100 Tz
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(the) Tj
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107 Tz
(integer) Tj
100 Tz
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1 0 0 1 371 287 Tm
(k) Tj
/F9 10 Tf
1 0 0 1 380 287 Tm
83 Tz
(is) Tj
100 Tz
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(any) Tj
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109 Tz
(number) Tj
100 Tz
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1 0 0 1 449 287 Tm
(k) Tj
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1 0 0 1 460 287 Tm
129 Tz
(\>) Tj
100 Tz
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80 Tz
(2) Tj
100 Tz
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(the) Tj
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106 Tz
(user) Tj
100 Tz
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103 Tz
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100 Tz
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106 Tz
(The) Tj
100 Tz
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(piecewise) Tj
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104 Tz
(polynomial) Tj
100 Tz
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108 Tz
(representation) Tj
100 Tz
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83 Tz
(is) Tj
100 Tz
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106 Tz
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100 Tz
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1 0 0 1 388 275 Tm
(satisfy) Tj
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104 Tz
(certain) Tj
100 Tz
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105 Tz
(continuity) Tj
100 Tz
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121 Tz
(res) Tj
100 Tz
(-) Tj
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106 Tz
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107 Tz
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104 Tz
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(each) Tj
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105 Tz
(mesh) Tj
100 Tz
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106 Tz
(sub-interval.) Tj
100 Tz
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(Specifically,) Tj
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(let) Tj
1 0 0 1 431 264 Tm
(ir) Tj
/F5 5 Tf
1 0 0 1 440 264 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 450 264 Tm
83 Tz
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(1) Tj
100 Tz
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1 0 0 1 474 264 Tm
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1 0 0 1 480 264 Tm
(.) Tj
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1 0 0 1 486 264 Tm
(.) Tj
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1 0 0 1 500 264 Tm
67 Tz
(x~,) Tj
100 Tz
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1 0 0 1 97 252 Tm
104 Tz
(where) Tj
100 Tz
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(L) Tj
1 0 0 1 137 252 Tm
(=x) Tj
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(1) Tj
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(2) Tj
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1 0 0 1 214 252 Tm
(.) Tj
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1 0 0 1 224 252 Tm
200 Tz
(=) Tj
100 Tz
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1 0 0 1 235 252 Tm
119 Tz
(xp.~R,) Tj
100 Tz
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1 0 0 1 271 252 Tm
(be) Tj
1 0 0 1 284 252 Tm
75 Tz
(a) Tj
100 Tz
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1 0 0 1 292 252 Tm
94 Tz
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100 Tz
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(on) Tj
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(the) Tj
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103 Tz
(interval) Tj
100 Tz
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108 Tz
(\(L,R\).) Tj
100 Tz
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(Let) Tj
/F11 10 Tf
1 0 0 1 434 252 Tm
89 Tz
(m,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 448 252 Tm
(be) Tj
1 0 0 1 461 252 Tm
(the) Tj
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(multipli) Tj
100 Tz
(-) Tj
/F9 10 Tf
1 0 0 1 97 240 Tm
106 Tz
(city) Tj
100 Tz
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(of) Tj
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90 Tz
(x,,) Tj
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109 Tz
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105 Tz
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86 Tz
(x,) Tj
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103 Tz
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(in) Tj
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(the) Tj
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(~r.) Tj
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106 Tz
(The) Tj
100 Tz
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96 Tz
(order) Tj
100 Tz
/F11 10 Tf
1 0 0 1 514 240 Tm
(k) Tj
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106 Tz
(defined) Tj
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(on) Tj
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(the) Tj
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83 Tz
(ir) Tj
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(is) Tj
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106 Tz
(defined) Tj
100 Tz
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(to) Tj
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(be) Tj
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(the) Tj
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102 Tz
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111 Tz
(of) Tj
100 Tz
1 0 0 1 345 228 Tm
80 Tz
(all) Tj
100 Tz
1 0 0 1 359 228 Tm
105 Tz
(functions) Tj
100 Tz
/F5 11 Tf
1 0 0 1 403 228 Tm
83 Tz
(/,) Tj
100 Tz
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1 0 0 1 98 213 Tm
(\(2.1\)) Tj
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(which) Tj
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(polynomials) Tj
100 Tz
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111 Tz
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100 Tz
1 0 0 1 244 213 Tm
104 Tz
(degree) Tj
100 Tz
1 0 0 1 277 213 Tm
117 Tz
(\<) Tj
100 Tz
/F11 10 Tf
1 0 0 1 289 213 Tm
(k) Tj
/F9 10 Tf
1 0 0 1 297 213 Tm
(on) Tj
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(each) Tj
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103 Tz
(interval) Tj
100 Tz
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109 Tz
(\(x,) Tj
100 Tz
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110 Tz
(x~) Tj
-5 Ts

(1) Tj
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(\)) Tj
100 Tz
1 0 0 1 418 213 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 434 213 Tm
107 Tz
(/=1) Tj
100 Tz
1 0 0 1 480 213 Tm
(N\205i,) Tj
/F5 7 Tf
1 0 0 1 185 202 Tm
92 Tz
(k\205I\205rn) Tj
100 Tz
1 0 0 1 252 202 Tm
50 Tz
(A) Tj
100 Tz
/F9 7 Tf
1 0 0 1 257 202 Tm
82 Tz
(\205I\205rn) Tj
100 Tz
/F1 11 Tf
1 0 0 1 98 194 Tm
58 Tz
(\(2.2\)) Tj
100 Tz
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63 Tz
(for) Tj
100 Tz
1 0 0 1 148 194 Tm
76 Tz
(which) Tj
100 Tz
/F11 10 Tf
1 0 0 1 179 194 Tm
(d) Tj
1 0 0 1 209 194 Tm
109 Tz
(\2011\(x,\)) Tj
100 Tz
1 0 0 1 235 194 Tm
120 Tz
(/dx) Tj
100 Tz
/F9 10 Tf
1 0 0 1 282 194 Tm
104 Tz
(exists) Tj
100 Tz
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107 Tz
(and) Tj
100 Tz
1 0 0 1 331 194 Tm
(is) Tj
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109 Tz
(continuous) Tj
100 Tz
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(at) Tj
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(each) Tj
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90 Tz
(x,,) Tj
100 Tz
1 0 0 1 444 194 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 462 194 Tm
72 Tz
(,==1) Tj
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96 Tz
(when) Tj
100 Tz
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94 Tz
(viewed) Tj
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89 Tz
(as) Tj
100 Tz
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75 Tz
(a) Tj
100 Tz
1 0 0 1 211 185 Tm
97 Tz
(function) Tj
100 Tz
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97 Tz
(defined) Tj
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95 Tz
(only) Tj
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91 Tz
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100 Tz
/F11 10 Tf
1 0 0 1 323 185 Tm
(\[L.RJ,) Tj
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93 Tz
(and) Tj
100 Tz
/F9 10 Tf
1 0 0 1 98 169 Tm
(\(2.3\)) Tj
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108 Tz
(for) Tj
100 Tz
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(which) Tj
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1 0 0 1 176 169 Tm
150 Tz
(I) Tj
100 Tz
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1 0 0 1 205 169 Tm
67 Tz
(0) Tj
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65 Tz
(outside) Tj
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1 0 0 1 249 169 Tm
(\[L,R\].) Tj
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94 Tz
(The) Tj
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1 0 0 1 143 154 Tm
94 Tz
(multiplicity) Tj
100 Tz
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1 0 0 1 197 154 Tm
89 Tz
(m,) Tj
100 Tz
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1 0 0 1 211 154 Tm
69 Tz
(of) Tj
100 Tz
1 0 0 1 224 154 Tm
67 Tz
(a) Tj
100 Tz
1 0 0 1 233 154 Tm
64 Tz
(point) Tj
100 Tz
/F9 11 Tf
1 0 0 1 259 154 Tm
75 Tz
(x,) Tj
100 Tz
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71 Tz
(is) Tj
100 Tz
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98 Tz
(restricted) Tj
100 Tz
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88 Tz
(to) Tj
100 Tz
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90 Tz
(be) Tj
100 Tz
1 0 0 1 354 154 Tm
88 Tz
(in) Tj
100 Tz
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92 Tz
(the) Tj
100 Tz
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92 Tz
(range) Tj
100 Tz
1 0 0 1 413 154 Tm
40 Tz
(1) Tj
100 Tz
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1 0 0 1 423 154 Tm
200 Tz
(=) Tj
100 Tz
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1 0 0 1 449 154 Tm
(~) Tj
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1 0 0 1 461 154 Tm
(k.) Tj
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1 0 0 1 476 154 Tm
93 Tz
(For) Tj
100 Tz
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1 0 0 1 495 154 Tm
110 Tz
(m=1) Tj
100 Tz
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92 Tz
(we) Tj
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95 Tz
(have) Tj
100 Tz
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1 0 0 1 137 142 Tm
125 Tz
(dk) Tj
5 Ts

(2) Tj
5 Ts

(f/dx&) Tj
5 Ts

(2) Tj
100 Tz
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1 0 0 1 197 142 Tm
5 Ts
98 Tz
(continuous) Tj
100 Tz
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(at) Tj
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82 Tz
(x,.) Tj
100 Tz
1 0 0 1 277 142 Tm
95 Tz
(This) Tj
100 Tz
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71 Tz
(is) Tj
100 Tz
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92 Tz
(the) Tj
100 Tz
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95 Tz
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100 Tz
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98 Tz
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92 Tz
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87 Tz
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90 Tz
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95 Tz
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(at) Tj
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75 Tz
(x,) Tj
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97 Tz
(without) Tj
100 Tz
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94 Tz
(making) Tj
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/F11 12 Tf
1 0 0 1 171 131 Tm
150 Tz
(f) Tj
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/F9 10 Tf
1 0 0 1 180 131 Tm
(a) Tj
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1 0 0 1 189 131 Tm
92 Tz
(polynomial) Tj
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(of) Tj
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93 Tz
(degree) Tj
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1 0 0 1 289 131 Tm
106 Tz
(k\205I) Tj
100 Tz
/F9 10 Tf
1 0 0 1 312 131 Tm
(on) Tj
1 0 0 1 329 131 Tm
135 Tz
(\(x) Tj
-5 Ts

(1) Tj
-5 Ts

(,) Tj
100 Tz
1 0 0 1 356 131 Tm
96 Tz
(x,+) Tj
-5 Ts

(1) Tj
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(\).) Tj
100 Tz
1 0 0 1 388 131 Tm
107 Tz
(For) Tj
100 Tz
/F11 10 Tf
1 0 0 1 408 131 Tm
89 Tz
(m,) Tj
100 Tz
/F5 5 Tf
1 0 0 1 420 131 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 429 131 Tm
120 Tz
(k) Tj
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1 0 0 1 438 131 Tm
(the) Tj
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107 Tz
(condition) Tj
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106 Tz
(that) Tj
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1 0 0 1 97 119 Tm
143 Tz
(d\202f/dx\202) Tj
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1 0 0 1 150 119 Tm
69 Tz
(be) Tj
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1 0 0 1 164 119 Tm
71 Tz
(continuous) Tj
100 Tz
1 0 0 1 217 119 Tm
46 Tz
(is) Tj
100 Tz
/F9 11 Tf
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98 Tz
(interpreted) Tj
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88 Tz
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(that) Tj
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(I) Tj
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86 Tz
(is) Tj
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98 Tz
(continuous) Tj
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92 Tz
(the) Tj
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(right) Tj
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94 Tz
(\(but) Tj
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93 Tz
(not) Tj
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(necessarily) Tj
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92 Tz
(the) Tj
100 Tz
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94 Tz
(left\)) Tj
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(at) Tj
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(x) Tj
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1 0 0 1 236 107 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 246 107 Tm
(x,,) Tj
/F1 11 Tf
1 0 0 1 261 107 Tm
63 Tz
(for) Tj
100 Tz
1 0 0 1 278 107 Tm
46 Tz
(x,) Tj
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1 0 0 1 290 107 Tm
(\<) Tj
1 0 0 1 303 107 Tm
(R,) Tj
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1 0 0 1 316 107 Tm
93 Tz
(and) Tj
100 Tz
/F1 11 Tf
1 0 0 1 336 107 Tm
71 Tz
(continuous) Tj
100 Tz
1 0 0 1 388 107 Tm
77 Tz
(from) Tj
100 Tz
1 0 0 1 413 107 Tm
68 Tz
(the) Tj
100 Tz
1 0 0 1 431 107 Tm
50 Tz
(left) Tj
100 Tz
1 0 0 1 450 107 Tm
46 Tz
(if) Tj
100 Tz
1 0 0 1 460 107 Tm
46 Tz
(x,) Tj
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100 Tz
1 0 0 1 438 448 Tm
107 Tz
(due) Tj
100 Tz
1 0 0 1 459 448 Tm
(to) Tj
1 0 0 1 471 448 Tm
(the) Tj
1 0 0 1 489 448 Tm
108 Tz
(rather) Tj
100 Tz
1 0 0 1 92 436 Tm
107 Tz
(natural) Tj
100 Tz
1 0 0 1 127 436 Tm
94 Tz
(way) Tj
100 Tz
1 0 0 1 148 436 Tm
(in) Tj
1 0 0 1 161 436 Tm
(which) Tj
1 0 0 1 190 436 Tm
105 Tz
(such) Tj
100 Tz
1 0 0 1 214 436 Tm
107 Tz
(splines) Tj
100 Tz
1 0 0 1 248 436 Tm
105 Tz
(arise) Tj
100 Tz
1 0 0 1 273 436 Tm
(and) Tj
1 0 0 1 293 436 Tm
105 Tz
(their) Tj
100 Tz
1 0 0 1 317 436 Tm
107 Tz
(smoothing) Tj
100 Tz
1 0 0 1 367 436 Tm
107 Tz
(properties) Tj
100 Tz
1 0 0 1 414 436 Tm
105 Tz
(when) Tj
100 Tz
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(used) Tj
100 Tz
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(to) Tj
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114 Tz
(approxi) Tj
100 Tz
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1 0 0 1 92 424 Tm
105 Tz
(mate) Tj
100 Tz
1 0 0 1 117 424 Tm
107 Tz
(functions) Tj
100 Tz
1 0 0 1 162 424 Tm
105 Tz
(described) Tj
100 Tz
1 0 0 1 207 424 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 221 424 Tm
103 Tz
(discrete) Tj
100 Tz
1 0 0 1 259 424 Tm
106 Tz
(data) Tj
100 Tz
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102 Tz
(Typically,) Tj
100 Tz
1 0 0 1 349 424 Tm
(the) Tj
1 0 0 1 367 424 Tm
105 Tz
(more) Tj
100 Tz
1 0 0 1 393 424 Tm
103 Tz
(accuracy) Tj
100 Tz
1 0 0 1 434 424 Tm
106 Tz
(desired,) Tj
100 Tz
1 0 0 1 473 424 Tm
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(larger) Tj
1 0 0 1 92 412 Tm
(the) Tj
1 0 0 1 110 412 Tm
105 Tz
(value) Tj
100 Tz
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107 Tz
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100 Tz
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(be.) Tj
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107 Tz
(For) Tj
100 Tz
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108 Tz
(example,) Tj
100 Tz
1 0 0 1 272 412 Tm
(if) Tj
1 0 0 1 282 412 Tm
117 Tz
(k=8) Tj
100 Tz
1 0 0 1 304 412 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 324 412 Tm
(the) Tj
1 0 0 1 341 412 Tm
105 Tz
(mesh) Tj
100 Tz
1 0 0 1 367 412 Tm
104 Tz
(length) Tj
100 Tz
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1 0 0 1 398 412 Tm
80 Tz
(h) Tj
100 Tz
/F9 10 Tf
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83 Tz
(is) Tj
100 Tz
1 0 0 1 417 412 Tm
103 Tz
(halved,) Tj
100 Tz
1 0 0 1 452 412 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 475 412 Tm
108 Tz
(Theorem) Tj
100 Tz
1 0 0 1 93 400 Tm
92 Tz
(2.1) Tj
100 Tz
1 0 0 1 110 400 Tm
105 Tz
(indicates) Tj
100 Tz
1 0 0 1 151 400 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 172 400 Tm
(the) Tj
1 0 0 1 190 400 Tm
110 Tz
(error) Tj
100 Tz
1 0 0 1 215 400 Tm
107 Tz
(should) Tj
100 Tz
1 0 0 1 247 400 Tm
106 Tz
(decrease) Tj
100 Tz
1 0 0 1 288 400 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 302 400 Tm
(a) Tj
1 0 0 1 311 400 Tm
104 Tz
(factor) Tj
100 Tz
1 0 0 1 339 400 Tm
(of) Tj
1 0 0 1 352 400 Tm
70 Tz
(28) Tj
100 Tz
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133 Tz
(\205) Tj
100 Tz
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1 0 0 1 375 400 Tm
89 Tz
(256.) Tj
100 Tz
1 0 0 1 400 400 Tm
95 Tz
(However,) Tj
100 Tz
1 0 0 1 446 400 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 459 400 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 474 400 Tm
90 Tz
(shall) Tj
100 Tz
1 0 0 1 498 400 Tm
(see,) Tj
/F9 10 Tf
1 0 0 1 92 388 Tm
(the) Tj
1 0 0 1 110 388 Tm
(work) Tj
1 0 0 1 135 388 Tm
107 Tz
(needed) Tj
100 Tz
1 0 0 1 170 388 Tm
(to) Tj
1 0 0 1 182 388 Tm
(solve) Tj
1 0 0 1 208 388 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 216 388 Tm
106 Tz
(problem) Tj
100 Tz
1 0 0 1 256 388 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 282 388 Tm
103 Tz
(B-splines) Tj
100 Tz
1 0 0 1 325 388 Tm
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/F7 10 Tf
1 0 0 1 336 388 Tm
109 Tz
(0\(Nk) Tj
5 Ts

(3) Tj
5 Ts

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100 Tz
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1 0 0 1 380 388 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 409 388 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 418 388 Tm
117 Tz
(k=8) Tj
100 Tz
1 0 0 1 440 388 Tm
106 Tz
(solution) Tj
100 Tz
1 0 0 1 479 388 Tm
93 Tz
(will) Tj
100 Tz
1 0 0 1 497 388 Tm
(cost) Tj
1 0 0 1 92 377 Tm
80 Tz
(8) Tj
100 Tz
1 0 0 1 103 377 Tm
105 Tz
(times) Tj
100 Tz
1 0 0 1 131 377 Tm
(as) Tj
1 0 0 1 145 377 Tm
104 Tz
(much) Tj
100 Tz
1 0 0 1 174 377 Tm
(as) Tj
1 0 0 1 187 377 Tm
(a) Tj
/F7 10 Tf
1 0 0 1 197 377 Tm
111 Tz
(k=4) Tj
100 Tz
/F9 10 Tf
1 0 0 1 221 377 Tm
106 Tz
(solution) Tj
100 Tz
1 0 0 1 261 377 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 278 377 Tm
(the) Tj
/F10 10 Tf
1 0 0 1 297 377 Tm
105 Tz
(same) Tj
100 Tz
/F9 10 Tf
1 0 0 1 325 377 Tm
104 Tz
(mesh.) Tj
100 Tz
1 0 0 1 359 377 Tm
103 Tz
(Hence,) Tj
100 Tz
1 0 0 1 394 377 Tm
(the) Tj
1 0 0 1 413 377 Tm
103 Tz
(optimal) Tj
100 Tz
/F7 10 Tf
1 0 0 1 451 377 Tm
80 Tz
(k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 460 377 Tm
104 Tz
(results) Tj
100 Tz
1 0 0 1 493 377 Tm
105 Tz
(from) Tj
100 Tz
1 0 0 1 92 365 Tm
104 Tz
(minimizing) Tj
100 Tz
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1 0 0 1 146 365 Tm
107 Tz
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5 Ts

(3) Tj
5 Ts

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100 Tz
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1 0 0 1 192 365 Tm
96 Tz
(where) Tj
100 Tz
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1 0 0 1 223 365 Tm
69 Tz
(N~) Tj
100 Tz
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1 0 0 1 239 365 Tm
(is) Tj
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(the) Tj
1 0 0 1 267 365 Tm
109 Tz
(number) Tj
100 Tz
1 0 0 1 305 365 Tm
(of) Tj
1 0 0 1 318 365 Tm
105 Tz
(mesh) Tj
100 Tz
1 0 0 1 345 365 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 375 365 Tm
107 Tz
(needed) Tj
100 Tz
1 0 0 1 410 365 Tm
(to) Tj
1 0 0 1 422 365 Tm
(solve) Tj
1 0 0 1 449 365 Tm
(the) Tj
1 0 0 1 466 365 Tm
106 Tz
(problem) Tj
100 Tz
1 0 0 1 506 365 Tm
(to) Tj
1 0 0 1 92 353 Tm
(the) Tj
1 0 0 1 109 353 Tm
107 Tz
(desired) Tj
100 Tz
1 0 0 1 144 353 Tm
103 Tz
(accuracy) Tj
100 Tz
1 0 0 1 184 353 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 210 353 Tm
(a) Tj
/F7 10 Tf
1 0 0 1 218 353 Tm
80 Tz
(k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 226 353 Tm
105 Tz
(order) Tj
100 Tz
1 0 0 1 253 353 Tm
103 Tz
(B-spline.) Tj
100 Tz
1 0 0 1 297 353 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 320 353 Tm
106 Tz
(optimization) Tj
100 Tz
1 0 0 1 377 353 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 387 353 Tm
(highly) Tj
1 0 0 1 418 353 Tm
103 Tz
(problem) Tj
100 Tz
1 0 0 1 457 353 Tm
107 Tz
(dependent.) Tj
100 Tz
1 0 0 1 117 338 Tm
86 Tz
(A) Tj
100 Tz
1 0 0 1 128 338 Tm
104 Tz
(computationally) Tj
100 Tz
1 0 0 1 201 338 Tm
109 Tz
(convenient) Tj
100 Tz
1 0 0 1 254 338 Tm
(basis) Tj
1 0 0 1 278 338 Tm
108 Tz
(exists) Tj
100 Tz
1 0 0 1 307 338 Tm
(for) Tj
1 0 0 1 324 338 Tm
(the) Tj
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83 Tz
(BRA.) Tj
100 Tz
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106 Tz
(The) Tj
100 Tz
1 0 0 1 423 338 Tm
107 Tz
(dimension) Tj
100 Tz
1 0 0 1 472 338 Tm
111 Tz
(of) Tj
100 Tz
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1 0 0 1 485 338 Tm
85 Tz
(B~) Tj
100 Tz
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1 0 0 1 508 338 Tm
83 Tz
(is) Tj
100 Tz
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1 0 0 1 93 326 Tm
91 Tz
(N\205k) Tj
100 Tz
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107 Tz
(and) Tj
100 Tz
1 0 0 1 137 326 Tm
(the) Tj
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(basis) Tj
1 0 0 1 178 326 Tm
106 Tz
(consists) Tj
100 Tz
1 0 0 1 215 326 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 228 326 Tm
108 Tz
(elements) Tj
100 Tz
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(B,\(x\),) Tj
/F9 10 Tf
1 0 0 1 307 326 Tm
107 Tz
(i=1) Tj
100 Tz
/F11 10 Tf
1 0 0 1 353 326 Tm
(N\205k.) Tj
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86 Tz
(A) Tj
100 Tz
1 0 0 1 394 326 Tm
108 Tz
(complete) Tj
100 Tz
1 0 0 1 437 326 Tm
104 Tz
(description) Tj
100 Tz
1 0 0 1 488 326 Tm
(of) Tj
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1 0 0 1 93 314 Tm
70 Tz
(B,) Tj
100 Tz
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(is) Tj
1 0 0 1 116 314 Tm
104 Tz
(given) Tj
100 Tz
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(in) Tj
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107 Tz
(and) Tj
100 Tz
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(Briefly,) Tj
1 0 0 1 250 314 Tm
105 Tz
(when) Tj
100 Tz
1 0 0 1 278 314 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 296 314 Tm
105 Tz
(multiplicities) Tj
100 Tz
1 0 0 1 355 314 Tm
111 Tz
(of) Tj
100 Tz
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(the) Tj
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(first) Tj
1 0 0 1 407 314 Tm
(and) Tj
1 0 0 1 426 314 Tm
107 Tz
(last) Tj
100 Tz
1 0 0 1 445 314 Tm
105 Tz
(mesh) Tj
100 Tz
1 0 0 1 472 314 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 501 314 Tm
(are) Tj
1 0 0 1 93 302 Tm
106 Tz
(both) Tj
100 Tz
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1 0 0 1 116 302 Tm
114 Tz
(k,) Tj
100 Tz
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1 0 0 1 127 302 Tm
(so) Tj
1 0 0 1 140 302 Tm
106 Tz
(that) Tj
100 Tz
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1 0 0 1 190 285 Tm
122 Tz
(=xA) Tj
100 Tz
/F9 11 Tf
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(and) Tj
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1 0 0 1 175 249 Tm
167 Tz
(=) Tj
100 Tz
1 0 0 1 210 249 Tm
167 Tz
(=) Tj
100 Tz
/F8 8 Tf
1 0 0 1 221 249 Tm
(xN) Tj
/F9 10 Tf
1 0 0 1 93 231 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 116 231 Tm
(the) Tj
1 0 0 1 133 231 Tm
(main) Tj
1 0 0 1 158 231 Tm
107 Tz
(properties) Tj
100 Tz
1 0 0 1 204 231 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 216 231 Tm
(the) Tj
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(B,\(x\)) Tj
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1 0 0 1 261 231 Tm
90 Tz
(follow:) Tj
100 Tz
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1 0 0 1 94 204 Tm
(\(2.4\)) Tj
1 0 0 1 127 204 Tm
105 Tz
(Each) Tj
100 Tz
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1 0 0 1 153 204 Tm
70 Tz
(B,) Tj
100 Tz
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1 0 0 1 167 204 Tm
(is) Tj
1 0 0 1 178 204 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 222 204 Tm
(only) Tj
1 0 0 1 244 204 Tm
(on) Tj
1 0 0 1 261 204 Tm
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/F5 8 Tf
1 0 0 1 280 204 Tm
110 Tz
(x,+A\]) Tj
100 Tz
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1 0 0 1 306 204 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 326 204 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 337 204 Tm
102 Tz
(identically) Tj
100 Tz
1 0 0 1 386 204 Tm
(zero) Tj
1 0 0 1 408 204 Tm
107 Tz
(elsewhere,) Tj
100 Tz
1 0 0 1 459 204 Tm
(as) Tj
1 0 0 1 472 204 Tm
94 Tz
(well) Tj
100 Tz
1 0 0 1 493 204 Tm
(as) Tj
1 0 0 1 507 204 Tm
(at) Tj
1 0 0 1 166 193 Tm
120 Tz
(x,) Tj
-5 Ts

(1) Tj
100 Tz
1 0 0 1 186 193 Tm
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107 Tz
(and) Tj
100 Tz
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1 0 0 1 205 193 Tm
104 Tz
(x,+A+I) Tj
100 Tz
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1 0 0 1 260 193 Tm
(xN,) Tj
/F9 10 Tf
1 0 0 1 278 193 Tm
110 Tz
(even) Tj
100 Tz
1 0 0 1 302 193 Tm
(if) Tj
1 0 0 1 312 193 Tm
106 Tz
(they) Tj
100 Tz
1 0 0 1 333 193 Tm
(are) Tj
1 0 0 1 350 193 Tm
(in) Tj
1 0 0 1 363 193 Tm
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/F9 11 Tf
1 0 0 1 381 193 Tm
75 Tz
(x,+) Tj
-5 Ts

(1) Tj
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(1.) Tj
100 Tz
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1 0 0 1 94 178 Tm
95 Tz
(\(2.5\)) Tj
100 Tz
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1 0 0 1 127 178 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 148 178 Tm
106 Tz
(sum) Tj
100 Tz
/F8 10 Tf
1 0 0 1 170 178 Tm
106 Tz
(B) Tj
-5 Ts

(1) Tj
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(\(x\)+) Tj
100 Tz
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1 0 0 1 219 178 Tm
(.) Tj
/F8 8 Tf
1 0 0 1 225 178 Tm
121 Tz
(+B~v~\(x\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 273 178 Tm
(is) Tj
1 0 0 1 284 178 Tm
(identically) Tj
1 0 0 1 331 178 Tm
106 Tz
(one.) Tj
100 Tz
1 0 0 1 94 162 Tm
(\(2.6\)) Tj
1 0 0 1 128 162 Tm
105 Tz
(Each) Tj
100 Tz
/F8 10 Tf
1 0 0 1 152 162 Tm
70 Tz
(B,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 165 162 Tm
92 Tz
(obeys) Tj
100 Tz
1 0 0 1 192 162 Tm
80 Tz
(0) Tj
100 Tz
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1 0 0 1 203 162 Tm
200 Tz
(=) Tj
100 Tz
/F8 10 Tf
1 0 0 1 214 162 Tm
(B,\(x\)) Tj
/F9 11 Tf
1 0 0 1 243 162 Tm
200 Tz
(=) Tj
100 Tz
1 0 0 1 255 162 Tm
40 Tz
(1) Tj
100 Tz
1 0 0 1 263 162 Tm
96 Tz
(everywhere) Tj
100 Tz
1 0 0 1 316 162 Tm
(and) Tj
1 0 0 1 335 162 Tm
95 Tz
(possesses) Tj
100 Tz
1 0 0 1 379 162 Tm
95 Tz
(only) Tj
100 Tz
1 0 0 1 401 162 Tm
(one) Tj
1 0 0 1 420 162 Tm
96 Tz
(maximum.) Tj
100 Tz
1 0 0 1 93 147 Tm
96 Tz
(These) Tj
100 Tz
1 0 0 1 122 147 Tm
92 Tz
(are) Tj
100 Tz
1 0 0 1 139 147 Tm
89 Tz
(very) Tj
100 Tz
1 0 0 1 161 147 Tm
89 Tz
(nice) Tj
100 Tz
1 0 0 1 182 147 Tm
98 Tz
(properties) Tj
100 Tz
1 0 0 1 228 147 Tm
(and) Tj
1 0 0 1 247 147 Tm
82 Tz
(will) Tj
100 Tz
1 0 0 1 265 147 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 279 147 Tm
95 Tz
(used) Tj
100 Tz
1 0 0 1 302 147 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 314 147 Tm
95 Tz
(later) Tj
100 Tz
1 0 0 1 336 147 Tm
95 Tz
(sections.) Tj
100 Tz
1 0 0 1 117 132 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 138 132 Tm
96 Tz
(convergence) Tj
100 Tz
1 0 0 1 196 132 Tm
96 Tz
(result) Tj
100 Tz
1 0 0 1 223 132 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 235 132 Tm
(Theorem) Tj
1 0 0 1 279 132 Tm
85 Tz
(2.1) Tj
100 Tz
1 0 0 1 295 132 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 306 132 Tm
98 Tz
(independent) Tj
100 Tz
1 0 0 1 362 132 Tm
(of) Tj
1 0 0 1 375 132 Tm
(the) Tj
1 0 0 1 392 132 Tm
95 Tz
(multiplicities) Tj
100 Tz
/F8 10 Tf
1 0 0 1 452 132 Tm
73 Tz
(m,) Tj
100 Tz
/F1 11 Tf
1 0 0 1 465 132 Tm
62 Tz
(of) Tj
100 Tz
1 0 0 1 477 132 Tm
68 Tz
(the) Tj
100 Tz
1 0 0 1 495 132 Tm
69 Tz
(inte) Tj
100 Tz
(-) Tj
1 0 0 1 93 121 Tm
79 Tz
(nor) Tj
100 Tz
1 0 0 1 113 121 Tm
64 Tz
(points) Tj
100 Tz
1 0 0 1 143 121 Tm
46 Tz
(x,) Tj
100 Tz
1 0 0 1 156 121 Tm
50 Tz
(\() Tj
100 Tz
/F11 10 Tf
1 0 0 1 164 121 Tm
120 Tz
(k) Tj
100 Tz
1 0 0 1 175 121 Tm
(\<) Tj
1 0 0 1 187 121 Tm
50 Tz
(i) Tj
100 Tz
/F1 11 Tf
1 0 0 1 195 121 Tm
114 Tz
(\() Tj
100 Tz
/F11 10 Tf
1 0 0 1 208 121 Tm
(N\205k) Tj
/F1 11 Tf
1 0 0 1 232 121 Tm
33 Tz
(\)) Tj
100 Tz
1 0 0 1 241 121 Tm
69 Tz
(of) Tj
100 Tz
1 0 0 1 254 121 Tm
63 Tz
(the) Tj
100 Tz
1 0 0 1 271 121 Tm
73 Tz
(mesh.) Tj
100 Tz
1 0 0 1 304 121 Tm
65 Tz
(Usually,) Tj
100 Tz
1 0 0 1 344 121 Tm
63 Tz
(for) Tj
100 Tz
1 0 0 1 360 121 Tm
79 Tz
(smooth) Tj
100 Tz
1 0 0 1 397 121 Tm
68 Tz
(functions) Tj
100 Tz
/F7 12 Tf
1 0 0 1 441 121 Tm
114 Tz
(f,) Tj
100 Tz
/F8 10 Tf
1 0 0 1 454 121 Tm
87 Tz
(m,=1) Tj
100 Tz
/F9 10 Tf
1 0 0 1 480 121 Tm
(is) Tj
1 0 0 1 491 121 Tm
105 Tz
(taken) Tj
100 Tz
1 0 0 1 93 108 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 109 108 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 123 108 Tm
105 Tz
(these) Tj
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(the) Tj
1 0 0 1 490 691 Tm
106 Tz
(only) Tj
100 Tz
/F11 10 Tf
1 0 0 1 514 691 Tm
88 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 99 679 Tm
104 Tz
(which) Tj
100 Tz
1 0 0 1 130 679 Tm
(are) Tj
1 0 0 1 149 679 Tm
108 Tz
(not) Tj
100 Tz
1 0 0 1 169 679 Tm
(zero) Tj
1 0 0 1 192 679 Tm
(at) Tj
/F11 10 Tf
1 0 0 1 206 679 Tm
(L) Tj
/F9 10 Tf
1 0 0 1 217 679 Tm
107 Tz
(and) Tj
100 Tz
/F11 10 Tf
1 0 0 1 239 679 Tm
(R) Tj
/F9 10 Tf
1 0 0 1 250 679 Tm
(are) Tj
/F11 10 Tf
1 0 0 1 270 679 Tm
73 Tz
(B) Tj
-5 Ts

(1) Tj
100 Tz
/F9 10 Tf
1 0 0 1 285 679 Tm
-5 Ts
107 Tz
(and) Tj
100 Tz
/F7 8 Tf
1 0 0 1 307 679 Tm
125 Tz
(BVA,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 337 679 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 359 679 Tm
105 Tz
(these) Tj
100 Tz
1 0 0 1 386 679 Tm
104 Tz
(values) Tj
100 Tz
1 0 0 1 419 679 Tm
(are) Tj
1 0 0 1 438 679 Tm
104 Tz
(respectively) Tj
100 Tz
1 0 0 1 494 679 Tm
(simply) Tj
/F11 10 Tf
1 0 0 1 100 667 Tm
108 Tz
(B) Tj
-5 Ts

(1) Tj
-5 Ts

(\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 131 667 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 142 667 Tm
40 Tz
(1) Tj
100 Tz
1 0 0 1 150 667 Tm
107 Tz
(and) Tj
100 Tz
/F11 10 Tf
1 0 0 1 170 667 Tm
93 Tz
(BV~~A\(R\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 214 667 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 225 667 Tm
71 Tz
(1.) Tj
100 Tz
1 0 0 1 124 652 Tm
114 Tz
(If) Tj
100 Tz
1 0 0 1 135 652 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 153 652 Tm
106 Tz
(function) Tj
100 Tz
1 0 0 1 193 652 Tm
(\[has) Tj
1 0 0 1 218 652 Tm
(a) Tj
1 0 0 1 226 652 Tm
105 Tz
(discontinuity) Tj
100 Tz
1 0 0 1 286 652 Tm
(in) Tj
1 0 0 1 298 652 Tm
92 Tz
(say) Tj
100 Tz
1 0 0 1 315 652 Tm
(its) Tj
1 0 0 1 327 652 Tm
(~) Tj
1 0 0 1 341 652 Tm
107 Tz
(derivative,) Tj
100 Tz
1 0 0 1 391 652 Tm
(at) Tj
1 0 0 1 402 652 Tm
(x,,) Tj
1 0 0 1 417 652 Tm
106 Tz
(then) Tj
100 Tz
/F7 10 Tf
1 0 0 1 440 652 Tm
73 Tz
(m,) Tj
100 Tz
/F5 4 Tf
1 0 0 1 452 652 Tm
167 Tz
(=) Tj
100 Tz
/F7 10 Tf
1 0 0 1 461 652 Tm
(k\205i) Tj
/F9 13 Tf
1 0 0 1 482 652 Tm
63 Tz
(is) Tj
100 Tz
/F9 10 Tf
1 0 0 1 492 652 Tm
107 Tz
(chosen) Tj
100 Tz
/F9 11 Tf
1 0 0 1 100 640 Tm
97 Tz
(because) Tj
100 Tz
1 0 0 1 138 640 Tm
(this) Tj
1 0 0 1 158 640 Tm
89 Tz
(allows) Tj
100 Tz
1 0 0 1 189 640 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 207 640 Tm
97 Tz
(elements) Tj
100 Tz
1 0 0 1 250 640 Tm
(of) Tj
/F7 9 Tf
1 0 0 1 264 640 Tm
89 Tz
(Bff,A) Tj
100 Tz
/F9 10 Tf
1 0 0 1 288 640 Tm
(to) Tj
1 0 0 1 301 640 Tm
105 Tz
(have) Tj
100 Tz
1 0 0 1 325 640 Tm
(the) Tj
1 0 0 1 343 640 Tm
105 Tz
(same) Tj
100 Tz
1 0 0 1 369 640 Tm
105 Tz
(behavior.) Tj
100 Tz
1 0 0 1 418 640 Tm
114 Tz
(If) Tj
100 Tz
1 0 0 1 429 640 Tm
(a) Tj
1 0 0 1 437 640 Tm
103 Tz
(smaller) Tj
100 Tz
1 0 0 1 473 640 Tm
104 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 99 628 Tm
105 Tz
(were) Tj
100 Tz
1 0 0 1 124 628 Tm
106 Tz
(chosen,) Tj
100 Tz
1 0 0 1 162 628 Tm
(the) Tj
/F7 6 Tf
1 0 0 1 179 628 Tm
(~h) Tj
/F9 10 Tf
1 0 0 1 194 628 Tm
107 Tz
(derivative) Tj
100 Tz
1 0 0 1 241 628 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 254 628 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 268 628 Tm
(the) Tj
1 0 0 1 286 628 Tm
108 Tz
(elements) Tj
100 Tz
1 0 0 1 329 628 Tm
(of) Tj
/F8 8 Tf
1 0 0 1 343 628 Tm
106 Tz
(B~A) Tj
100 Tz
/F9 10 Tf
1 0 0 1 366 628 Tm
(would) Tj
1 0 0 1 396 628 Tm
(be) Tj
1 0 0 1 410 628 Tm
109 Tz
(continuous) Tj
100 Tz
1 0 0 1 462 628 Tm
(at) Tj
1 0 0 1 474 628 Tm
90 Tz
(x,,) Tj
100 Tz
1 0 0 1 489 628 Tm
(and) Tj
1 0 0 1 508 628 Tm
(the) Tj
1 0 0 1 100 616 Tm
(best) Tj
1 0 0 1 120 616 Tm
(fit) Tj
1 0 0 1 133 616 Tm
(to) Tj
1 0 0 1 145 616 Tm
(\[from) Tj
/F11 10 Tf
1 0 0 1 178 616 Tm
(Bff) Tj
/F7 7 Tf
1 0 0 1 196 616 Tm
50 Tz
(A) Tj
100 Tz
/F9 11 Tf
1 0 0 1 203 616 Tm
93 Tz
(would) Tj
100 Tz
1 0 0 1 233 616 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 251 616 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 264 616 Tm
89 Tz
(very) Tj
100 Tz
1 0 0 1 286 616 Tm
91 Tz
(good) Tj
100 Tz
1 0 0 1 310 616 Tm
(at) Tj
1 0 0 1 322 616 Tm
82 Tz
(x,.) Tj
100 Tz
1 0 0 1 124 601 Tm
97 Tz
(Another) Tj
100 Tz
1 0 0 1 165 601 Tm
98 Tz
(important) Tj
100 Tz
1 0 0 1 210 601 Tm
95 Tz
(property) Tj
100 Tz
1 0 0 1 250 601 Tm
(of) Tj
1 0 0 1 262 601 Tm
93 Tz
(B-splines) Tj
100 Tz
1 0 0 1 305 601 Tm
71 Tz
(is) Tj
100 Tz
1 0 0 1 315 601 Tm
95 Tz
(their) Tj
100 Tz
1 0 0 1 339 601 Tm
98 Tz
(numerical) Tj
100 Tz
1 0 0 1 385 601 Tm
94 Tz
(stability) Tj
100 Tz
1 0 0 1 422 601 Tm
89 Tz
(or) Tj
100 Tz
/F11 10 Tf
1 0 0 1 435 601 Tm
97 Tz
(condiuon.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 480 601 Tm
(Since) Tj
1 0 0 1 507 601 Tm
(any) Tj
/F5 7 Tf
1 0 0 1 228 595 Tm
(V) Tj
/F5 4 Tf
1 0 0 1 234 595 Tm
(\205) Tj
/F7 7 Tf
1 0 0 1 239 595 Tm
(k) Tj
/F9 11 Tf
1 0 0 1 100 584 Tm
92 Tz
(B-spline) Tj
100 Tz
/F11 12 Tf
1 0 0 1 138 584 Tm
150 Tz
(I) Tj
100 Tz
/F9 10 Tf
1 0 0 1 147 584 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 156 584 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 169 584 Tm
(the) Tj
1 0 0 1 186 584 Tm
105 Tz
(form) Tj
100 Tz
/F1 12 Tf
1 0 0 1 210 584 Tm
86 Tz
(f) Tj
100 Tz
/F5 5 Tf
1 0 0 1 219 584 Tm
167 Tz
(=) Tj
100 Tz
/F10 10 Tf
1 0 0 1 246 584 Tm
86 Tz
(a,) Tj
100 Tz
/F11 10 Tf
1 0 0 1 256 584 Tm
88 Tz
(B,) Tj
100 Tz
/F1 11 Tf
1 0 0 1 268 584 Tm
79 Tz
(and) Tj
100 Tz
1 0 0 1 287 584 Tm
69 Tz
(each) Tj
100 Tz
/F11 10 Tf
1 0 0 1 310 584 Tm
88 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 322 584 Tm
104 Tz
(obeys) Tj
100 Tz
1 0 0 1 350 584 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 360 584 Tm
200 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 372 584 Tm
88 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 387 584 Tm
200 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 395 584 Tm
40 Tz
(1) Tj
100 Tz
1 0 0 1 403 584 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 418 584 Tm
108 Tz
(see) Tj
100 Tz
1 0 0 1 435 584 Tm
(that) Tj
1 0 0 1 455 584 Tm
(if) Tj
/F5 15 Tf
1 0 0 1 466 584 Tm
0 Tz
(I) Tj
100 Tz
/F5 15 Tf
1 0 0 1 469 584 Tm
109 Tz
(Ill) Tj
100 Tz
/F5 15 Tf
1 0 0 1 483 584 Tm
25 Tz
(I) Tj
100 Tz
/F9 13 Tf
1 0 0 1 489 584 Tm
63 Tz
(is) Tj
100 Tz
/F9 10 Tf
1 0 0 1 499 584 Tm
105 Tz
(small) Tj
100 Tz
/F9 11 Tf
1 0 0 1 99 565 Tm
95 Tz
(compared) Tj
100 Tz
1 0 0 1 150 565 Tm
95 Tz
(with) Tj
100 Tz
/F5 15 Tf
1 0 0 1 179 565 Tm
25 Tz
(I) Tj
100 Tz
/F9 13 Tf
1 0 0 1 183 565 Tm
70 Tz
(Ia) Tj
100 Tz
/F5 15 Tf
1 0 0 1 193 565 Tm
47 Tz
(II~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 212 565 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 240 565 Tm
(many) Tj
1 0 0 1 272 565 Tm
102 Tz
(significant) Tj
100 Tz
1 0 0 1 324 565 Tm
(digits) Tj
1 0 0 1 355 565 Tm
(are) Tj
1 0 0 1 377 565 Tm
(lost) Tj
1 0 0 1 401 565 Tm
105 Tz
(when) Tj
100 Tz
1 0 0 1 433 565 Tm
107 Tz
(computing) Tj
100 Tz
/F11 12 Tf
1 0 0 1 488 565 Tm
140 Tz
(f) Tj
100 Tz
/F9 10 Tf
1 0 0 1 501 565 Tm
105 Tz
(from) Tj
100 Tz
/F10 10 Tf
1 0 0 1 100 553 Tm
80 Tz
(a) Tj
-5 Ts

(1) Tj
100 Tz
1 0 0 1 147 553 Tm
-5 Ts
68 Tz
(a\%~/~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 172 553 Tm
(in) Tj
1 0 0 1 184 553 Tm
104 Tz
(floating-point) Tj
100 Tz
1 0 0 1 245 553 Tm
107 Tz
(arithmetic) Tj
100 Tz
1 0 0 1 293 553 Tm
(\[25\].) Tj
1 0 0 1 319 553 Tm
98 Tz
(Specifically,) Tj
100 Tz
/F5 7 Tf
1 0 0 1 238 541 Tm
87 Tz
(V\205k) Tj
100 Tz
/F11 10 Tf
1 0 0 1 149 529 Tm
(d) Tj
/F9 10 Tf
1 0 0 1 159 529 Tm
200 Tz
(=) Tj
100 Tz
/F11 12 Tf
1 0 0 1 172 529 Tm
83 Tz
(Log) Tj
-5 Ts

(1) Tj
-5 Ts

(o\(IIa) Tj
100 Tz
/F9 13 Tf
1 0 0 1 216 529 Tm
(Il/Il) Tj
/F5 17 Tf
1 0 0 1 240 529 Tm
82 Tz
(X) Tj
100 Tz
/F1 12 Tf
1 0 0 1 255 529 Tm
58 Tz
(a,B,II\)) Tj
100 Tz
/F9 9 Tf
1 0 0 1 501 529 Tm
111 Tz
(\(2.7\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 99 504 Tm
103 Tz
(decimal) Tj
100 Tz
1 0 0 1 136 504 Tm
(digits) Tj
1 0 0 1 163 504 Tm
(are) Tj
1 0 0 1 180 504 Tm
106 Tz
(lost,) Tj
100 Tz
1 0 0 1 201 504 Tm
107 Tz
(due) Tj
100 Tz
1 0 0 1 221 504 Tm
(to) Tj
1 0 0 1 232 504 Tm
106 Tz
(cancellation,) Tj
100 Tz
1 0 0 1 290 504 Tm
(in) Tj
1 0 0 1 302 504 Tm
105 Tz
(evaluating) Tj
100 Tz
/F11 12 Tf
1 0 0 1 349 504 Tm
150 Tz
(f) Tj
100 Tz
/F9 11 Tf
1 0 0 1 363 504 Tm
89 Tz
(In) Tj
100 Tz
/F9 10 Tf
1 0 0 1 376 504 Tm
(\[61) Tj
/F9 10 Tf
1 0 0 1 391 504 Tm
(de) Tj
1 0 0 1 405 504 Tm
95 Tz
(Boor) Tj
100 Tz
1 0 0 1 429 504 Tm
(shows) Tj
1 0 0 1 458 504 Tm
106 Tz
(that) Tj
100 Tz
/F8 8 Tf
1 0 0 1 158 492 Tm
68 Tz
(V\205A) Tj
100 Tz
/F5 15 Tf
1 0 0 1 153 480 Tm
107 Tz
(IX) Tj
100 Tz
/F1 12 Tf
1 0 0 1 175 480 Tm
58 Tz
(a,B,II) Tj
100 Tz
/F7 10 Tf
1 0 0 1 206 480 Tm
117 Tz
(\>) Tj
100 Tz
/F5 14 Tf
1 0 0 1 218 480 Tm
78 Tz
(CkIIaII) Tj
100 Tz
/F9 10 Tf
1 0 0 1 500 480 Tm
(\(2.8\)) Tj
/F11 5 Tf
1 0 0 1 158 472 Tm
122 Tz
(1=I) Tj
100 Tz
/F1 11 Tf
1 0 0 1 99 456 Tm
76 Tz
(where) Tj
100 Tz
/F11 10 Tf
1 0 0 1 130 456 Tm
82 Tz
(Ck) Tj
100 Tz
/F9 10 Tf
1 0 0 1 144 456 Tm
(is) Tj
1 0 0 1 154 456 Tm
(a) Tj
1 0 0 1 162 456 Tm
106 Tz
(constant) Tj
100 Tz
1 0 0 1 202 456 Tm
107 Tz
(depending) Tj
100 Tz
/F11 12 Tf
1 0 0 1 250 456 Tm
80 Tz
(only) Tj
100 Tz
/F9 10 Tf
1 0 0 1 270 456 Tm
105 Tz
(upon) Tj
100 Tz
/F11 10 Tf
1 0 0 1 295 456 Tm
(k,) Tj
/F1 11 Tf
1 0 0 1 307 456 Tm
74 Tz
(and) Tj
100 Tz
1 0 0 1 325 456 Tm
66 Tz
(therefore) Tj
100 Tz
/F7 10 Tf
1 0 0 1 149 438 Tm
(d) Tj
/F7 11 Tf
1 0 0 1 159 438 Tm
200 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 169 438 Tm
104 Tz
(Log) Tj
-5 Ts

(10) Tj
-5 Ts

(\(CJ\202\).) Tj
100 Tz
/F1 11 Tf
1 0 0 1 99 421 Tm
62 Tz
(In) Tj
100 Tz
1 0 0 1 113 421 Tm
58 Tz
(particular,) Tj
100 Tz
1 0 0 1 162 421 Tm
69 Tz
(he) Tj
100 Tz
1 0 0 1 176 421 Tm
76 Tz
(shows) Tj
100 Tz
1 0 0 1 207 421 Tm
63 Tz
(for) Tj
100 Tz
1 0 0 1 224 421 Tm
67 Tz
(a) Tj
100 Tz
1 0 0 1 233 421 Tm
74 Tz
(uniform) Tj
100 Tz
1 0 0 1 273 421 Tm
76 Tz
(mesh,) Tj
100 Tz
1 0 0 1 304 421 Tm
79 Tz
(one) Tj
100 Tz
1 0 0 1 324 421 Tm
76 Tz
(where) Tj
100 Tz
1 0 0 1 354 421 Tm
47 Tz
(all) Tj
100 Tz
1 0 0 1 369 421 Tm
63 Tz
(the) Tj
100 Tz
1 0 0 1 387 421 Tm
85 Tz
(mesh) Tj
100 Tz
1 0 0 1 415 421 Tm
61 Tz
(intervals) Tj
100 Tz
1 0 0 1 456 421 Tm
73 Tz
(have) Tj
100 Tz
1 0 0 1 481 421 Tm
63 Tz
(the) Tj
100 Tz
1 0 0 1 499 421 Tm
77 Tz
(same) Tj
100 Tz
/F9 10 Tf
1 0 0 1 99 409 Tm
107 Tz
(length,) Tj
100 Tz
1 0 0 1 133 409 Tm
106 Tz
(that) Tj
100 Tz
/F7 8 Tf
1 0 0 1 149 391 Tm
73 Tz
(CA) Tj
100 Tz
/F7 6 Tf
1 0 0 1 177 391 Tm
-5 Ts
116 Tz
(10) Tj
-5 Ts

(\205A/5) Tj
100 Tz
/F9 10 Tf
1 0 0 1 500 391 Tm
(\(2.8\)) Tj
/F1 11 Tf
1 0 0 1 99 373 Tm
73 Tz
(Thus,) Tj
100 Tz
1 0 0 1 129 373 Tm
85 Tz
(when) Tj
100 Tz
1 0 0 1 157 373 Tm
67 Tz
(evaluating) Tj
100 Tz
1 0 0 1 206 373 Tm
67 Tz
(a) Tj
100 Tz
1 0 0 1 215 373 Tm
65 Tz
(B-spline) Tj
100 Tz
1 0 0 1 255 373 Tm
67 Tz
(defined) Tj
100 Tz
1 0 0 1 291 373 Tm
77 Tz
(on) Tj
100 Tz
/F9 10 Tf
1 0 0 1 307 373 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 316 373 Tm
108 Tz
(uniform,) Tj
100 Tz
1 0 0 1 359 373 Tm
(or) Tj
1 0 0 1 373 373 Tm
104 Tz
(nearly) Tj
100 Tz
1 0 0 1 404 373 Tm
108 Tz
(uniform,) Tj
100 Tz
1 0 0 1 447 373 Tm
108 Tz
(mesh,) Tj
100 Tz
1 0 0 1 478 373 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 494 373 Tm
(would) Tj
/F1 11 Tf
1 0 0 1 99 361 Tm
69 Tz
(expect) Tj
100 Tz
1 0 0 1 131 361 Tm
54 Tz
(to) Tj
100 Tz
1 0 0 1 144 361 Tm
62 Tz
(lose) Tj
100 Tz
1 0 0 1 164 361 Tm
77 Tz
(no) Tj
100 Tz
1 0 0 1 179 361 Tm
81 Tz
(more) Tj
100 Tz
1 0 0 1 205 361 Tm
69 Tz
(than) Tj
100 Tz
/F9 11 Tf
1 0 0 1 228 361 Tm
96 Tz
(about) Tj
100 Tz
/F11 10 Tf
1 0 0 1 256 361 Tm
114 Tz
(k/S) Tj
100 Tz
/F9 10 Tf
1 0 0 1 276 361 Tm
103 Tz
(decimal) Tj
100 Tz
1 0 0 1 313 361 Tm
(digits.) Tj
1 0 0 1 345 361 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 368 361 Tm
(is) Tj
1 0 0 1 378 361 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 387 361 Tm
106 Tz
(very) Tj
100 Tz
1 0 0 1 409 361 Tm
102 Tz
(satisfactory) Tj
100 Tz
1 0 0 1 461 361 Tm
104 Tz
(result) Tj
100 Tz
1 0 0 1 489 361 Tm
(since) Tj
/F9 9 Tf
1 0 0 1 515 361 Tm
(it) Tj
/F9 10 Tf
1 0 0 1 99 349 Tm
105 Tz
(indicates) Tj
100 Tz
1 0 0 1 141 349 Tm
106 Tz
(that,) Tj
100 Tz
1 0 0 1 164 349 Tm
(at) Tj
1 0 0 1 175 349 Tm
(least) Tj
1 0 0 1 198 349 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 214 349 Tm
106 Tz
(uniform) Tj
100 Tz
1 0 0 1 253 349 Tm
106 Tz
(meshes,) Tj
100 Tz
1 0 0 1 291 349 Tm
(the) Tj
1 0 0 1 308 349 Tm
106 Tz
(conditioning) Tj
100 Tz
1 0 0 1 365 349 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 378 349 Tm
(the) Tj
1 0 0 1 395 349 Tm
(B-spline) Tj
1 0 0 1 433 349 Tm
(basis) Tj
1 0 0 1 457 349 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 467 349 Tm
108 Tz
(independent) Tj
100 Tz
1 0 0 1 99 337 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 112 337 Tm
(the) Tj
1 0 0 1 128 337 Tm
(size) Tj
1 0 0 1 148 337 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 161 337 Tm
(the) Tj
1 0 0 1 178 337 Tm
104 Tz
(mesh.) Tj
100 Tz
1 0 0 1 99 314 Tm
(3.) Tj
/F10 10 Tf
1 0 0 1 114 314 Tm
105 Tz
(Rayleigh-Ritz) Tj
100 Tz
1 0 0 1 180 314 Tm
(Method) Tj
/F1 11 Tf
1 0 0 1 124 298 Tm
84 Tz
(The) Tj
100 Tz
1 0 0 1 146 298 Tm
68 Tz
(Rayleigh-Ritz) Tj
100 Tz
1 0 0 1 209 298 Tm
82 Tz
(method) Tj
100 Tz
1 0 0 1 247 298 Tm
63 Tz
(for) Tj
100 Tz
1 0 0 1 264 298 Tm
65 Tz
(solving) Tj
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(original) Tj
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109 Tz
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100 Tz
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104 Tz
(whose) Tj
100 Tz
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105 Tz
(generalizations) Tj
100 Tz
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103 Tz
(nobody) Tj
100 Tz
1 0 0 1 461 601 Tm
(knows) Tj
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(how) Tj
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1 0 0 1 100 589 Tm
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(directly) Tj
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1 0 0 1 262 589 Tm
(a) Tj
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(problem) Tj
100 Tz
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100 Tz
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100 Tz
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100 Tz
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(well) Tj
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106 Tz
(chosen,) Tj
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(function) Tj
100 Tz
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75 Tz
(v) Tj
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(which) Tj
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104 Tz
(results) Tj
100 Tz
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(from) Tj
100 Tz
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(minimizing) Tj
100 Tz
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(B-splines) Tj
100 Tz
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100 Tz
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(they) Tj
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(approximate) Tj
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(smooth) Tj
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(function) Tj
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(w) Tj
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(to) Tj
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(within) Tj
100 Tz
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(0\(h) Tj
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(that) Tj
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(question) Tj
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(method) Tj
100 Tz
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(produce) Tj
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(a) Tj
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(be) Tj
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(achieved.) Tj
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(answer) Tj
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(course,) Tj
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(yes.) Tj
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(proof) Tj
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(of) Tj
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(such) Tj
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(results) Tj
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(general) Tj
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1 0 0 1 449 484 Tm
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(com) Tj
100 Tz
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(plicated) Tj
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(matter.) Tj
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(However,) Tj
100 Tz
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(a) Tj
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(weak) Tj
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(form) Tj
100 Tz
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(of) Tj
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(the) Tj
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(convergence) Tj
100 Tz
1 0 0 1 353 472 Tm
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(result) Tj
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1 0 0 1 380 472 Tm
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(be) Tj
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(easily) Tj
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(obtained) Tj
100 Tz
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(for) Tj
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(the) Tj
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(simple) Tj
100 Tz
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(example) Tj
100 Tz
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(under) Tj
100 Tz
1 0 0 1 187 460 Tm
107 Tz
(consideration) Tj
100 Tz
1 0 0 1 249 460 Tm
108 Tz
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100 Tz
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100 Tz
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(any) Tj
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(mesh) Tj
100 Tz
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(on) Tj
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(with) Tj
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(multiplicity) Tj
100 Tz
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(of) Tj
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(xv) Tj
100 Tz
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(both) Tj
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86 Tz
(k.) Tj
100 Tz
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96 Tz
(Then,) Tj
100 Tz
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95 Tz
(since) Tj
100 Tz
1 0 0 1 495 445 Tm
92 Tz
(the) Tj
100 Tz
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1 0 0 1 268 438 Tm
167 Tz
(\\) Tj
100 Tz
/F5 7 Tf
1 0 0 1 274 438 Tm
64 Tz
(\205A) Tj
100 Tz
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1 0 0 1 88 428 Tm
95 Tz
(Rayleigh-Ritz) Tj
100 Tz
1 0 0 1 151 428 Tm
96 Tz
(approximate) Tj
100 Tz
1 0 0 1 208 428 Tm
97 Tz
(solution) Tj
100 Tz
/F11 11 Tf
1 0 0 1 246 428 Tm
(~) Tj
/F5 17 Tf
1 0 0 1 270 428 Tm
82 Tz
(X) Tj
100 Tz
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1 0 0 1 285 428 Tm
107 Tz
(aB,) Tj
100 Tz
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(which) Tj
1 0 0 1 336 428 Tm
105 Tz
(minimizes) Tj
100 Tz
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1 0 0 1 385 428 Tm
116 Tz
(I\(a) Tj
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(1) Tj
100 Tz
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(a~,A) Tj
100 Tz
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(\)) Tj
100 Tz
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105 Tz
(must) Tj
100 Tz
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1 0 0 1 272 419 Tm
133 Tz
(=) Tj
100 Tz
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(I) Tj
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1 0 0 1 87 408 Tm
(satisfy) Tj
1 0 0 1 119 408 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 137 408 Tm
105 Tz
(boundary) Tj
100 Tz
1 0 0 1 181 408 Tm
107 Tz
(conditions) Tj
100 Tz
1 0 0 1 231 408 Tm
109 Tz
(\(3.2\),) Tj
100 Tz
1 0 0 1 259 408 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 275 408 Tm
(see) Tj
1 0 0 1 293 408 Tm
(that) Tj
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(a) Tj
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(1) Tj
100 Tz
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1 0 0 1 327 408 Tm
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167 Tz
(=) Tj
100 Tz
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1 0 0 1 337 408 Tm
80 Tz
(0) Tj
100 Tz
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167 Tz
(=) Tj
100 Tz
/F7 8 Tf
1 0 0 1 356 408 Tm
136 Tz
(a\\A.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 387 408 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 416 408 Tm
(the) Tj
1 0 0 1 434 408 Tm
103 Tz
(subspace) Tj
100 Tz
1 0 0 1 476 408 Tm
111 Tz
(of) Tj
100 Tz
/F11 11 Tf
1 0 0 1 490 408 Tm
92 Tz
(B~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 87 397 Tm
104 Tz
(which) Tj
100 Tz
1 0 0 1 117 397 Tm
104 Tz
(obeys) Tj
100 Tz
1 0 0 1 146 397 Tm
(\(3.2\)) Tj
1 0 0 1 172 397 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 181 397 Tm
112 Tz
(just) Tj
100 Tz
1 0 0 1 201 397 Tm
(the) Tj
1 0 0 1 219 397 Tm
(span) Tj
1 0 0 1 242 397 Tm
(of) Tj
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82 Tz
(B) Tj
-5 Ts

(2) Tj
100 Tz
/F7 8 Tf
1 0 0 1 296 397 Tm
-5 Ts
131 Tz
(BWAI) Tj
100 Tz
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1 0 0 1 331 397 Tm
(and) Tj
1 0 0 1 350 397 Tm
(the) Tj
/F9 10 Tf
1 0 0 1 368 397 Tm
103 Tz
(Rayleigh-Ritz) Tj
100 Tz
1 0 0 1 430 397 Tm
(basis) Tj
/F9 10 Tf
1 0 0 1 455 397 Tm
107 Tz
(functions) Tj
100 Tz
/F7 9 Tf
1 0 0 1 501 397 Tm
67 Tz
(w,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 87 385 Tm
(are) Tj
1 0 0 1 104 385 Tm
(simply) Tj
1 0 0 1 136 385 Tm
(~) Tj
1 0 0 1 157 385 Tm
108 Tz
(for) Tj
100 Tz
/F11 11 Tf
1 0 0 1 174 385 Tm
87 Tz
(i=1) Tj
100 Tz
1 0 0 1 219 385 Tm
90 Tz
(N\205k\2052.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 112 370 Tm
83 Tz
(It) Tj
100 Tz
/F9 10 Tf
1 0 0 1 122 370 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 132 370 Tm
106 Tz
(assumed) Tj
100 Tz
1 0 0 1 173 370 Tm
(that) Tj
1 0 0 1 192 370 Tm
(any) Tj
1 0 0 1 211 370 Tm
105 Tz
(mesh) Tj
100 Tz
/F9 7 Tf
1 0 0 1 238 370 Tm
133 Tz
(if) Tj
100 Tz
/F9 10 Tf
1 0 0 1 248 370 Tm
(is) Tj
1 0 0 1 258 370 Tm
107 Tz
(chosen) Tj
100 Tz
1 0 0 1 292 370 Tm
(so) Tj
1 0 0 1 304 370 Tm
(that) Tj
/F7 10 Tf
1 0 0 1 137 352 Tm
(Max) Tj
/F11 11 Tf
1 0 0 1 162 352 Tm
111 Tz
(Ix,\261) Tj
-5 Ts

(1) Tj
-5 Ts

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100 Tz
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1 0 0 1 233 352 Tm
71 Tz
(Ix,.,.r\205x,I) Tj
100 Tz
/F1 11 Tf
1 0 0 1 278 352 Tm
124 Tz
(\(~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 482 352 Tm
(\(3.10\)) Tj
/F1 11 Tf
1 0 0 1 88 334 Tm
76 Tz
(where) Tj
100 Tz
1 0 0 1 117 334 Tm
(~) Tj
1 0 0 1 128 334 Tm
38 Tz
(is) Tj
100 Tz
1 0 0 1 139 334 Tm
85 Tz
(some) Tj
100 Tz
1 0 0 1 165 334 Tm
61 Tz
(fixed) Tj
100 Tz
1 0 0 1 190 334 Tm
63 Tz
(constant.) Tj
100 Tz
1 0 0 1 236 334 Tm
69 Tz
(This) Tj
100 Tz
1 0 0 1 259 334 Tm
60 Tz
(relation) Tj
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1 0 0 1 296 334 Tm
69 Tz
(simply) Tj
100 Tz
1 0 0 1 327 334 Tm
65 Tz
(assures) Tj
100 Tz
1 0 0 1 362 334 Tm
62 Tz
(that) Tj
100 Tz
1 0 0 1 383 334 Tm
63 Tz
(the) Tj
100 Tz
1 0 0 1 400 334 Tm
85 Tz
(mesh) Tj
100 Tz
1 0 0 1 427 334 Tm
(w) Tj
1 0 0 1 438 334 Tm
72 Tz
(cannot) Tj
100 Tz
1 0 0 1 471 334 Tm
58 Tz
(stray) Tj
100 Tz
1 0 0 1 495 334 Tm
63 Tz
(too) Tj
100 Tz
/F9 10 Tf
1 0 0 1 88 322 Tm
91 Tz
(far) Tj
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1 0 0 1 103 322 Tm
105 Tz
(from) Tj
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1 0 0 1 128 322 Tm
(being) Tj
1 0 0 1 155 322 Tm
106 Tz
(uniform.) Tj
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1 0 0 1 199 322 Tm
106 Tz
(This) Tj
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1 0 0 1 222 322 Tm
105 Tz
(restriction) Tj
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(on) Tj
1 0 0 1 284 322 Tm
(the) Tj
1 0 0 1 301 322 Tm
105 Tz
(mesh) Tj
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1 0 0 1 327 322 Tm
(is) Tj
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107 Tz
(important) Tj
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1 0 0 1 383 322 Tm
(and) Tj
1 0 0 1 402 322 Tm
93 Tz
(will) Tj
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1 0 0 1 421 322 Tm
89 Tz
(be) Tj
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1 0 0 1 434 322 Tm
105 Tz
(used) Tj
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1 0 0 1 458 322 Tm
(in) Tj
1 0 0 1 469 322 Tm
106 Tz
(later) Tj
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1 0 0 1 492 322 Tm
125 Tz
(sec) Tj
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(-) Tj
1 0 0 1 88 310 Tm
105 Tz
(tions.) Tj
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1 0 0 1 112 296 Tm
(Now) Tj
1 0 0 1 136 296 Tm
106 Tz
(note) Tj
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1 0 0 1 159 296 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 181 296 Tm
(\(3.3\)) Tj
1 0 0 1 206 296 Tm
(is) Tj
1 0 0 1 217 296 Tm
107 Tz
(equivalent) Tj
100 Tz
1 0 0 1 266 296 Tm
(to) Tj
1 0 0 1 280 296 Tm
105 Tz
(\(3.4\),) Tj
100 Tz
1 0 0 1 308 296 Tm
(which) Tj
1 0 0 1 337 296 Tm
(in) Tj
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106 Tz
(turn) Tj
100 Tz
1 0 0 1 372 296 Tm
(is) Tj
1 0 0 1 382 296 Tm
104 Tz
(equivalent,) Tj
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1 0 0 1 435 296 Tm
105 Tz
(using) Tj
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1 0 0 1 462 296 Tm
104 Tz
(integration) Tj
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1 0 0 1 88 283 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 101 283 Tm
105 Tz
(parts,) Tj
100 Tz
1 0 0 1 129 283 Tm
(to) Tj
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/F12 23 Tf
1 0 0 1 137 254 Tm
(I) Tj
1 0 0 1 139 258 Tm
/F11 10 Tf
1 0 0 1 150 258 Tm
133 Tz
(\(y4+frj\)) Tj
100 Tz
/F7 10 Tf
1 0 0 1 202 258 Tm
90 Tz
(dx) Tj
100 Tz
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1 0 0 1 216 258 Tm
124 Tz
(=0,) Tj
100 Tz
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1 0 0 1 482 258 Tm
(\(3.11\)) Tj
/F5 7 Tf
1 0 0 1 139 249 Tm
(p) Tj
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1 0 0 1 88 232 Tm
106 Tz
(But,) Tj
100 Tz
1 0 0 1 109 232 Tm
105 Tz
(from) Tj
100 Tz
1 0 0 1 135 232 Tm
109 Tz
(\(3.7\),) Tj
100 Tz
1 0 0 1 162 232 Tm
(the) Tj
1 0 0 1 180 232 Tm
103 Tz
(Rayleigh-Ritz) Tj
100 Tz
1 0 0 1 242 232 Tm
106 Tz
(solution) Tj
100 Tz
1 0 0 1 280 232 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 289 232 Tm
107 Tz
(just) Tj
100 Tz
1 0 0 1 308 232 Tm
(the) Tj
1 0 0 1 325 232 Tm
103 Tz
(solution) Tj
100 Tz
/F9 10 Tf
1 0 0 1 363 232 Tm
80 Tz
(5) Tj
100 Tz
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1 0 0 1 371 232 Tm
111 Tz
(of) Tj
100 Tz
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1 0 0 1 407 643 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 427 643 Tm
(the) Tj
1 0 0 1 445 643 Tm
106 Tz
(equation) Tj
100 Tz
1 0 0 1 487 643 Tm
108 Tz
(for) Tj
100 Tz
/F11 10 Tf
1 0 0 1 504 643 Tm
80 Tz
(a) Tj
-5 Ts

(1) Tj
100 Tz
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1 0 0 1 518 643 Tm
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(is) Tj
/F9 10 Tf
1 0 0 1 102 631 Tm
(simply) Tj
/F7 10 Tf
1 0 0 1 152 613 Tm
73 Tz
(a) Tj
-5 Ts

(1) Tj
100 Tz
/F5 5 Tf
1 0 0 1 165 613 Tm
-5 Ts
167 Tz
(=) Tj
100 Tz
/F7 7 Tf
1 0 0 1 176 613 Tm
137 Tz
(YL/aL.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 499 613 Tm
(\(4.14\)) Tj
/F9 10 Tf
1 0 0 1 126 592 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 148 592 Tm
107 Tz
(second) Tj
100 Tz
1 0 0 1 182 592 Tm
105 Tz
(boundary) Tj
100 Tz
1 0 0 1 227 592 Tm
(case) Tj
1 0 0 1 249 592 Tm
(is) Tj
1 0 0 1 259 592 Tm
105 Tz
(when) Tj
100 Tz
/F11 10 Tf
1 0 0 1 287 592 Tm
67 Tz
(1) Tj
5 Ts

(3) Tj
5 Ts

(L) Tj
100 Tz
1 0 0 1 303 592 Tm
200 Tz
(!=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 315 592 Tm
88 Tz
(0.) Tj
100 Tz
1 0 0 1 331 592 Tm
95 Tz
(Here) Tj
100 Tz
1 0 0 1 356 592 Tm
(the) Tj
1 0 0 1 374 592 Tm
96 Tz
(value) Tj
100 Tz
1 0 0 1 401 592 Tm
(of) Tj
/F11 10 Tf
1 0 0 1 415 592 Tm
115 Tz
(5\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 441 592 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 451 592 Tm
80 Tz
(a) Tj
-5 Ts

(1) Tj
100 Tz
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83 Tz
(is) Tj
100 Tz
1 0 0 1 475 592 Tm
105 Tz
(unknown) Tj
100 Tz
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80 Tz
(a) Tj
100 Tz
1 0 0 1 102 580 Tm
92 Tz
(priori.) Tj
100 Tz
/F9 11 Tf
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86 Tz
(A) Tj
100 Tz
/F9 10 Tf
1 0 0 1 146 580 Tm
107 Tz
(natural) Tj
100 Tz
1 0 0 1 181 580 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 201 580 Tm
(logical,) Tj
1 0 0 1 236 580 Tm
108 Tz
(but) Tj
100 Tz
1 0 0 1 255 580 Tm
105 Tz
(improper,) Tj
100 Tz
1 0 0 1 301 580 Tm
(way) Tj
1 0 0 1 322 580 Tm
(to) Tj
1 0 0 1 335 580 Tm
103 Tz
(specify) Tj
100 Tz
/F11 8 Tf
1 0 0 1 369 580 Tm
125 Tz
(al) Tj
100 Tz
/F9 10 Tf
1 0 0 1 383 580 Tm
(would) Tj
1 0 0 1 413 580 Tm
(be) Tj
1 0 0 1 428 580 Tm
(to) Tj
1 0 0 1 441 580 Tm
(force) Tj
1 0 0 1 466 580 Tm
(the) Tj
1 0 0 1 484 580 Tm
105 Tz
(boundary) Tj
100 Tz
/F9 10 Tf
1 0 0 1 102 568 Tm
107 Tz
(condition) Tj
100 Tz
1 0 0 1 147 568 Tm
(\(1.2\)) Tj
1 0 0 1 172 568 Tm
(to) Tj
1 0 0 1 184 568 Tm
106 Tz
(hold) Tj
100 Tz
/F11 10 Tf
1 0 0 1 207 568 Tm
96 Tz
(exactly) Tj
100 Tz
/F9 10 Tf
1 0 0 1 238 568 Tm
(at) Tj
/F11 10 Tf
1 0 0 1 249 568 Tm
114 Tz
(x=L:) Tj
100 Tz
/F11 10 Tf
1 0 0 1 152 550 Tm
118 Tz
(aLy\(L\)) Tj
100 Tz
1 0 0 1 190 550 Tm
(+) Tj
/F11 8 Tf
1 0 0 1 201 550 Tm
116 Tz
(PLY\202) Tj
100 Tz
/F11 10 Tf
1 0 0 1 223 550 Tm
114 Tz
(\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 242 550 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 253 550 Tm
77 Tz
(-y~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 499 550 Tm
(\(4.15\)) Tj
/F9 10 Tf
1 0 0 1 103 533 Tm
(It) Tj
1 0 0 1 113 533 Tm
(is) Tj
1 0 0 1 123 533 Tm
(easily) Tj
1 0 0 1 150 533 Tm
106 Tz
(seen) Tj
100 Tz
1 0 0 1 173 533 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 193 533 Tm
(this) Tj
1 0 0 1 212 533 Tm
107 Tz
(cannot) Tj
100 Tz
1 0 0 1 244 533 Tm
(be) Tj
1 0 0 1 258 533 Tm
(a) Tj
1 0 0 1 266 533 Tm
104 Tz
(proper) Tj
100 Tz
1 0 0 1 297 533 Tm
105 Tz
(thing) Tj
100 Tz
1 0 0 1 323 533 Tm
(to) Tj
1 0 0 1 335 533 Tm
(do,) Tj
1 0 0 1 352 533 Tm
108 Tz
(for) Tj
100 Tz
/F7 10 Tf
1 0 0 1 369 533 Tm
92 Tz
(\(4.15\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 398 533 Tm
(would) Tj
1 0 0 1 428 533 Tm
105 Tz
(mean) Tj
100 Tz
1 0 0 1 455 533 Tm
(that) Tj
/F11 10 Tf
1 0 0 1 152 515 Tm
118 Tz
(aLe\(L\)) Tj
100 Tz
1 0 0 1 190 515 Tm
113 Tz
(+/3Le\202\(L\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 242 515 Tm
129 Tz
(=0) Tj
100 Tz
/F5 10 Tf
1 0 0 1 498 515 Tm
92 Tz
(\(4.16\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 102 498 Tm
104 Tz
(where) Tj
100 Tz
/F11 10 Tf
1 0 0 1 133 498 Tm
(e) Tj
/F5 5 Tf
1 0 0 1 142 498 Tm
167 Tz
(=) Tj
100 Tz
/F11 11 Tf
1 0 0 1 152 498 Tm
120 Tz
(y) Tj
100 Tz
/F9 10 Tf
1 0 0 1 160 498 Tm
81 Tz
(\205f\202) Tj
100 Tz
1 0 0 1 177 498 Tm
(is) Tj
1 0 0 1 188 498 Tm
(the) Tj
1 0 0 1 205 498 Tm
109 Tz
(error.) Tj
100 Tz
1 0 0 1 237 498 Tm
(But) Tj
1 0 0 1 256 498 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 271 498 Tm
105 Tz
(want,) Tj
100 Tz
1 0 0 1 299 498 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 318 498 Tm
107 Tz
(expect,) Tj
100 Tz
/F11 10 Tf
1 0 0 1 354 498 Tm
(e) Tj
/F9 10 Tf
1 0 0 1 361 498 Tm
(to) Tj
1 0 0 1 374 498 Tm
(be) Tj
/F11 10 Tf
1 0 0 1 388 498 Tm
116 Tz
(0\(hk\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 419 498 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 439 498 Tm
(know) Tj
1 0 0 1 466 498 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 486 498 Tm
108 Tz
(e\202cannot) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 485 Tm
(in) Tj
1 0 0 1 114 485 Tm
106 Tz
(general) Tj
100 Tz
1 0 0 1 149 485 Tm
(be) Tj
1 0 0 1 163 485 Tm
108 Tz
(better) Tj
100 Tz
1 0 0 1 192 485 Tm
106 Tz
(than) Tj
100 Tz
/F11 10 Tf
1 0 0 1 214 485 Tm
112 Tz
(O\(h) Tj
100 Tz
/F7 7 Tf
1 0 0 1 233 485 Tm
106 Tz
(A\205I\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 260 485 Tm
108 Tz
(Thus,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 289 485 Tm
105 Tz
(since) Tj
100 Tz
1 0 0 1 314 485 Tm
(in) Tj
1 0 0 1 326 485 Tm
103 Tz
(general) Tj
100 Tz
/F11 8 Tf
1 0 0 1 361 485 Tm
120 Tz
(aL) Tj
100 Tz
/F1 10 Tf
1 0 0 1 377 485 Tm
200 Tz
(!=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 389 485 Tm
(0,) Tj
1 0 0 1 402 485 Tm
(\(4.16\)) Tj
1 0 0 1 431 485 Tm
103 Tz
(implies) Tj
100 Tz
1 0 0 1 466 485 Tm
(that) Tj
/F11 10 Tf
1 0 0 1 152 461 Tm
120 Tz
(e\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 178 461 Tm
167 Tz
(=) Tj
100 Tz
/F5 4 Tf
1 0 0 1 189 461 Tm
133 Tz
(\205) Tj
100 Tz
/F1 10 Tf
1 0 0 1 499 461 Tm
67 Tz
(\(4.17\)) Tj
100 Tz
/F7 10 Tf
1 0 0 1 198 461 Tm
126 Tz
(\205e\202) Tj
100 Tz
/F11 10 Tf
1 0 0 1 225 461 Tm
114 Tz
(\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 244 461 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 256 461 Tm
123 Tz
(0\(hA) Tj
5 Ts

(1) Tj
5 Ts

(\)) Tj
100 Tz
/F11 8 Tf
1 0 0 1 200 455 Tm
120 Tz
(aL) Tj
100 Tz
/F9 10 Tf
1 0 0 1 102 437 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 122 437 Tm
103 Tz
(forcing) Tj
100 Tz
1 0 0 1 156 437 Tm
(the) Tj
1 0 0 1 174 437 Tm
105 Tz
(boundary) Tj
100 Tz
1 0 0 1 218 437 Tm
107 Tz
(conditions) Tj
100 Tz
1 0 0 1 267 437 Tm
112 Tz
(to) Tj
100 Tz
1 0 0 1 279 437 Tm
106 Tz
(hold) Tj
100 Tz
1 0 0 1 302 437 Tm
103 Tz
(exactly) Tj
100 Tz
1 0 0 1 335 437 Tm
(at) Tj
/F11 10 Tf
1 0 0 1 347 437 Tm
120 Tz
(x=L) Tj
100 Tz
/F9 10 Tf
1 0 0 1 371 437 Tm
104 Tz
(results) Tj
100 Tz
1 0 0 1 403 437 Tm
(in) Tj
1 0 0 1 415 437 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 423 437 Tm
104 Tz
(lower) Tj
100 Tz
1 0 0 1 450 437 Tm
106 Tz
(convergence) Tj
100 Tz
1 0 0 1 508 437 Tm
107 Tz
(rate) Tj
100 Tz
1 0 0 1 103 425 Tm
106 Tz
(than) Tj
100 Tz
1 0 0 1 125 425 Tm
(is) Tj
1 0 0 1 135 425 Tm
105 Tz
(possible,) Tj
100 Tz
1 0 0 1 176 425 Tm
105 Tz
(expected) Tj
100 Tz
1 0 0 1 218 425 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 237 425 Tm
103 Tz
(optimal.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 127 410 Tm
86 Tz
(A) Tj
100 Tz
/F9 10 Tf
1 0 0 1 140 410 Tm
104 Tz
(similar) Tj
100 Tz
1 0 0 1 174 410 Tm
107 Tz
(argument) Tj
100 Tz
1 0 0 1 220 410 Tm
107 Tz
(can) Tj
100 Tz
/F9 10 Tf
1 0 0 1 241 410 Tm
(be) Tj
1 0 0 1 256 410 Tm
105 Tz
(made) Tj
100 Tz
1 0 0 1 284 410 Tm
103 Tz
(against) Tj
100 Tz
1 0 0 1 318 410 Tm
104 Tz
(doing) Tj
100 Tz
1 0 0 1 348 410 Tm
(the) Tj
1 0 0 1 367 410 Tm
106 Tz
(next) Tj
100 Tz
1 0 0 1 391 410 Tm
105 Tz
(most) Tj
100 Tz
1 0 0 1 417 410 Tm
(\203obvious\204) Tj
1 0 0 1 463 410 Tm
(thing,) Tj
1 0 0 1 493 410 Tm
103 Tz
(namely) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 397 Tm
103 Tz
(forcing) Tj
100 Tz
1 0 0 1 137 397 Tm
(the) Tj
1 0 0 1 154 397 Tm
107 Tz
(boundary) Tj
100 Tz
1 0 0 1 197 397 Tm
107 Tz
(conditions) Tj
100 Tz
1 0 0 1 246 397 Tm
(to) Tj
1 0 0 1 258 397 Tm
106 Tz
(hold) Tj
100 Tz
1 0 0 1 280 397 Tm
(in) Tj
1 0 0 1 292 397 Tm
(the) Tj
1 0 0 1 309 397 Tm
102 Tz
(\203Galerkin\204) Tj
100 Tz
1 0 0 1 357 397 Tm
109 Tz
(sense) Tj
100 Tz
1 0 0 1 384 397 Tm
80 Tz
(by) Tj
100 Tz
1 0 0 1 398 397 Tm
105 Tz
(requiring) Tj
100 Tz
/F7 7 Tf
1 0 0 1 154 385 Tm
80 Tz
(R) Tj
100 Tz
/F11 23 Tf
1 0 0 1 152 372 Tm
73 Tz
(5) Tj
100 Tz
/F11 8 Tf
1 0 0 1 165 372 Tm
139 Tz
(\(aJ\202+f3L\202\205YL\)BIdx) Tj
100 Tz
/F9 10 Tf
1 0 0 1 276 372 Tm
129 Tz
(=0) Tj
100 Tz
/F9 11 Tf
1 0 0 1 498 372 Tm
92 Tz
(\(4.18\)) Tj
100 Tz
/F11 8 Tf
1 0 0 1 154 362 Tm
75 Tz
(L) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 347 Tm
(to) Tj
1 0 0 1 115 347 Tm
(hold.) Tj
1 0 0 1 143 347 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 172 347 Tm
(the) Tj
1 0 0 1 189 347 Tm
105 Tz
(boundary) Tj
100 Tz
1 0 0 1 232 347 Tm
107 Tz
(conditions) Tj
100 Tz
1 0 0 1 280 347 Tm
104 Tz
(alone,) Tj
100 Tz
1 0 0 1 311 347 Tm
(in) Tj
1 0 0 1 322 347 Tm
(a) Tj
1 0 0 1 330 347 Tm
106 Tz
(vacuum,) Tj
100 Tz
1 0 0 1 371 347 Tm
(are) Tj
1 0 0 1 388 347 Tm
108 Tz
(not) Tj
100 Tz
1 0 0 1 405 347 Tm
105 Tz
(sufficient) Tj
100 Tz
1 0 0 1 448 347 Tm
(to) Tj
1 0 0 1 459 347 Tm
109 Tz
(determine) Tj
100 Tz
/F11 11 Tf
1 0 0 1 507 347 Tm
85 Tz
(a) Tj
-5 Ts

(1) Tj
-5 Ts

(.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 127 331 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 149 331 Tm
106 Tz
(only) Tj
100 Tz
1 0 0 1 173 331 Tm
109 Tz
(other) Tj
100 Tz
1 0 0 1 201 331 Tm
106 Tz
(information) Tj
100 Tz
1 0 0 1 257 331 Tm
103 Tz
(available) Tj
100 Tz
1 0 0 1 299 331 Tm
(is) Tj
1 0 0 1 311 331 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 333 331 Tm
105 Tz
(provided) Tj
100 Tz
1 0 0 1 375 331 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 390 331 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 409 331 Tm
104 Tz
(differential) Tj
100 Tz
1 0 0 1 460 331 Tm
106 Tz
(equation) Tj
100 Tz
1 0 0 1 503 331 Tm
(\(1.1\)) Tj
/F9 10 Tf
1 0 0 1 103 319 Tm
(itself.) Tj
1 0 0 1 133 319 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 156 319 Tm
108 Tz
(observation) Tj
100 Tz
1 0 0 1 210 319 Tm
104 Tz
(results) Tj
100 Tz
1 0 0 1 242 319 Tm
(in) Tj
1 0 0 1 254 319 Tm
(the) Tj
1 0 0 1 271 319 Tm
103 Tz
(following) Tj
100 Tz
1 0 0 1 314 319 Tm
106 Tz
(formulation) Tj
100 Tz
1 0 0 1 368 319 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 381 319 Tm
(the) Tj
1 0 0 1 398 319 Tm
106 Tz
(equation) Tj
100 Tz
1 0 0 1 439 319 Tm
108 Tz
(for) Tj
100 Tz
/F11 10 Tf
1 0 0 1 455 319 Tm
92 Tz
(a) Tj
-5 Ts

(1) Tj
-5 Ts

(,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 471 319 Tm
(and) Tj
1 0 0 1 490 319 Tm
103 Tz
(requires) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 308 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 111 308 Tm
106 Tz
(detailed) Tj
100 Tz
1 0 0 1 148 308 Tm
106 Tz
(look) Tj
100 Tz
1 0 0 1 170 308 Tm
(at) Tj
1 0 0 1 181 308 Tm
(the) Tj
1 0 0 1 198 308 Tm
107 Tz
(derivation) Tj
100 Tz
1 0 0 1 245 308 Tm
(of) Tj
1 0 0 1 258 308 Tm
(the) Tj
1 0 0 1 274 308 Tm
108 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 316 308 Tm
107 Tz
(equations.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 127 293 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 148 293 Tm
103 Tz
(starting) Tj
100 Tz
1 0 0 1 183 293 Tm
105 Tz
(point) Tj
100 Tz
1 0 0 1 208 293 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 224 293 Tm
105 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 273 293 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 318 293 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 335 293 Tm
(\(1.1\)) Tj
1 0 0 1 359 293 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 369 293 Tm
(the) Tj
1 0 0 1 386 293 Tm
106 Tz
(relation) Tj
100 Tz
/F7 9 Tf
1 0 0 1 173 280 Tm
67 Tz
(R) Tj
100 Tz
/F9 11 Tf
1 0 0 1 152 267 Tm
80 Tz
(0) Tj
100 Tz
/F5 5 Tf
1 0 0 1 161 267 Tm
167 Tz
(=) Tj
100 Tz
/F11 23 Tf
1 0 0 1 171 267 Tm
73 Tz
(5) Tj
100 Tz
/F7 10 Tf
1 0 0 1 184 267 Tm
113 Tz
(\(\(a53\202\)\202+b~\202+c\)B,dx) Tj
100 Tz
/F11 8 Tf
1 0 0 1 174 257 Tm
75 Tz
(L) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 242 Tm
104 Tz
(which,) Tj
100 Tz
1 0 0 1 135 242 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 162 242 Tm
104 Tz
(integration) Tj
100 Tz
1 0 0 1 212 242 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 225 242 Tm
105 Tz
(parts,) Tj
100 Tz
1 0 0 1 252 242 Tm
108 Tz
(becomes) Tj
100 Tz
/F7 9 Tf
1 0 0 1 174 229 Tm
67 Tz
(R) Tj
100 Tz
/F11 23 Tf
1 0 0 1 152 215 Tm
71 Tz
(o=5) Tj
100 Tz
/F11 10 Tf
1 0 0 1 184 215 Tm
112 Tz
(\(\205afl\202B,\202+b5B,+cB,\)dx) Tj
100 Tz
/F9 10 Tf
1 0 0 1 297 215 Tm
117 Tz
(+) Tj
100 Tz
/F11 11 Tf
1 0 0 1 308 215 Tm
92 Tz
(aB,5\204I~.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 498 215 Tm
(\(4.19\)) Tj
/F11 8 Tf
1 0 0 1 174 206 Tm
75 Tz
(L) Tj
100 Tz
/F9 11 Tf
1 0 0 1 104 190 Tm
93 Tz
(For) Tj
100 Tz
1 0 0 1 130 190 Tm
(i=2) Tj
/F9 10 Tf
1 0 0 1 176 190 Tm
94 Tz
(N\205k\205i) Tj
100 Tz
1 0 0 1 221 190 Tm
(we) Tj
1 0 0 1 244 190 Tm
105 Tz
(have) Tj
100 Tz
/F11 10 Tf
1 0 0 1 275 190 Tm
117 Tz
(B,\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 305 190 Tm
167 Tz
(=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 315 190 Tm
80 Tz
(0) Tj
100 Tz
/F11 10 Tf
1 0 0 1 335 190 Tm
118 Tz
(B,\(R\),) Tj
100 Tz
/F9 10 Tf
1 0 0 1 375 190 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 397 190 Tm
108 Tz
(\(2.2\)-\(2.6\),) Tj
100 Tz
1 0 0 1 457 190 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 483 190 Tm
106 Tz
(thus) Tj
100 Tz
1 0 0 1 512 190 Tm
(for) Tj
1 0 0 1 104 179 Tm
107 Tz
(i=2) Tj
100 Tz
1 0 0 1 150 179 Tm
94 Tz
(N\205k\205i) Tj
100 Tz
/F9 10 Tf
1 0 0 1 189 179 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 236 179 Tm
(\(4.19\)) Tj
1 0 0 1 267 179 Tm
(are) Tj
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/F7 10 Tf
1 0 0 1 268 586 Tm
116 Tz
(\2058,I8,Ia\(L\)\205-\205+8,vA~,VAa\(R\)\205\]) Tj
100 Tz
/F5 5 Tf
1 0 0 1 455 586 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 327 573 Tm
175 Tz
(PL\202) Tj
100 Tz
/F7 10 Tf
1 0 0 1 431 573 Tm
77 Tz
(PR) Tj
100 Tz
/F11 10 Tf
1 0 0 1 154 557 Tm
(\(c,B,\)) Tj
1 0 0 1 182 557 Tm
(+) Tj
1 0 0 1 192 557 Tm
106 Tz
(8,.V~Aa\(R\)) Tj
100 Tz
1 0 0 1 244 557 Tm
92 Tz
(PR) Tj
100 Tz
1 0 0 1 499 557 Tm
(\(4.25\)) Tj
/F11 8 Tf
1 0 0 1 230 533 Tm
(-IL) Tj
/F9 10 Tf
1 0 0 1 129 498 Tm
105 Tz
(Equations) Tj
100 Tz
1 0 0 1 176 498 Tm
107 Tz
(\(4.23\)-\(4.25\)) Tj
100 Tz
1 0 0 1 235 498 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 249 498 Tm
105 Tz
(have) Tj
100 Tz
1 0 0 1 273 498 Tm
(the) Tj
1 0 0 1 290 498 Tm
105 Tz
(form) Tj
100 Tz
/F7 10 Tf
1 0 0 1 154 480 Tm
85 Tz
(Ga) Tj
100 Tz
1 0 0 1 181 480 Tm
80 Tz
(b) Tj
100 Tz
/F9 10 Tf
1 0 0 1 499 480 Tm
(\(4.26\)) Tj
/F9 10 Tf
1 0 0 1 104 463 Tm
104 Tz
(where) Tj
100 Tz
/F11 10 Tf
1 0 0 1 135 463 Tm
86 Tz
(G) Tj
100 Tz
/F9 10 Tf
1 0 0 1 147 463 Tm
(is) Tj
1 0 0 1 158 463 Tm
(an) Tj
/F11 11 Tf
1 0 0 1 174 463 Tm
91 Tz
(N\205k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 199 463 Tm
90 Tz
(by) Tj
100 Tz
/F11 11 Tf
1 0 0 1 214 463 Tm
91 Tz
(N\205k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 239 463 Tm
104 Tz
(matrix) Tj
100 Tz
1 0 0 1 272 463 Tm
(and) Tj
/F11 10 Tf
1 0 0 1 293 463 Tm
80 Tz
(b) Tj
100 Tz
/F9 10 Tf
1 0 0 1 302 463 Tm
(is) Tj
1 0 0 1 313 463 Tm
(a) Tj
1 0 0 1 322 463 Tm
104 Tz
(known) Tj
100 Tz
/F11 11 Tf
1 0 0 1 357 463 Tm
91 Tz
(N\205k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 382 463 Tm
104 Tz
(vector.) Tj
100 Tz
1 0 0 1 420 463 Tm
(Such) Tj
1 0 0 1 445 463 Tm
106 Tz
(systems) Tj
100 Tz
1 0 0 1 483 463 Tm
(are) Tj
1 0 0 1 501 463 Tm
(easily) Tj
/F9 10 Tf
1 0 0 1 104 451 Tm
103 Tz
(solved,) Tj
100 Tz
1 0 0 1 140 451 Tm
108 Tz
(see) Tj
100 Tz
/F9 11 Tf
1 0 0 1 160 451 Tm
68 Tz
(1131) Tj
100 Tz
/F9 10 Tf
1 0 0 1 182 451 Tm
(for) Tj
1 0 0 1 199 451 Tm
103 Tz
(example.) Tj
100 Tz
1 0 0 1 246 451 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 269 451 Tm
104 Tz
(system) Tj
100 Tz
1 0 0 1 303 451 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 317 451 Tm
108 Tz
(linear) Tj
100 Tz
/F9 10 Tf
1 0 0 1 346 451 Tm
103 Tz
(algebraic) Tj
100 Tz
1 0 0 1 389 451 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 435 451 Tm
(given) Tj
1 0 0 1 464 451 Tm
90 Tz
(by) Tj
100 Tz
/F9 10 Tf
1 0 0 1 480 451 Tm
(\(4.26\)) Tj
1 0 0 1 511 451 Tm
(has) Tj
/F9 10 Tf
1 0 0 1 104 439 Tm
105 Tz
(some) Tj
100 Tz
1 0 0 1 131 439 Tm
106 Tz
(very) Tj
100 Tz
1 0 0 1 155 439 Tm
(nice) Tj
1 0 0 1 176 439 Tm
105 Tz
(properties.) Tj
100 Tz
1 0 0 1 229 439 Tm
109 Tz
(First,) Tj
100 Tz
1 0 0 1 257 439 Tm
(it) Tj
1 0 0 1 267 439 Tm
(is) Tj
/F11 8 Tf
1 0 0 1 278 439 Tm
119 Tz
(symmetric,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 326 439 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 346 439 Tm
89 Tz
(is,) Tj
100 Tz
/F11 10 Tf
1 0 0 1 361 439 Tm
83 Tz
(G,,) Tj
100 Tz
/F5 5 Tf
1 0 0 1 377 439 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 388 439 Tm
83 Tz
(G,,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 404 439 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 420 439 Tm
90 Tz
(all) Tj
100 Tz
/F11 10 Tf
1 0 0 1 435 439 Tm
(i) Tj
/F9 10 Tf
1 0 0 1 441 439 Tm
(and) Tj
/F11 11 Tf
1 0 0 1 460 439 Tm
(j.) Tj
/F9 10 Tf
1 0 0 1 474 439 Tm
106 Tz
(This) Tj
100 Tz
/F9 10 Tf
1 0 0 1 497 439 Tm
104 Tz
(means) Tj
100 Tz
/F9 10 Tf
1 0 0 1 104 427 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 127 427 Tm
(we) Tj
1 0 0 1 144 427 Tm
106 Tz
(only) Tj
100 Tz
1 0 0 1 169 427 Tm
110 Tz
(need) Tj
100 Tz
1 0 0 1 196 427 Tm
112 Tz
(to) Tj
100 Tz
1 0 0 1 210 427 Tm
105 Tz
(store) Tj
100 Tz
/F11 10 Tf
1 0 0 1 238 427 Tm
83 Tz
(G,,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 256 427 Tm
108 Tz
(for) Tj
100 Tz
/F11 11 Tf
1 0 0 1 274 427 Tm
125 Tz
(I) Tj
100 Tz
/F9 11 Tf
1 0 0 1 281 427 Tm
200 Tz
(=) Tj
100 Tz
/F11 8 Tf
1 0 0 1 290 427 Tm
75 Tz
(i.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 303 427 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 326 427 Tm
104 Tz
(matrix) Tj
100 Tz
/F11 10 Tf
1 0 0 1 361 427 Tm
(G) Tj
/F9 10 Tf
1 0 0 1 374 427 Tm
(is) Tj
1 0 0 1 386 427 Tm
(also) Tj
/F11 10 Tf
1 0 0 1 410 427 Tm
(banded) Tj
/F9 10 Tf
1 0 0 1 444 427 Tm
(in) Tj
1 0 0 1 459 427 Tm
106 Tz
(that) Tj
100 Tz
/F11 10 Tf
1 0 0 1 482 427 Tm
83 Tz
(G,,) Tj
100 Tz
/F5 5 Tf
1 0 0 1 497 427 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 507 427 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 518 427 Tm
(if) Tj
/F7 15 Tf
1 0 0 1 105 416 Tm
75 Tz
(li\205il) Tj
100 Tz
/F1 10 Tf
1 0 0 1 133 416 Tm
(\>) Tj
/F11 10 Tf
1 0 0 1 144 416 Tm
114 Tz
(k,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 156 416 Tm
107 Tz
(this) Tj
100 Tz
1 0 0 1 176 416 Tm
106 Tz
(property) Tj
100 Tz
1 0 0 1 216 416 Tm
103 Tz
(coming) Tj
100 Tz
1 0 0 1 251 416 Tm
105 Tz
(from) Tj
100 Tz
1 0 0 1 276 416 Tm
(the) Tj
1 0 0 1 293 416 Tm
107 Tz
(fact) Tj
100 Tz
/F9 10 Tf
1 0 0 1 313 416 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 333 416 Tm
(each) Tj
1 0 0 1 356 416 Tm
(B-spline) Tj
1 0 0 1 395 416 Tm
(basis) Tj
1 0 0 1 419 416 Tm
106 Tz
(function) Tj
100 Tz
/F11 10 Tf
1 0 0 1 460 416 Tm
88 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 472 416 Tm
(is) Tj
1 0 0 1 482 416 Tm
(only) Tj
1 0 0 1 505 416 Tm
121 Tz
(non) Tj
100 Tz
(-) Tj
1 0 0 1 104 403 Tm
106 Tz
(zero) Tj
100 Tz
1 0 0 1 127 403 Tm
(on) Tj
1 0 0 1 142 403 Tm
(the) Tj
1 0 0 1 159 403 Tm
106 Tz
(interval) Tj
100 Tz
/F11 8 Tf
1 0 0 1 198 403 Tm
(\[x,,x,+k) Tj
/F9 11 Tf
1 0 0 1 230 403 Tm
63 Tz
(1.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 244 403 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 265 403 Tm
104 Tz
(matrix) Tj
100 Tz
/F11 10 Tf
1 0 0 1 298 403 Tm
86 Tz
(G) Tj
100 Tz
/F9 10 Tf
1 0 0 1 309 403 Tm
(is) Tj
1 0 0 1 320 403 Tm
(also) Tj
/F11 8 Tf
1 0 0 1 340 403 Tm
118 Tz
(positive-definite.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 413 403 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 437 403 Tm
(is) Tj
1 0 0 1 448 403 Tm
(easily) Tj
1 0 0 1 475 403 Tm
106 Tz
(seen) Tj
100 Tz
1 0 0 1 499 403 Tm
(in) Tj
1 0 0 1 511 403 Tm
(the) Tj
1 0 0 1 104 392 Tm
(case) Tj
1 0 0 1 125 392 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 139 392 Tm
(\(1.1\)) Tj
1 0 0 1 164 392 Tm
103 Tz
(subject) Tj
100 Tz
1 0 0 1 198 392 Tm
(to) Tj
/F11 10 Tf
1 0 0 1 209 392 Tm
127 Tz
(y\(L\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 236 392 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 246 392 Tm
80 Tz
(0) Tj
100 Tz
/F5 5 Tf
1 0 0 1 255 392 Tm
167 Tz
(=) Tj
100 Tz
/F11 10 Tf
1 0 0 1 265 392 Tm
123 Tz
(y\(R\):) Tj
100 Tz
/F9 10 Tf
1 0 0 1 295 392 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 316 392 Tm
107 Tz
(statement) Tj
100 Tz
1 0 0 1 362 392 Tm
106 Tz
(that) Tj
100 Tz
/F11 10 Tf
1 0 0 1 384 392 Tm
86 Tz
(G) Tj
100 Tz
/F9 10 Tf
1 0 0 1 393 392 Tm
(is) Tj
1 0 0 1 404 392 Tm
103 Tz
(positive) Tj
100 Tz
1 0 0 1 441 392 Tm
103 Tz
(definite) Tj
100 Tz
1 0 0 1 477 392 Tm
104 Tz
(means) Tj
100 Tz
1 0 0 1 508 392 Tm
(that) Tj
/F9 10 Tf
1 0 0 1 104 380 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 120 380 Tm
(any) Tj
1 0 0 1 139 380 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 182 380 Tm
104 Tz
(vector) Tj
100 Tz
/F11 8 Tf
1 0 0 1 212 380 Tm
120 Tz
(a) Tj
100 Tz
/F9 10 Tf
1 0 0 1 221 380 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 237 380 Tm
105 Tz
(have) Tj
100 Tz
/F11 10 Tf
1 0 0 1 261 380 Tm
(a\202Ga) Tj
/F9 10 Tf
1 0 0 1 286 380 Tm
117 Tz
(\>) Tj
100 Tz
1 0 0 1 295 380 Tm
(0.) Tj
1 0 0 1 310 380 Tm
114 Tz
(If) Tj
100 Tz
1 0 0 1 322 380 Tm
110 Tz
(there) Tj
100 Tz
1 0 0 1 347 380 Tm
(is) Tj
1 0 0 1 358 380 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 366 380 Tm
108 Tz
(vector) Tj
100 Tz
/F11 10 Tf
1 0 0 1 397 380 Tm
80 Tz
(a) Tj
100 Tz
/F9 11 Tf
1 0 0 1 405 380 Tm
89 Tz
(so) Tj
100 Tz
1 0 0 1 418 380 Tm
(that) Tj
/F11 10 Tf
1 0 0 1 439 380 Tm
(a\202Ga) Tj
/F1 10 Tf
1 0 0 1 465 380 Tm
200 Tz
(=) Tj
100 Tz
1 0 0 1 477 380 Tm
58 Tz
(0,) Tj
100 Tz
1 0 0 1 490 380 Tm
75 Tz
(then) Tj
100 Tz
1 0 0 1 512 380 Tm
67 Tz
(for) Tj
100 Tz
/F7 8 Tf
1 0 0 1 124 370 Tm
80 Tz
(N) Tj
100 Tz
1 0 0 1 130 370 Tm
77 Tz
(\205k\205I) Tj
100 Tz
1 0 0 1 296 370 Tm
80 Tz
(R) Tj
100 Tz
/F5 5 Tf
1 0 0 1 113 357 Tm
167 Tz
(=) Tj
100 Tz
/F11 17 Tf
1 0 0 1 131 357 Tm
90 Tz
(X) Tj
100 Tz
/F11 10 Tf
1 0 0 1 152 357 Tm
110 Tz
(a,B,\(x\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 191 357 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 209 357 Tm
105 Tz
(have) Tj
100 Tz
1 0 0 1 235 357 Tm
(0) Tj
/F9 10 Tf
1 0 0 1 245 357 Tm
129 Tz
(\>) Tj
100 Tz
/F11 10 Tf
1 0 0 1 257 357 Tm
(a\202Ga) Tj
/F5 5 Tf
1 0 0 1 283 357 Tm
167 Tz
(=) Tj
100 Tz
/F11 23 Tf
1 0 0 1 293 357 Tm
73 Tz
(5) Tj
100 Tz
/F11 10 Tf
1 0 0 1 306 357 Tm
107 Tz
(\(av\202B,\202\205bvB,\)dx,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 390 357 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 409 357 Tm
129 Tz
(#2) Tj
100 Tz
/F11 11 Tf
1 0 0 1 455 357 Tm
(N\205k\205l,) Tj
/F9 10 Tf
1 0 0 1 499 357 Tm
(which) Tj
/F5 7 Tf
1 0 0 1 133 350 Tm
(=2) Tj
/F11 8 Tf
1 0 0 1 296 350 Tm
75 Tz
(L) Tj
100 Tz
1 0 0 1 224 346 Tm
(R) Tj
/F9 10 Tf
1 0 0 1 104 333 Tm
(simply) Tj
1 0 0 1 140 333 Tm
107 Tz
(means) Tj
100 Tz
1 0 0 1 176 333 Tm
(that) Tj
1 0 0 1 199 333 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 209 333 Tm
117 Tz
(\>) Tj
100 Tz
/F11 23 Tf
1 0 0 1 221 333 Tm
73 Tz
(5) Tj
100 Tz
/F11 8 Tf
1 0 0 1 234 333 Tm
131 Tz
(\(a\(v\202\)) Tj
5 Ts

(2) Tj
5 Ts

(\205bv) Tj
5 Ts

(2) Tj
5 Ts

(\)) Tj
100 Tz
/F11 10 Tf
1 0 0 1 300 333 Tm
(dx.) Tj
/F9 10 Tf
1 0 0 1 323 333 Tm
(But) Tj
1 0 0 1 346 333 Tm
105 Tz
(since) Tj
100 Tz
1 0 0 1 376 333 Tm
150 Tz
(b=0) Tj
100 Tz
1 0 0 1 405 333 Tm
(on) Tj
/F11 10 Tf
1 0 0 1 424 333 Tm
117 Tz
(\(L.R\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 458 333 Tm
(we) Tj
1 0 0 1 477 333 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 504 333 Tm
105 Tz
(have) Tj
100 Tz
/F11 8 Tf
1 0 0 1 224 323 Tm
75 Tz
(L) Tj
100 Tz
/F7 8 Tf
1 0 0 1 128 314 Tm
80 Tz
(R) Tj
100 Tz
/F9 10 Tf
1 0 0 1 104 305 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 114 305 Tm
129 Tz
(\>) Tj
100 Tz
/F11 23 Tf
1 0 0 1 125 305 Tm
73 Tz
(5) Tj
100 Tz
/F11 10 Tf
1 0 0 1 138 305 Tm
117 Tz
(a\(v\202\)) Tj
5 Ts

(2) Tj
5 Ts

(dx) Tj
100 Tz
/F9 10 Tf
1 0 0 1 182 305 Tm
(which) Tj
1 0 0 1 210 305 Tm
103 Tz
(implies) Tj
100 Tz
1 0 0 1 245 305 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 266 305 Tm
75 Tz
(v\202) Tj
100 Tz
1 0 0 1 289 305 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 298 305 Tm
(on) Tj
/F11 10 Tf
1 0 0 1 313 305 Tm
(\[L,RI) Tj
/F9 10 Tf
1 0 0 1 342 305 Tm
105 Tz
(since) Tj
100 Tz
/F11 10 Tf
1 0 0 1 367 305 Tm
120 Tz
(a\(x\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 394 305 Tm
129 Tz
(\>) Tj
100 Tz
/F9 10 Tf
1 0 0 1 405 305 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 414 305 Tm
104 Tz
(there.) Tj
100 Tz
1 0 0 1 446 305 Tm
(Having) Tj
/F11 10 Tf
1 0 0 1 481 305 Tm
116 Tz
(v\(L\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 519 305 Tm
(0) Tj
/F11 8 Tf
1 0 0 1 128 295 Tm
75 Tz
(L) Tj
100 Tz
/F9 10 Tf
1 0 0 1 104 285 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 125 285 Tm
75 Tz
(v\202) Tj
100 Tz
/F11 10 Tf
1 0 0 1 135 285 Tm
108 Tz
(\(x\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 165 285 Tm
(0) Tj
1 0 0 1 175 285 Tm
91 Tz
(on) Tj
100 Tz
/F11 10 Tf
1 0 0 1 191 285 Tm
117 Tz
(\(L,R\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 221 285 Tm
(gives) Tj
/F11 10 Tf
1 0 0 1 249 285 Tm
117 Tz
(v\(x\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 285 285 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 295 285 Tm
91 Tz
(on) Tj
100 Tz
/F11 10 Tf
1 0 0 1 311 285 Tm
(\[L,RI) Tj
/F9 10 Tf
1 0 0 1 340 285 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 360 285 Tm
104 Tz
(hence) Tj
100 Tz
/F11 10 Tf
1 0 0 1 390 285 Tm
80 Tz
(a) Tj
-5 Ts

(1) Tj
100 Tz
/F5 5 Tf
1 0 0 1 403 285 Tm
-5 Ts
167 Tz
(=) Tj
100 Tz
1 0 0 1 439 285 Tm
167 Tz
(=) Tj
100 Tz
/F7 8 Tf
1 0 0 1 449 285 Tm
153 Tz
(a~) Tj
100 Tz
/F5 5 Tf
1 0 0 1 474 285 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 484 285 Tm
86 Tz
(0.) Tj
100 Tz
1 0 0 1 499 285 Tm
108 Tz
(Thus,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 104 274 Tm
(the) Tj
1 0 0 1 121 274 Tm
106 Tz
(only) Tj
100 Tz
1 0 0 1 143 274 Tm
108 Tz
(vector) Tj
100 Tz
/F11 10 Tf
1 0 0 1 173 274 Tm
(a) Tj
/F9 10 Tf
1 0 0 1 182 274 Tm
105 Tz
(such) Tj
100 Tz
1 0 0 1 205 274 Tm
106 Tz
(that) Tj
100 Tz
/F11 10 Tf
1 0 0 1 225 274 Tm
(a\202Ga) Tj
1 0 0 1 251 274 Tm
157 Tz
(\() Tj
100 Tz
/F9 11 Tf
1 0 0 1 263 274 Tm
(0) Tj
1 0 0 1 272 274 Tm
71 Tz
(is) Tj
100 Tz
1 0 0 1 282 274 Tm
111 Tz
(a=0) Tj
100 Tz
/F9 10 Tf
1 0 0 1 304 274 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 323 274 Tm
(G) Tj
1 0 0 1 335 274 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 344 274 Tm
104 Tz
(positive-definite.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 128 259 Tm
(Now) Tj
1 0 0 1 152 259 Tm
(a) Tj
1 0 0 1 161 259 Tm
106 Tz
(banded,) Tj
100 Tz
1 0 0 1 199 259 Tm
105 Tz
(symmetric) Tj
100 Tz
1 0 0 1 247 259 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 267 259 Tm
104 Tz
(positive-definite) Tj
100 Tz
1 0 0 1 340 259 Tm
104 Tz
(matrix) Tj
100 Tz
1 0 0 1 372 259 Tm
(is) Tj
1 0 0 1 383 259 Tm
(an) Tj
1 0 0 1 397 259 Tm
102 Tz
(especially) Tj
100 Tz
1 0 0 1 442 259 Tm
(nice) Tj
1 0 0 1 463 259 Tm
104 Tz
(matrix) Tj
100 Tz
1 0 0 1 495 259 Tm
(to) Tj
1 0 0 1 507 259 Tm
(deal) Tj
/F9 10 Tf
1 0 0 1 104 247 Tm
(with) Tj
1 0 0 1 126 247 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 145 247 Tm
108 Tz
(the) Tj
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100 Tz
1 0 0 1 381 641 Tm
94 Tz
(number) Tj
100 Tz
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(of) Tj
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(G) Tj
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1 0 0 1 443 641 Tm
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(and) Tj
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1 0 0 1 464 641 Tm
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(is) Tj
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1 0 0 1 476 641 Tm
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(denoted) Tj
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1 0 0 1 516 641 Tm
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(by) Tj
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1 0 0 1 104 630 Tm
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(Cond\(G\).) Tj
100 Tz
/F9 10 Tf
1 0 0 1 154 630 Tm
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(The) Tj
100 Tz
1 0 0 1 175 630 Tm
107 Tz
(condition) Tj
100 Tz
1 0 0 1 220 630 Tm
109 Tz
(number) Tj
100 Tz
1 0 0 1 257 630 Tm
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(of) Tj
100 Tz
/F12 10 Tf
1 0 0 1 271 630 Tm
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(G) Tj
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/F9 10 Tf
1 0 0 1 281 630 Tm
(clearly) Tj
1 0 0 1 313 630 Tm
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(relates) Tj
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1 0 0 1 344 630 Tm
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(the) Tj
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1 0 0 1 362 630 Tm
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(relative) Tj
100 Tz
1 0 0 1 397 630 Tm
103 Tz
(change) Tj
100 Tz
1 0 0 1 432 630 Tm
(in) Tj
1 0 0 1 444 630 Tm
(the) Tj
1 0 0 1 461 630 Tm
105 Tz
(right-hand-side) Tj
100 Tz
/F12 10 Tf
1 0 0 1 104 618 Tm
80 Tz
(b) Tj
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/F9 11 Tf
1 0 0 1 111 618 Tm
(to) Tj
1 0 0 1 124 618 Tm
92 Tz
(the) Tj
100 Tz
/F9 10 Tf
1 0 0 1 141 618 Tm
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(relative) Tj
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1 0 0 1 176 618 Tm
103 Tz
(change) Tj
100 Tz
1 0 0 1 210 618 Tm
(in) Tj
1 0 0 1 222 618 Tm
(the) Tj
1 0 0 1 239 618 Tm
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(solution) Tj
100 Tz
/F12 10 Tf
1 0 0 1 278 618 Tm
86 Tz
(a.) Tj
100 Tz
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1 0 0 1 292 618 Tm
(If) Tj
/F12 10 Tf
1 0 0 1 303 618 Tm
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(Cond\(G\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 348 618 Tm
83 Tz
(is) Tj
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1 0 0 1 358 618 Tm
108 Tz
(not) Tj
100 Tz
1 0 0 1 375 618 Tm
108 Tz
(too) Tj
100 Tz
1 0 0 1 393 618 Tm
(large,) Tj
1 0 0 1 420 618 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 442 618 Tm
(a) Tj
1 0 0 1 451 618 Tm
(small) Tj
1 0 0 1 476 618 Tm
103 Tz
(change) Tj
100 Tz
1 0 0 1 510 618 Tm
(in) Tj
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(b) Tj
100 Tz
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1 0 0 1 103 606 Tm
93 Tz
(will) Tj
100 Tz
1 0 0 1 122 606 Tm
108 Tz
(result) Tj
100 Tz
1 0 0 1 151 606 Tm
(in) Tj
1 0 0 1 163 606 Tm
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(only) Tj
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1 0 0 1 185 606 Tm
(a) Tj
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(small) Tj
1 0 0 1 220 606 Tm
103 Tz
(change) Tj
100 Tz
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(in) Tj
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86 Tz
(a.) Tj
100 Tz
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(Similarly) Tj
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1 0 0 1 324 606 Tm
(it) Tj
1 0 0 1 333 606 Tm
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(can) Tj
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(be) Tj
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(shown) Tj
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(that) Tj
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(perturbing) Tj
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(G) Tj
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(by) Tj
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(~G) Tj
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(as) Tj
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1 0 0 1 115 595 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 127 595 Tm
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(the) Tj
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1 0 0 1 144 595 Tm
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(equations) Tj
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1 0 0 1 153 576 Tm
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(Ga=b) Tj
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1 0 0 1 194 576 Tm
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(and) Tj
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1 0 0 1 217 576 Tm
133 Tz
(\(G+8G\)\(a+~a\)=b) Tj
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1 0 0 1 103 559 Tm
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(gives,) Tj
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1 0 0 1 132 559 Tm
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(for) Tj
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1 0 0 1 148 559 Tm
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(sufficiently) Tj
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1 0 0 1 198 559 Tm
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(small) Tj
100 Tz
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(\(\(BG) Tj
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1 0 0 1 250 559 Tm
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(\(I,) Tj
100 Tz
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1 0 0 1 103 514 Tm
108 Tz
(Thus,) Tj
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1 0 0 1 133 514 Tm
107 Tz
(perturbing) Tj
100 Tz
1 0 0 1 182 514 Tm
108 Tz
(either) Tj
100 Tz
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86 Tz
(G) Tj
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(or) Tj
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1 0 0 1 236 514 Tm
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(b) Tj
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(by) Tj
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1 0 0 1 259 514 Tm
(a) Tj
1 0 0 1 267 514 Tm
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(relative) Tj
100 Tz
1 0 0 1 303 514 Tm
103 Tz
(change) Tj
100 Tz
1 0 0 1 337 514 Tm
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(of) Tj
100 Tz
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(results) Tj
100 Tz
1 0 0 1 395 514 Tm
(in) Tj
1 0 0 1 407 514 Tm
(a) Tj
1 0 0 1 415 514 Tm
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(relative) Tj
100 Tz
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103 Tz
(change) Tj
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(a) Tj
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(at) Tj
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(most) Tj
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(Cond\(G\)E.) Tj
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106 Tz
(This) Tj
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1 0 0 1 207 503 Tm
(tells) Tj
1 0 0 1 229 503 Tm
(us) Tj
1 0 0 1 242 503 Tm
(how) Tj
1 0 0 1 263 503 Tm
105 Tz
(sensitive) Tj
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1 0 0 1 305 503 Tm
(the) Tj
1 0 0 1 322 503 Tm
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(solution) Tj
100 Tz
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(a) Tj
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1 0 0 1 369 503 Tm
(of) Tj
1 0 0 1 382 503 Tm
(\(4.26\)) Tj
1 0 0 1 412 503 Tm
(is) Tj
1 0 0 1 423 503 Tm
(to) Tj
1 0 0 1 434 503 Tm
103 Tz
(changes) Tj
100 Tz
1 0 0 1 472 503 Tm
(in) Tj
1 0 0 1 484 503 Tm
108 Tz
(either) Tj
100 Tz
1 0 0 1 512 503 Tm
(the) Tj
1 0 0 1 104 491 Tm
104 Tz
(matrix) Tj
100 Tz
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(G) Tj
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1 0 0 1 146 491 Tm
(or) Tj
1 0 0 1 158 491 Tm
(the) Tj
1 0 0 1 175 491 Tm
106 Tz
(right-hand-side) Tj
100 Tz
/F11 10 Tf
1 0 0 1 245 491 Tm
71 Tz
(b.) Tj
100 Tz
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1 0 0 1 128 475 Tm
(We) Tj
1 0 0 1 147 475 Tm
(now) Tj
1 0 0 1 169 475 Tm
105 Tz
(study) Tj
100 Tz
1 0 0 1 197 475 Tm
(the) Tj
1 0 0 1 214 475 Tm
109 Tz
(perturbations) Tj
100 Tz
1 0 0 1 276 475 Tm
109 Tz
(introduced) Tj
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1 0 0 1 327 475 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 341 475 Tm
108 Tz
(approximate) Tj
100 Tz
1 0 0 1 399 475 Tm
106 Tz
(integration.) Tj
100 Tz
1 0 0 1 456 475 Tm
91 Tz
(By) Tj
100 Tz
1 0 0 1 471 475 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 498 475 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 507 475 Tm
(k\205i) Tj
1 0 0 1 104 463 Tm
105 Tz
(point) Tj
100 Tz
1 0 0 1 130 463 Tm
108 Tz
(Gauss-Legendre) Tj
100 Tz
1 0 0 1 205 463 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 257 463 Tm
105 Tz
(rule,) Tj
100 Tz
1 0 0 1 281 463 Tm
(we) Tj
1 0 0 1 297 463 Tm
104 Tz
(obtain) Tj
100 Tz
1 0 0 1 329 463 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 344 463 Tm
103 Tz
(integrals) Tj
100 Tz
1 0 0 1 384 463 Tm
106 Tz
(accurate) Tj
100 Tz
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(to) Tj
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120 Tz
(0\(h) Tj
5 Ts

(2) Tj
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(\(kIi\)) Tj
100 Tz
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(and) Tj
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(thus) Tj
100 Tz
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(the) Tj
100 Tz
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109 Tz
(perturbations) Tj
100 Tz
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(are) Tj
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(all) Tj
100 Tz
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115 Tz
(0\(h) Tj
5 Ts

(2) Tj
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(\(k~\)\).) Tj
100 Tz
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(Thus,) Tj
100 Tz
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(may) Tj
1 0 0 1 333 451 Tm
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(write) Tj
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200 Tz
(=) Tj
100 Tz
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113 Tz
(Cond\(G\)0\(h) Tj
5 Ts

(2) Tj
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100 Tz
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95 Tz
(\(5.5\)) Tj
100 Tz
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83 Tz
(\(\(all) Tj
100 Tz
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(if) Tj
1 0 0 1 135 406 Tm
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(we) Tj
100 Tz
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(can) Tj
100 Tz
1 0 0 1 169 406 Tm
(show) Tj
1 0 0 1 196 406 Tm
106 Tz
(that) Tj
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/F12 10 Tf
1 0 0 1 218 406 Tm
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(Cond\(G\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 263 406 Tm
106 Tz
(doesn\202t) Tj
100 Tz
1 0 0 1 299 406 Tm
(grow) Tj
1 0 0 1 324 406 Tm
108 Tz
(too) Tj
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1 0 0 1 342 406 Tm
107 Tz
(fast) Tj
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1 0 0 1 362 406 Tm
106 Tz
(with) Tj
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1 0 0 1 386 406 Tm
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(h\2050,) Tj
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1 0 0 1 415 406 Tm
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(then) Tj
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(\(5.5\)) Tj
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93 Tz
(will) Tj
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1 0 0 1 483 406 Tm
(show) Tj
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(that) Tj
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(the) Tj
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1 0 0 1 124 395 Tm
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(computed) Tj
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1 0 0 1 174 395 Tm
108 Tz
(Galerkin) Tj
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1 0 0 1 218 395 Tm
106 Tz
(solution) Tj
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1 0 0 1 260 395 Tm
104 Tz
(coefficients) Tj
100 Tz
/F11 10 Tf
1 0 0 1 315 395 Tm
(a\202) Tj
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1 0 0 1 331 395 Tm
94 Tz
(stay) Tj
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(close) Tj
1 0 0 1 382 395 Tm
(to) Tj
1 0 0 1 398 395 Tm
(the) Tj
1 0 0 1 418 395 Tm
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(exact) Tj
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1 0 0 1 447 395 Tm
108 Tz
(Galerkin) Tj
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1 0 0 1 491 395 Tm
106 Tz
(solution) Tj
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1 0 0 1 104 382 Tm
102 Tz
(coefficients.) Tj
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/F9 10 Tf
1 0 0 1 129 367 Tm
(As) Tj
1 0 0 1 144 367 Tm
(a) Tj
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106 Tz
(start) Tj
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1 0 0 1 176 367 Tm
91 Tz
(we) Tj
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1 0 0 1 192 367 Tm
(first) Tj
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(consider) Tj
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(Poisson\202s) Tj
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108 Tz
(equation) Tj
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(\(5.1\).) Tj
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(To) Tj
1 0 0 1 388 367 Tm
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(estimate) Tj
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1 0 0 1 429 367 Tm
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(Cond\(G\)) Tj
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1 0 0 1 473 367 Tm
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(we) Tj
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1 0 0 1 488 367 Tm
(first) Tj
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(find) Tj
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(an) Tj
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(upper) Tj
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1 0 0 1 146 355 Tm
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(bound) Tj
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1 0 0 1 176 355 Tm
(on) Tj
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1 0 0 1 192 355 Tm
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(\(I) Tj
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1 0 0 1 199 355 Tm
(G) Tj
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1 0 0 1 208 355 Tm
62 Tz
(\(\(.) Tj
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1 0 0 1 224 355 Tm
106 Tz
(This) Tj
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1 0 0 1 246 355 Tm
(is) Tj
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(obtained) Tj
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1 0 0 1 297 355 Tm
90 Tz
(by) Tj
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1 0 0 1 311 355 Tm
107 Tz
(noting) Tj
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1 0 0 1 342 355 Tm
106 Tz
(that) Tj
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1 0 0 1 362 355 Tm
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(for) Tj
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1 0 0 1 378 355 Tm
(any) Tj
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(vector) Tj
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1 0 0 1 427 355 Tm
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(a) Tj
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(we) Tj
1 0 0 1 450 355 Tm
105 Tz
(have) Tj
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1 0 0 1 154 324 Tm
107 Tz
(\(XG,,a,\(=\(XaiSB,\202B,\202dx11) Tj
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1 0 0 1 323 324 Tm
90 Tz
(X) Tj
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/F7 11 Tf
1 0 0 1 344 324 Tm
126 Tz
(a,5B,\202B\>lxI=\(IaIIO\(h\202\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 104 298 Tm
95 Tz
(since) Tj
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/F12 10 Tf
1 0 0 1 130 298 Tm
88 Tz
(\(\(B,\202\(\() Tj
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1 0 0 1 157 298 Tm
167 Tz
(=) Tj
100 Tz
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1 0 0 1 169 298 Tm
132 Tz
(0\(h\202\),) Tj
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1 0 0 1 207 298 Tm
93 Tz
(see) Tj
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1 0 0 1 225 298 Tm
95 Tz
(Appendix) Tj
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1 0 0 1 270 298 Tm
75 Tz
(2.) Tj
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1 0 0 1 284 298 Tm
95 Tz
(This) Tj
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1 0 0 1 307 298 Tm
91 Tz
(gives) Tj
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78 Tz
(\(\(GIl) Tj
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(=) Tj
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1 0 0 1 192 281 Tm
130 Tz
(0\(h\202\).) Tj
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1 0 0 1 505 281 Tm
(\(5.6\)) Tj
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93 Tz
(We) Tj
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(next) Tj
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(need) Tj
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(an) Tj
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(upper) Tj
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1 0 0 1 212 265 Tm
96 Tz
(bound) Tj
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1 0 0 1 243 265 Tm
91 Tz
(on) Tj
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1 0 0 1 259 265 Tm
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(\(I) Tj
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1 0 0 1 267 265 Tm
133 Tz
(G\202) Tj
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1 0 0 1 287 265 Tm
54 Tz
(\(I.) Tj
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1 0 0 1 303 265 Tm
106 Tz
(This) Tj
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1 0 0 1 325 265 Tm
(is) Tj
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(obtained) Tj
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90 Tz
(by) Tj
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1 0 0 1 390 265 Tm
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(noting) Tj
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1 0 0 1 421 265 Tm
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(that) Tj
100 Tz
1 0 0 1 441 265 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 457 265 Tm
(any) Tj
1 0 0 1 476 265 Tm
104 Tz
(vector) Tj
100 Tz
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1 0 0 1 506 265 Tm
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(a) Tj
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(we) Tj
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95 Tz
(have) Tj
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1 0 0 1 154 223 Tm
97 Tz
(Xa,aiG,,) Tj
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167 Tz
(=) Tj
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67 Tz
(5) Tj
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1 0 0 1 221 223 Tm
(~a,a,B,\202B) Tj
1 0 0 1 272 223 Tm
(dx) Tj
/F5 5 Tf
1 0 0 1 286 223 Tm
167 Tz
(=) Tj
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1 0 0 1 297 223 Tm
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(5) Tj
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1 0 0 1 310 223 Tm
88 Tz
(\(X) Tj
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1 0 0 1 328 223 Tm
86 Tz
(a,B,\202\)) Tj
5 Ts

(2) Tj
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5 Ts
90 Tz
(dx) Tj
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1 0 0 1 373 223 Tm
129 Tz
(\>) Tj
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57 Tz
(\(I) Tj
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90 Tz
(X) Tj
100 Tz
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1 0 0 1 408 223 Tm
75 Tz
(a,) Tj
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78 Tz
(B,) Tj
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1 0 0 1 430 223 Tm
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(112) Tj
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1 0 0 1 505 223 Tm
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(\(5.7\)) Tj
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1 0 0 1 211 216 Tm
(0) Tj
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1 0 0 1 223 216 Tm
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(,.j) Tj
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1 0 0 1 299 216 Tm
(0) Tj
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1 0 0 1 105 201 Tm
96 Tz
(using) Tj
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(the) Tj
1 0 0 1 148 201 Tm
95 Tz
(same) Tj
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1 0 0 1 173 201 Tm
(argument) Tj
1 0 0 1 218 201 Tm
(that) Tj
1 0 0 1 238 201 Tm
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(led) Tj
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1 0 0 1 255 201 Tm
88 Tz
(to) Tj
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1 0 0 1 161 635 Tm
(less) Tj
1 0 0 1 180 635 Tm
106 Tz
(than) Tj
100 Tz
1 0 0 1 204 635 Tm
(k\205i) Tj
1 0 0 1 227 635 Tm
105 Tz
(point) Tj
100 Tz
1 0 0 1 252 635 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 303 635 Tm
106 Tz
(rule) Tj
100 Tz
1 0 0 1 324 635 Tm
107 Tz
(cannot) Tj
100 Tz
1 0 0 1 356 635 Tm
(in) Tj
1 0 0 1 368 635 Tm
106 Tz
(general) Tj
100 Tz
1 0 0 1 404 635 Tm
(be) Tj
1 0 0 1 418 635 Tm
105 Tz
(sufficient) Tj
100 Tz
1 0 0 1 461 635 Tm
(to) Tj
1 0 0 1 473 635 Tm
108 Tz
(compute) Tj
100 Tz
1 0 0 1 515 635 Tm
(an) Tj
1 0 0 1 102 624 Tm
103 Tz
(accurate) Tj
100 Tz
1 0 0 1 141 624 Tm
108 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 183 624 Tm
103 Tz
(matrix.) Tj
100 Tz
1 0 0 1 220 624 Tm
(To) Tj
1 0 0 1 236 624 Tm
(show) Tj
1 0 0 1 261 624 Tm
106 Tz
(that) Tj
100 Tz
/F11 10 Tf
1 0 0 1 282 624 Tm
86 Tz
(G) Tj
100 Tz
/F9 10 Tf
1 0 0 1 297 624 Tm
(is) Tj
1 0 0 1 307 624 Tm
106 Tz
(singular,) Tj
100 Tz
1 0 0 1 347 624 Tm
(it) Tj
1 0 0 1 357 624 Tm
(is) Tj
/F9 10 Tf
1 0 0 1 367 624 Tm
106 Tz
(enough) Tj
100 Tz
1 0 0 1 403 624 Tm
(to) Tj
1 0 0 1 414 624 Tm
107 Tz
(exhibit) Tj
100 Tz
1 0 0 1 447 624 Tm
(a) Tj
1 0 0 1 456 624 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 498 624 Tm
108 Tz
(vector) Tj
100 Tz
/F7 9 Tf
1 0 0 1 102 604 Tm
(a) Tj
/F5 5 Tf
1 0 0 1 112 604 Tm
167 Tz
(=) Tj
100 Tz
/F7 9 Tf
1 0 0 1 124 604 Tm
105 Tz
(\(0,a) Tj
-5 Ts

(2) Tj
100 Tz
1 0 0 1 175 604 Tm
-5 Ts
121 Tz
(a~A.I,O\)) Tj
100 Tz
/F5 13 Tf
1 0 0 1 223 604 Tm
70 Tz
(such) Tj
100 Tz
/F9 10 Tf
1 0 0 1 246 604 Tm
(that) Tj
/F11 10 Tf
1 0 0 1 267 604 Tm
120 Tz
(Ga) Tj
100 Tz
/F5 5 Tf
1 0 0 1 288 604 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 298 604 Tm
(0.) Tj
1 0 0 1 312 604 Tm
(Now) Tj
1 0 0 1 336 604 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 351 604 Tm
(know) Tj
1 0 0 1 378 604 Tm
106 Tz
(that) Tj
100 Tz
/F11 10 Tf
1 0 0 1 399 604 Tm
83 Tz
(G,,) Tj
100 Tz
/F5 5 Tf
1 0 0 1 414 604 Tm
167 Tz
(=) Tj
100 Tz
/F11 23 Tf
1 0 0 1 424 604 Tm
73 Tz
(5) Tj
100 Tz
/F11 10 Tf
1 0 0 1 437 604 Tm
103 Tz
(B,\202B;dx) Tj
100 Tz
/F9 10 Tf
1 0 0 1 471 604 Tm
(and) Tj
1 0 0 1 491 604 Tm
111 Tz
(thus) Tj
100 Tz
/F5 7 Tf
1 0 0 1 427 595 Tm
(0) Tj
/F11 14 Tf
1 0 0 1 152 566 Tm
111 Tz
(X) Tj
100 Tz
/F7 10 Tf
1 0 0 1 166 566 Tm
90 Tz
(G,,a,) Tj
100 Tz
/F11 23 Tf
1 0 0 1 190 566 Tm
70 Tz
(=5) Tj
100 Tz
/F11 14 Tf
1 0 0 1 214 566 Tm
79 Tz
(B,\202\(X) Tj
100 Tz
/F11 10 Tf
1 0 0 1 242 566 Tm
124 Tz
(a,B\)) Tj
100 Tz
1 0 0 1 267 566 Tm
110 Tz
(dx) Tj
100 Tz
/F11 10 Tf
1 0 0 1 498 566 Tm
105 Tz
(\(5.16) Tj
100 Tz
/F5 7 Tf
1 0 0 1 203 559 Tm
(0) Tj
/F9 10 Tf
1 0 0 1 102 545 Tm
(We) Tj
1 0 0 1 120 545 Tm
107 Tz
(can) Tj
100 Tz
1 0 0 1 139 545 Tm
107 Tz
(exhibit) Tj
100 Tz
1 0 0 1 173 545 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 191 545 Tm
104 Tz
(appropriate) Tj
100 Tz
1 0 0 1 243 545 Tm
108 Tz
(vector) Tj
100 Tz
/F11 10 Tf
1 0 0 1 274 545 Tm
80 Tz
(a) Tj
100 Tz
/F9 10 Tf
1 0 0 1 282 545 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 296 545 Tm
106 Tz
(constructing) Tj
100 Tz
1 0 0 1 353 545 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 361 545 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 403 545 Tm
106 Tz
(function) Tj
100 Tz
1 0 0 1 444 545 Tm
60 Tz
(v) Tj
100 Tz
/F11 14 Tf
1 0 0 1 464 545 Tm
111 Tz
(X) Tj
100 Tz
/F11 10 Tf
1 0 0 1 478 545 Tm
119 Tz
(aB,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 499 545 Tm
(which) Tj
/F9 10 Tf
1 0 0 1 102 525 Tm
(is) Tj
1 0 0 1 112 525 Tm
(a) Tj
1 0 0 1 121 525 Tm
104 Tz
(polynomial) Tj
100 Tz
1 0 0 1 173 525 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 185 525 Tm
104 Tz
(degree) Tj
100 Tz
1 0 0 1 219 525 Tm
124 Tz
(=k\2051,) Tj
100 Tz
1 0 0 1 253 525 Tm
104 Tz
(whose) Tj
100 Tz
1 0 0 1 284 525 Tm
(first) Tj
1 0 0 1 304 525 Tm
107 Tz
(derivative) Tj
100 Tz
1 0 0 1 350 525 Tm
106 Tz
(vanishes) Tj
100 Tz
1 0 0 1 391 525 Tm
(at) Tj
1 0 0 1 402 525 Tm
(the) Tj
/F11 10 Tf
1 0 0 1 419 525 Tm
91 Tz
(x,,,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 434 525 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 451 525 Tm
105 Tz
(m=1) Tj
100 Tz
1 0 0 1 502 525 Tm
95 Tz
(k\2052,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 102 513 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 122 513 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 138 513 Tm
(which) Tj
/F11 10 Tf
1 0 0 1 168 513 Tm
75 Tz
(v) Tj
100 Tz
/F9 11 Tf
1 0 0 1 174 513 Tm
92 Tz
(\(0\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 191 513 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 201 513 Tm
80 Tz
(0) Tj
100 Tz
/F5 5 Tf
1 0 0 1 210 513 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 221 513 Tm
60 Tz
(v) Tj
100 Tz
1 0 0 1 227 513 Tm
(\(1\).) Tj
1 0 0 1 250 513 Tm
86 Tz
(A) Tj
100 Tz
1 0 0 1 261 513 Tm
104 Tz
(polynomial) Tj
100 Tz
1 0 0 1 312 513 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 324 513 Tm
104 Tz
(degree) Tj
100 Tz
/F11 10 Tf
1 0 0 1 357 513 Tm
120 Tz
(k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 363 513 Tm
73 Tz
(\2052) Tj
100 Tz
1 0 0 1 379 513 Tm
(which) Tj
1 0 0 1 408 513 Tm
106 Tz
(vanishes) Tj
100 Tz
1 0 0 1 449 513 Tm
(at) Tj
1 0 0 1 460 513 Tm
(the) Tj
1 0 0 1 477 513 Tm
107 Tz
(quadrature) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 500 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 132 500 Tm
(is) Tj
1 0 0 1 142 500 Tm
(given) Tj
1 0 0 1 169 500 Tm
90 Tz
(by) Tj
100 Tz
/F9 11 Tf
1 0 0 1 102 465 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 123 465 Tm
94 Tz
(polynomial) Tj
100 Tz
1 0 0 1 174 465 Tm
(of) Tj
1 0 0 1 186 465 Tm
93 Tz
(degree) Tj
100 Tz
1 0 0 1 218 465 Tm
89 Tz
(k\205i) Tj
100 Tz
1 0 0 1 240 465 Tm
96 Tz
(given) Tj
100 Tz
1 0 0 1 267 465 Tm
82 Tz
(by) Tj
100 Tz
/F5 7 Tf
1 0 0 1 192 430 Tm
(0) Tj
/F9 11 Tf
1 0 0 1 103 410 Tm
93 Tz
(has) Tj
100 Tz
/F11 10 Tf
1 0 0 1 121 410 Tm
86 Tz
(v\202) Tj
100 Tz
/F7 10 Tf
1 0 0 1 131 410 Tm
85 Tz
(\(xrn\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 154 410 Tm
167 Tz
(=) Tj
100 Tz
/F1 11 Tf
1 0 0 1 164 410 Tm
59 Tz
(z\(x~,\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 193 410 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 203 410 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 212 410 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 229 410 Tm
105 Tz
(m=1) Tj
100 Tz
1 0 0 1 279 410 Tm
95 Tz
(k\2052,) Tj
100 Tz
1 0 0 1 306 410 Tm
110 Tz
(v\(0\)=0) Tj
100 Tz
1 0 0 1 341 410 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 361 410 Tm
117 Tz
(v\(1\)=fz\(.r,\)) Tj
100 Tz
/F11 10 Tf
1 0 0 1 419 410 Tm
73 Tz
(d-r\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 435 410 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 445 410 Tm
80 Tz
(0) Tj
100 Tz
/F9 10 Tf
1 0 0 1 454 410 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 468 410 Tm
104 Tz
(virtue) Tj
100 Tz
1 0 0 1 497 410 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 510 410 Tm
(the) Tj
/F5 7 Tf
1 0 0 1 390 398 Tm
(0) Tj
/F9 11 Tf
1 0 0 1 103 386 Tm
88 Tz
(fact) Tj
100 Tz
1 0 0 1 122 386 Tm
(that) Tj
1 0 0 1 143 386 Tm
92 Tz
(the) Tj
100 Tz
/F11 10 Tf
1 0 0 1 160 386 Tm
110 Tz
(x~) Tj
100 Tz
/F9 10 Tf
1 0 0 1 175 386 Tm
(are) Tj
1 0 0 1 192 386 Tm
(the) Tj
1 0 0 1 210 386 Tm
105 Tz
(Gaussian) Tj
100 Tz
1 0 0 1 252 386 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 303 386 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 333 386 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 350 386 Tm
(the) Tj
1 0 0 1 367 386 Tm
106 Tz
(interval) Tj
100 Tz
1 0 0 1 405 386 Tm
(\[0,) Tj
/F9 12 Tf
1 0 0 1 418 386 Tm
60 Tz
(11.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 435 386 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 457 386 Tm
92 Tz
(polynomial) Tj
100 Tz
/F11 10 Tf
1 0 0 1 509 386 Tm
75 Tz
(v) Tj
100 Tz
/F9 10 Tf
1 0 0 1 516 386 Tm
83 Tz
(is) Tj
100 Tz
/F9 11 Tf
1 0 0 1 102 374 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 121 374 Tm
(zero) Tj
1 0 0 1 144 374 Tm
95 Tz
(since) Tj
100 Tz
1 0 0 1 170 374 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 187 374 Tm
91 Tz
(coefficient) Tj
100 Tz
/F9 10 Tf
1 0 0 1 236 374 Tm
111 Tz
(of) Tj
100 Tz
/F7 8 Tf
1 0 0 1 249 374 Tm
144 Tz
(xkI) Tj
100 Tz
/F9 10 Tf
1 0 0 1 273 374 Tm
(is) Tj
1 0 0 1 285 374 Tm
40 Tz
(1) Tj
100 Tz
/F9 12 Tf
1 0 0 1 290 374 Tm
(/) Tj
/F9 10 Tf
1 0 0 1 297 374 Tm
104 Tz
(\(k\205I\).) Tj
100 Tz
1 0 0 1 334 374 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 364 374 Tm
(the) Tj
1 0 0 1 382 374 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 425 374 Tm
102 Tz
(polynomial) Tj
100 Tz
1 0 0 1 478 374 Tm
60 Tz
(v) Tj
100 Tz
1 0 0 1 485 374 Tm
(given) Tj
1 0 0 1 513 374 Tm
90 Tz
(by) Tj
100 Tz
/F11 10 Tf
1 0 0 1 103 362 Tm
(\(5.17\)) Tj
/F9 10 Tf
1 0 0 1 133 362 Tm
106 Tz
(provides) Tj
100 Tz
/F9 10 Tf
1 0 0 1 174 362 Tm
(the) Tj
1 0 0 1 191 362 Tm
103 Tz
(necessary) Tj
100 Tz
1 0 0 1 235 362 Tm
108 Tz
(non-zero) Tj
100 Tz
1 0 0 1 277 362 Tm
102 Tz
(coefficient) Tj
100 Tz
1 0 0 1 325 362 Tm
108 Tz
(vector) Tj
100 Tz
/F11 10 Tf
1 0 0 1 356 362 Tm
80 Tz
(a) Tj
100 Tz
/F9 11 Tf
1 0 0 1 365 362 Tm
(for) Tj
1 0 0 1 381 362 Tm
92 Tz
(which) Tj
100 Tz
/F11 10 Tf
1 0 0 1 410 362 Tm
120 Tz
(Ga) Tj
100 Tz
/F5 5 Tf
1 0 0 1 431 362 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 442 362 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 450 362 Tm
(and) Tj
/F11 10 Tf
1 0 0 1 471 362 Tm
86 Tz
(G) Tj
100 Tz
/F9 10 Tf
1 0 0 1 485 362 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 495 362 Tm
(in) Tj
1 0 0 1 507 362 Tm
(fact) Tj
/F9 10 Tf
1 0 0 1 102 351 Tm
103 Tz
(singular.) Tj
100 Tz
1 0 0 1 146 351 Tm
108 Tz
(These) Tj
100 Tz
1 0 0 1 175 351 Tm
104 Tz
(results) Tj
100 Tz
1 0 0 1 207 351 Tm
(may) Tj
1 0 0 1 228 351 Tm
(be) Tj
1 0 0 1 241 351 Tm
108 Tz
(summed) Tj
100 Tz
1 0 0 1 282 351 Tm
(up) Tj
1 0 0 1 296 351 Tm
(as) Tj
1 0 0 1 308 351 Tm
(follows:) Tj
/F7 10 Tf
1 0 0 1 127 335 Tm
88 Tz
(Precisely) Tj
100 Tz
/F5 10 Tf
1 0 0 1 167 335 Tm
(k\205i) Tj
/F7 10 Tf
1 0 0 1 190 335 Tm
88 Tz
(Gauss-Legendre) Tj
100 Tz
1 0 0 1 260 335 Tm
92 Tz
(quadrature) Tj
100 Tz
1 0 0 1 308 335 Tm
92 Tz
(points) Tj
100 Tz
1 0 0 1 337 335 Tm
90 Tz
(should) Tj
100 Tz
1 0 0 1 367 335 Tm
73 Tz
(be) Tj
100 Tz
1 0 0 1 380 335 Tm
86 Tz
(used) Tj
100 Tz
1 0 0 1 402 335 Tm
86 Tz
(per) Tj
100 Tz
1 0 0 1 419 335 Tm
83 Tz
(mesh) Tj
100 Tz
1 0 0 1 444 335 Tm
94 Tz
(interval) Tj
100 Tz
1 0 0 1 478 335 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 490 335 Tm
87 Tz
(compute) Tj
100 Tz
/F7 10 Tf
1 0 0 1 127 323 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 144 323 Tm
88 Tz
(integrals.) Tj
100 Tz
1 0 0 1 191 323 Tm
82 Tz
(The) Tj
100 Tz
1 0 0 1 210 323 Tm
81 Tz
(use) Tj
100 Tz
1 0 0 1 228 323 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 240 323 Tm
91 Tz
(more) Tj
100 Tz
1 0 0 1 265 323 Tm
88 Tz
(points) Tj
100 Tz
1 0 0 1 294 323 Tm
88 Tz
(would) Tj
100 Tz
1 0 0 1 322 323 Tm
86 Tz
(increase) Tj
100 Tz
1 0 0 1 359 323 Tm
92 Tz
(the) Tj
100 Tz
/F7 10 Tf
1 0 0 1 376 323 Tm
81 Tz
(cost,) Tj
100 Tz
1 0 0 1 399 323 Tm
92 Tz
(but) Tj
100 Tz
1 0 0 1 416 323 Tm
92 Tz
(not) Tj
100 Tz
1 0 0 1 434 323 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 451 323 Tm
84 Tz
(convergence) Tj
100 Tz
1 0 0 1 504 323 Tm
85 Tz
(rate,) Tj
100 Tz
/F7 10 Tf
1 0 0 1 128 312 Tm
91 Tz
(while) Tj
100 Tz
1 0 0 1 152 312 Tm
85 Tz
(the) Tj
100 Tz
1 0 0 1 167 312 Tm
75 Tz
(use) Tj
100 Tz
1 0 0 1 184 312 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 193 312 Tm
96 Tz
(fewer) Tj
100 Tz
1 0 0 1 219 312 Tm
92 Tz
(quadrature) Tj
100 Tz
1 0 0 1 267 312 Tm
92 Tz
(points) Tj
100 Tz
1 0 0 1 294 312 Tm
91 Tz
(could) Tj
100 Tz
1 0 0 1 319 312 Tm
88 Tz
(result) Tj
100 Tz
1 0 0 1 344 312 Tm
86 Tz
(in) Tj
100 Tz
1 0 0 1 355 312 Tm
80 Tz
(a) Tj
100 Tz
1 0 0 1 363 312 Tm
91 Tz
(singular) Tj
100 Tz
1 0 0 1 399 312 Tm
90 Tz
(matrix.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 127 297 Tm
87 Tz
(We) Tj
100 Tz
1 0 0 1 145 297 Tm
94 Tz
(now) Tj
100 Tz
1 0 0 1 167 297 Tm
95 Tz
(have) Tj
100 Tz
1 0 0 1 190 297 Tm
(a) Tj
1 0 0 1 198 297 Tm
95 Tz
(complete) Tj
100 Tz
1 0 0 1 241 297 Tm
98 Tz
(formulation) Tj
100 Tz
1 0 0 1 296 297 Tm
(of) Tj
1 0 0 1 308 297 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 357 297 Tm
97 Tz
(method) Tj
100 Tz
/F11 14 Tf
1 0 0 1 394 297 Tm
50 Tz
(-) Tj
100 Tz
/F9 10 Tf
1 0 0 1 401 297 Tm
(the) Tj
1 0 0 1 418 297 Tm
105 Tz
(equations) Tj
100 Tz
1 0 0 1 462 297 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 481 297 Tm
(a) Tj
1 0 0 1 490 297 Tm
106 Tz
(method) Tj
100 Tz
/F9 10 Tf
1 0 0 1 103 285 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 120 285 Tm
106 Tz
(forming) Tj
100 Tz
1 0 0 1 159 285 Tm
(the) Tj
1 0 0 1 178 285 Tm
103 Tz
(integrals) Tj
100 Tz
1 0 0 1 219 285 Tm
(which) Tj
1 0 0 1 249 285 Tm
105 Tz
(make) Tj
100 Tz
1 0 0 1 277 285 Tm
(up) Tj
1 0 0 1 293 285 Tm
(the) Tj
1 0 0 1 311 285 Tm
105 Tz
(equations.) Tj
100 Tz
1 0 0 1 362 285 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 385 285 Tm
105 Tz
(most) Tj
100 Tz
1 0 0 1 410 285 Tm
106 Tz
(obvious) Tj
100 Tz
1 0 0 1 449 285 Tm
94 Tz
(way) Tj
100 Tz
1 0 0 1 470 285 Tm
(to) Tj
1 0 0 1 483 285 Tm
105 Tz
(form) Tj
100 Tz
1 0 0 1 509 285 Tm
(the) Tj
/F9 11 Tf
1 0 0 1 102 273 Tm
97 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 145 273 Tm
98 Tz
(equations) Tj
100 Tz
/F9 10 Tf
1 0 0 1 191 273 Tm
(is) Tj
/F9 10 Tf
1 0 0 1 203 273 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 227 273 Tm
(basically) Tj
1 0 0 1 268 273 Tm
(to) Tj
1 0 0 1 280 273 Tm
106 Tz
(compute) Tj
100 Tz
1 0 0 1 323 273 Tm
83 Tz
(G,,) Tj
100 Tz
1 0 0 1 340 273 Tm
108 Tz
(for) Tj
100 Tz
/F11 10 Tf
1 0 0 1 356 273 Tm
162 Tz
(1=) Tj
100 Tz
/F7 12 Tf
1 0 0 1 372 273 Tm
(i) Tj
/F9 10 Tf
1 0 0 1 378 273 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 393 273 Tm
104 Tz
(doing) Tj
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(way) Tj
100 Tz
1 0 0 1 341 654 Tm
(would) Tj
1 0 0 1 370 654 Tm
(be) Tj
1 0 0 1 384 654 Tm
(to) Tj
1 0 0 1 396 654 Tm
106 Tz
(evaluate) Tj
100 Tz
1 0 0 1 435 654 Tm
(and) Tj
1 0 0 1 455 654 Tm
105 Tz
(store) Tj
100 Tz
1 0 0 1 479 654 Tm
108 Tz
(the) Tj
100 Tz
/F11 10 Tf
1 0 0 1 497 654 Tm
80 Tz
(a) Tj
100 Tz
/F9 10 Tf
1 0 0 1 505 654 Tm
(and) Tj
/F12 10 Tf
1 0 0 1 99 642 Tm
80 Tz
(b) Tj
100 Tz
/F9 11 Tf
1 0 0 1 106 642 Tm
96 Tz
(values) Tj
100 Tz
1 0 0 1 137 642 Tm
(and) Tj
1 0 0 1 156 642 Tm
82 Tz
(all) Tj
100 Tz
1 0 0 1 170 642 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 188 642 Tm
97 Tz
(non-zero) Tj
100 Tz
/F12 10 Tf
1 0 0 1 230 642 Tm
89 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 244 642 Tm
(and) Tj
/F12 10 Tf
1 0 0 1 263 642 Tm
(B) Tj
/F9 10 Tf
1 0 0 1 276 642 Tm
(at) Tj
1 0 0 1 287 642 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 301 642 Tm
(the) Tj
1 0 0 1 318 642 Tm
108 Tz
(Gaussian) Tj
100 Tz
1 0 0 1 361 642 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 411 642 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 441 642 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 453 642 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 467 642 Tm
(the) Tj
1 0 0 1 484 642 Tm
106 Tz
(intervals) Tj
100 Tz
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1 0 0 1 99 630 Tm
(\(x,,x,+) Tj
-5 Ts

(1) Tj
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1 0 0 1 165 630 Tm
109 Tz
(\(x,+k1,x,+A\)) Tj
100 Tz
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1 0 0 1 224 630 Tm
(at) Tj
1 0 0 1 236 630 Tm
(the) Tj
1 0 0 1 254 630 Tm
(top) Tj
1 0 0 1 271 630 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 285 630 Tm
(the) Tj
1 0 0 1 302 630 Tm
109 Tz
(outer) Tj
100 Tz
/F9 10 Tf
1 0 0 1 329 630 Tm
(loop.) Tj
1 0 0 1 358 630 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 380 630 Tm
(would) Tj
1 0 0 1 411 630 Tm
107 Tz
(require) Tj
100 Tz
1 0 0 1 446 630 Tm
105 Tz
(2k\(k\2051\)) Tj
100 Tz
1 0 0 1 490 630 Tm
107 Tz
(storage) Tj
100 Tz
/F9 10 Tf
1 0 0 1 98 618 Tm
105 Tz
(locations) Tj
100 Tz
1 0 0 1 141 618 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 157 618 Tm
107 Tz
(storing) Tj
100 Tz
1 0 0 1 191 618 Tm
(the) Tj
/F12 10 Tf
1 0 0 1 208 618 Tm
(a) Tj
/F9 10 Tf
1 0 0 1 217 618 Tm
107 Tz
(and) Tj
100 Tz
/F11 10 Tf
1 0 0 1 237 618 Tm
80 Tz
(b) Tj
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/F9 10 Tf
1 0 0 1 246 618 Tm
103 Tz
(values,) Tj
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1 0 0 1 280 618 Tm
114 Tz
(but) Tj
100 Tz
1 0 0 1 298 618 Tm
106 Tz
(requires) Tj
100 Tz
1 0 0 1 337 618 Tm
(2k) Tj
5 Ts

(2) Tj
5 Ts

(\(k\2051\)) Tj
1 0 0 1 383 618 Tm
104 Tz
(words) Tj
100 Tz
1 0 0 1 413 618 Tm
(to) Tj
1 0 0 1 425 618 Tm
105 Tz
(store) Tj
100 Tz
1 0 0 1 450 618 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 464 618 Tm
(the) Tj
1 0 0 1 482 618 Tm
108 Tz
(non-zero) Tj
100 Tz
/F12 10 Tf
1 0 0 1 99 606 Tm
89 Tz
(B,) Tj
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/F9 11 Tf
1 0 0 1 111 606 Tm
(and) Tj
/F12 10 Tf
1 0 0 1 131 606 Tm
(B) Tj
/F9 11 Tf
1 0 0 1 144 606 Tm
97 Tz
(values.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 122 592 Tm
91 Tz
(By) Tj
100 Tz
1 0 0 1 139 592 Tm
108 Tz
(re-ordering) Tj
100 Tz
1 0 0 1 193 592 Tm
(the) Tj
1 0 0 1 212 592 Tm
105 Tz
(loops) Tj
100 Tz
1 0 0 1 240 592 Tm
(to) Tj
1 0 0 1 254 592 Tm
(do) Tj
1 0 0 1 270 592 Tm
(the) Tj
1 0 0 1 289 592 Tm
107 Tz
(operations) Tj
100 Tz
/F7 10 Tf
1 0 0 1 339 592 Tm
94 Tz
(interval) Tj
100 Tz
/F11 10 Tf
1 0 0 1 375 592 Tm
89 Tz
(by) Tj
100 Tz
/F7 9 Tf
1 0 0 1 389 592 Tm
103 Tz
(interval) Tj
100 Tz
/F9 10 Tf
1 0 0 1 424 592 Tm
(we) Tj
1 0 0 1 442 592 Tm
(can) Tj
1 0 0 1 461 592 Tm
108 Tz
(construct) Tj
100 Tz
1 0 0 1 507 592 Tm
(the) Tj
/F9 10 Tf
1 0 0 1 98 580 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 144 580 Tm
(in) Tj
1 0 0 1 156 580 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 173 580 Tm
105 Tz
(same) Tj
100 Tz
1 0 0 1 199 580 Tm
109 Tz
(number) Tj
100 Tz
1 0 0 1 236 580 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 249 580 Tm
107 Tz
(operations) Tj
100 Tz
1 0 0 1 297 580 Tm
(and) Tj
1 0 0 1 316 580 Tm
(yet) Tj
1 0 0 1 333 580 Tm
(use) Tj
1 0 0 1 351 580 Tm
93 Tz
(less) Tj
100 Tz
1 0 0 1 370 580 Tm
103 Tz
(storage.) Tj
100 Tz
1 0 0 1 410 580 Tm
(On) Tj
1 0 0 1 427 580 Tm
(any) Tj
1 0 0 1 446 580 Tm
103 Tz
(interval) Tj
100 Tz
/F11 10 Tf
1 0 0 1 484 580 Tm
129 Tz
(\(x,x,\261) Tj
-5 Ts

(1) Tj
-5 Ts

(\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 98 568 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 116 568 Tm
106 Tz
(only) Tj
100 Tz
/F12 10 Tf
1 0 0 1 139 568 Tm
89 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 152 568 Tm
(which) Tj
1 0 0 1 182 568 Tm
(are) Tj
1 0 0 1 199 568 Tm
105 Tz
(non-zero) Tj
100 Tz
1 0 0 1 241 568 Tm
110 Tz
(there) Tj
100 Tz
1 0 0 1 267 568 Tm
(are) Tj
1 0 0 1 284 568 Tm
(for) Tj
/F11 10 Tf
1 0 0 1 300 568 Tm
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(j1k) Tj
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/F9 10 Tf
1 0 0 1 333 568 Tm
(+1) Tj
/F12 10 Tf
1 0 0 1 375 568 Tm
71 Tz
(1.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 387 568 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 416 568 Tm
(we) Tj
1 0 0 1 432 568 Tm
107 Tz
(can) Tj
100 Tz
1 0 0 1 450 568 Tm
105 Tz
(form) Tj
100 Tz
1 0 0 1 475 568 Tm
(the) Tj
1 0 0 1 492 568 Tm
115 Tz
(Galer) Tj
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(-) Tj
/F9 10 Tf
1 0 0 1 98 556 Tm
108 Tz
(kin) Tj
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1 0 0 1 115 556 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 161 556 Tm
90 Tz
(by) Tj
100 Tz
/F9 11 Tf
1 0 0 1 122 540 Tm
93 Tz
(For) Tj
100 Tz
1 0 0 1 142 540 Tm
88 Tz
(1=1) Tj
100 Tz
/F11 10 Tf
1 0 0 1 189 540 Tm
105 Tz
(N\205k) Tj
100 Tz
/F9 10 Tf
1 0 0 1 154 513 Tm
106 Tz
(Evaluate) Tj
100 Tz
1 0 0 1 194 513 Tm
(and) Tj
1 0 0 1 213 513 Tm
105 Tz
(store) Tj
100 Tz
/F11 10 Tf
1 0 0 1 238 513 Tm
(a,b) Tj
/F9 10 Tf
1 0 0 1 254 513 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 273 513 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 286 513 Tm
105 Tz
(non-zero) Tj
100 Tz
/F12 10 Tf
1 0 0 1 329 513 Tm
78 Tz
(B,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 341 513 Tm
(and) Tj
/F12 10 Tf
1 0 0 1 360 513 Tm
67 Tz
(B,\202) Tj
100 Tz
/F9 10 Tf
1 0 0 1 153 501 Tm
(at) Tj
1 0 0 1 165 501 Tm
(the) Tj
1 0 0 1 182 501 Tm
94 Tz
(k\205I) Tj
100 Tz
1 0 0 1 204 501 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 255 501 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 284 501 Tm
111 Tz
(of) Tj
100 Tz
/F12 10 Tf
1 0 0 1 297 501 Tm
95 Tz
(\(x,,x) Tj
-5 Ts

(1) Tj
-5 Ts

(+) Tj
-5 Ts

(1) Tj
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(\).) Tj
100 Tz
/F9 11 Tf
1 0 0 1 154 489 Tm
93 Tz
(For) Tj
100 Tz
/F7 10 Tf
1 0 0 1 173 489 Tm
110 Tz
(i=I\205k+1) Tj
100 Tz
1 0 0 1 246 489 Tm
60 Tz
(1) Tj
100 Tz
/F9 10 Tf
1 0 0 1 184 461 Tm
107 Tz
(For) Tj
100 Tz
/F11 10 Tf
1 0 0 1 202 461 Tm
112 Tz
(j=I\205k+1) Tj
100 Tz
/F7 6 Tf
1 0 0 1 259 438 Tm
109 Tz
(\2011,-I) Tj
100 Tz
/F9 10 Tf
1 0 0 1 215 425 Tm
108 Tz
(Compute) Tj
100 Tz
/F11 23 Tf
1 0 0 1 261 425 Tm
73 Tz
(5) Tj
100 Tz
/F7 10 Tf
1 0 0 1 273 425 Tm
(\(aB,\202B,\205bB,B,\)dx) Tj
/F9 10 Tf
1 0 0 1 215 403 Tm
105 Tz
(using) Tj
100 Tz
/F9 10 Tf
1 0 0 1 241 403 Tm
(a) Tj
1 0 0 1 250 403 Tm
94 Tz
(k\205i) Tj
100 Tz
1 0 0 1 272 403 Tm
105 Tz
(point) Tj
100 Tz
1 0 0 1 297 403 Tm
105 Tz
(Gaussian) Tj
100 Tz
1 0 0 1 339 403 Tm
109 Tz
(quadrature) Tj
100 Tz
1 0 0 1 390 403 Tm
(rule.) Tj
1 0 0 1 98 351 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 120 351 Tm
106 Tz
(uses) Tj
100 Tz
1 0 0 1 142 351 Tm
106 Tz
(only) Tj
100 Tz
/F7 10 Tf
1 0 0 1 148 333 Tm
138 Tz
(!\(k21\)\(Nk\)) Tj
100 Tz
1 0 0 1 492 333 Tm
92 Tz
(\(5.18\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 149 323 Tm
80 Tz
(2) Tj
100 Tz
/F9 10 Tf
1 0 0 1 97 307 Tm
106 Tz
(operations,) Tj
100 Tz
/F7 10 Tf
1 0 0 1 152 307 Tm
102 Tz
(\(N\205k\)\(k\2051\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 217 307 Tm
106 Tz
(evaluations) Tj
100 Tz
1 0 0 1 271 307 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 286 307 Tm
120 Tz
(a) Tj
100 Tz
1 0 0 1 296 307 Tm
(and) Tj
/F11 10 Tf
1 0 0 1 318 307 Tm
86 Tz
(b,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 331 307 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 352 307 Tm
108 Tz
(2\(k\2051\)+2\(k\2051\)k) Tj
100 Tz
/F5 5 Tf
1 0 0 1 436 307 Tm
167 Tz
(=) Tj
100 Tz
/F5 10 Tf
1 0 0 1 447 307 Tm
89 Tz
(2\(k) Tj
5 Ts

(2) Tj
5 Ts

(\2051\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 488 307 Tm
(scratch) Tj
/F9 11 Tf
1 0 0 1 97 295 Tm
97 Tz
(storage) Tj
100 Tz
1 0 0 1 132 295 Tm
93 Tz
(locations.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 122 280 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 143 280 Tm
104 Tz
(above) Tj
100 Tz
1 0 0 1 172 280 Tm
107 Tz
(outline) Tj
100 Tz
1 0 0 1 206 280 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 221 280 Tm
105 Tz
(evaluating) Tj
100 Tz
1 0 0 1 269 280 Tm
(the) Tj
1 0 0 1 286 280 Tm
108 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 328 280 Tm
103 Tz
(matrix,) Tj
100 Tz
1 0 0 1 362 280 Tm
(as) Tj
1 0 0 1 374 280 Tm
94 Tz
(well) Tj
100 Tz
1 0 0 1 394 280 Tm
(as) Tj
1 0 0 1 407 280 Tm
(the) Tj
1 0 0 1 424 280 Tm
105 Tz
(right-hand-side) Tj
100 Tz
/F11 10 Tf
1 0 0 1 493 280 Tm
86 Tz
(b,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 504 280 Tm
(can) Tj
/F9 11 Tf
1 0 0 1 98 268 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 113 268 Tm
98 Tz
(implemented) Tj
100 Tz
1 0 0 1 175 268 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 188 268 Tm
88 Tz
(less) Tj
100 Tz
1 0 0 1 209 268 Tm
(than) Tj
1 0 0 1 233 268 Tm
82 Tz
(35) Tj
100 Tz
1 0 0 1 248 268 Tm
96 Tz
(FORTRAN) Tj
100 Tz
1 0 0 1 303 268 Tm
98 Tz
(statements,) Tj
100 Tz
1 0 0 1 357 268 Tm
89 Tz
(with) Tj
100 Tz
1 0 0 1 381 268 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 399 268 Tm
(treatment) Tj
1 0 0 1 446 268 Tm
(of) Tj
1 0 0 1 459 268 Tm
(the) Tj
1 0 0 1 478 268 Tm
95 Tz
(boundary) Tj
100 Tz
/F9 10 Tf
1 0 0 1 97 257 Tm
107 Tz
(conditions) Tj
100 Tz
1 0 0 1 146 257 Tm
105 Tz
(requiring) Tj
100 Tz
1 0 0 1 189 257 Tm
93 Tz
(less) Tj
100 Tz
1 0 0 1 208 257 Tm
106 Tz
(than) Tj
100 Tz
1 0 0 1 231 257 Tm
90 Tz
(35) Tj
100 Tz
1 0 0 1 245 257 Tm
106 Tz
(FORTRAN) Tj
100 Tz
1 0 0 1 298 257 Tm
107 Tz
(statements) Tj
100 Tz
1 0 0 1 348 257 Tm
88 Tz
(as) Tj
100 Tz
1 0 0 1 361 257 Tm
95 Tz
(well.) Tj
100 Tz
1 0 0 1 387 257 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 410 257 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 420 257 Tm
(roughly) Tj
1 0 0 1 456 257 Tm
(twice) Tj
1 0 0 1 482 257 Tm
(as) Tj
1 0 0 1 494 257 Tm
(much) Tj
/F9 10 Tf
1 0 0 1 97 245 Tm
106 Tz
(FORTRAN) Tj
100 Tz
1 0 0 1 154 245 Tm
105 Tz
(code) Tj
100 Tz
1 0 0 1 179 245 Tm
(as) Tj
1 0 0 1 194 245 Tm
(the) Tj
1 0 0 1 213 245 Tm
107 Tz
(implementation) Tj
100 Tz
1 0 0 1 287 245 Tm
(of) Tj
1 0 0 1 301 245 Tm
(a) Tj
1 0 0 1 312 245 Tm
108 Tz
(centered) Tj
100 Tz
1 0 0 1 355 245 Tm
(finite) Tj
1 0 0 1 382 245 Tm
105 Tz
(difference) Tj
100 Tz
1 0 0 1 431 245 Tm
109 Tz
(scheme) Tj
100 Tz
1 0 0 1 469 245 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 487 245 Tm
(solving) Tj
1 0 0 1 98 234 Tm
108 Tz
(\(1.1\)-\(1.2\).) Tj
100 Tz
1 0 0 1 157 234 Tm
108 Tz
(Thus,) Tj
100 Tz
1 0 0 1 189 234 Tm
(the) Tj
1 0 0 1 208 234 Tm
106 Tz
(benefits) Tj
100 Tz
1 0 0 1 248 234 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 262 234 Tm
105 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 314 234 Tm
103 Tz
(method) Tj
100 Tz
/F12 12 Tf
1 0 0 1 352 234 Tm
67 Tz
(-) Tj
100 Tz
/F9 10 Tf
1 0 0 1 362 234 Tm
104 Tz
(higher) Tj
100 Tz
/F9 10 Tf
1 0 0 1 395 234 Tm
105 Tz
(order) Tj
100 Tz
1 0 0 1 424 234 Tm
107 Tz
(rate) Tj
100 Tz
1 0 0 1 446 234 Tm
(of) Tj
1 0 0 1 461 234 Tm
104 Tz
(convergence,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 97 221 Tm
(greater) Tj
1 0 0 1 131 221 Tm
93 Tz
(accuracy,) Tj
100 Tz
1 0 0 1 175 221 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 193 221 Tm
89 Tz
(ability) Tj
100 Tz
1 0 0 1 222 221 Tm
(to) Tj
1 0 0 1 235 221 Tm
93 Tz
(use) Tj
100 Tz
1 0 0 1 253 221 Tm
(a) Tj
1 0 0 1 261 221 Tm
98 Tz
(non-uniform) Tj
100 Tz
1 0 0 1 320 221 Tm
96 Tz
(mesh) Tj
100 Tz
/F12 12 Tf
1 0 0 1 347 221 Tm
67 Tz
(-) Tj
100 Tz
/F9 10 Tf
1 0 0 1 354 221 Tm
105 Tz
(come) Tj
100 Tz
1 0 0 1 381 221 Tm
(at) Tj
1 0 0 1 392 221 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 400 221 Tm
105 Tz
(price) Tj
100 Tz
1 0 0 1 425 221 Tm
(which) Tj
1 0 0 1 454 221 Tm
(is) Tj
1 0 0 1 464 221 Tm
108 Tz
(not) Tj
100 Tz
1 0 0 1 482 221 Tm
(too) Tj
1 0 0 1 499 221 Tm
(high) Tj
/F9 10 Tf
1 0 0 1 97 210 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 114 210 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 131 210 Tm
107 Tz
(implementor) Tj
100 Tz
1 0 0 1 190 210 Tm
(of) Tj
1 0 0 1 202 210 Tm
(such) Tj
1 0 0 1 225 210 Tm
106 Tz
(techniques.) Tj
100 Tz
1 0 0 1 122 194 Tm
107 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 171 194 Tm
106 Tz
(method,) Tj
100 Tz
1 0 0 1 211 194 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 237 194 Tm
102 Tz
(B-splines,) Tj
100 Tz
1 0 0 1 283 194 Tm
(has) Tj
1 0 0 1 301 194 Tm
105 Tz
(been) Tj
100 Tz
1 0 0 1 325 194 Tm
105 Tz
(implemented) Tj
100 Tz
1 0 0 1 384 194 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 400 194 Tm
103 Tz
(solving) Tj
100 Tz
1 0 0 1 435 194 Tm
(the) Tj
1 0 0 1 452 194 Tm
105 Tz
(boundary) Tj
100 Tz
1 0 0 1 495 194 Tm
(value) Tj
/F9 10 Tf
1 0 0 1 97 183 Tm
106 Tz
(problem) Tj
100 Tz
1 0 0 1 138 183 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 154 183 Tm
106 Tz
(systems) Tj
100 Tz
1 0 0 1 191 183 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 204 183 Tm
104 Tz
(linear) Tj
100 Tz
1 0 0 1 232 183 Tm
104 Tz
(differential) Tj
100 Tz
1 0 0 1 283 183 Tm
107 Tz
(equations) Tj
100 Tz
1 0 0 1 328 183 Tm
(in) Tj
1 0 0 1 340 183 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 348 183 Tm
104 Tz
(single) Tj
100 Tz
1 0 0 1 377 183 Tm
103 Tz
(variable.) Tj
100 Tz
1 0 0 1 420 183 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 443 183 Tm
103 Tz
(package) Tj
100 Tz
1 0 0 1 481 183 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 492 183 Tm
(in) Tj
1 0 0 1 504 183 Tm
(use) Tj
/F9 10 Tf
1 0 0 1 97 171 Tm
(as) Tj
1 0 0 1 109 171 Tm
108 Tz
(the) Tj
100 Tz
1 0 0 1 126 171 Tm
106 Tz
(core) Tj
100 Tz
1 0 0 1 148 171 Tm
111 Tz
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100 Tz
1 0 0 1 263 535 Tm
(demand) Tj
1 0 0 1 301 535 Tm
96 Tz
(some) Tj
100 Tz
1 0 0 1 328 535 Tm
(sort) Tj
1 0 0 1 348 535 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 360 535 Tm
(statement) Tj
1 0 0 1 407 535 Tm
96 Tz
(about) Tj
100 Tz
1 0 0 1 435 535 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 452 535 Tm
90 Tz
(accuracy) Tj
100 Tz
1 0 0 1 492 535 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 505 535 Tm
92 Tz
(the) Tj
100 Tz
/F9 11 Tf
1 0 0 1 96 524 Tm
98 Tz
(computed) Tj
100 Tz
1 0 0 1 143 524 Tm
96 Tz
(approximate) Tj
100 Tz
1 0 0 1 201 524 Tm
95 Tz
(solutions.) Tj
100 Tz
1 0 0 1 249 524 Tm
95 Tz
(This) Tj
100 Tz
1 0 0 1 273 524 Tm
94 Tz
(section) Tj
100 Tz
1 0 0 1 307 524 Tm
93 Tz
(makes) Tj
100 Tz
1 0 0 1 338 524 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 346 524 Tm
93 Tz
(\203reasonable\204) Tj
100 Tz
1 0 0 1 403 524 Tm
97 Tz
(attempt) Tj
100 Tz
1 0 0 1 440 524 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 452 524 Tm
97 Tz
(respond) Tj
100 Tz
1 0 0 1 490 524 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 503 524 Tm
93 Tz
(this) Tj
100 Tz
1 0 0 1 96 512 Tm
97 Tz
(unanswerable) Tj
100 Tz
1 0 0 1 159 512 Tm
95 Tz
(question.) Tj
100 Tz
1 0 0 1 121 497 Tm
93 Tz
(We) Tj
100 Tz
1 0 0 1 139 497 Tm
95 Tz
(want) Tj
100 Tz
1 0 0 1 163 497 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 175 497 Tm
97 Tz
(estimate) Tj
100 Tz
1 0 0 1 215 497 Tm
92 Tz
(the) Tj
100 Tz
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1 0 0 1 232 497 Tm
105 Tz
(error) Tj
100 Tz
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1 0 0 1 257 497 Tm
(in) Tj
1 0 0 1 268 497 Tm
(a) Tj
1 0 0 1 276 497 Tm
(given) Tj
1 0 0 1 303 497 Tm
107 Tz
(computed) Tj
100 Tz
1 0 0 1 349 497 Tm
108 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 390 497 Tm
106 Tz
(solution) Tj
100 Tz
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1 0 0 1 428 497 Tm
83 Tz
(5) Tj
100 Tz
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1 0 0 1 437 497 Tm
(to) Tj
1 0 0 1 449 497 Tm
107 Tz
(\(1.1\)-\(1.2\)) Tj
100 Tz
1 0 0 1 499 497 Tm
106 Tz
(over) Tj
100 Tz
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1 0 0 1 96 485 Tm
(a) Tj
1 0 0 1 105 485 Tm
96 Tz
(mcsh) Tj
100 Tz
/F7 8 Tf
1 0 0 1 132 485 Tm
125 Tz
(ir.) Tj
100 Tz
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1 0 0 1 148 485 Tm
93 Tz
(We) Tj
100 Tz
1 0 0 1 167 485 Tm
92 Tz
(know) Tj
100 Tz
1 0 0 1 194 485 Tm
(that) Tj
1 0 0 1 214 485 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 231 485 Tm
95 Tz
(error) Tj
100 Tz
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1 0 0 1 258 485 Tm
65 Tz
(II~) Tj
100 Tz
/F5 15 Tf
1 0 0 1 272 485 Tm
61 Tz
(\205SII) Tj
100 Tz
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1 0 0 1 298 485 Tm
167 Tz
(=) Tj
100 Tz
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1 0 0 1 309 485 Tm
111 Tz
(0\(hA\),) Tj
100 Tz
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1 0 0 1 342 485 Tm
(which) Tj
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1 0 0 1 371 485 Tm
(simply) Tj
1 0 0 1 404 485 Tm
104 Tz
(means) Tj
100 Tz
1 0 0 1 435 485 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 456 485 Tm
105 Tz
(there) Tj
100 Tz
1 0 0 1 481 485 Tm
(is) Tj
1 0 0 1 491 485 Tm
(a) Tj
1 0 0 1 499 485 Tm
122 Tz
(con) Tj
100 Tz
(-) Tj
1 0 0 1 96 472 Tm
110 Tz
(stant) Tj
100 Tz
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114 Tz
(C) Tj
100 Tz
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1 0 0 1 131 472 Tm
89 Tz
(so) Tj
100 Tz
1 0 0 1 144 472 Tm
(that) Tj
1 0 0 1 164 472 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 176 472 Tm
91 Tz
(h\2050) Tj
100 Tz
1 0 0 1 201 472 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 216 472 Tm
95 Tz
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100 Tz
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106 Tz
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100 Tz
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200 Tz
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100 Tz
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86 Tz
(Ch) Tj
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(4) Tj
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100 Tz
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1 0 0 1 497 454 Tm
95 Tz
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100 Tz
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1 0 0 1 108 437 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 124 437 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 143 437 Tm
97 Tz
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100 Tz
1 0 0 1 184 437 Tm
92 Tz
(the) Tj
100 Tz
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97 Tz
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100 Tz
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1 0 0 1 243 437 Tm
89 Tz
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100 Tz
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1 0 0 1 257 437 Tm
106 Tz
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100 Tz
1 0 0 1 280 437 Tm
90 Tz
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100 Tz
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1 0 0 1 321 437 Tm
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82 Tz
(will) Tj
100 Tz
1 0 0 1 357 437 Tm
95 Tz
(have) Tj
100 Tz
1 0 0 1 381 437 Tm
90 Tz
(an) Tj
100 Tz
1 0 0 1 396 437 Tm
97 Tz
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100 Tz
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63 Tz
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100 Tz
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1 0 0 1 469 437 Tm
64 Tz
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100 Tz
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1 0 0 1 501 437 Tm
106 Tz
(The) Tj
100 Tz
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1 0 0 1 96 420 Tm
93 Tz
(actual,) Tj
100 Tz
1 0 0 1 128 420 Tm
97 Tz
(observed) Tj
100 Tz
1 0 0 1 171 420 Tm
97 Tz
(behavior) Tj
100 Tz
1 0 0 1 212 420 Tm
(of) Tj
1 0 0 1 224 420 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 241 420 Tm
(error) Tj
1 0 0 1 265 420 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 278 420 Tm
91 Tz
(h\2050) Tj
100 Tz
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1 0 0 1 303 420 Tm
83 Tz
(is) Tj
100 Tz
/F7 9 Tf
1 0 0 1 146 407 Tm
114 Tz
(\(\(y\2055\(\() Tj
100 Tz
/F5 5 Tf
1 0 0 1 186 407 Tm
167 Tz
(=) Tj
100 Tz
/F7 11 Tf
1 0 0 1 197 407 Tm
75 Tz
(ChA.) Tj
100 Tz
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1 0 0 1 497 407 Tm
90 Tz
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100 Tz
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1 0 0 1 96 390 Tm
95 Tz
(This) Tj
100 Tz
1 0 0 1 120 390 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 131 390 Tm
90 Tz
(an) Tj
100 Tz
1 0 0 1 145 390 Tm
94 Tz
(asymptotic) Tj
100 Tz
1 0 0 1 195 390 Tm
(statement) Tj
1 0 0 1 242 390 Tm
(about) Tj
1 0 0 1 270 390 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 288 390 Tm
97 Tz
(observed) Tj
100 Tz
1 0 0 1 331 390 Tm
97 Tz
(behavior) Tj
100 Tz
1 0 0 1 373 390 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 386 390 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 404 390 Tm
(error) Tj
1 0 0 1 429 390 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 442 390 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 460 390 Tm
91 Tz
(mesh) Tj
100 Tz
1 0 0 1 486 390 Tm
91 Tz
(spacing) Tj
100 Tz
1 0 0 1 96 378 Tm
96 Tz
(approaches) Tj
100 Tz
1 0 0 1 150 378 Tm
95 Tz
(zero,) Tj
100 Tz
1 0 0 1 177 378 Tm
93 Tz
(but) Tj
100 Tz
1 0 0 1 196 378 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 214 378 Tm
95 Tz
(shall) Tj
100 Tz
1 0 0 1 239 378 Tm
97 Tz
(assume) Tj
100 Tz
1 0 0 1 276 378 Tm
(that) Tj
1 0 0 1 298 378 Tm
83 Tz
(it) Tj
100 Tz
1 0 0 1 309 378 Tm
96 Tz
(holds) Tj
100 Tz
1 0 0 1 338 378 Tm
(for) Tj
1 0 0 1 356 378 Tm
82 Tz
(all) Tj
100 Tz
1 0 0 1 371 378 Tm
94 Tz
(meshes.) Tj
100 Tz
1 0 0 1 415 378 Tm
93 Tz
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100 Tz
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1 0 0 1 435 378 Tm
(if) Tj
-5 Ts

(1) Tj
/F9 10 Tf
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82 Tz
(iT) Tj
-5 Ts

(2) Tj
100 Tz
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1 0 0 1 487 378 Tm
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1 0 0 1 502 378 Tm
93 Tz
(two) Tj
100 Tz
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1 0 0 1 97 366 Tm
109 Tz
(meshes,) Tj
100 Tz
1 0 0 1 136 366 Tm
106 Tz
(with) Tj
100 Tz
/F12 10 Tf
1 0 0 1 160 366 Tm
75 Tz
(\(7r) Tj
-5 Ts

(2) Tj
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(\() Tj
100 Tz
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1 0 0 1 182 366 Tm
117 Tz
(\<) Tj
100 Tz
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1 0 0 1 195 366 Tm
126 Tz
(\(if) Tj
-5 Ts

(1) Tj
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(\(,) Tj
100 Tz
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1 0 0 1 219 366 Tm
(and) Tj
1 0 0 1 239 366 Tm
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/F11 10 Tf
1 0 0 1 253 366 Tm
(y,) Tj
/F9 10 Tf
1 0 0 1 265 366 Tm
(be) Tj
1 0 0 1 279 366 Tm
(the) Tj
1 0 0 1 296 366 Tm
107 Tz
(computed) Tj
100 Tz
1 0 0 1 343 366 Tm
108 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 385 366 Tm
106 Tz
(approximate) Tj
100 Tz
1 0 0 1 442 366 Tm
103 Tz
(solution) Tj
100 Tz
1 0 0 1 481 366 Tm
106 Tz
(over) Tj
100 Tz
1 0 0 1 504 366 Tm
(the) Tj
/F9 11 Tf
1 0 0 1 97 353 Tm
96 Tz
(mesh) Tj
100 Tz
/F7 8 Tf
1 0 0 1 124 353 Tm
125 Tz
(ir,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 137 353 Tm
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1 0 0 1 153 353 Tm
(~=1,2.) Tj
1 0 0 1 188 353 Tm
(If) Tj
1 0 0 1 199 353 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 214 353 Tm
92 Tz
(set) Tj
100 Tz
/F12 10 Tf
1 0 0 1 229 353 Tm
60 Tz
(o~) Tj
100 Tz
/F5 5 Tf
1 0 0 1 240 353 Tm
167 Tz
(=) Tj
100 Tz
/F1 10 Tf
1 0 0 1 251 353 Tm
70 Tz
(h~) Tj
-5 Ts

(2) Tj
-5 Ts

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100 Tz
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1 0 0 1 295 353 Tm
117 Tz
(\<) Tj
100 Tz
/F9 10 Tf
1 0 0 1 308 353 Tm
86 Tz
(1,) Tj
100 Tz
1 0 0 1 319 353 Tm
106 Tz
(then) Tj
100 Tz
1 0 0 1 342 353 Tm
(we) Tj
1 0 0 1 357 353 Tm
104 Tz
(obtain) Tj
100 Tz
/F5 5 Tf
1 0 0 1 190 336 Tm
167 Tz
(=) Tj
100 Tz
/F11 13 Tf
1 0 0 1 201 336 Tm
69 Tz
(C\(n~) Tj
-5 Ts

(2) Tj
-5 Ts

(lk) Tj
100 Tz
/F5 5 Tf
1 0 0 1 234 336 Tm
167 Tz
(=) Tj
100 Tz
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1 0 0 1 244 336 Tm
(~.k\(~y~~y) Tj
-5 Ts

(1) Tj
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1 0 0 1 496 336 Tm
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/F9 11 Tf
1 0 0 1 96 318 Tm
(and) Tj
1 0 0 1 116 318 Tm
(thus) Tj
/F7 12 Tf
1 0 0 1 191 299 Tm
200 Tz
(=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 97 284 Tm
87 Tz
(We) Tj
100 Tz
1 0 0 1 114 284 Tm
94 Tz
(also) Tj
100 Tz
1 0 0 1 135 284 Tm
95 Tz
(have) Tj
100 Tz
1 0 0 1 159 284 Tm
95 Tz
(from) Tj
100 Tz
1 0 0 1 183 284 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 200 284 Tm
94 Tz
(triangle) Tj
100 Tz
1 0 0 1 236 284 Tm
96 Tz
(inequality,) Tj
100 Tz
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1 0 0 1 147 261 Tm
108 Tz
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(1) Tj
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100 Tz
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1 0 0 1 191 261 Tm
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110 Tz
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(2) Tj
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(2) Tj
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100 Tz
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1 0 0 1 301 261 Tm
167 Tz
(=) Tj
100 Tz
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1 0 0 1 97 248 Tm
96 Tz
(which) Tj
100 Tz
1 0 0 1 125 248 Tm
96 Tz
(when) Tj
100 Tz
1 0 0 1 152 248 Tm
98 Tz
(combined) Tj
100 Tz
1 0 0 1 198 248 Tm
95 Tz
(with) Tj
100 Tz
1 0 0 1 220 248 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 237 248 Tm
97 Tz
(previous) Tj
100 Tz
1 0 0 1 277 248 Tm
93 Tz
(inequality) Tj
100 Tz
1 0 0 1 323 248 Tm
91 Tz
(gives) Tj
100 Tz
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1 0 0 1 96 213 Tm
106 Tz
(This) Tj
100 Tz
1 0 0 1 119 213 Tm
106 Tz
(relation) Tj
100 Tz
1 0 0 1 155 213 Tm
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1 0 0 1 149 195 Tm
61 Tz
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(2) Tj
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(11) Tj
100 Tz
ET
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149 193 m
190 193 l
S
BT
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200 Tz
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100 Tz
1 0 0 1 212 195 Tm
105 Tz
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100 Tz
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1 0 0 1 256 195 Tm
(.\<) Tj
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1 0 0 1 271 195 Tm
33 Tz
(I) Tj
100 Tz
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1 0 0 1 275 195 Tm
83 Tz
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(2) Tj
100 Tz
1 0 0 1 289 195 Tm
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(1) Tj
100 Tz
ET
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304 193 l
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1 0 0 1 496 195 Tm
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100 Tz
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1 0 0 1 159 182 Tm
123 Tz
(1\261crk) Tj
100 Tz
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1 0 0 1 97 166 Tm
92 Tz
(Since) Tj
100 Tz
1 0 0 1 125 166 Tm
92 Tz
(typically) Tj
100 Tz
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1 0 0 1 167 166 Tm
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120 Tz
(0) Tj
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(A) Tj
100 Tz
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1 0 0 1 182 166 Tm
162 Tz
(\253) Tj
100 Tz
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40 Tz
(1) Tj
100 Tz
1 0 0 1 211 166 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 228 166 Tm
93 Tz
(see) Tj
100 Tz
1 0 0 1 247 166 Tm
95 Tz
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100 Tz
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1 0 0 1 274 166 Tm
95 Tz
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100 Tz
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1 0 0 1 301 166 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 322 166 Tm
(a) Tj
1 0 0 1 332 166 Tm
106 Tz
(very) Tj
100 Tz
1 0 0 1 356 166 Tm
(good) Tj
1 0 0 1 382 166 Tm
106 Tz
(estimate) Tj
100 Tz
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1 0 0 1 425 166 Tm
67 Tz
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100 Tz
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1 0 0 1 433 166 Tm
104 Tz
(upper) Tj
100 Tz
1 0 0 1 463 166 Tm
104 Tz
(bound) Tj
100 Tz
/F9 12 Tf
1 0 0 1 495 166 Tm
67 Tz
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100 Tz
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1 0 0 1 505 166 Tm
(for) Tj
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(1) Tj
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90 Tz
(by) Tj
100 Tz
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1 0 0 1 212 136 Tm
33 Tz
(I) Tj
100 Tz
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71 Tz
(\(y~) Tj
100 Tz
1 0 0 1 238 136 Tm
80 Tz
(y) Tj
-5 Ts

(1) Tj
100 Tz
ET
1 w
212 134 m
246 134 l
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100 Tz
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97 Tz
(estimate) Tj
100 Tz
1 0 0 1 158 107 Tm
92 Tz
(given) Tj
100 Tz
1 0 0 1 185 107 Tm
82 Tz
(by) Tj
100 Tz
1 0 0 1 200 107 Tm
90 Tz
(\(6.6\)) Tj
100 Tz
1 0 0 1 225 107 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 235 107 Tm
95 Tz
(obviously) Tj
100 Tz
1 0 0 1 280 107 Tm
96 Tz
(computable) Tj
100 Tz
1 0 0 1 334 107 Tm
95 Tz
(since) Tj
100 Tz
1 0 0 1 359 107 Tm
95 Tz
(both) Tj
100 Tz
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1 0 0 1 382 107 Tm
89 Tz
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(1) Tj
100 Tz
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89 Tz
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(2) Tj
100 Tz
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(known) Tj
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1 0 0 1 97 95 Tm
94 Tz
(The) Tj
100 Tz
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97 Tz
(estimate) Tj
100 Tz
1 0 0 1 160 95 Tm
92 Tz
(given) Tj
100 Tz
1 0 0 1 188 95 Tm
82 Tz
(by) Tj
100 Tz
1 0 0 1 204 95 Tm
90 Tz
(\(6.6\)) Tj
100 Tz
1 0 0 1 230 95 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 241 95 Tm
89 Tz
(very) Tj
100 Tz
1 0 0 1 264 95 Tm
94 Tz
(accurate) Tj
100 Tz
1 0 0 1 304 95 Tm
93 Tz
(and) Tj
100 Tz
1 0 0 1 324 95 Tm
94 Tz
(reliable,) Tj
100 Tz
1 0 0 1 364 95 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 377 95 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 393 95 Tm
90 Tz
(shall) Tj
100 Tz
1 0 0 1 418 95 Tm
93 Tz
(see) Tj
100 Tz
1 0 0 1 436 95 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 449 95 Tm
94 Tz
(section) Tj
100 Tz
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1 0 0 1 484 95 Tm
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1 0 0 1 339 415 Tm
(cos\(O,\),) Tj
1 0 0 1 376 415 Tm
108 Tz
(for) Tj
100 Tz
/F11 10 Tf
1 0 0 1 393 415 Tm
(i) Tj
/F5 5 Tf
1 0 0 1 400 415 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 411 415 Tm
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(1) Tj
100 Tz
/F11 10 Tf
1 0 0 1 445 415 Tm
89 Tz
(m,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 459 415 Tm
91 Tz
(we) Tj
100 Tz
1 0 0 1 474 415 Tm
104 Tz
(obtain) Tj
100 Tz
/F4 11 Tf
1 0 0 1 153 397 Tm
(Max) Tj
/F12 10 Tf
1 0 0 1 185 397 Tm
97 Tz
(\(P~\(x,\)\() Tj
100 Tz
/F9 12 Tf
1 0 0 1 224 397 Tm
200 Tz
(=) Tj
100 Tz
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1 0 0 1 236 397 Tm
92 Tz
(I\(PJ\() Tj
100 Tz
1 0 0 1 266 397 Tm
200 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 277 397 Tm
60 Tz
(2) Tj
100 Tz
/F12 10 Tf
1 0 0 1 294 397 Tm
(Max) Tj
/F7 10 Tf
1 0 0 1 325 397 Tm
97 Tz
(IP,,\(x,\)I) Tj
100 Tz
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1 0 0 1 147 391 Tm
150 Tz
(=) Tj
100 Tz
/F5 7 Tf
1 0 0 1 155 391 Tm
80 Tz
(1.) Tj
100 Tz
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67 Tz
(.rn) Tj
100 Tz
/F7 11 Tf
1 0 0 1 282 391 Tm
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(=) Tj
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(the) Tj
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(norm) Tj
100 Tz
1 0 0 1 156 374 Tm
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(of) Tj
100 Tz
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1 0 0 1 168 374 Tm
82 Tz
(P~) Tj
100 Tz
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1 0 0 1 182 374 Tm
94 Tz
(may) Tj
100 Tz
1 0 0 1 204 374 Tm
(be) Tj
1 0 0 1 217 374 Tm
107 Tz
(estimated) Tj
100 Tz
1 0 0 1 262 374 Tm
(to) Tj
1 0 0 1 274 374 Tm
104 Tz
(within) Tj
100 Tz
1 0 0 1 304 374 Tm
(a) Tj
1 0 0 1 312 374 Tm
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(factor) Tj
100 Tz
1 0 0 1 340 374 Tm
111 Tz
(of) Tj
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(2) Tj
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1 0 0 1 361 374 Tm
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(by) Tj
100 Tz
1 0 0 1 375 374 Tm
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(using) Tj
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1 0 0 1 401 374 Tm
(a) Tj
1 0 0 1 409 374 Tm
104 Tz
(search) Tj
100 Tz
1 0 0 1 440 374 Tm
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(pattern) Tj
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1 0 0 1 474 374 Tm
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1 0 0 1 93 362 Tm
93 Tz
(points.) Tj
100 Tz
1 0 0 1 128 362 Tm
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(Such) Tj
100 Tz
1 0 0 1 154 362 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 161 362 Tm
97 Tz
(search,) Tj
100 Tz
1 0 0 1 195 362 Tm
95 Tz
(over) Tj
100 Tz
1 0 0 1 218 362 Tm
90 Tz
(each) Tj
100 Tz
1 0 0 1 241 362 Tm
97 Tz
(interval) Tj
100 Tz
1 0 0 1 277 362 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 290 362 Tm
(the) Tj
1 0 0 1 307 362 Tm
96 Tz
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100 Tz
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1 0 0 1 334 362 Tm
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(if,) Tj
100 Tz
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1 0 0 1 347 362 Tm
83 Tz
(is) Tj
100 Tz
1 0 0 1 357 362 Tm
103 Tz
(sufficient) Tj
100 Tz
1 0 0 1 400 362 Tm
108 Tz
(for) Tj
100 Tz
1 0 0 1 416 362 Tm
107 Tz
(our) Tj
100 Tz
1 0 0 1 435 362 Tm
104 Tz
(needs,) Tj
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1 0 0 1 466 362 Tm
(and) Tj
1 0 0 1 485 362 Tm
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(results) Tj
100 Tz
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1 0 0 1 93 351 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 105 351 Tm
92 Tz
(the) Tj
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1 0 0 1 121 351 Tm
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(computation) Tj
100 Tz
1 0 0 1 179 351 Tm
(of) Tj
1 0 0 1 191 351 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 208 351 Tm
(error) Tj
1 0 0 1 233 351 Tm
98 Tz
(estimates) Tj
100 Tz
1 0 0 1 276 351 Tm
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(to) Tj
100 Tz
1 0 0 1 288 351 Tm
93 Tz
(within) Tj
100 Tz
1 0 0 1 318 351 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 326 351 Tm
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(factor) Tj
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(of) Tj
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(2.) Tj
100 Tz
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1 0 0 1 117 336 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 138 336 Tm
106 Tz
(very) Tj
100 Tz
1 0 0 1 160 336 Tm
(first) Tj
1 0 0 1 181 336 Tm
106 Tz
(user) Tj
100 Tz
1 0 0 1 202 336 Tm
105 Tz
(complaint) Tj
100 Tz
1 0 0 1 248 336 Tm
104 Tz
(about) Tj
100 Tz
1 0 0 1 276 336 Tm
105 Tz
(using) Tj
100 Tz
1 0 0 1 303 336 Tm
(\(6.6\)) Tj
1 0 0 1 328 336 Tm
(to) Tj
1 0 0 1 340 336 Tm
106 Tz
(estimate) Tj
100 Tz
1 0 0 1 379 336 Tm
(the) Tj
1 0 0 1 396 336 Tm
110 Tz
(error) Tj
100 Tz
1 0 0 1 422 336 Tm
(in) Tj
1 0 0 1 433 336 Tm
(the) Tj
1 0 0 1 450 336 Tm
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(Galerkin) Tj
100 Tz
1 0 0 1 492 336 Tm
120 Tz
(solu) Tj
100 Tz
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1 0 0 1 93 323 Tm
94 Tz
(tion) Tj
100 Tz
1 0 0 1 115 323 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 127 323 Tm
(that) Tj
1 0 0 1 149 323 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 167 323 Tm
97 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 210 323 Tm
95 Tz
(equations) Tj
100 Tz
1 0 0 1 256 323 Tm
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(must) Tj
100 Tz
1 0 0 1 283 323 Tm
90 Tz
(be) Tj
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1 0 0 1 298 323 Tm
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(formed) Tj
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93 Tz
(and) Tj
100 Tz
1 0 0 1 355 323 Tm
96 Tz
(solved) Tj
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(for) Tj
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(two) Tj
100 Tz
1 0 0 1 425 323 Tm
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(different) Tj
100 Tz
1 0 0 1 468 323 Tm
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100 Tz
1 0 0 1 505 323 Tm
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(in) Tj
100 Tz
1 0 0 1 93 312 Tm
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(order) Tj
100 Tz
1 0 0 1 120 312 Tm
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(to) Tj
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(estimate) Tj
100 Tz
1 0 0 1 172 312 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 189 312 Tm
(error) Tj
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(in) Tj
100 Tz
1 0 0 1 225 312 Tm
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(of) Tj
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(them.) Tj
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(Just) Tj
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(how) Tj
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(big) Tj
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(an) Tj
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1 0 0 1 362 312 Tm
98 Tz
(overhead) Tj
100 Tz
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(does) Tj
100 Tz
1 0 0 1 429 312 Tm
90 Tz
(\(6.6\)) Tj
100 Tz
1 0 0 1 454 312 Tm
94 Tz
(require) Tj
100 Tz
1 0 0 1 488 312 Tm
92 Tz
(above) Tj
100 Tz
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1 0 0 1 92 300 Tm
106 Tz
(just) Tj
100 Tz
1 0 0 1 112 300 Tm
94 Tz
(solving) Tj
100 Tz
1 0 0 1 146 300 Tm
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(y) Tj
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(2) Tj
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1 0 0 1 178 300 Tm
(the) Tj
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(numerical) Tj
100 Tz
1 0 0 1 242 300 Tm
106 Tz
(solution) Tj
100 Tz
1 0 0 1 280 300 Tm
(which) Tj
1 0 0 1 309 300 Tm
93 Tz
(will) Tj
100 Tz
1 0 0 1 328 300 Tm
(be) Tj
1 0 0 1 342 300 Tm
105 Tz
(used) Tj
100 Tz
1 0 0 1 365 300 Tm
103 Tz
(because) Tj
100 Tz
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(it) Tj
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(is) Tj
1 0 0 1 422 300 Tm
(the) Tj
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(more) Tj
100 Tz
1 0 0 1 465 300 Tm
103 Tz
(accurate) Tj
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100 Tz
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(typically) Tj
100 Tz
1 0 0 1 213 287 Tm
115 Tz
(k\>4) Tj
100 Tz
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(and) Tj
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1 0 0 1 257 287 Tm
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(o~) Tj
100 Tz
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1 0 0 1 282 287 Tm
142 Tz
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(2) Tj
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(1) Tj
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100 Tz
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1 0 0 1 337 287 Tm
(2/3,) Tj
1 0 0 1 359 287 Tm
(and) Tj
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(thus) Tj
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1 0 0 1 401 287 Tm
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(cr~) Tj
100 Tz
/F9 11 Tf
1 0 0 1 428 287 Tm
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(1/5.) Tj
100 Tz
1 0 0 1 452 287 Tm
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(Since) Tj
100 Tz
1 0 0 1 479 287 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 496 287 Tm
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(cost) Tj
100 Tz
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1 0 0 1 92 276 Tm
(of) Tj
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(obtaining) Tj
100 Tz
1 0 0 1 148 276 Tm
(a) Tj
1 0 0 1 156 276 Tm
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(Galerkin) Tj
100 Tz
1 0 0 1 198 276 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 236 276 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 246 276 Tm
96 Tz
(proportional) Tj
100 Tz
1 0 0 1 303 276 Tm
92 Tz
(\(5.18\)) Tj
100 Tz
1 0 0 1 333 276 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 345 276 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 362 276 Tm
(number) Tj
1 0 0 1 399 276 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 411 276 Tm
96 Tz
(points) Tj
100 Tz
1 0 0 1 441 276 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 452 276 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 470 276 Tm
96 Tz
(mesh,) Tj
100 Tz
1 0 0 1 499 276 Tm
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(the) Tj
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1 0 0 1 92 265 Tm
105 Tz
(ratio) Tj
100 Tz
1 0 0 1 116 265 Tm
111 Tz
(of) Tj
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1 0 0 1 129 265 Tm
(the) Tj
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(cost) Tj
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(of) Tj
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(computing) Tj
100 Tz
1 0 0 1 229 265 Tm
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(both) Tj
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100 Tz
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(to) Tj
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1 0 0 1 308 265 Tm
(just) Tj
1 0 0 1 328 265 Tm
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(computing) Tj
100 Tz
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1 0 0 1 378 265 Tm
89 Tz
(y) Tj
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100 Tz
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1 0 0 1 391 265 Tm
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(5/3.) Tj
100 Tz
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100 Tz
1 0 0 1 451 265 Tm
(at) Tj
1 0 0 1 462 265 Tm
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(first) Tj
100 Tz
1 0 0 1 482 265 Tm
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(glance,) Tj
100 Tz
1 0 0 1 92 253 Tm
94 Tz
(appears) Tj
100 Tz
1 0 0 1 129 253 Tm
(to) Tj
1 0 0 1 142 253 Tm
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(be) Tj
100 Tz
1 0 0 1 157 253 Tm
(a) Tj
1 0 0 1 167 253 Tm
(rather) Tj
1 0 0 1 197 253 Tm
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(large) Tj
100 Tz
1 0 0 1 222 253 Tm
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(price) Tj
100 Tz
1 0 0 1 247 253 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 260 253 Tm
93 Tz
(pay) Tj
100 Tz
1 0 0 1 280 253 Tm
(for) Tj
1 0 0 1 297 253 Tm
(error) Tj
1 0 0 1 323 253 Tm
95 Tz
(estimates.) Tj
100 Tz
1 0 0 1 373 253 Tm
93 Tz
(However,) Tj
100 Tz
1 0 0 1 420 253 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 438 253 Tm
96 Tz
(alternative) Tj
100 Tz
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1 0 0 1 488 253 Tm
(is) Tj
1 0 0 1 499 253 Tm
(the) Tj
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1 0 0 1 92 242 Tm
105 Tz
(relatively) Tj
100 Tz
1 0 0 1 136 242 Tm
104 Tz
(cheap) Tj
100 Tz
1 0 0 1 165 242 Tm
106 Tz
(computation) Tj
100 Tz
1 0 0 1 222 242 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 235 242 Tm
(a) Tj
1 0 0 1 244 242 Tm
93 Tz
(pile) Tj
100 Tz
1 0 0 1 263 242 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 276 242 Tm
108 Tz
(numbers) Tj
100 Tz
1 0 0 1 318 242 Tm
(of) Tj
1 0 0 1 331 242 Tm
102 Tz
(absolutely) Tj
100 Tz
1 0 0 1 378 242 Tm
105 Tz
(unknown) Tj
100 Tz
1 0 0 1 422 242 Tm
(accuracy.) Tj
1 0 0 1 468 242 Tm
106 Tz
(The) Tj
100 Tz
1 0 0 1 488 242 Tm
118 Tz
(situa) Tj
100 Tz
(-) Tj
1 0 0 1 92 229 Tm
106 Tz
(tion) Tj
100 Tz
1 0 0 1 113 229 Tm
94 Tz
(may) Tj
100 Tz
1 0 0 1 134 229 Tm
(be) Tj
1 0 0 1 148 229 Tm
108 Tz
(summed) Tj
100 Tz
1 0 0 1 188 229 Tm
(up) Tj
1 0 0 1 202 229 Tm
91 Tz
(as:) Tj
100 Tz
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1 0 0 1 119 215 Tm
92 Tz
(We) Tj
100 Tz
1 0 0 1 135 215 Tm
(can) Tj
1 0 0 1 152 215 Tm
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1 0 0 1 167 215 Tm
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(the) Tj
100 Tz
1 0 0 1 184 215 Tm
92 Tz
(\(potentially\)) Tj
100 Tz
1 0 0 1 236 215 Tm
89 Tz
(wrong) Tj
100 Tz
1 0 0 1 263 215 Tm
88 Tz
(answer) Tj
100 Tz
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84 Tz
(cheaply,) Tj
100 Tz
1 0 0 1 331 215 Tm
85 Tz
(the) Tj
100 Tz
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(correct) Tj
100 Tz
1 0 0 1 377 215 Tm
88 Tz
(answer) Tj
100 Tz
1 0 0 1 409 215 Tm
78 Tz
(costs) Tj
100 Tz
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91 Tz
(more) Tj
100 Tz
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100 Tz
1 0 0 1 466 215 Tm
87 Tz
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100 Tz
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1 0 0 1 117 199 Tm
109 Tz
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100 Tz
1 0 0 1 157 199 Tm
110 Tz
(error) Tj
100 Tz
1 0 0 1 183 199 Tm
107 Tz
(estimation) Tj
100 Tz
1 0 0 1 232 199 Tm
106 Tz
(scheme) Tj
100 Tz
1 0 0 1 269 199 Tm
106 Tz
(that) Tj
100 Tz
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107 Tz
(can) Tj
100 Tz
1 0 0 1 309 199 Tm
(be) Tj
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(used) Tj
100 Tz
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(is) Tj
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(obtained) Tj
100 Tz
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(from) Tj
100 Tz
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(by) Tj
100 Tz
1 0 0 1 464 199 Tm
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(noting) Tj
100 Tz
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(1) Tj
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(\(6.7\)) Tj
1 0 0 1 220 157 Tm
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(1) Tj
100 Tz
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100 Tz
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(relation) Tj
100 Tz
1 0 0 1 153 140 Tm
93 Tz
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100 Tz
1 0 0 1 172 140 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 187 140 Tm
95 Tz
(used) Tj
100 Tz
1 0 0 1 211 140 Tm
88 Tz
(to) Tj
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1 0 0 1 224 140 Tm
97 Tz
(estimate) Tj
100 Tz
1 0 0 1 264 140 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 282 140 Tm
(error) Tj
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100 Tz
1 0 0 1 321 140 Tm
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(the) Tj
100 Tz
1 0 0 1 339 140 Tm
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(most) Tj
100 Tz
1 0 0 1 364 140 Tm
94 Tz
(accurate) Tj
100 Tz
1 0 0 1 403 140 Tm
98 Tz
(approximate) Tj
100 Tz
1 0 0 1 461 140 Tm
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(solution) Tj
100 Tz
1 0 0 1 500 140 Tm
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(we) Tj
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100 Tz
1 0 0 1 116 129 Tm
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100 Tz
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(y) Tj
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(the) Tj
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90 Tz
(least) Tj
100 Tz
1 0 0 1 276 129 Tm
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(accurate,) Tj
100 Tz
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91 Tz
(y) Tj
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(1) Tj
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100 Tz
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106 Tz
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100 Tz
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(seems) Tj
100 Tz
1 0 0 1 390 129 Tm
93 Tz
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(it) Tj
100 Tz
/F9 11 Tf
1 0 0 1 269 574 Tm
97 Tz
(consists) Tj
100 Tz
1 0 0 1 307 574 Tm
(of) Tj
1 0 0 1 320 574 Tm
60 Tz
(2) Tj
100 Tz
1 0 0 1 329 574 Tm
96 Tz
(differential) Tj
100 Tz
1 0 0 1 381 574 Tm
(equations,) Tj
1 0 0 1 430 574 Tm
90 Tz
(each) Tj
100 Tz
1 0 0 1 454 574 Tm
94 Tz
(holding) Tj
100 Tz
1 0 0 1 490 574 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 506 574 Tm
93 Tz
(one) Tj
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/F9 11 Tf
1 0 0 1 99 562 Tm
94 Tz
(side) Tj
100 Tz
1 0 0 1 120 562 Tm
(of) Tj
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(an) Tj
1 0 0 1 149 562 Tm
92 Tz
(\203interface\204.) Tj
100 Tz
1 0 0 1 203 562 Tm
91 Tz
(Such) Tj
100 Tz
1 0 0 1 229 562 Tm
97 Tz
(interface) Tj
100 Tz
1 0 0 1 271 562 Tm
98 Tz
(problems) Tj
100 Tz
1 0 0 1 315 562 Tm
96 Tz
(occur) Tj
100 Tz
1 0 0 1 343 562 Tm
96 Tz
(frequently) Tj
100 Tz
1 0 0 1 392 562 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 405 562 Tm
94 Tz
(practice) Tj
100 Tz
1 0 0 1 442 562 Tm
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(it) Tj
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(is) Tj
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104 Tz
(useful) Tj
100 Tz
1 0 0 1 513 562 Tm
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1 0 0 1 99 550 Tm
93 Tz
(see) Tj
100 Tz
1 0 0 1 117 550 Tm
94 Tz
(how) Tj
100 Tz
1 0 0 1 138 550 Tm
98 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 188 550 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 224 550 Tm
97 Tz
(handles) Tj
100 Tz
1 0 0 1 261 550 Tm
(them.) Tj
1 0 0 1 292 550 Tm
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(Second,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 330 550 Tm
83 Tz
(it) Tj
100 Tz
1 0 0 1 340 550 Tm
86 Tz
(is) Tj
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1 0 0 1 350 550 Tm
(a) Tj
1 0 0 1 358 550 Tm
97 Tz
(problem) Tj
100 Tz
1 0 0 1 398 550 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 409 550 Tm
96 Tz
(which) Tj
100 Tz
1 0 0 1 439 550 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 456 550 Tm
96 Tz
(exact) Tj
100 Tz
1 0 0 1 481 550 Tm
94 Tz
(answer) Tj
100 Tz
1 0 0 1 515 550 Tm
71 Tz
(is) Tj
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1 0 0 1 99 538 Tm
97 Tz
(known) Tj
100 Tz
1 0 0 1 132 538 Tm
(and) Tj
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92 Tz
(we) Tj
100 Tz
1 0 0 1 168 538 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 187 538 Tm
97 Tz
(examine) Tj
100 Tz
1 0 0 1 228 538 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 246 538 Tm
98 Tz
(convergence) Tj
100 Tz
1 0 0 1 304 538 Tm
94 Tz
(rate) Tj
100 Tz
1 0 0 1 325 538 Tm
(of) Tj
1 0 0 1 337 538 Tm
98 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 387 538 Tm
(method,) Tj
1 0 0 1 427 538 Tm
(and) Tj
1 0 0 1 447 538 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 464 538 Tm
(error) Tj
1 0 0 1 490 538 Tm
107 Tz
(estima) Tj
100 Tz
(-) Tj
1 0 0 1 99 526 Tm
94 Tz
(tion) Tj
100 Tz
1 0 0 1 120 526 Tm
97 Tz
(schemes) Tj
100 Tz
1 0 0 1 161 526 Tm
(of) Tj
1 0 0 1 174 526 Tm
94 Tz
(section) Tj
100 Tz
1 0 0 1 207 526 Tm
88 Tz
(6,) Tj
100 Tz
1 0 0 1 220 526 Tm
(for) Tj
1 0 0 1 237 526 Tm
93 Tz
(this) Tj
100 Tz
1 0 0 1 256 526 Tm
93 Tz
(simple) Tj
100 Tz
1 0 0 1 288 526 Tm
95 Tz
(problem.) Tj
100 Tz
1 0 0 1 333 526 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 355 526 Tm
93 Tz
(last) Tj
100 Tz
1 0 0 1 373 526 Tm
97 Tz
(purpose) Tj
100 Tz
1 0 0 1 411 526 Tm
96 Tz
(served) Tj
100 Tz
1 0 0 1 443 526 Tm
82 Tz
(by) Tj
100 Tz
1 0 0 1 457 526 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 474 526 Tm
94 Tz
(first) Tj
100 Tz
1 0 0 1 494 526 Tm
112 Tz
(exam) Tj
100 Tz
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(ple) Tj
100 Tz
1 0 0 1 116 514 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 127 514 Tm
(to) Tj
1 0 0 1 139 514 Tm
(exhibit) Tj
1 0 0 1 174 514 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 182 514 Tm
94 Tz
(\203superconvergence\204) Tj
100 Tz
1 0 0 1 271 514 Tm
95 Tz
(property) Tj
100 Tz
1 0 0 1 312 514 Tm
(\[10,11,241) Tj
1 0 0 1 360 514 Tm
(of) Tj
1 0 0 1 373 514 Tm
98 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 423 514 Tm
97 Tz
(method.) Tj
100 Tz
1 0 0 1 465 514 Tm
87 Tz
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100 Tz
1 0 0 1 483 514 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 502 514 Tm
(then) Tj
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(note) Tj
1 0 0 1 122 503 Tm
(that) Tj
1 0 0 1 143 503 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 159 503 Tm
(error) Tj
1 0 0 1 185 503 Tm
98 Tz
(estimation) Tj
100 Tz
1 0 0 1 233 503 Tm
97 Tz
(schemes) Tj
100 Tz
1 0 0 1 273 503 Tm
(of) Tj
1 0 0 1 286 503 Tm
94 Tz
(section) Tj
100 Tz
1 0 0 1 320 503 Tm
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(6) Tj
100 Tz
1 0 0 1 329 503 Tm
92 Tz
(are) Tj
100 Tz
1 0 0 1 345 503 Tm
96 Tz
(robust) Tj
100 Tz
1 0 0 1 376 503 Tm
97 Tz
(enough) Tj
100 Tz
1 0 0 1 412 503 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 424 503 Tm
96 Tz
(detect) Tj
100 Tz
1 0 0 1 453 503 Tm
93 Tz
(and) Tj
100 Tz
1 0 0 1 472 503 Tm
97 Tz
(exhibit) Tj
100 Tz
1 0 0 1 506 503 Tm
93 Tz
(this) Tj
100 Tz
1 0 0 1 99 491 Tm
95 Tz
(property) Tj
100 Tz
1 0 0 1 139 491 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 152 491 Tm
90 Tz
(well.) Tj
100 Tz
1 0 0 1 179 491 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 200 491 Tm
97 Tz
(second) Tj
100 Tz
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(example) Tj
100 Tz
1 0 0 1 274 491 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 284 491 Tm
97 Tz
(cooked) Tj
100 Tz
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91 Tz
(up) Tj
100 Tz
1 0 0 1 333 491 Tm
88 Tz
(to) Tj
100 Tz
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88 Tz
(\203break\204) Tj
100 Tz
1 0 0 1 381 491 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 398 491 Tm
(error) Tj
1 0 0 1 424 491 Tm
98 Tz
(estimation) Tj
100 Tz
1 0 0 1 472 491 Tm
95 Tz
(schemes) Tj
100 Tz
1 0 0 1 512 491 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 99 478 Tm
94 Tz
(section) Tj
100 Tz
1 0 0 1 133 478 Tm
88 Tz
(6.) Tj
100 Tz
1 0 0 1 148 478 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 169 478 Tm
104 Tz
(extent) Tj
100 Tz
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(to) Tj
100 Tz
1 0 0 1 212 478 Tm
92 Tz
(which) Tj
100 Tz
1 0 0 1 242 478 Tm
96 Tz
(those) Tj
100 Tz
1 0 0 1 268 478 Tm
97 Tz
(schemes) Tj
100 Tz
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92 Tz
(break) Tj
100 Tz
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96 Tz
(down) Tj
100 Tz
1 0 0 1 363 478 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 373 478 Tm
95 Tz
(discussed) Tj
100 Tz
1 0 0 1 418 478 Tm
(and) Tj
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96 Tz
(compared,) Tj
100 Tz
1 0 0 1 486 478 Tm
92 Tz
(showing) Tj
100 Tz
1 0 0 1 99 466 Tm
(that) Tj
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(the) Tj
100 Tz
1 0 0 1 138 466 Tm
97 Tz
(second) Tj
100 Tz
1 0 0 1 171 466 Tm
(error) Tj
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98 Tz
(estimation) Tj
100 Tz
1 0 0 1 246 466 Tm
97 Tz
(scheme) Tj
100 Tz
1 0 0 1 282 466 Tm
(of) Tj
1 0 0 1 295 466 Tm
94 Tz
(section) Tj
100 Tz
1 0 0 1 329 466 Tm
80 Tz
(6) Tj
100 Tz
1 0 0 1 338 466 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 349 466 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 361 466 Tm
94 Tz
(fact) Tj
100 Tz
1 0 0 1 380 466 Tm
(rather) Tj
1 0 0 1 410 466 Tm
96 Tz
(unreliable,) Tj
100 Tz
1 0 0 1 460 466 Tm
92 Tz
(while) Tj
100 Tz
1 0 0 1 487 466 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 504 466 Tm
94 Tz
(first) Tj
100 Tz
1 0 0 1 99 455 Tm
97 Tz
(scheme) Tj
100 Tz
1 0 0 1 135 455 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 145 455 Tm
95 Tz
(quite) Tj
100 Tz
1 0 0 1 171 455 Tm
94 Tz
(reliable.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 124 439 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 146 439 Tm
97 Tz
(previous) Tj
100 Tz
1 0 0 1 189 439 Tm
94 Tz
(sections) Tj
100 Tz
1 0 0 1 229 439 Tm
(have) Tj
1 0 0 1 256 439 Tm
95 Tz
(described) Tj
100 Tz
1 0 0 1 302 439 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 354 439 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 392 439 Tm
(for) Tj
1 0 0 1 410 439 Tm
(a) Tj
1 0 0 1 421 439 Tm
(rather) Tj
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96 Tz
(clean,) Tj
100 Tz
1 0 0 1 483 439 Tm
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(textbook) Tj
100 Tz
1 0 0 1 100 427 Tm
97 Tz
(problem,) Tj
100 Tz
1 0 0 1 143 427 Tm
108 Tz
(\(l.I\)-\(1.2\).) Tj
100 Tz
1 0 0 1 199 427 Tm
98 Tz
(Problems) Tj
100 Tz
1 0 0 1 244 427 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 256 427 Tm
94 Tz
(real) Tj
100 Tz
1 0 0 1 276 427 Tm
86 Tz
(life) Tj
100 Tz
1 0 0 1 293 427 Tm
92 Tz
(are) Tj
100 Tz
1 0 0 1 309 427 Tm
(often) Tj
1 0 0 1 336 427 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 354 427 Tm
(that) Tj
1 0 0 1 374 427 Tm
94 Tz
(simple.) Tj
100 Tz
1 0 0 1 412 427 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 432 427 Tm
94 Tz
(first) Tj
100 Tz
1 0 0 1 452 427 Tm
94 Tz
(sample) Tj
100 Tz
1 0 0 1 486 427 Tm
94 Tz
(problem) Tj
100 Tz
1 0 0 1 100 416 Tm
98 Tz
(illustrates) Tj
100 Tz
1 0 0 1 145 416 Tm
(a) Tj
1 0 0 1 154 416 Tm
97 Tz
(common) Tj
100 Tz
1 0 0 1 197 416 Tm
93 Tz
(\203unclean\204) Tj
100 Tz
1 0 0 1 242 416 Tm
97 Tz
(problem) Tj
100 Tz
1 0 0 1 283 416 Tm
96 Tz
(which) Tj
100 Tz
1 0 0 1 312 416 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 331 416 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 346 416 Tm
98 Tz
(laundered) Tj
100 Tz
1 0 0 1 393 416 Tm
94 Tz
(into) Tj
100 Tz
1 0 0 1 415 416 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 433 416 Tm
95 Tz
(form) Tj
100 Tz
1 0 0 1 458 416 Tm
(of) Tj
1 0 0 1 472 416 Tm
98 Tz
(\(1.1\)-\(1.2\).) Tj
100 Tz
1 0 0 1 99 405 Tm
97 Tz
(Consider) Tj
100 Tz
1 0 0 1 142 405 Tm
(the) Tj
1 0 0 1 159 405 Tm
97 Tz
(problem) Tj
100 Tz
1 0 0 1 148 386 Tm
129 Tz
(y\202=0) Tj
100 Tz
1 0 0 1 193 386 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 211 386 Tm
88 Tz
(\(\2051,0\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 500 366 Tm
(\(7.1\)) Tj
/F1 11 Tf
1 0 0 1 147 338 Tm
(~) Tj
1 0 0 1 226 338 Tm
107 Tz
(=0) Tj
100 Tz
1 0 0 1 258 338 Tm
85 Tz
(on\(0,+1\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 99 322 Tm
106 Tz
(with) Tj
100 Tz
1 0 0 1 121 322 Tm
94 Tz
(y) Tj
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(1) Tj
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(\(\2051\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 155 322 Tm
167 Tz
(=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 166 322 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 174 322 Tm
93 Tz
(and) Tj
100 Tz
1 0 0 1 192 322 Tm
97 Tz
(y) Tj
-5 Ts

(2) Tj
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(\(+1\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 227 322 Tm
167 Tz
(=) Tj
100 Tz
1 0 0 1 256 322 Tm
167 Tz
(=) Tj
100 Tz
/F9 12 Tf
1 0 0 1 267 322 Tm
61 Tz
(.~/~777i7,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 302 322 Tm
94 Tz
(subject) Tj
100 Tz
1 0 0 1 335 322 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 347 322 Tm
97 Tz
(interface) Tj
100 Tz
1 0 0 1 388 322 Tm
98 Tz
(conditions) Tj
100 Tz
/F10 11 Tf
1 0 0 1 148 305 Tm
96 Tz
(y) Tj
-5 Ts

(1) Tj
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(\(0\)) Tj
100 Tz
/F1 11 Tf
1 0 0 1 175 305 Tm
79 Tz
(=y) Tj
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(2) Tj
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(\(0\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 495 305 Tm
104 Tz
(\(7.2a\)) Tj
100 Tz
/F11 10 Tf
1 0 0 1 148 287 Tm
112 Tz
(yj) Tj
100 Tz
/F9 11 Tf
1 0 0 1 159 287 Tm
92 Tz
(\(0\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 176 287 Tm
167 Tz
(=) Tj
100 Tz
/F1 11 Tf
1 0 0 1 186 287 Tm
74 Tz
(2y~) Tj
100 Tz
/F9 11 Tf
1 0 0 1 204 287 Tm
93 Tz
(\(0\).) Tj
100 Tz
/F9 10 Tf
1 0 0 1 495 287 Tm
(\(7.) Tj
1 0 0 1 507 287 Tm
92 Tz
(2b\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 99 269 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 120 269 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 158 269 Tm
(of) Tj
1 0 0 1 171 269 Tm
(this) Tj
1 0 0 1 190 269 Tm
97 Tz
(problem) Tj
100 Tz
1 0 0 1 229 269 Tm
86 Tz
(is) Tj
100 Tz
/F7 9 Tf
1 0 0 1 148 250 Tm
117 Tz
(y) Tj
-5 Ts

(1) Tj
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(\(x\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 176 250 Tm
167 Tz
(=) Tj
100 Tz
/F9 10 Tf
1 0 0 1 186 250 Tm
112 Tz
(x+1) Tj
100 Tz
1 0 0 1 218 250 Tm
(on) Tj
1 0 0 1 237 250 Tm
97 Tz
(\(\2051,0\)) Tj
100 Tz
1 0 0 1 500 234 Tm
(\(7.3\)) Tj
/F9 10 Tf
1 0 0 1 218 217 Tm
(on) Tj
1 0 0 1 237 217 Tm
132 Tz
(\(0,\2611\).) Tj
100 Tz
/F9 11 Tf
1 0 0 1 100 199 Tm
97 Tz
(Interface) Tj
100 Tz
1 0 0 1 142 199 Tm
98 Tz
(conditions) Tj
100 Tz
1 0 0 1 191 199 Tm
90 Tz
(\(7.2\)) Tj
100 Tz
1 0 0 1 215 199 Tm
91 Tz
(do) Tj
100 Tz
1 0 0 1 229 199 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 247 199 Tm
89 Tz
(fit) Tj
100 Tz
1 0 0 1 259 199 Tm
94 Tz
(into) Tj
100 Tz
1 0 0 1 280 199 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 297 199 Tm
97 Tz
(problem) Tj
100 Tz
1 0 0 1 336 199 Tm
98 Tz
(formulation) Tj
100 Tz
/F9 10 Tf
1 0 0 1 391 199 Tm
108 Tz
(\(1.1\)-\(1.2\),) Tj
100 Tz
1 0 0 1 444 199 Tm
(which) Tj
1 0 0 1 473 199 Tm
106 Tz
(only) Tj
100 Tz
1 0 0 1 495 199 Tm
96 Tz
(allows) Tj
100 Tz
/F9 11 Tf
1 0 0 1 100 187 Tm
98 Tz
(conditions) Tj
100 Tz
1 0 0 1 149 187 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 162 187 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 176 187 Tm
95 Tz
(imposed) Tj
100 Tz
1 0 0 1 217 187 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 233 187 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 250 187 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 289 187 Tm
(of) Tj
1 0 0 1 303 187 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 320 187 Tm
96 Tz
(differential) Tj
100 Tz
1 0 0 1 372 187 Tm
97 Tz
(equation) Tj
100 Tz
1 0 0 1 413 187 Tm
(at) Tj
1 0 0 1 426 187 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 444 187 Tm
(end) Tj
1 0 0 1 464 187 Tm
96 Tz
(points) Tj
100 Tz
1 0 0 1 494 187 Tm
(of) Tj
1 0 0 1 508 187 Tm
92 Tz
(the) Tj
100 Tz
/F9 11 Tf
1 0 0 1 100 175 Tm
94 Tz
(interval.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 124 160 Tm
92 Tz
(To) Tj
100 Tz
1 0 0 1 140 160 Tm
93 Tz
(see) Tj
100 Tz
1 0 0 1 158 160 Tm
94 Tz
(how) Tj
100 Tz
1 0 0 1 180 160 Tm
96 Tz
(Galerkin\202s) Tj
100 Tz
1 0 0 1 230 160 Tm
97 Tz
(method) Tj
100 Tz
1 0 0 1 267 160 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 286 160 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 300 160 Tm
95 Tz
(used) Tj
100 Tz
1 0 0 1 324 160 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 336 160 Tm
96 Tz
(solve) Tj
100 Tz
1 0 0 1 363 160 Tm
(\(7.1\)-\(7.2\)) Tj
1 0 0 1 414 160 Tm
97 Tz
(requires) Tj
100 Tz
1 0 0 1 452 160 Tm
(a) Tj
1 0 0 1 461 160 Tm
90 Tz
(little) Tj
100 Tz
1 0 0 1 484 160 Tm
92 Tz
(analysis,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 100 148 Tm
107 Tz
(and) Tj
100 Tz
1 0 0 1 119 148 Tm
104 Tz
(handwaving.) Tj
100 Tz
1 0 0 1 181 148 Tm
83 Tz
(It) Tj
100 Tz
1 0 0 1 191 148 Tm
(is) Tj
1 0 0 1 201 148 Tm
105 Tz
(instructive) Tj
100 Tz
1 0 0 1 251 148 Tm
(to) Tj
1 0 0 1 263 148 Tm
108 Tz
(see) Tj
100 Tz
1 0 0 1 280 148 Tm
(how) Tj
1 0 0 1 302 148 Tm
105 Tz
(such) Tj
100 Tz
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1 0 0 1 383 619 Tm
107 Tz
(smoothly,) Tj
100 Tz
1 0 0 1 431 619 Tm
108 Tz
(not) Tj
100 Tz
1 0 0 1 449 619 Tm
105 Tz
(abruptly,) Tj
100 Tz
1 0 0 1 491 619 Tm
105 Tz
(from) Tj
100 Tz
/F8 8 Tf
1 0 0 1 516 619 Tm
89 Tz
(aL) Tj
100 Tz
/F9 11 Tf
1 0 0 1 105 607 Tm
88 Tz
(to) Tj
100 Tz
/F7 10 Tf
1 0 0 1 117 607 Tm
75 Tz
(aR) Tj
100 Tz
/F9 11 Tf
1 0 0 1 132 607 Tm
97 Tz
(between) Tj
100 Tz
1 0 0 1 172 607 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 189 607 Tm
98 Tz
(Gaussian) Tj
100 Tz
1 0 0 1 233 607 Tm
96 Tz
(quadrature) Tj
100 Tz
1 0 0 1 283 607 Tm
96 Tz
(points) Tj
100 Tz
1 0 0 1 312 607 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 327 607 Tm
96 Tz
(either) Tj
100 Tz
1 0 0 1 356 607 Tm
94 Tz
(side) Tj
100 Tz
1 0 0 1 377 607 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 390 607 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 407 607 Tm
93 Tz
(origin.) Tj
100 Tz
1 0 0 1 442 607 Tm
89 Tz
(In) Tj
100 Tz
1 0 0 1 455 607 Tm
93 Tz
(this) Tj
100 Tz
1 0 0 1 474 607 Tm
90 Tz
(case,) Tj
100 Tz
1 0 0 1 498 607 Tm
83 Tz
(we) Tj
100 Tz
1 0 0 1 514 607 Tm
92 Tz
(are) Tj
100 Tz
/F9 10 Tf
1 0 0 1 105 595 Tm
104 Tz
(trying) Tj
100 Tz
1 0 0 1 133 595 Tm
(to) Tj
1 0 0 1 145 595 Tm
108 Tz
(approximate) Tj
100 Tz
1 0 0 1 202 595 Tm
109 Tz
(another,) Tj
100 Tz
1 0 0 1 242 595 Tm
(slightly) Tj
1 0 0 1 276 595 Tm
108 Tz
(different,) Tj
100 Tz
1 0 0 1 320 595 Tm
106 Tz
(solution.) Tj
100 Tz
/F7 10 Tf
1 0 0 1 130 580 Tm
95 Tz
(How) Tj
100 Tz
1 0 0 1 152 580 Tm
81 Tz
(does) Tj
100 Tz
1 0 0 1 174 580 Tm
92 Tz
(Galerkin) Tj
100 Tz
1 0 0 1 211 580 Tm
57 Tz
(\201s) Tj
100 Tz
1 0 0 1 219 580 Tm
88 Tz
(method) Tj
100 Tz
1 0 0 1 252 580 Tm
91 Tz
(know) Tj
100 Tz
1 0 0 1 278 580 Tm
84 Tz
(which) Tj
100 Tz
1 0 0 1 304 580 Tm
91 Tz
(solution) Tj
100 Tz
1 0 0 1 339 580 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 349 580 Tm
92 Tz
(approximate?) Tj
100 Tz
/F9 11 Tf
1 0 0 1 105 565 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 127 565 Tm
91 Tz
(main) Tj
100 Tz
1 0 0 1 155 565 Tm
95 Tz
(difference) Tj
100 Tz
1 0 0 1 203 565 Tm
94 Tz
(between) Tj
100 Tz
1 0 0 1 244 565 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 263 565 Tm
94 Tz
(two) Tj
100 Tz
1 0 0 1 284 565 Tm
98 Tz
(solutions,) Tj
100 Tz
1 0 0 1 332 565 Tm
(for) Tj
/F7 10 Tf
1 0 0 1 350 565 Tm
80 Tz
(a) Tj
100 Tz
1 0 0 1 358 565 Tm
108 Tz
(\(x\)) Tj
100 Tz
/F9 11 Tf
1 0 0 1 377 565 Tm
97 Tz
(smooth) Tj
100 Tz
1 0 0 1 415 565 Tm
(and) Tj
1 0 0 1 437 565 Tm
97 Tz
(discontinuous,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 505 565 Tm
(is) Tj
1 0 0 1 517 565 Tm
111 Tz
(of) Tj
100 Tz
/F9 10 Tf
1 0 0 1 105 553 Tm
104 Tz
(course) Tj
100 Tz
1 0 0 1 137 553 Tm
(in) Tj
1 0 0 1 149 553 Tm
(the) Tj
1 0 0 1 167 553 Tm
107 Tz
(continuity) Tj
100 Tz
1 0 0 1 214 553 Tm
(of) Tj
1 0 0 1 226 553 Tm
112 Tz
(y\202at) Tj
100 Tz
1 0 0 1 248 553 Tm
(0,) Tj
/F9 10 Tf
1 0 0 1 260 553 Tm
(the) Tj
1 0 0 1 277 553 Tm
107 Tz
(former) Tj
100 Tz
1 0 0 1 311 553 Tm
(being) Tj
1 0 0 1 337 553 Tm
106 Tz
(smooth) Tj
100 Tz
1 0 0 1 374 553 Tm
(and) Tj
1 0 0 1 393 553 Tm
(the) Tj
1 0 0 1 410 553 Tm
105 Tz
(latter) Tj
100 Tz
1 0 0 1 437 553 Tm
(being) Tj
1 0 0 1 463 553 Tm
105 Tz
(discontinuous.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 105 541 Tm
109 Tz
(Remember) Tj
100 Tz
1 0 0 1 157 541 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 178 541 Tm
(the) Tj
1 0 0 1 195 541 Tm
105 Tz
(continuity) Tj
100 Tz
1 0 0 1 242 541 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 255 541 Tm
(a) Tj
1 0 0 1 263 541 Tm
103 Tz
(B-spline) Tj
100 Tz
1 0 0 1 302 541 Tm
(at) Tj
1 0 0 1 313 541 Tm
(a) Tj
1 0 0 1 321 541 Tm
111 Tz
(knot) Tj
100 Tz
/F7 10 Tf
1 0 0 1 344 541 Tm
86 Tz
(x,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 356 541 Tm
(is) Tj
1 0 0 1 366 541 Tm
108 Tz
(determined) Tj
100 Tz
1 0 0 1 419 541 Tm
90 Tz
(by) Tj
100 Tz
1 0 0 1 433 541 Tm
(the) Tj
1 0 0 1 451 541 Tm
102 Tz
(multiplicity) Tj
100 Tz
/F7 10 Tf
1 0 0 1 503 541 Tm
73 Tz
(m,) Tj
100 Tz
/F9 10 Tf
1 0 0 1 517 541 Tm
111 Tz
(of) Tj
100 Tz
/F9 10 Tf
1 0 0 1 105 528 Tm
106 Tz
(that) Tj
100 Tz
1 0 0 1 125 528 Tm
111 Tz
(knot) Tj
100 Tz
1 0 0 1 148 528 Tm
(in) Tj
1 0 0 1 160 528 Tm
(the) Tj
1 0 0 1 178 528 Tm
104 Tz
(mesh.) Tj
100 Tz
1 0 0 1 210 528 Tm
(Specifically,) Tj
1 0 0 1 264 528 Tm
(for) Tj
1 0 0 1 280 528 Tm
(any) Tj
1 0 0 1 299 528 Tm
103 Tz
(B-spline) Tj
100 Tz
/F7 10 Tf
1 0 0 1 338 528 Tm
75 Tz
(u,) Tj
100 Tz
/F9 11 Tf
1 0 0 1 349 528 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 365 528 Tm
95 Tz
(have) Tj
100 Tz
1 0 0 1 388 528 Tm
(that) Tj
/F7 10 Tf
1 0 0 1 409 528 Tm
83 Tz
(u) Tj
5 Ts

(101) Tj
5 Ts

(\(x,\)) Tj
100 Tz
1 0 0 1 471 528 Tm
80 Tz
(u) Tj
100 Tz
/F7 11 Tf
1 0 0 1 478 528 Tm
55 Tz
(iA\205tn,) Tj
100 Tz
1 0 0 1 504 528 Tm
(~) Tj
/F7 10 Tf
1 0 0 1 510 528 Tm
93 Tz
(\(x,\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 105 516 Tm
(are) Tj
1 0 0 1 122 516 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 135 516 Tm
110 Tz
(continuous) Tj
100 Tz
1 0 0 1 188 516 Tm
(at) Tj
/F7 10 Tf
1 0 0 1 199 516 Tm
(x,,) Tj
/F9 10 Tf
1 0 0 1 214 516 Tm
108 Tz
(but) Tj
100 Tz
1 0 0 1 232 516 Tm
90 Tz
(all) Tj
100 Tz
1 0 0 1 246 516 Tm
107 Tz
(higher) Tj
100 Tz
1 0 0 1 278 516 Tm
107 Tz
(derivatives) Tj
100 Tz
1 0 0 1 329 516 Tm
(of) Tj
/F7 10 Tf
1 0 0 1 342 516 Tm
80 Tz
(u) Tj
100 Tz
/F9 10 Tf
1 0 0 1 350 516 Tm
94 Tz
(may) Tj
100 Tz
1 0 0 1 372 516 Tm
(be) Tj
1 0 0 1 386 516 Tm
107 Tz
(discontinuous) Tj
100 Tz
1 0 0 1 450 516 Tm
(at) Tj
/F7 10 Tf
1 0 0 1 461 516 Tm
90 Tz
(x,.) Tj
100 Tz
/F9 10 Tf
1 0 0 1 479 516 Tm
(Recall) Tj
1 0 0 1 509 516 Tm
(that) Tj
/F9 10 Tf
1 0 0 1 105 504 Tm
107 Tz
(this) Tj
100 Tz
1 0 0 1 125 504 Tm
103 Tz
(forced,) Tj
100 Tz
1 0 0 1 161 504 Tm
(in) Tj
1 0 0 1 173 504 Tm
106 Tz
(general,) Tj
100 Tz
1 0 0 1 213 504 Tm
(the) Tj
1 0 0 1 231 504 Tm
102 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 284 504 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 298 504 Tm
(the) Tj
1 0 0 1 316 504 Tm
(first) Tj
1 0 0 1 338 504 Tm
(and) Tj
1 0 0 1 358 504 Tm
(last) Tj
1 0 0 1 377 504 Tm
105 Tz
(mesh) Tj
100 Tz
1 0 0 1 405 504 Tm
104 Tz
(points) Tj
100 Tz
1 0 0 1 436 504 Tm
(to) Tj
1 0 0 1 449 504 Tm
89 Tz
(be) Tj
100 Tz
/F7 10 Tf
1 0 0 1 464 504 Tm
(k) Tj
/F9 10 Tf
1 0 0 1 473 504 Tm
(and) Tj
1 0 0 1 493 504 Tm
(allowed) Tj
/F5 5 Tf
1 0 0 1 119 492 Tm
167 Tz
(=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 130 492 Tm
67 Tz
(I) Tj
100 Tz
1 0 0 1 139 492 Tm
(at) Tj
1 0 0 1 152 492 Tm
82 Tz
(all) Tj
100 Tz
1 0 0 1 166 492 Tm
(other) Tj
1 0 0 1 194 492 Tm
(knots) Tj
1 0 0 1 222 492 Tm
89 Tz
(of) Tj
100 Tz
/F7 10 Tf
1 0 0 1 236 492 Tm
(u) Tj
/F9 10 Tf
1 0 0 1 246 492 Tm
104 Tz
(where) Tj
100 Tz
/F7 10 Tf
1 0 0 1 276 492 Tm
80 Tz
(u) Tj
100 Tz
/F9 10 Tf
1 0 0 1 286 492 Tm
(is) Tj
/F7 10 Tf
1 0 0 1 297 492 Tm
86 Tz
(smooth.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 337 492 Tm
95 Tz
(Here) Tj
100 Tz
1 0 0 1 363 492 Tm
97 Tz
(however) Tj
100 Tz
1 0 0 1 404 492 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 421 492 Tm
95 Tz
(have) Tj
100 Tz
1 0 0 1 446 492 Tm
75 Tz
(a) Tj
100 Tz
1 0 0 1 455 492 Tm
95 Tz
(point) Tj
100 Tz
1 0 0 1 482 492 Tm
96 Tz
(where) Tj
100 Tz
1 0 0 1 512 492 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 105 480 Tm
97 Tz
(solution) Tj
100 Tz
1 0 0 1 143 480 Tm
86 Tz
(is) Tj
100 Tz
1 0 0 1 154 480 Tm
93 Tz
(not) Tj
100 Tz
1 0 0 1 171 480 Tm
97 Tz
(smooth) Tj
100 Tz
1 0 0 1 206 480 Tm
(at) Tj
1 0 0 1 217 480 Tm
(a) Tj
1 0 0 1 225 480 Tm
96 Tz
(mesh) Tj
100 Tz
1 0 0 1 252 480 Tm
92 Tz
(point.) Tj
100 Tz
1 0 0 1 283 480 Tm
(Thus,) Tj
/F7 8 Tf
1 0 0 1 130 465 Tm
133 Tz
(it) Tj
100 Tz
/F7 10 Tf
1 0 0 1 141 465 Tm
71 Tz
(is) Tj
100 Tz
1 0 0 1 152 465 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 169 465 Tm
93 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 218 465 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 231 465 Tm
85 Tz
(the) Tj
100 Tz
1 0 0 1 248 465 Tm
(knot) Tj
1 0 0 1 271 465 Tm
88 Tz
(at) Tj
100 Tz
/F5 10 Tf
1 0 0 1 283 465 Tm
80 Tz
(0) Tj
100 Tz
/F7 10 Tf
1 0 0 1 294 465 Tm
88 Tz
(which) Tj
100 Tz
1 0 0 1 322 465 Tm
88 Tz
(determines) Tj
100 Tz
1 0 0 1 370 465 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 387 465 Tm
83 Tz
(type) Tj
100 Tz
1 0 0 1 408 465 Tm
111 Tz
(of) Tj
100 Tz
1 0 0 1 421 465 Tm
91 Tz
(solution) Tj
100 Tz
1 0 0 1 458 465 Tm
75 Tz
(we) Tj
100 Tz
1 0 0 1 473 465 Tm
86 Tz
(are) Tj
100 Tz
1 0 0 1 491 465 Tm
92 Tz
(trying) Tj
100 Tz
1 0 0 1 519 465 Tm
88 Tz
(to) Tj
100 Tz
/F7 10 Tf
1 0 0 1 129 453 Tm
91 Tz
(approximate.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 105 437 Tm
97 Tz
(Instead) Tj
100 Tz
1 0 0 1 139 437 Tm
(of) Tj
1 0 0 1 152 437 Tm
95 Tz
(choosing) Tj
100 Tz
1 0 0 1 194 437 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 211 437 Tm
94 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 263 437 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 275 437 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 284 437 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 296 437 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 310 437 Tm
75 Tz
(1,) Tj
100 Tz
1 0 0 1 322 437 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 337 437 Tm
94 Tz
(take) Tj
100 Tz
1 0 0 1 358 437 Tm
83 Tz
(it) Tj
100 Tz
1 0 0 1 367 437 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 379 437 Tm
90 Tz
(be) Tj
100 Tz
1 0 0 1 393 437 Tm
91 Tz
(k\205I,) Tj
100 Tz
1 0 0 1 418 437 Tm
89 Tz
(giving) Tj
100 Tz
1 0 0 1 448 437 Tm
95 Tz
(only) Tj
100 Tz
/F7 10 Tf
1 0 0 1 471 437 Tm
64 Tz
(u) Tj
5 Ts

(101) Tj
100 Tz
/F9 11 Tf
1 0 0 1 489 437 Tm
5 Ts
106 Tz
(continu) Tj
100 Tz
(-) Tj
1 0 0 1 105 425 Tm
89 Tz
(ous.) Tj
100 Tz
1 0 0 1 129 425 Tm
91 Tz
(With) Tj
100 Tz
1 0 0 1 154 425 Tm
93 Tz
(this) Tj
100 Tz
1 0 0 1 173 425 Tm
96 Tz
(choice) Tj
100 Tz
1 0 0 1 205 425 Tm
92 Tz
(for) Tj
100 Tz
1 0 0 1 220 425 Tm
(the) Tj
1 0 0 1 238 425 Tm
92 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 290 425 Tm
(of) Tj
1 0 0 1 302 425 Tm
(0) Tj
1 0 0 1 311 425 Tm
88 Tz
(in) Tj
100 Tz
1 0 0 1 323 425 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 340 425 Tm
(mesh,) Tj
1 0 0 1 370 425 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 385 425 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 403 425 Tm
96 Tz
(expect) Tj
100 Tz
1 0 0 1 435 425 Tm
88 Tz
(to) Tj
100 Tz
1 0 0 1 446 425 Tm
88 Tz
(find) Tj
100 Tz
1 0 0 1 466 425 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 483 425 Tm
94 Tz
(solution) Tj
100 Tz
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1 0 0 1 520 425 Tm
120 Tz
(y) Tj
100 Tz
/F9 11 Tf
1 0 0 1 105 414 Tm
89 Tz
(of) Tj
100 Tz
1 0 0 1 118 414 Tm
90 Tz
(\(7.4\)) Tj
100 Tz
1 0 0 1 142 414 Tm
97 Tz
(accurate) Tj
100 Tz
1 0 0 1 181 414 Tm
88 Tz
(to) Tj
100 Tz
/F7 10 Tf
1 0 0 1 193 414 Tm
118 Tz
(0\(h) Tj
100 Tz
/F7 6 Tf
1 0 0 1 212 414 Tm
(A\)) Tj
/F9 11 Tf
1 0 0 1 129 399 Tm
87 Tz
(We) Tj
100 Tz
1 0 0 1 147 399 Tm
93 Tz
(can) Tj
100 Tz
1 0 0 1 165 399 Tm
94 Tz
(now) Tj
100 Tz
1 0 0 1 186 399 Tm
98 Tz
(re-formulate) Tj
100 Tz
1 0 0 1 245 399 Tm
98 Tz
(\(7.1\)-\(7.2\)) Tj
100 Tz
1 0 0 1 294 399 Tm
89 Tz
(as) Tj
100 Tz
/F10 10 Tf
1 0 0 1 153 380 Tm
130 Tz
(y\202=0) Tj
100 Tz
1 0 0 1 195 380 Tm
109 Tz
(on\(\2051,0\)) Tj
100 Tz
/F9 10 Tf
1 0 0 1 504 361 Tm
(\(7,7\)) Tj
/F7 10 Tf
1 0 0 1 210 340 Tm
94 Tz
(e\202) Tj
5 Ts

(12) Tj
100 Tz
ET
1 w
210 338 m
227 338 l
S
BT
/F9 10 Tf
1 0 0 1 235 340 Tm
5 Ts
129 Tz
(=0) Tj
100 Tz
1 0 0 1 266 340 Tm
134 Tz
(on\(0,\2611\),) Tj
100 Tz
/F9 8 Tf
1 0 0 1 217 334 Tm
(2) Tj
/F9 11 Tf
1 0 0 1 105 317 Tm
95 Tz
(with) Tj
100 Tz
1 0 0 1 127 317 Tm
91 Tz
(y) Tj
-5 Ts

(1) Tj
-5 Ts

(\(\205I\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 162 317 Tm
167 Tz
(=) Tj
100 Tz
/F9 11 Tf
1 0 0 1 172 317 Tm
88 Tz
(0,) Tj
100 Tz
1 0 0 1 185 317 Tm
94 Tz
(\)\2012\(+1\)) Tj
100 Tz
/F5 5 Tf
1 0 0 1 220 317 Tm
167 Tz
(=) Tj
100 Tz
/F11 12 Tf
1 0 0 1 230 317 Tm
54 Tz
(-\201./~i) Tj
100 Tz
/F9 10 Tf
1 0 0 1 248 317 Tm
(and) Tj
1 0 0 1 269 317 Tm
(the) Tj
1 0 0 1 288 317 Tm
111 Tz
(knot) Tj
100 Tz
1 0 0 1 312 317 Tm
(at) Tj
1 0 0 1 325 317 Tm
(0) Tj
1 0 0 1 335 317 Tm
(of) Tj
1 0 0 1 349 317 Tm
104 Tz
(multiplicity) Tj
100 Tz
1 0 0 1 404 317 Tm
94 Tz
(k\205I) Tj
100 Tz
1 0 0 1 427 317 Tm
(in) Tj
1 0 0 1 441 317 Tm
(the) Tj
1 0 0 1 459 317 Tm
(B-spline) Tj
1 0 0 1 499 317 Tm
104 Tz
(mesh.) Tj
100 Tz
/F9 11 Tf
1 0 0 1 104 306 Tm
95 Tz
(This) Tj
100 Tz
1 0 0 1 128 306 Tm
91 Tz
(gives) Tj
100 Tz
1 0 0 1 155 306 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 173 306 Tm
90 Tz
(ratio) Tj
100 Tz
1 0 0 1 196 306 Tm
(of) Tj
1 0 0 1 210 306 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 228 306 Tm
92 Tz
(coefficients) Tj
100 Tz
1 0 0 1 280 306 Tm
(of) Tj
/F7 10 Tf
1 0 0 1 293 306 Tm
111 Tz
(y\204) Tj
100 Tz
/F9 11 Tf
1 0 0 1 306 306 Tm
91 Tz
(on) Tj
100 Tz
1 0 0 1 322 306 Tm
92 Tz
(the) Tj
100 Tz
1 0 0 1 340 306 Tm
93 Tz
(left) Tj
100 Tz
1 0 0 1 358 306 Tm
(and) Tj
1 0 0 1 378 306 Tm
95 Tz
(right) Tj
100 Tz
1 0 0 1 402 306 Tm
91 Tz
(sides) Tj
100 Tz
1 0 0 1 427 306 Tm
(of) Tj
1 0 0 1 441 306 Tm
80 Tz
(0) Tj
100 Tz
1 0 0 1 450 306 Tm
89 Tz
(as) Tj
100 Tz
1 0 0 1 463 306 Tm
75 Tz
(2.) Tj
100 Tz
1 0 0 1 478 306 Tm
94 Tz
(The) Tj
100 Tz
1 0 0 1 499 306 Tm
92 Tz
(above) Tj
100 Tz
/F9 11 Tf
1 0 0 1 105 294 Tm
94 Tz
(handwaving) Tj
100 Tz
1 0 0 1 160 294 Tm
105 Tz
(then) Tj
100 Tz
1 0 0 1 183 294 Tm
95 Tz
(indicates) Tj
100 Tz
1 0 0 1 225 294 Tm
94 Tz
(that) Tj
100 Tz
1 0 0 1 244 294 Tm
92 Tz
(we) Tj
100 Tz
1 0 0 1 260 294 Tm
97 Tz
(should) Tj
100 Tz
1 0 0 1 293 294 Tm
95 Tz
(have) Tj
100 Tz
/F7 10 Tf
1 0 0 1 316 294 Tm
67 Tz
(y) Tj
-5 Ts

(1) Tj
-5 Ts

(\202) Tj
100 Tz
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(2~41~) Tj
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/F9 10 Tf
1 0 0 1 330 99 Tm
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