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343 0 R +/Parent 347 0 R /Prev 403 0 R /Next 411 0 R >> endobj 403 0 obj << /Title 404 0 R /A 401 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 399 0 R /Next 407 0 R >> endobj 399 0 obj << /Title 400 0 R /A 397 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 395 0 R /Next 403 0 R >> endobj 395 0 obj << /Title 396 0 R /A 393 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 391 0 R /Next 399 0 R >> endobj 391 0 obj << /Title 392 0 R /A 389 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 387 0 R /Next 395 0 R >> endobj 387 0 obj << /Title 388 0 R /A 385 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 383 0 R /Next 391 0 R >> endobj 383 0 obj << /Title 384 0 R /A 381 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 379 0 R /Next 387 0 R >> endobj 379 0 obj << /Title 380 0 R /A 377 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 375 0 R /Next 383 0 R >> endobj 375 0 obj << /Title 376 0 R /A 373 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 371 0 R /Next 379 0 R >> endobj 371 0 obj << /Title 372 0 R /A 369 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 367 0 R /Next 375 0 R >> endobj 367 0 obj << /Title 368 0 R /A 365 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 363 0 R /Next 371 0 R >> endobj 363 0 obj << /Title 364 0 R /A 361 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 359 0 R /Next 367 0 R >> endobj 359 0 obj << /Title 360 0 R /A 357 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 355 0 R /Next 363 0 R >> endobj 355 0 obj << /Title 356 0 R /A 353 0 R -/Parent 343 0 R +/Parent 347 0 R /Prev 351 0 R /Next 359 0 R >> endobj 351 0 obj << /Title 352 0 R /A 349 0 R -/Parent 343 0 R -/Prev 347 0 R +/Parent 347 0 R /Next 355 0 R >> endobj 347 0 obj << /Title 348 0 R /A 345 0 R -/Parent 343 0 R -/Next 351 0 R +/Parent 139 0 R +/Prev 331 0 R +/Next 431 0 R +/First 351 0 R +/Last 427 0 R +/Count -20 >> endobj 343 0 obj << /Title 344 0 R /A 341 0 R -/Parent 135 0 R -/Prev 327 0 R -/Next 419 0 R -/First 347 0 R -/Last 415 0 R -/Count -18 +/Parent 331 0 R +/Prev 339 0 R >> endobj 339 0 obj << /Title 340 0 R /A 337 0 R -/Parent 327 0 R +/Parent 331 0 R /Prev 335 0 R +/Next 343 0 R >> endobj 335 0 obj << /Title 336 0 R /A 333 0 R -/Parent 327 0 R -/Prev 331 0 R +/Parent 331 0 R /Next 339 0 R >> endobj 331 0 obj << /Title 332 0 R /A 329 0 R -/Parent 327 0 R -/Next 335 0 R +/Parent 139 0 R +/Prev 267 0 R +/Next 347 0 R +/First 335 0 R +/Last 343 0 R +/Count -3 >> endobj 327 0 obj << /Title 328 0 R /A 325 0 R -/Parent 135 0 R -/Prev 263 0 R -/Next 343 0 R -/First 331 0 R -/Last 339 0 R -/Count -3 +/Parent 267 0 R +/Prev 323 0 R >> endobj 323 0 obj << /Title 324 0 R /A 321 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 319 0 R +/Next 327 0 R >> endobj 319 0 obj << /Title 320 0 R /A 317 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 315 0 R /Next 323 0 R >> endobj 315 0 obj << /Title 316 0 R /A 313 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 311 0 R /Next 319 0 R >> endobj 311 0 obj << /Title 312 0 R /A 309 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 307 0 R /Next 315 0 R >> endobj 307 0 obj << /Title 308 0 R /A 305 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 303 0 R /Next 311 0 R >> endobj 303 0 obj << /Title 304 0 R /A 301 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 299 0 R /Next 307 0 R >> endobj 299 0 obj << /Title 300 0 R /A 297 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 295 0 R /Next 303 0 R >> endobj 295 0 obj << /Title 296 0 R /A 293 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 291 0 R /Next 299 0 R >> endobj 291 0 obj << /Title 292 0 R /A 289 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 287 0 R /Next 295 0 R >> endobj 287 0 obj << /Title 288 0 R /A 285 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 283 0 R /Next 291 0 R >> endobj 283 0 obj << /Title 284 0 R /A 281 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 279 0 R /Next 287 0 R >> endobj 279 0 obj << /Title 280 0 R /A 277 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 275 0 R /Next 283 0 R >> endobj 275 0 obj << /Title 276 0 R /A 273 0 R -/Parent 263 0 R +/Parent 267 0 R /Prev 271 0 R /Next 279 0 R >> endobj 271 0 obj << /Title 272 0 R /A 269 0 R -/Parent 263 0 R -/Prev 267 0 R +/Parent 267 0 R /Next 275 0 R >> endobj 267 0 obj << /Title 268 0 R /A 265 0 R -/Parent 263 0 R -/Next 271 0 R +/Parent 139 0 R +/Prev 227 0 R +/Next 331 0 R +/First 271 0 R +/Last 327 0 R +/Count -15 >> endobj 263 0 obj << /Title 264 0 R /A 261 0 R -/Parent 135 0 R -/Prev 223 0 R -/Next 327 0 R -/First 267 0 R -/Last 323 0 R -/Count -15 +/Parent 227 0 R +/Prev 259 0 R >> endobj 259 0 obj << /Title 260 0 R /A 257 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 255 0 R +/Next 263 0 R >> endobj 255 0 obj << /Title 256 0 R /A 253 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 251 0 R /Next 259 0 R >> endobj 251 0 obj << /Title 252 0 R /A 249 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 247 0 R /Next 255 0 R >> endobj 247 0 obj << /Title 248 0 R /A 245 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 243 0 R /Next 251 0 R >> endobj 243 0 obj << /Title 244 0 R /A 241 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 239 0 R /Next 247 0 R >> endobj 239 0 obj << /Title 240 0 R /A 237 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 235 0 R /Next 243 0 R >> endobj 235 0 obj << /Title 236 0 R /A 233 0 R -/Parent 223 0 R +/Parent 227 0 R /Prev 231 0 R /Next 239 0 R >> endobj 231 0 obj << /Title 232 0 R /A 229 0 R -/Parent 223 0 R -/Prev 227 0 R +/Parent 227 0 R /Next 235 0 R >> endobj 227 0 obj << /Title 228 0 R /A 225 0 R -/Parent 223 0 R -/Next 231 0 R +/Parent 139 0 R +/Prev 167 0 R +/Next 267 0 R +/First 231 0 R +/Last 263 0 R +/Count -9 >> endobj 223 0 obj << /Title 224 0 R /A 221 0 R -/Parent 135 0 R -/Prev 163 0 R -/Next 263 0 R -/First 227 0 R -/Last 259 0 R -/Count -9 +/Parent 167 0 R +/Prev 219 0 R >> endobj 219 0 obj << /Title 220 0 R /A 217 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 215 0 R +/Next 223 0 R >> endobj 215 0 obj << /Title 216 0 R /A 213 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 211 0 R /Next 219 0 R >> endobj 211 0 obj << /Title 212 0 R /A 209 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 207 0 R /Next 215 0 R >> endobj 207 0 obj << /Title 208 0 R /A 205 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 203 0 R /Next 211 0 R >> endobj 203 0 obj << /Title 204 0 R /A 201 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 199 0 R /Next 207 0 R >> endobj 199 0 obj << /Title 200 0 R /A 197 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 195 0 R /Next 203 0 R >> endobj 195 0 obj << /Title 196 0 R /A 193 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 191 0 R /Next 199 0 R >> endobj 191 0 obj << /Title 192 0 R /A 189 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 187 0 R /Next 195 0 R >> endobj 187 0 obj << /Title 188 0 R /A 185 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 183 0 R /Next 191 0 R >> endobj 183 0 obj << /Title 184 0 R /A 181 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 179 0 R /Next 187 0 R >> endobj 179 0 obj << /Title 180 0 R /A 177 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 175 0 R /Next 183 0 R >> endobj 175 0 obj << /Title 176 0 R /A 173 0 R -/Parent 163 0 R +/Parent 167 0 R /Prev 171 0 R /Next 179 0 R >> endobj 171 0 obj << 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+/Next 139 0 R +/First 119 0 R +/Last 127 0 R +/Count -3 >> endobj 111 0 obj << /Title 112 0 R /A 109 0 R -/Parent 994 0 R +/Parent 1017 0 R /Prev 107 0 R -/Next 135 0 R -/First 115 0 R -/Last 123 0 R -/Count -3 +/Next 115 0 R >> endobj 107 0 obj << /Title 108 0 R /A 105 0 R -/Parent 994 0 R -/Prev 103 0 R +/Parent 1017 0 R +/Prev 67 0 R /Next 111 0 R >> endobj 103 0 obj << /Title 104 0 R /A 101 0 R -/Parent 994 0 R -/Prev 63 0 R -/Next 107 0 R +/Parent 91 0 R +/Prev 99 0 R >> endobj 99 0 obj << /Title 100 0 R /A 97 0 R -/Parent 87 0 R +/Parent 91 0 R /Prev 95 0 R +/Next 103 0 R >> endobj 95 0 obj << /Title 96 0 R /A 93 0 R -/Parent 87 0 R -/Prev 91 0 R +/Parent 91 0 R /Next 99 0 R >> endobj 91 0 obj << /Title 92 0 R /A 89 0 R -/Parent 87 0 R -/Next 95 0 R +/Parent 67 0 R +/Prev 71 0 R +/First 95 0 R +/Last 103 0 R +/Count -3 >> endobj 87 0 obj << /Title 88 0 R /A 85 0 R -/Parent 63 0 R -/Prev 67 0 R -/First 91 0 R -/Last 99 0 R -/Count -3 +/Parent 71 0 R +/Prev 83 0 R >> endobj 83 0 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0 R +/Last 63 0 R +/Count -2 >> endobj 51 0 obj << /Title 52 0 R /A 49 0 R -/Parent 23 0 R +/Parent 27 0 R /Prev 47 0 R -/First 55 0 R -/Last 59 0 R -/Count -2 +/Next 55 0 R >> endobj 47 0 obj << /Title 48 0 R /A 45 0 R -/Parent 23 0 R +/Parent 27 0 R /Prev 43 0 R /Next 51 0 R >> endobj 43 0 obj << /Title 44 0 R /A 41 0 R -/Parent 23 0 R +/Parent 27 0 R /Prev 39 0 R /Next 47 0 R >> endobj 39 0 obj << /Title 40 0 R /A 37 0 R -/Parent 23 0 R +/Parent 27 0 R /Prev 35 0 R /Next 43 0 R >> endobj 35 0 obj << /Title 36 0 R /A 33 0 R -/Parent 23 0 R +/Parent 27 0 R /Prev 31 0 R /Next 39 0 R >> endobj 31 0 obj << /Title 32 0 R /A 29 0 R -/Parent 23 0 R -/Prev 27 0 R +/Parent 27 0 R /Next 35 0 R >> endobj 27 0 obj << /Title 28 0 R /A 25 0 R -/Parent 23 0 R -/Next 31 0 R +/Parent 1017 0 R +/Prev 23 0 R +/Next 67 0 R +/First 31 0 R +/Last 55 0 R +/Count -7 >> endobj 23 0 obj << /Title 24 0 R /A 21 0 R -/Parent 994 0 R +/Parent 1017 0 R /Prev 19 0 R -/Next 63 0 R -/First 27 0 R -/Last 51 0 R 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(subsubsection.4.1.2)] +/Names [(equation.9.17) 821 0 R (equation.9.18) 828 0 R (equation.9.19) 829 0 R (equation.9.20) 838 0 R (equation.9.21) 839 0 R (equation.9.22) 840 0 R] +/Limits [(equation.9.17) (equation.9.22)] >> endobj 1021 0 obj << -/Names [(subsubsection.4.1.3) 78 0 R (subsubsection.4.1.4) 82 0 R (subsubsection.4.2.1) 90 0 R (subsubsection.4.2.2) 94 0 R (subsubsection.4.2.3) 98 0 R (subsubsection.7.3.1) 126 0 R] -/Limits [(subsubsection.4.1.3) (subsubsection.7.3.1)] +/Names [(equation.9.23) 841 0 R (equation.9.24) 842 0 R (equation.9.25) 851 0 R (equation.9.26) 852 0 R (equation.9.27) 853 0 R (equation.9.28) 863 0 R] +/Limits [(equation.9.23) (equation.9.28)] >> endobj 1022 0 obj << -/Names [(subsubsection.7.3.2) 130 0 R (subsubsection.8.1.1) 142 0 R (subsubsection.8.1.2) 146 0 R (subsubsection.8.1.3) 150 0 R (subsubsection.8.1.4) 154 0 R (subsubsection.8.1.5) 158 0 R] -/Limits [(subsubsection.7.3.2) (subsubsection.8.1.5)] +/Names [(equation.9.29) 865 0 R (equation.9.30) 866 0 R (equation.9.31) 867 0 R (equation.9.32) 876 0 R (equation.9.33) 877 0 R (equation.9.34) 878 0 R] +/Limits [(equation.9.29) (equation.9.34)] >> endobj 1023 0 obj << -/Names [(subsubsection.8.2.1) 166 0 R (subsubsection.8.2.10) 202 0 R (subsubsection.8.2.11) 206 0 R (subsubsection.8.2.12) 210 0 R (subsubsection.8.2.13) 214 0 R (subsubsection.8.2.14) 218 0 R] -/Limits [(subsubsection.8.2.1) (subsubsection.8.2.14)] +/Names [(equation.9.35) 888 0 R (equation.9.36) 889 0 R (equation.9.37) 901 0 R (equation.9.38) 902 0 R (equation.9.39) 915 0 R (equation.9.40) 916 0 R] +/Limits [(equation.9.35) (equation.9.40)] >> endobj 1024 0 obj << -/Names [(subsubsection.8.2.2) 170 0 R (subsubsection.8.2.3) 174 0 R (subsubsection.8.2.4) 178 0 R (subsubsection.8.2.5) 182 0 R (subsubsection.8.2.6) 186 0 R (subsubsection.8.2.7) 190 0 R] -/Limits [(subsubsection.8.2.2) (subsubsection.8.2.7)] +/Names [(equation.9.41) 917 0 R (equation.9.42) 926 0 R (equation.9.43) 927 0 R (equation.9.44) 938 0 R (equation.9.45) 954 0 R (equation.9.46) 955 0 R] +/Limits [(equation.9.41) (equation.9.46)] >> endobj 1025 0 obj << -/Names [(subsubsection.8.2.8) 194 0 R (subsubsection.8.2.9) 198 0 R (subsubsection.8.3.1) 226 0 R (subsubsection.8.3.2) 230 0 R (subsubsection.8.3.3) 234 0 R (subsubsection.8.3.4) 238 0 R] -/Limits [(subsubsection.8.2.8) (subsubsection.8.3.4)] +/Names [(equation.9.47) 956 0 R (equation.9.48) 964 0 R (equation.9.49) 965 0 R (equation.9.5) 793 0 R (equation.9.50) 966 0 R (equation.9.51) 967 0 R] +/Limits [(equation.9.47) (equation.9.51)] >> endobj 1026 0 obj << -/Names [(subsubsection.8.3.5) 242 0 R (subsubsection.8.3.6) 246 0 R (subsubsection.8.3.7) 250 0 R (subsubsection.8.3.8) 254 0 R (subsubsection.8.3.9) 258 0 R (subsubsection.8.4.1) 266 0 R] -/Limits [(subsubsection.8.3.5) (subsubsection.8.4.1)] +/Names [(equation.9.6) 794 0 R (equation.9.7) 795 0 R (equation.9.8) 796 0 R (equation.9.9) 805 0 R (figure.1) 717 0 R (figure.2) 733 0 R] +/Limits [(equation.9.6) 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(subsubsection.8.5.2) 334 0 R (subsubsection.8.5.3) 338 0 R (subsubsection.8.6.1) 346 0 R] -/Limits [(subsubsection.8.4.8) (subsubsection.8.6.1)] +/Names [(page.17) 732 0 R (page.18) 737 0 R (page.19) 745 0 R (page.2) 585 0 R (page.20) 753 0 R (page.21) 759 0 R] +/Limits [(page.17) (page.21)] >> endobj 1030 0 obj << -/Names [(subsubsection.8.6.10) 382 0 R (subsubsection.8.6.11) 386 0 R (subsubsection.8.6.12) 390 0 R (subsubsection.8.6.13) 394 0 R (subsubsection.8.6.14) 398 0 R (subsubsection.8.6.15) 402 0 R] -/Limits [(subsubsection.8.6.10) (subsubsection.8.6.15)] +/Names [(page.22) 766 0 R (page.23) 772 0 R (page.24) 777 0 R (page.25) 781 0 R (page.26) 792 0 R (page.27) 804 0 R] +/Limits [(page.22) (page.27)] >> endobj 1031 0 obj << -/Names [(subsubsection.8.6.16) 406 0 R (subsubsection.8.6.17) 410 0 R (subsubsection.8.6.18) 414 0 R (subsubsection.8.6.2) 350 0 R (subsubsection.8.6.3) 354 0 R (subsubsection.8.6.4) 358 0 R] -/Limits [(subsubsection.8.6.16) (subsubsection.8.6.4)] +/Names [(page.28) 818 0 R (page.29) 827 0 R (page.3) 627 0 R (page.30) 837 0 R (page.31) 850 0 R (page.32) 862 0 R] +/Limits [(page.28) (page.32)] >> endobj 1032 0 obj << -/Names [(subsubsection.8.6.5) 362 0 R (subsubsection.8.6.6) 366 0 R (subsubsection.8.6.7) 370 0 R (subsubsection.8.6.8) 374 0 R (subsubsection.8.6.9) 378 0 R (subsubsection.8.7.1) 422 0 R] -/Limits [(subsubsection.8.6.5) (subsubsection.8.7.1)] +/Names [(page.33) 875 0 R (page.34) 887 0 R (page.35) 899 0 R (page.36) 914 0 R (page.37) 925 0 R (page.38) 936 0 R] +/Limits [(page.33) (page.38)] >> endobj 1033 0 obj << -/Names [(subsubsection.8.7.2) 426 0 R (subsubsection.8.7.3) 430 0 R (subsubsection.8.7.4) 434 0 R (subsubsection.8.7.5) 438 0 R (subsubsection.8.7.6) 442 0 R (subsubsection.8.7.7) 446 0 R] -/Limits [(subsubsection.8.7.2) (subsubsection.8.7.7)] +/Names [(page.39) 946 0 R (page.4) 635 0 R (page.40) 953 0 R (page.41) 963 0 R (page.42) 971 0 R (page.5) 641 0 R] +/Limits [(page.39) (page.5)] >> endobj 1034 0 obj << -/Names [(subsubsection.8.7.8) 450 0 R (subsubsection.8.8.1) 458 0 R (subsubsection.8.8.2) 462 0 R (subsubsection.8.8.3) 466 0 R] -/Limits [(subsubsection.8.7.8) (subsubsection.8.8.3)] +/Names [(page.6) 648 0 R (page.7) 658 0 R (page.8) 666 0 R (page.9) 673 0 R (section*.1) 525 0 R (section*.10) 683 0 R] +/Limits [(page.6) (section*.10)] >> endobj 1035 0 obj << -/Kids [995 0 R 996 0 R 997 0 R 998 0 R 999 0 R 1000 0 R] -/Limits [(Doc-Start) (equation.8.40)] +/Names [(section*.11) 684 0 R (section*.12) 685 0 R (section*.13) 686 0 R (section*.14) 687 0 R (section*.15) 688 0 R (section*.16) 689 0 R] +/Limits [(section*.11) (section*.16)] >> endobj 1036 0 obj << -/Kids [1001 0 R 1002 0 R 1003 0 R 1004 0 R 1005 0 R 1006 0 R] -/Limits [(equation.8.41) (page.21)] +/Names [(section*.17) 690 0 R (section*.18) 695 0 R (section*.19) 700 0 R (section*.2) 636 0 R (section*.20) 705 0 R (section*.21) 722 0 R] +/Limits [(section*.17) (section*.21)] >> endobj 1037 0 obj << -/Kids [1007 0 R 1008 0 R 1009 0 R 1010 0 R 1011 0 R 1012 0 R] -/Limits [(page.22) (section*.18)] +/Names [(section*.22) 746 0 R (section*.23) 761 0 R (section*.24) 768 0 R (section*.3) 642 0 R (section*.4) 649 0 R (section*.5) 659 0 R] +/Limits [(section*.22) (section*.5)] >> endobj 1038 0 obj << -/Kids [1013 0 R 1014 0 R 1015 0 R 1016 0 R 1017 0 R 1018 0 R] -/Limits [(section*.19) (subsection.7.3)] +/Names [(section*.6) 660 0 R (section*.7) 668 0 R (section*.8) 680 0 R (section*.9) 682 0 R (section.1) 6 0 R (section.2) 18 0 R] +/Limits [(section*.6) (section.2)] >> endobj 1039 0 obj << -/Kids [1019 0 R 1020 0 R 1021 0 R 1022 0 R 1023 0 R 1024 0 R] -/Limits [(subsection.8.1) (subsubsection.8.2.7)] +/Names [(section.3) 22 0 R (section.4) 26 0 R (section.5) 66 0 R (section.6) 106 0 R (section.7) 110 0 R (section.8) 114 0 R] +/Limits [(section.3) (section.8)] >> endobj 1040 0 obj << -/Kids [1025 0 R 1026 0 R 1027 0 R 1028 0 R 1029 0 R 1030 0 R] -/Limits [(subsubsection.8.2.8) (subsubsection.8.6.15)] +/Names [(section.9) 138 0 R (subsection.1.1) 10 0 R (subsection.1.2) 14 0 R (subsection.4.1) 30 0 R (subsection.4.2) 34 0 R (subsection.4.3) 38 0 R] +/Limits [(section.9) (subsection.4.3)] >> endobj 1041 0 obj << -/Kids [1031 0 R 1032 0 R 1033 0 R 1034 0 R] -/Limits [(subsubsection.8.6.16) (subsubsection.8.8.3)] +/Names [(subsection.4.4) 42 0 R (subsection.4.5) 46 0 R (subsection.4.6) 50 0 R (subsection.4.7) 54 0 R (subsection.5.1) 70 0 R (subsection.5.2) 90 0 R] +/Limits [(subsection.4.4) (subsection.5.2)] >> endobj 1042 0 obj << -/Kids [1035 0 R 1036 0 R 1037 0 R 1038 0 R 1039 0 R 1040 0 R] -/Limits [(Doc-Start) (subsubsection.8.6.15)] +/Names [(subsection.8.1) 118 0 R (subsection.8.2) 122 0 R (subsection.8.3) 126 0 R (subsection.9.1) 142 0 R (subsection.9.2) 166 0 R (subsection.9.3) 226 0 R] +/Limits [(subsection.8.1) (subsection.9.3)] >> endobj 1043 0 obj << -/Kids [1041 0 R] -/Limits [(subsubsection.8.6.16) (subsubsection.8.8.3)] +/Names [(subsection.9.4) 266 0 R (subsection.9.5) 330 0 R (subsection.9.6) 346 0 R (subsection.9.7) 430 0 R (subsection.9.8) 466 0 R (subsubsection.4.7.1) 58 0 R] +/Limits [(subsection.9.4) (subsubsection.4.7.1)] >> endobj 1044 0 obj << -/Kids [1042 0 R 1043 0 R] -/Limits [(Doc-Start) (subsubsection.8.8.3)] +/Names [(subsubsection.4.7.2) 62 0 R (subsubsection.5.1.1) 74 0 R (subsubsection.5.1.2) 78 0 R (subsubsection.5.1.3) 82 0 R (subsubsection.5.1.4) 86 0 R (subsubsection.5.2.1) 94 0 R] +/Limits [(subsubsection.4.7.2) (subsubsection.5.2.1)] >> endobj 1045 0 obj << -/Dests 1044 0 R +/Names [(subsubsection.5.2.2) 98 0 R (subsubsection.5.2.3) 102 0 R (subsubsection.8.3.1) 130 0 R (subsubsection.8.3.2) 134 0 R (subsubsection.9.1.1) 146 0 R (subsubsection.9.1.2) 150 0 R] +/Limits [(subsubsection.5.2.2) (subsubsection.9.1.2)] >> endobj 1046 0 obj << +/Names [(subsubsection.9.1.3) 154 0 R (subsubsection.9.1.4) 158 0 R (subsubsection.9.1.5) 162 0 R (subsubsection.9.2.1) 170 0 R (subsubsection.9.2.10) 206 0 R (subsubsection.9.2.11) 210 0 R] +/Limits [(subsubsection.9.1.3) (subsubsection.9.2.11)] +>> endobj +1047 0 obj << +/Names [(subsubsection.9.2.12) 214 0 R (subsubsection.9.2.13) 218 0 R (subsubsection.9.2.14) 222 0 R (subsubsection.9.2.2) 174 0 R (subsubsection.9.2.3) 178 0 R (subsubsection.9.2.4) 182 0 R] +/Limits [(subsubsection.9.2.12) (subsubsection.9.2.4)] +>> endobj +1048 0 obj << +/Names [(subsubsection.9.2.5) 186 0 R (subsubsection.9.2.6) 190 0 R (subsubsection.9.2.7) 194 0 R (subsubsection.9.2.8) 198 0 R (subsubsection.9.2.9) 202 0 R (subsubsection.9.3.1) 230 0 R] +/Limits [(subsubsection.9.2.5) (subsubsection.9.3.1)] +>> endobj +1049 0 obj << +/Names [(subsubsection.9.3.2) 234 0 R (subsubsection.9.3.3) 238 0 R (subsubsection.9.3.4) 242 0 R (subsubsection.9.3.5) 246 0 R (subsubsection.9.3.6) 250 0 R (subsubsection.9.3.7) 254 0 R] +/Limits [(subsubsection.9.3.2) (subsubsection.9.3.7)] +>> endobj +1050 0 obj << +/Names [(subsubsection.9.3.8) 258 0 R (subsubsection.9.3.9) 262 0 R (subsubsection.9.4.1) 270 0 R (subsubsection.9.4.10) 306 0 R (subsubsection.9.4.11) 310 0 R (subsubsection.9.4.12) 314 0 R] +/Limits [(subsubsection.9.3.8) (subsubsection.9.4.12)] +>> endobj +1051 0 obj << +/Names [(subsubsection.9.4.13) 318 0 R (subsubsection.9.4.14) 322 0 R (subsubsection.9.4.15) 326 0 R (subsubsection.9.4.2) 274 0 R (subsubsection.9.4.3) 278 0 R (subsubsection.9.4.4) 282 0 R] +/Limits [(subsubsection.9.4.13) (subsubsection.9.4.4)] +>> endobj +1052 0 obj << +/Names [(subsubsection.9.4.5) 286 0 R (subsubsection.9.4.6) 290 0 R (subsubsection.9.4.7) 294 0 R (subsubsection.9.4.8) 298 0 R (subsubsection.9.4.9) 302 0 R (subsubsection.9.5.1) 334 0 R] +/Limits [(subsubsection.9.4.5) (subsubsection.9.5.1)] +>> endobj +1053 0 obj << +/Names [(subsubsection.9.5.2) 338 0 R (subsubsection.9.5.3) 342 0 R (subsubsection.9.6.1) 350 0 R (subsubsection.9.6.10) 386 0 R (subsubsection.9.6.11) 390 0 R (subsubsection.9.6.12) 394 0 R] +/Limits [(subsubsection.9.5.2) (subsubsection.9.6.12)] +>> endobj +1054 0 obj << 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(subsubsection.9.7.8) 462 0 R] +/Limits [(subsubsection.9.7.3) (subsubsection.9.7.8)] +>> endobj +1058 0 obj << +/Names [(subsubsection.9.8.1) 470 0 R (subsubsection.9.8.2) 474 0 R (subsubsection.9.8.3) 478 0 R] +/Limits [(subsubsection.9.8.1) (subsubsection.9.8.3)] +>> endobj +1059 0 obj << +/Kids [1018 0 R 1019 0 R 1020 0 R 1021 0 R 1022 0 R 1023 0 R] +/Limits [(Doc-Start) (equation.9.40)] +>> endobj +1060 0 obj << +/Kids [1024 0 R 1025 0 R 1026 0 R 1027 0 R 1028 0 R 1029 0 R] +/Limits [(equation.9.41) (page.21)] +>> endobj +1061 0 obj << +/Kids [1030 0 R 1031 0 R 1032 0 R 1033 0 R 1034 0 R 1035 0 R] +/Limits [(page.22) (section*.16)] +>> endobj +1062 0 obj << +/Kids [1036 0 R 1037 0 R 1038 0 R 1039 0 R 1040 0 R 1041 0 R] +/Limits [(section*.17) (subsection.5.2)] +>> endobj +1063 0 obj << +/Kids [1042 0 R 1043 0 R 1044 0 R 1045 0 R 1046 0 R 1047 0 R] +/Limits [(subsection.8.1) (subsubsection.9.2.4)] +>> endobj +1064 0 obj << +/Kids [1048 0 R 1049 0 R 1050 0 R 1051 0 R 1052 0 R 1053 0 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scheme and convention} @@ -91,6 +92,29 @@ For each routine, there is a generic interface (simply remove \verb'_*' in the n There's a general module called ``\verb'fvn''' that include all fvn submodules. So whatever part of the library is used in the program a ``\verb'use fvn''' will be sufficient instead of specifying the specific module. +\section{kind specification} +In \verb'fvn' the following definitions are made (this is done in module \verb'fvn_common') to ease portability: + +\begin{verbatim} +integer, parameter :: ip_kind = kind(1) +integer, parameter :: sp_kind = kind(1.0E0) +integer, parameter :: dp_kind = kind(1.0D0) +\end{verbatim} + +Although not mandatory, it is a good idea to use these definitions when programming with fvn, that is for example : +\begin{verbatim} +real(kind=sp_kind) :: x +real(kind=dp_kind) :: y +complex(kind=dp_kind) :: z +\end{verbatim} +instead of +\begin{verbatim} +real :: x +double precision :: y +complex(8) :: z +\end{verbatim} + + \section{Linear algebra} The linear algebra routines of fvn are an interface to lapack, which make it easier to use. \subsection{Matrix inversion} @@ -1692,7 +1716,7 @@ This relation is directly used in upward direction to compute $J_n(x)$ for $x>n$ \label{bsjnnorm} 1=J_0+2J_2+2J_4+2J_6+... \end{equation} -The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$ +The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40 for single precision and 150 for double precision. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$ By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsj0(x)$ or $bsj1(x)$ is actually performed. @@ -1707,6 +1731,22 @@ By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsj Specific interfaces : \verb'besjn,dbesjn' + +\subsubsection{besrj} +\begin{verbatim} + call besrj(x,n,b) +\end{verbatim} +\begin{itemize} + \item x (in) is a real + \item n (in) is an integer + \item b (out) is a real array of dimension n +\end{itemize} +This subroutine evaluates Bessel function of the first kind of order \verb'0' to \verb'n-1' for argument x and return the result in array b, which then contain \verb'b(1)='$J_0(x)$,\verb'b(2)='$J_1(x)$,...,\verb'b(n)='$J_{n-1}(x)$. + +The algorithm is different from the one used in \verb'bsjn', the choice between the two depends on accuracy and timing considerations, there are test programs (\verb'test_besrj' and \verb'test_bestime') in the \verb'fvn_test' directory which can help choosing the good one. + +Specific interfaces : \verb'besrj,dbesrj' + \subsubsection{bsy0} \begin{verbatim} bsy0(x) @@ -1817,7 +1857,7 @@ This relation is unstable in the upward direction, therefore a Miller's Algorith \label{bsinnorm} 1=I_0-2I_2+2I_4-2I_6+.... \end{equation} -The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$ +The optional parameters \verb'factor' and \verb'big' can be used to modify the behaviour of the algorithm. \verb'factor' is used in determining the arbitrary starting order ( an even integer near $n+\sqrt{factor~n}$), the default $factor$ value is 40 for single precision and 150 for double precision. \verb'big' is a real determining the threshold for which anti-overflow counter measures has to be taken, default value is $1.10^{10}$ By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsi0(x)$ or $bsi1(x)$ is actually performed. @@ -1833,6 +1873,22 @@ By convenience, the routine accept $n=0$ and $n=1$, in that cases a call to $bsi Specific interfaces : \verb'besin,dbesin' +\subsubsection{besri} +\begin{verbatim} + call besri(x,n,b) +\end{verbatim} +\begin{itemize} + \item x (in) is a real + \item n (in) is an integer + \item b (out) is a real array of dimension n +\end{itemize} +This subroutine evaluates Bessel function of the third kind of order \verb'0' to \verb'n-1' for argument x and return the result in array b, which then contain \verb'b(1)='$I_0(x)$,\verb'b(2)='$I_1(x)$,...,\verb'b(n)='$I_{n-1}(x)$. + +The algorithm is different from the one used in \verb'bsin' and tends to be more accurate, the choice between the two depends on accuracy and timing considerations, there are test programs (\verb'test_besri' and \verb'test_bestime') in the \verb'fvn_test' directory which can help choosing the good one. + +Specific interfaces : \verb'besri,dbesri' + + \subsubsection{bsk0} \begin{verbatim} bsk0(x) diff --git a/fvn_test/init_random_seed.f90 b/fvn_test/init_random_seed.f90 index 4cc1d78..13fe8a3 100644 --- a/fvn_test/init_random_seed.f90 +++ b/fvn_test/init_random_seed.f90 @@ -11,7 +11,7 @@ SUBROUTINE init_random_seed() INTEGER :: i, n, clock INTEGER, DIMENSION(:), ALLOCATABLE :: seed - logical :: gfortran=.false. + logical :: gfortran=.true. ! ! Set gfortran to .true. if using gfortran to obtain the desired behaviour ! diff --git a/fvn_test/test_besri.f90 b/fvn_test/test_besri.f90 index d0fae93..34e5088 100644 --- a/fvn_test/test_besri.f90 +++ b/fvn_test/test_besri.f90 @@ -1,4 +1,4 @@ -program test_dbesri +program test_besri use fvn_fnlib implicit none ! Variables locales ----------------------------- @@ -8,8 +8,8 @@ real(kind=dp_kind), dimension(:,:), allocatable:: bessvec1,bessvec2 real(kind=dp_kind) :: x,xstep,xmax -open (unit=1, file='dbesri.dat') -write (1,*) '# n, x, bsin(n,x), b(n) in dbesri(x,norder+1,b)' +open (unit=1, file='besri.dat') +write (1,*) '# n, x, bsin(n,x), b(n) in besri(x,norder+1,b)' write (1,*) '# arg x from 0 to 20, 201 points' write (1,*) '# order n from 0 to 20' @@ -22,7 +22,7 @@ allocate(bessvec0(0:norder)) do i=0,nstep !loop on x x=i*xstep - call dbesri(x,norder+1,bessvec0) + call besri(x,norder+1,bessvec0) bessvec2(i,:)=bessvec0 do n=0,norder !loop on rank, for dbesjn only bessvec1(i,n)=bsin(n,x) @@ -40,5 +40,13 @@ enddo close(1) deallocate(bessvec0,bessvec1,bessvec2) -END +write(*,*) "Exemple use of generated file besri.dat with gnuplot :" +write(*,*) "pl 'besri.dat' u 2:($1==5 ? $3 : 1/0) w l" +write(*,*) " will plot I5 according to bsin" +write(*,*) "pl 'besri.dat' u 2:($1==5 ? $4 : 1/0) w l" +write(*,*) " will plot I5 according to besri" +write(*,*) "pl 'besri.dat' u 2:($1==5 ? ($4-$3)/$3 : 1/0) w l" +write(*,*) " will plot relative shift between besri and bsin" + +end program diff --git a/fvn_test/test_besrj.f90 b/fvn_test/test_besrj.f90 index db2e14a..6cecec5 100644 --- a/fvn_test/test_besrj.f90 +++ b/fvn_test/test_besrj.f90 @@ -1,4 +1,4 @@ -program test_dbesrj +program test_besrj use fvn_fnlib implicit none ! Variables locales ----------------------------- @@ -8,8 +8,8 @@ real(kind=dp_kind), dimension(:,:), allocatable:: bessvec1,bessvec2 real(kind=dp_kind) :: x,xstep,xmax -open (unit=1, file='dbesrj.dat') -write (1,*) '# n, x, bsjn(n,x), b(n) in dbesrj(x,norder+1,b)' +open (unit=1, file='besrj.dat') +write (1,*) '# n, x, bsjn(n,x), b(n) in besrj(x,norder+1,b)' write (1,*) '# arg x from 0 to 50, 501 points' write (1,*) '# order n from 0 to 50' @@ -22,7 +22,7 @@ allocate(bessvec0(0:norder)) do i=0,nstep !loop on x x=i*xstep - call dbesrj(x,norder+1,bessvec0) + call besrj(x,norder+1,bessvec0) bessvec2(i,:)=bessvec0 do n=0,norder !loop on rank, for dbesjn only bessvec1(i,n)=bsjn(n,x) @@ -40,5 +40,15 @@ enddo close(1) deallocate(bessvec0,bessvec1,bessvec2) -END +write(*,*) "Exemple use of generated file besrj.dat with gnuplot :" +write(*,*) "pl 'besrj.dat' u 2:($1==5 ? $3 : 1/0) w l" +write(*,*) " will plot J5 according to bsjn" +write(*,*) "pl 'besrj.dat' u 2:($1==5 ? $4 : 1/0) w l" +write(*,*) " will plot J5 according to besrj" +write(*,*) "pl 'besrj.dat' u 2:($1==5 ? ($4-$3)/$3 : 1/0) w l" +write(*,*) " will plot relative shift between besrj and bsjn" + + + +end program diff --git a/fvn_test/test_bestime.f90 b/fvn_test/test_bestime.f90 index 34c98c8..256c0d7 100644 --- a/fvn_test/test_bestime.f90 +++ b/fvn_test/test_bestime.f90 @@ -56,6 +56,53 @@ write (*,'("Calculating "(I10)" values of J_n for n=0 to "(I3)" with besrj :")') write(*,*) t2-t1 +write(*,*) +write(*,*) +write(*,*) +write(*,*) "Computation time comparison between bsin and besri" +write(*,*) "bsin is faster when computing I_n for only one value of n" +write(*,*) "besri is faster when computing I_n for n=0 to n" + +call cpu_time(t1) +do it=1,iter + do i=1,npoints + x=xmin+i*xstep + do nn=0,n + bes=bsin(nn,x) + end do + end do +end do +call cpu_time(t2) +write (*,'("Calculating "(I10)" values of I_n for n=0 to "(I3)" with bsin :")') iter*npoints,n +write(*,*) t2-t1 + +call cpu_time(t1) +do it=1,iter + do i=1,npoints + x=xmin+i*xstep + bes=bsin(n,x) + end do +end do +call cpu_time(t2) +write (*,'("Calculating "(I10)" values of I_n for n="(I3)" with bsin :")') iter*npoints,n +write(*,*) t2-t1 + + +! Calcultate J_n(x) on 200 points for n=0 to 50 with besrj +! do it iter times to have a sufficient time value +call cpu_time(t1) +do it=1,iter + do i=1,npoints + x=xmin+i*xstep + call besri(x,n+1,b) + end do +end do +call cpu_time(t2) +write (*,'("Calculating "(I10)" values of I_n for n=0 to "(I3)" with besri :")') iter*npoints,n +write(*,*) t2-t1 + + + end program bestime \ No newline at end of file