Numerical recipes

Recommendations

Info

692 Chapter 15. Modeling of Data set DS (0) . Because of the replacement, you do not simply get back your original data set each time. You get sets in which a random fraction of the original points, typically ∼ 1/e ≈ 37%, are replaced by duplicated original points. Now, exactly as in the previous discussion, you subject these data sets to the same estimation procedure as was performed on the actual data, giving a set of simulated measured parameters aS (1) , aS (2) ,.... These will be distributed around a (0) in close to the same way that a (0) is distributed around atrue. Sounds like getting something for nothing, doesn’t it? In fact, it has taken more than a decade for the bootstrap method to become accepted by statisticians. By now, however, enough theorems have been proved to render the bootstrap reputable (see [2] for references). The basic idea behind the bootstrap is that the actual data set, viewed as a probability distribution consisting of delta functions at the measured values, is in most cases the best — or only — available estimator of the underlying probability distribution. It takes courage, but one can often simply use that distribution as the basis for Monte Carlo simulations. Watch out for cases where the bootstrap’s “iid” assumption is violated. For example, if you have made measurements at evenly spaced intervals of some control variable, then you can usually get away with pretending that these are “iid,” uniformly distributed over the measured range. However, some estimators of a (e.g., ones involving Fourier methods) might be particularly sensitive to all the points on a grid being present. In that case, the bootstrap is going to give a wrong distribution. Also watch out for estimators that look at anything like small-scale clumpiness within the N data points, or estimators that sort the data and look at sequential differences. Obviously the bootstrap will fail on these, too. (The theorems justifying the method are still true, but some of their technical assumptions are violated by these examples.) For a large class of problems, however, the bootstrap does yield easy, very quick, Monte Carlo estimates of the errors in an estimated parameter set. Confidence Limits Rather than present all details of the probability distribution of errors in parameter estimation, it is common practice to summarize the distribution in the form of confidence limits. The full probability distribution is a function defined on the M-dimensional space of parameters a. A confidence region (or confidence interval) is just a region of that M-dimensional space (hopefully a small region) that contains a certain (hopefully large) percentage of the total probability distribution. You point to a confidence region and say, e.g., “there is a 99 percent chance that the true parameter values fall within this region around the measured value.” It is worth emphasizing that you, the experimenter, get to pick both the confidence level (99 percent in the above example), and the shape of the confidence region. The only requirement is that your region does include the stated percentage of probability. Certain percentages are, however, customary in scientific usage: 68.3 percent (the lowest confidence worthy of quoting), 90 percent, 95.4 percent, 99 percent, and 99.73 percent. Higher confidence levels are conventionally “ninety-nine point nine ... nine.” As for shape, obviously you want a region that is compact and reasonably centered on your measurement a (0), since the whole purpose of a confidence limit is to inspire confidence in that measured value. In one dimension, the convention is to use a line segment centered on the measured value; in higher dimensions, ellipses or ellipsoids are most frequently used. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).
15.6 Confidence Limits on Estimated Model Parameters 693 (s) a (i)2 − a (0)2 68% confidence interval on a1 bias 68% confidence region on a1 and a2 jointly 68% confidence interval on a2 (s) a (i)1 − a (0)1 Figure 15.6.3. Confidence intervals in 1 and 2 dimensions. The same fraction of measured points (here 68%) lies (i) between the two vertical lines, (ii) between the two horizontal lines, (iii) within the ellipse. You might suspect, correctly, that the numbers 68.3 percent, 95.4 percent, and 99.73 percent, and the use of ellipsoids, have some connection with a normal distribution. That is true historically, but not always relevant nowadays. In general, the probability distribution of the parameters will not be normal, and the above numbers, used as levels of confidence, are purely matters of convention. Figure 15.6.3 sketches a possible probability distribution for the case M =2. Shown are three different confidence regions which might usefully be given, all at the same confidence level. The two vertical lines enclose a band (horizontal interval) which represents the 68 percent confidence interval for the variable a 1 without regard to the value of a2. Similarly the horizontal lines enclose a 68 percent confidence interval for a2. The ellipse shows a 68 percent confidence interval for a 1 and a2 jointly. Notice that to enclose the same probability as the two bands, the ellipse must necessarily extend outside of both of them (a point we will return to below). Constant Chi-Square Boundaries as Confidence Limits When the method used to estimate the parameters a (0) is chi-square minimization, as in the previous sections of this chapter, then there is a natural choice for the shape of confidence intervals, whose use is almost universal. For the observed data set D (0), the value of χ2 is a minimum at a (0). Call this minimum value χ2 min . If Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copying of machinereadable files (including this one) to any server computer, is strictly prohibited. To order Numerical Recipes books or CDROMs, visit website http://www.nr.com or call 1-800-872-7423 (North America only), or send email to directcustserv@cambridge.org (outside North America).
Page 1 and 2:
Numerical Recipes in C The Art of S
Page 3 and 4:
Contents Preface to the Second Edit
Page 5 and 6:
Contents vii 7.2 Transformation Met
Page 7 and 8:
Contents ix 15.6 Confidence Limits
Page 9 and 10:
Preface to the Second Edition Our a
Page 11 and 12:
Preface to the Second Edition xiii
Page 13 and 14:
Preface to the First Edition xv lin
Page 15 and 16:
Types of License Offered License In
Page 17 and 18:
Computer Programs by Chapter and Se
Page 19 and 20:
Page 21 and 22:
Page 23 and 24:
Page 25 and 26:
Chapter 1. Preliminaries 1.0 Introd
Page 27 and 28:
1.0 Introduction 3 Tested Machines
Page 29 and 30:
1.1 Program Organization and Contro
Page 31 and 32:
Page 33 and 34:
true 1.1 Program Organization and C
Page 35 and 36:
Page 37 and 38:
Page 39 and 40:
} 1.2 Some C Conventions for Scient
Page 41 and 42:
1.2 Some C Conventions for Scientif
Page 43 and 44:
Page 45 and 46:
(a) **m (b) **m 1.2 Some C Conventi
Page 47 and 48:
Page 49 and 50:
Page 51 and 52:
Page 53 and 54:
= 0 3 = 0 = 0 = 0 = 0 = 0 1 ⁄ 2 1
Page 55 and 56:
1.3 Error, Accuracy, and Stability
Page 57 and 58:
2.0 Introduction 33 Accumulated rou
Page 59 and 60:
2.0 Introduction 35 between singula
Page 61 and 62:
2.1 Gauss-Jordan Elimination 37 Eli
Page 63 and 64:
2.1 Gauss-Jordan Elimination 39 pre
Page 65 and 66:
2.2 Gaussian Elimination with Backs
Page 67 and 68:
2.3 LU Decomposition and Its Applic
Page 69 and 70:
a b c d 2.3 LU Decomposition and It
Page 71 and 72:
} 2.3 LU Decomposition and Its Appl
Page 73 and 74:
2.3 LU Decomposition and Its Applic
Page 75 and 76:
2.4 Tridiagonal and Band Diagonal S
Page 77 and 78:
2.4 Tridiagonal and Band Diagonal S
Page 79 and 80:
δx 2.5 Iterative Improvement of a
Page 81 and 82:
2.5 Iterative Improvement of a Solu
Page 83 and 84:
2.6 Singular Value Decomposition 59
Page 85 and 86:
SVD of a Square Matrix 2.6 Singular
Page 87 and 88:
solutions of A ⋅ x = d null space
Page 89 and 90:
Page 91 and 92:
Page 93 and 94:
} 2.6 Singular Value Decomposition
Page 95 and 96:
2.7 Sparse Linear Systems 2.7 Spars
Page 97 and 98:
2.7 Sparse Linear Systems 73 prescr
Page 99 and 100:
2.7 Sparse Linear Systems 75 Here
Page 101 and 102:
2.7 Sparse Linear Systems 77 Finall
Page 103 and 104:
2.7 Sparse Linear Systems 79 In row
Page 105 and 106:
} 2.7 Sparse Linear Systems 81 jp=i
Page 107 and 108:
2.7 Sparse Linear Systems 83 Matrix
Page 109 and 110:
2.7 Sparse Linear Systems 85 while
Page 111 and 112:
p=dvector(1,n); pp=dvector(1,n); r=
Page 113 and 114:
} 2.7 Sparse Linear Systems 89 void
Page 115 and 116:
2.8 Vandermonde Matrices and Toepli
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
2.9 Cholesky Decomposition 97 If yo
Page 123 and 124:
2.10 QR Decomposition 99 The standa
Page 125 and 126:
#include #include "nrutil.h" 2.10
Page 127 and 128:
in terms of which 2.11 Is Matrix In
Page 129 and 130:
Chapter 3. Interpolation and Extrap
Page 131 and 132:
(a) (b) 3.0 Introduction 107 Figure
Page 133 and 134:
3.1 Polynomial Interpolation and Ex
Page 135 and 136:
3.2 Rational Function Interpolation
Page 137 and 138:
3.3 Cubic Spline Interpolation 113
Page 139 and 140:
3.3 Cubic Spline Interpolation 115
Page 141 and 142:
3.4 How to Search an Ordered Table
Page 143 and 144:
} 3.4 How to Search an Ordered Tabl
Page 145 and 146:
#include "nrutil.h" 3.5 Coefficient
Page 147 and 148:
3.6 Interpolation in Two or More Di
Page 149 and 150:
} int j; float *ymtmp; 3.6 Interpol
Page 151 and 152:
3.6 Interpolation in Two or More Di
Page 153 and 154:
Chapter 4. Integration of Functions
Page 155 and 156:
4.1 Classical Formulas for Equally
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
#define FUNC(x) ((*func)(x)) 4.2 El
Page 163 and 164:
4.2 Elementary Algorithms 139 trapz
Page 165 and 166:
4.4 Improper Integrals 141 which co
Page 167 and 168:
4.4 Improper Integrals 143 The rout
Page 169 and 170:
4.4 Improper Integrals 145 If you n
Page 171 and 172:
4.5 Gaussian Quadratures and Orthog
Page 173 and 174:
Page 175 and 176:
Gauss-Legendre: Gauss-Chebyshev: Ga
Page 177 and 178:
} 4.5 Gaussian Quadratures and Orth
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
4.6 Multidimensional Integrals 161
Page 187 and 188:
y 4.6 Multidimensional Integrals 16
Page 189 and 190:
Chapter 5. Evaluation of Functions
Page 191 and 192:
5.1 Series and Their Convergence 16
Page 193 and 194:
5.2 Evaluation of Continued Fractio
Page 195 and 196:
5.2 Evaluation of Continued Fractio
Page 197 and 198:
5.3 Polynomials and Rational Functi
Page 199 and 200:
5.3 Polynomials and Rational Functi
Page 201 and 202:
5.4 Complex Arithmetic 177 Actually
Page 203 and 204:
5.5 Recurrence Relations and Clensh
Page 205 and 206:
5.5 Recurrence Relations and Clensh
Page 207 and 208:
5.6 Quadratic and Cubic Equations 1
Page 209 and 210:
5.6 Quadratic and Cubic Equations 1
Page 211 and 212:
5.7 Numerical Derivatives 187 With
Page 213 and 214:
} 5.7 Numerical Derivatives 189 The
Page 215 and 216:
Chebyshev polynomials 1 .5 0 −.5
Page 217 and 218:
} 5.8 Chebyshev Approximation 193 f
Page 219 and 220:
5.9 Derivatives or Integrals of a C
Page 221 and 222:
5.10 Polynomial Approximation from
Page 223 and 224:
5.11 Economization of Power Series
Page 225 and 226:
5.12 Padé Approximants 201 then R(
Page 227 and 228:
5.12 Padé Approximants 203 cof[2*n
Page 229 and 230:
5.13 Rational Chebyshev Approximati
Page 231 and 232:
5.13 Rational Chebyshev Approximati
Page 233 and 234:
5.14 Evaluation of Functions by Pat
Page 235 and 236:
5.14 Evaluation of Functions by Pat
Page 237 and 238:
6.1 Gamma, Beta, and Related Functi
Page 239 and 240:
6.1 Gamma, Beta, and Related Functi
Page 241 and 242:
incomplete gamma function P(a,x) 1.
Page 243 and 244:
} void nrerror(char error_text[]);
Page 245 and 246:
#include 6.2 Incomplete Gamma Func
Page 247 and 248:
6.3 Exponential Integrals 223 where
Page 249 and 250:
6.3 Exponential Integrals 225 #incl
Page 251 and 252:
6.4 Incomplete Beta Function 227 Th
Page 253 and 254:
6.4 Incomplete Beta Function 229 So
Page 255 and 256:
Bessel functions 1 .5 0 −.5 −1
Page 257 and 258:
} 6.5 Bessel Functions of Integer O
Page 259 and 260:
6.5 Bessel Functions of Integer Ord
Page 261 and 262:
modified Bessel functions 4 3 2 1 6
Page 263 and 264:
6.6 Modified Bessel Functions of In
Page 265 and 266:
6.7 Bessel Functions of Fractional
Page 267 and 268:
Page 269 and 270:
} 6.7 Bessel Functions of Fractiona
Page 271 and 272:
Page 273 and 274:
Page 275 and 276:
Page 277 and 278:
6.8 Spherical Harmonics 253 The fir
Page 279 and 280:
6.9 Fresnel Integrals, Cosine and S
Page 281 and 282:
} 6.9 Fresnel Integrals, Cosine and
Page 283 and 284:
} 6.10 Dawson’s Integral 259 } if
Page 285 and 286:
6.11 Elliptic Integrals and Jacobia
Page 287 and 288:
where 6.11 Elliptic Integrals and J
Page 289 and 290:
#define TINY 1.0e-25 #define BIG 4.
Page 291 and 292:
6.11 Elliptic Integrals and Jacobia
Page 293 and 294:
#include #include "nrutil.h" 6.11
Page 295 and 296:
6.12 Hypergeometric Functions 271 C
Page 297 and 298:
#include "complex.h" #define ONE Co
Page 299 and 300:
7.1 Uniform Deviates 275 As for ref
Page 301 and 302:
7.1 Uniform Deviates 277 This corre
Page 303 and 304:
#define IA 16807 #define IM 2147483
Page 305 and 306:
RAN 3 1 7.1 Uniform Deviates 281 iy
Page 307 and 308:
7.1 Uniform Deviates 283 from the w
Page 309 and 310:
7.1 Uniform Deviates 285 Constants
Page 311 and 312:
7.2 Transformation Method: Exponent
Page 313 and 314:
7.2 Transformation Method: Exponent
Page 315 and 316:
first random deviate in 7.3 Rejecti
Page 317 and 318:
} 7.3 Rejection Method: Gamma, Pois
Page 319 and 320:
} 7.3 Rejection Method: Gamma, Pois
Page 321 and 322:
7.4 Generation of Random Bits 297 s
Page 323 and 324:
} } *iseed
Page 325 and 326:
7.5 Random Sequences Based on Data
Page 327 and 328:
} } 7.5 Random Sequences Based on D
Page 329 and 330:
7.6 Simple Monte Carlo Integration
Page 331 and 332:
7.6 Simple Monte Carlo Integration
Page 333 and 334:
7.7 Quasi- (that is, Sub-) Random S
Page 335 and 336:
Page 337 and 338:
} 7.7 Quasi- (that is, Sub-) Random
Page 339 and 340:
Page 341 and 342:
7.8 Adaptive and Recursive Monte Ca
Page 343 and 344:
Page 345 and 346:
Page 347 and 348:
} } 7.8 Adaptive and Recursive Mont
Page 349 and 350:
Page 351 and 352:
} 7.8 Adaptive and Recursive Monte
Page 353 and 354:
Chapter 8. Sorting 8.0 Introduction
Page 355 and 356:
8.1 Straight Insertion and Shell’
Page 357 and 358:
8.2 Quicksort 333 question of the b
Page 359 and 360:
8.2 Quicksort 335 istack=lvector(1,
Page 361 and 362:
a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
Page 363 and 364:
original array 1 2 3 4 5 6 14 8 32
Page 365 and 366:
8.5 Selecting the Mth Largest 341 w
Page 367 and 368:
8.5 Selecting the Mth Largest 343 r
Page 369 and 370:
} } } } 8.6 Determination of Equiva
Page 371 and 372:
Chapter 9. Root Finding and Nonline
Page 373 and 374:
9.0 Introduction 349 good first gue
Page 375 and 376:
(a) (b) (c) (d) 9.1 Bracketing and
Page 377 and 378:
Bisection Method 9.1 Bracketing and
Page 379 and 380:
9.2 Secant Method, False Position M
Page 381 and 382:
} 9.2 Secant Method, False Position
Page 383 and 384:
} 9.3 Van Wijngaarden-Dekker-Brent
Page 385 and 386:
9.3 Van Wijngaarden-Dekker-Brent Me
Page 387 and 388:
9.4 Newton-Raphson Method Using Der
Page 389 and 390:
9.4 Newton-Raphson Method Using Der
Page 391 and 392:
} 9.4 Newton-Raphson Method Using D
Page 393 and 394:
9.5 Roots of Polynomials 9.5 Roots
Page 395 and 396:
9.5 Roots of Polynomials 371 tentat
Page 397 and 398:
9.5 Roots of Polynomials 373 The me
Page 399 and 400:
Eigenvalue Methods 9.5 Roots of Pol
Page 401 and 402:
9.5 Roots of Polynomials 377 of goi
Page 403 and 404:
9.6 Newton-Raphson Method for Nonli
Page 405 and 406:
9.6 Newton-Raphson Method for Nonli
Page 407 and 408:
9.7 Globally Convergent Methods for
Page 409 and 410:
Page 411 and 412:
Page 413 and 414:
Page 415 and 416:
Page 417 and 418:
Page 419 and 420:
X1 A B C 10.0 Introduction 395 D Fi
Page 421 and 422:
10.1 Golden Section Search in One D
Page 423 and 424:
10.1 Golden Section Search in One D
Page 425 and 426:
} } 10.1 Golden Section Search in O
Page 427 and 428:
10.2 Parabolic Interpolation and Br
Page 429 and 430:
} 10.3 One-Dimensional Search with
Page 431 and 432:
10.3 One-Dimensional Search with Fi
Page 433 and 434:
(a) (b) (c) (d) 10.4 Downhill Simpl
Page 435 and 436:
10.4 Downhill Simplex Method in Mul
Page 437 and 438:
10.5 Direction Set (Powell’s) Met
Page 439 and 440:
Page 441 and 442:
Page 443 and 444:
Page 445 and 446:
(a) 10.6 Conjugate Gradient Methods
Page 447 and 448:
10.6 Conjugate Gradient Methods in
Page 449 and 450:
} 10.7 Variable Metric Methods in M
Page 451 and 452:
10.7 Variable Metric Methods in Mul
Page 453 and 454:
} 10.7 Variable Metric Methods in M
Page 455 and 456:
10.8 Linear Programming and the Sim
Page 457 and 458:
Page 459 and 460:
Page 461 and 462:
Page 463 and 464:
Page 465 and 466:
Page 467 and 468:
} 10.8 Linear Programming and the S
Page 469 and 470:
10.9 Simulated Annealing Methods 44
Page 471 and 472:
(a) (b) (c) 1 .5 0 1 .5 0 1 .5 0 10
Page 473 and 474:
} } 10.9 Simulated Annealing Method
Page 475 and 476:
} free_ivector(jorder,1,ncity); #in
Page 477 and 478:
Page 479 and 480:
Page 481 and 482:
Definitions and Basic Facts 11.0 In
Page 483 and 484:
11.0 Introduction 459 In both the d
Page 485 and 486:
11.0 Introduction 461 “Eigenpacka
Page 487 and 488:
11.1 Jacobi Transformations of a Sy
Page 489 and 490:
Page 491 and 492:
Page 493 and 494:
} } 11.2 Reduction of a Symmetric M
Page 495 and 496:
11.2 Reduction of a Symmetric Matri
Page 497 and 498:
11.2 Reduction of a Symmetric Matri
Page 499 and 500:
11.3 Eigenvalues and Eigenvectors o
Page 501 and 502:
Page 503 and 504:
Page 505 and 506:
} } 11.4 Hermitian Matrices 481 for
Page 507 and 508:
11.5 Reduction of a General Matrix
Page 509 and 510:
11.5 Reduction of a General Matrix
Page 511 and 512:
11.6 The QR Algorithm for Real Hess
Page 513 and 514:
Page 515 and 516:
Page 517 and 518:
11.7 Eigenvalues or Eigenvectors by
Page 519 and 520:
11.7 Eigenvalues or Eigenvectors by
Page 521 and 522:
and equation (12.0.1) looks like th
Page 523 and 524:
⏐h(t)⏐ 2 (a) Ph( f ) (one-sided
Page 525 and 526:
12.1 Fourier Transform of Discretel
Page 527 and 528:
12.1 Fourier Transform of Discretel
Page 529 and 530:
12.2 Fast Fourier Transform (FFT) 5
Page 531 and 532:
12.2 Fast Fourier Transform (FFT) 5
Page 533 and 534:
Other FFT Algorithms 12.2 Fast Four
Page 535 and 536:
12.3 FFT of Real Functions, Sine an
Page 537 and 538:
12.3 FFT of Real Functions, Sine an
Page 539 and 540:
(a) (b) (c) +1 0 −1 +1 0 −1 +1
Page 541 and 542:
#include 12.3 FFT of Real Function
Page 543 and 544:
} 12.3 FFT of Real Functions, Sine
Page 545 and 546:
} } } 12.4 FFT in Two or More Dimen
Page 547 and 548:
data [1] Re Im 12.4 FFT in Two or M
Page 549 and 550:
12.5 Fourier Transforms of Real Dat
Page 551 and 552:
Page 553 and 554:
Page 555 and 556:
Page 557 and 558:
12.6 External Storage or Memory-Loc
Page 559 and 560:
} } 12.6 External Storage or Memory
Page 561 and 562:
Chapter 13. Fourier and Spectral Ap
Page 563 and 564:
(t) r*s(t) 13.1 Convolution and Dec
Page 565 and 566:
13.1 Convolution and Deconvolution
Page 567 and 568:
13.1 Convolution and Deconvolution
Page 569 and 570:
13.2 Correlation and Autocorrelatio
Page 571 and 572:
13.3 Optimal (Wiener) Filtering wit
Page 573 and 574:
log scale 13.4 Power Spectrum Estim
Page 575 and 576:
13.4 Power Spectrum Estimation Usin
Page 577 and 578:
Page 579 and 580:
amplitude 1 .8 .6 .4 .2 0 13.4 Powe
Page 581 and 582:
Page 583 and 584:
13.5 Digital Filtering in the Time
Page 585 and 586:
IIR (Recursive) Filters 13.5 Digita
Page 587 and 588:
(a) (b) 13.5 Digital Filtering in t
Page 589 and 590:
13.6 Linear Prediction and Linear P
Page 591 and 592:
Page 593 and 594:
} 13.6 Linear Prediction and Linear
Page 595 and 596:
Page 597 and 598:
13.7 Maximum Entropy (All Poles) Me
Page 599 and 600:
13.8 Spectral Analysis of Unevenly
Page 601 and 602:
Page 603 and 604:
Page 605 and 606:
} 13.8 Spectral Analysis of Unevenl
Page 607 and 608:
} 13.8 Spectral Analysis of Unevenl
Page 609 and 610:
13.9 Computing Fourier Integrals Us
Page 611 and 612:
Cubic order: 13.9 Computing Fourier
Page 613 and 614:
13.9 Computing Fourier Integrals Us
Page 615 and 616:
13.10 Wavelet Transforms 591 The sp
Page 617 and 618:
13.10 Wavelet Transforms 593 p.”
Page 619 and 620:
13.10 Wavelet Transforms 595 comple
Page 621 and 622:
} 13.10 Wavelet Transforms 597 wfil
Page 623 and 624:
.2 0 −.2 .2 0 −.2 DAUB4 e10 + e
Page 625 and 626:
wavelet amplitude 1.5 1 .5 .0001 10
Page 627 and 628:
} nprev=nnew; } free_vector(wksp,1,
Page 629 and 630:
13.10 Wavelet Transforms 605 Figure
Page 631 and 632:
13.11 Numerical Use of the Sampling
Page 633 and 634:
Chapter 14. Statistical Description
Page 635 and 636:
14.1 Moments of a Distribution: Mea
Page 637 and 638:
14.1 Moments of a Distribution: Mea
Page 639 and 640:
14.2 Do Two Distributions Have the
Page 641 and 642:
Page 643 and 644:
Page 645 and 646:
14.3 Are Two Distributions Differen
Page 647 and 648:
Page 649 and 650:
Page 651 and 652:
Page 653 and 654:
1. male 2. female . 14.4 Contingenc
Page 655 and 656:
14.4 Contingency Table Analysis of
Page 657 and 658:
Page 659 and 660:
Page 661 and 662:
14.5 Linear Correlation 637 two dis
Page 663 and 664:
14.6 Nonparametric or Rank Correlat
Page 665 and 666: 14.6 Nonparametric or Rank Correlat
Page 667 and 668: 14.6 Nonparametric or Rank Correlat
Page 669 and 670: } } 14.7 Do Two-Dimensional Distrib
Page 671 and 672: 14.7 Do Two-Dimensional Distributio
Page 673 and 674: #include #include "nrutil.h" 14.7
Page 675 and 676: 14.8 Savitzky-Golay Smoothing Filte
Page 677 and 678: 8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 befor
Page 679 and 680: 14.8 Savitzky-Golay Smoothing Filte
Page 681 and 682: 15.1 Least Squares as a Maximum Lik
Page 683 and 684: 15.1 Least Squares as a Maximum Lik
Page 685 and 686: 15.2 Fitting Data to a Straight Lin
Page 687 and 688: 15.2 Fitting Data to a Straight Lin
Page 689 and 690: #include #include "nrutil.h" 15.2
Page 691 and 692: 15.3 Straight-Line Data with Errors
Page 693 and 694: 15.3 Straight-Line Data with Errors
Page 695 and 696: 15.4 General Linear Least Squares 6
Page 699 and 700: } 15.4 General Linear Least Squares
Page 705 and 706: 15.5 Nonlinear Models 681 Lawson, C
Page 707 and 708: 15.5 Nonlinear Models 683 Note that
Page 709 and 710: 15.5 Nonlinear Models 685 pivots, b
Page 711 and 712: #include "nrutil.h" 15.5 Nonlinear
Page 713 and 714: 15.6 Confidence Limits on Estimated
Page 715: actual data set 15.6 Confidence Lim
Page 723 and 724: 15.7 Robust Estimation 699 M� 1 C
Page 725 and 726: 15.7 Robust Estimation 701 where th
Page 727 and 728: 15.7 Robust Estimation 703 algorith
Page 729 and 730: } *a=aa; *b=bb; *abdev=abdevt/ndata
Page 731 and 732: Chapter 16. Integration of Ordinary
Page 733 and 734: 16.0 Introduction 709 problem where
Page 735 and 736: y(x) 16.1 Runge-Kutta Method 711 1
Page 737 and 738: } 16.1 Runge-Kutta Method 713 yt=ve
Page 739 and 740: 16.2 Adaptive Stepsize Control for
Page 747 and 748: 16.3 Modified Midpoint Method 723 H
Page 749 and 750: 16.4 Richardson Extrapolation and t
Page 757 and 758: 16.5 Second-Order Conservative Equa
Page 759 and 760: y 16.6 Stiff Sets of Equations 735
Page 761 and 762: 16.6 Stiff Sets of Equations 737 If
Page 763 and 764: 16.6 Stiff Sets of Equations 739 od
Page 765 and 766: } 16.6 Stiff Sets of Equations 741
Page 767 and 768:
16.6 Stiff Sets of Equations 743 Co
Page 769 and 770:
16.6 Stiff Sets of Equations 745 Se
Page 771 and 772:
} 16.7 Multistep, Multivalue, and P
Page 773 and 774:
16.7 Multistep, Multivalue, and Pre
Page 775 and 776:
16.7 Multistep, Multivalue, and Pre
Page 777 and 778:
Chapter 17. Two Point Boundary Valu
Page 779 and 780:
equired boundary value 17.0 Introdu
Page 781 and 782:
17.1 The Shooting Method 17.1 The S
Page 783 and 784:
} 17.1 The Shooting Method 759 floa
Page 785 and 786:
17.2 Shooting to a Fitting Point 76
Page 787 and 788:
17.3 Relaxation Methods 763 depende
Page 789 and 790:
X X X X X X X X X X X X X X X X X X
Page 791 and 792:
17.3 Relaxation Methods 767 Next co
Page 793 and 794:
17.3 Relaxation Methods 769 calcula
Page 795 and 796:
} 17.3 Relaxation Methods 771 for (
Page 797 and 798:
17.4 A Worked Example: Spheroidal H
Page 799 and 800:
Page 801 and 802:
Page 803 and 804:
} 17.4 A Worked Example: Spheroidal
Page 805 and 806:
Page 807 and 808:
17.5 Automated Allocation of Mesh P
Page 809 and 810:
17.6 Handling Internal Boundary Con
Page 811 and 812:
17.6 Handling Internal Boundary Con
Page 813 and 814:
18.0 Introduction 789 is called a F
Page 815 and 816:
18.1 Fredholm Equations of the Seco
Page 817 and 818:
#include "nrutil.h" 18.1 Fredholm E
Page 819 and 820:
18.2 Volterra Equations 795 in §18
Page 821 and 822:
18.3 Integral Equations with Singul
Page 823 and 824:
Page 825 and 826:
Page 827 and 828:
f (x) 1 .5 0 −.5 0 18.3 Integral
Page 829 and 830:
18.4 Inverse Problems and the Use o
Page 831 and 832:
18.4 Inverse Problems and the Use o
Page 833 and 834:
18.5 Linear Regularization Methods
Page 835 and 836:
Page 837 and 838:
Page 839 and 840:
18.6 Backus-Gilbert Method 815 nece
Page 841 and 842:
18.6 Backus-Gilbert Method 817 Subs
Page 843 and 844:
18.7 Maximum Entropy Image Restorat
Page 845 and 846:
Page 847 and 848:
Page 849 and 850:
Page 851 and 852:
Chapter 19. Partial Differential Eq
Page 853 and 854:
19.0 Introduction 829 possible stab
Page 855 and 856:
y L ∆ y1 19.0 Introduction 831 A
Page 857 and 858:
19.0 Introduction 833 So-called rap
Page 859 and 860:
19.1 Flux-Conservative Initial Valu
Page 861 and 862:
Page 863 and 864:
Page 865 and 866:
Page 867 and 868:
Page 869 and 870:
t or n 19.1 Flux-Conservative Initi
Page 871 and 872:
19.2 Diffusive Initial Value Proble
Page 873 and 874:
19.2 Diffusive Initial Value Proble
Page 875 and 876:
and the heuristic stability criteri
Page 877 and 878:
19.3 Initial Value Problems in Mult
Page 879 and 880:
19.3 Initial Value Problems in Mult
Page 881 and 882:
19.4 Fourier and Cyclic Reduction M
Page 883 and 884:
Page 885 and 886:
Page 887 and 888:
FACR Method 19.5 Relaxation Methods
Page 889 and 890:
19.5 Relaxation Methods for Boundar
Page 891 and 892:
Page 893 and 894:
Page 895 and 896:
19.6 Multigrid Methods for Boundary
Page 897 and 898:
Page 899 and 900:
S S S E γ = 1 19.6 Multigrid Metho
Page 901 and 902:
Page 903 and 904:
Page 905 and 906:
} 19.6 Multigrid Methods for Bounda
Page 907 and 908:
Page 909 and 910:
Page 911 and 912:
#include 19.6 Multigrid Methods fo
Page 913 and 914:
Chapter 20. Less-Numerical Algorith
Page 915 and 916:
20.1 Diagnosing Machine Parameters
Page 917 and 918:
20.1 Diagnosing Machine Parameters
Page 919 and 920:
(a) G(i) (b) MSB 4 3 2 1 0 LSB i MS
Page 921 and 922:
20.3 Cyclic Redundancy and Other Ch
Page 923 and 924:
Page 925 and 926:
Page 927 and 928:
} 20.4 Huffman Coding and Compressi
Page 929 and 930:
20.4 Huffman Coding and Compression
Page 931 and 932:
} 20.4 Huffman Coding and Compressi
Page 933 and 934:
} } } 20.4 Huffman Coding and Compr
Page 935 and 936:
1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2
Page 937 and 938:
20.5 Arithmetic Coding 913 Individu
Page 939 and 940:
20.6 Arithmetic at Arbitrary Precis
Page 941 and 942:
} for (j=n;j>=1;j--) { ireg=u[j]+HI
Page 943 and 944:
#include "nrutil.h" #define RX 256.
Page 945 and 946:
} 20.6 Arithmetic at Arbitrary Prec
Page 947 and 948:
Page 949:
show all

Numerical recipes

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?