| Numerical Recipes in C
Here we are... The art of scientific computing is now downloadable by
clicking HERE. About 1000 pages of routines in C and
smart solutions for scientific applications. It contains:
1 Preliminaries
1.1 Program
Organization and Control Structures 5
1.2 Some C
Conventions for Scientific Computing 15
1.3 Error,
Accuracy, and Stability 15
2 Solution of Linear Algebraic
Equations
2.1 Gauss-Jordan
Elimination 36
2.2 Gaussian
Elimination with Backsubstitution 41
2.3 LU
Decomposition and Its Applications 43
2.4 Tridiagonal and
Band Diagonal Systems of Equations 50
2.5 Iterative
Improvement of a Solution to Linear Equations 55
2.6 Singular Value
Decomposition 59
2.7 Sparse Linear
Systems 71
2.8 Vandermonde
Matrices and Toeplitz Matrices 90
2.9 Cholesky
Decomposition 96
2.10 QR
Decomposition 98
2.11 Is Matrix
Inversion an $N^3$ Process? 102
3 Interpolation and
Extrapolation
3.1 Polynomial
Interpolation and Extrapolation 108
3.2 Rational
Function Interpolation and Extrapolation 111
3.3 Cubic Spline
Interpolation 113
3.4 How to Search
an Ordered Table 117
3.5 Coefficients of
the Interpolating Polynomial 120
3.6 Interpolation
in Two or More Dimensions 123
4 Integration of Functions
4.1 Classical
Formulas for Equally Spaced Abscissas 130
4.2 Elementary
Algorithms 136
4.3 Romberg
Integration 140
4.4 Improper
Integrals 141
4.5 Gaussian
Quadratures and Orthogonal Polynomials 147
4.6
Multidimensional Integrals 161
5 Evaluation of Functions
5.1 Series and
Their Convergence 165
5.2 Evaluation of
Continued Fractions 169
5.3 Polynomials and
Rational Functions 173
5.4 Complex
Arithmetic 176
5.5 Recurrence
Relations and Clenshaw's Recurrence Formula 178
5.6 Quadratic and
Cubic Equations 183
5.7 Numerical
Derivatives 186
5.8 Chebyshev
Approximation 190
5.9 Derivatives or
Integrals of a Chebyshev-approximated Function 195
5.10 Polynomial
Approximation from Chebyshev Coefficients 197
5.11 Economization
of Power Series 198
5.12 Pad\'e
Approximants 200
5.13 Rational
Chebyshev Approximation 204
5.14 Evaluation of
Functions by Path Integration 208
6 Special Functions
6.1 Gamma Function,
Beta Function, Factorials, Binomial Coefficients 213
6.3 Exponential
Integrals 222
6.5 Bessel
Functions of Integer Order 230
6.6 Modified Bessel
Functions of Integer Order 236
6.7 Bessel
Functions of Fractional Order, Airy Functions, SphericalBessel Functions
240
6.8 Spherical
Harmonics 252
6.9 Fresnel
Integrals, Cosine and Sine Integrals 255
6.10 Dawson's
Integral 259
6.11 Elliptic
Integrals and Jacobian Elliptic Functions 261
6.12 Hypergeometric
Functions 271
7 Random Numbers
7.1 Uniform
Deviates 275
7.2 Transformation
Method: Exponential and Normal Deviates 287
7.3 Rejection
Method: Gamma, Poisson, Binomial Deviates 290
7.4 Generation of
Random Bits 296
7.5 Random
Sequences Based on Data Encryption 300
7.6 Simple Monte
Carlo Integration 304
7.7 Quasi- (that
is, Sub-) Random Sequences 309
7.8 Adaptive and
Recursive Monte Carlo Methods 316
8 Sorting
8.1 Straight
Insertion and Shell's Method 330
8.2 Quicksort
332
8.3 Heapsort
336
8.4 Indexing and
Ranking 338
8.5 Selecting the
$M$th Largest 341
8.6 Determination
of Equivalence Classes 345
9 Root Finding and Nonlinear Sets of
Equations
9.1 Bracketing and
Bisection 350
9.2 Secant Method,
False Position Method, and Ridders' Method 354
9.3 Van
Wijngaarden--Dekker--Brent Method 359
9.4 Newton-Raphson
Method Using Derivative 362
9.5 Roots of
Polynomials 369
9.6 Newton-Raphson
Method for Nonlinear Systems of Equations 379
9.7 Globally
Convergent Methods for Nonlinear Systems of Equations
383
10 Minimization or Maximization of
Functions
10.1 Golden Section
Search in One Dimension 397
10.2 Parabolic
Interpolation and Brent's Method in One Dimension 402
10.3
One-Dimensional Search with First Derivatives 305
10.4 Downhill
Simplex Method in Multidimensions 408
10.5 Direction Set
(Powell's) Methods in Multidimensions 412
10.6 Conjugate
Gradient Methods in Multidimensions 420
10.7 Variable
Metric Methods in Multidimensions 425
10.8 Linear
Programming and the Simplex Method 430
10.9 Simulated
Annealing Methods 444
11 Eigensystems
11.1 Jacobi
Transformations of a Symmetric Matrix 463
11.3 Eigenvalues
and Eigenvectors of a Tridiagonal Matrix 475
11.4 Hermitian
Matrices 481
11.5 Reduction of a
General Matrix to Hessenberg Form 482
11.6 The QR
Algorithm for Real Hessenberg Matrices 486
11.7 Improving
Eigenvalues and/or Finding Eigenvectors by Inverse Iteration
493
12 Fast Fourier Transform
12.1 Fourier
Transform of Discretely Sampled Data 500
12.2 Fast Fourier
Transform (FFT) 504
12.3 FFT of Real
Functions, Sine and Cosine Transforms 510
12.4 FFT in Two or
More Dimensions 521
12.5 Fourier
Transforms of Real Data in Two and Three Dimensions 525
12.6 External
Storage or Memory-Local FFTs 532
13 Fourier and Spectral
Applications
13.1 Convolution
and Deconvolution Using the FFT 538
13.2 Correlation
and Autocorrelation Using the FFT 545
13.3 Optimal
(Wiener) Filtering with the FFT 547
13.4 Power Spectrum
Estimation Using the FFT 549
13.5 Digital
Filtering in the Time Domain 558
13.6 Linear
Prediction and Linear Predictive Coding 564
13.7 Power Spectrum
Estimation by the Maximum Entropy (All Poles) Method 572
13.8 Spectral
Analysis of Unevenly Sampled Data 575
13.9 Computing
Fourier Integrals Using the FFT 584
13.10 Wavelet
Transforms 591
13.11 Numerical Use
of the Sampling Theorem 606
14 Statistical Description of
Data
14.1 Moments of a
Distribution: Mean, Variance, Skewness, and So Forth 610
14.2 Do Two
Distributions Have the Same Means or Variances? 615
14.3 Are Two
Distributions Different? 620
14.4 Contingency
Table Analysis of Two Distributions 628
14.5 Linear
Correlation 636
14.6 Nonparametric
or Rank Correlation 639
14.7 Do
Two-Dimensional Distributions Differ? 645
14.8 Savitzky-Golay
Smoothing Filters 650
15 Modeling of Data
15.1 Least Squares
as a Maximum Likelihood Estimator 657
15.2 Fitting Data
to a Straight Line 661
15.3 Straight-Line
Data with Errors in Both Coordinates 666
15.4 General Linear
Least Squares 671
15.5 Nonlinear
Models 681
15.6 Confidence
Limits on Estimated Model Parameters 689
15.7 Robust
Estimation 699
16 Integration of Ordinary Differential
Equations
16.1 Runge-Kutta
Method 710
16.2 Adaptive
Stepsize Control for Runge-Kutta 714
16.3 Modified
Midpoint Method 722
16.4 Richardson
Extrapolation and the Bulirsch-Stoer Method 724
16.5 Second-Order
Conservative Equations 732
16.6 Stiff Sets of
Equations 734
16.7 Multistep,
Multivalue, and Predictor-Corrector Methods 747
17 Two Point Boundary Value
Problems
17.1 The Shooting
Method 757
17.2 Shooting to a
Fitting Point 760
17.3 Relaxation
Methods 762
17.4 A Worked
Example: Spheroidal Harmonics 772
17.5 Automated
Allocation of Mesh Points 783
17.6 Handling
Internal Boundary Conditions or Singular Points
784
18 Integral Equations and Inverse
Theory
18.1 Fredholm
Equations of the Second Kind 791
18.2 Volterra
Equations 794
18.3 Integral
Equations with Singular Kernels 797
18.4 Inverse
Problems and the Use of A Priori Information 804
18.5 Linear
Regularization Methods 808
18.6 Backus-Gilbert
Method 815
18.7 Maximum
Entropy Image Restoration 818
19 Partial Differential
Equations
19.1
Flux-Conservative Initial Value Problems 834
19.2 Diffusive
Initial Value Problems 847
19.3 Initial Value
Problems in Multidimensions 853
19.4 Fourier and
Cyclic Reduction Methods for Boundary Value Problems 857
19.5 Relaxation
Methods for Boundary Value Problems 863
19.6 Multigrid
Methods for Boundary Value Problems 871
20 Less-Numerical
Algorithms
20.1 Diagnosing
Machine Parameters 889
20.2 Gray Codes
894
20.3 Cyclic
Redundancy and Other Checksums 896
20.4 Huffman Coding
and Compression of Data 903
20.5 Arithmetic
Coding 910
20.6 Arithmetic at
Arbitrary Precision 915
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