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922 Chapter 20. Less-<strong>Numerical</strong> Algorithms } fu=256.0*(fu-i); } for (;;) { Iterate Newton’s rule to convergence. mpmul(r,u,u,n,n); Construct S =(3−VU2 )/2. mplsh(r,n); mpmul(s,r,v,n,IMIN(m,n)); mplsh(s,n); mpneg(s,n); s[1] -= 253; mpsdv(s,s,n,2,&ir); for (j=2;j
20.6 Arithmetic at Arbitrary Precision 923 3.1415926535897932384626433832795028841971693993751058209749445923078164062 862089986280348253421170679821480865132823066470938446095505822317253594081 284811174502841027019385211055596446229489549303819644288109756659334461284 756482337867831652712019091456485669234603486104543266482133936072602491412 737245870066063155881748815209209628292540917153643678925903600113305305488 204665213841469519415116094330572703657595919530921861173819326117931051185 480744623799627495673518857527248912279381830119491298336733624406566430860 213949463952247371907021798609437027705392171762931767523846748184676694051 320005681271452635608277857713427577896091736371787214684409012249534301465 495853710507922796892589235420199561121290219608640344181598136297747713099 605187072113499999983729780499510597317328160963185950244594553469083026425 223082533446850352619311881710100031378387528865875332083814206171776691473 035982534904287554687311595628638823537875937519577818577805321712268066130 019278766111959092164201989380952572010654858632788659361533818279682303019 520353018529689957736225994138912497217752834791315155748572424541506959508 295331168617278558890750983817546374649393192550604009277016711390098488240 128583616035637076601047101819429555961989467678374494482553797747268471040 475346462080466842590694912933136770289891521047521620569660240580381501935 112533824300355876402474964732639141992726042699227967823547816360093417216 412199245863150302861829745557067498385054945885869269956909272107975093029 553211653449872027559602364806654991198818347977535663698074265425278625518 184175746728909777727938000816470600161452491921732172147723501414419735685 481613611573525521334757418494684385233239073941433345477624168625189835694 855620992192221842725502542568876717904946016534668049886272327917860857843 838279679766814541009538837863609506800642251252051173929848960841284886269 456042419652850222106611863067442786220391949450471237137869609563643719172 874677646575739624138908658326459958133904780275900994657640789512694683983 525957098258226205224894077267194782684826014769909026401363944374553050682 034962524517493996514314298091906592509372216964615157098583874105978859597 729754989301617539284681382686838689427741559918559252459539594310499725246 808459872736446958486538367362226260991246080512438843904512441365497627807 977156914359977001296160894416948685558484063534220722258284886481584560285 Figure 20.6.1. The first 2398 decimal digits of π, computed by the routines in this section. #include #include "nrutil.h" #define IAOFF 48 void mppi(int n) Demonstrate multiple precision routines by calculating and printing the first n bytes of π. { void mp2dfr(unsigned char a[], unsigned char s[], int n, int *m); void mpadd(unsigned char w[], unsigned char u[], unsigned char v[], int n); void mpinv(unsigned char u[], unsigned char v[], int n, int m); void mplsh(unsigned char u[], int n); void mpmov(unsigned char u[], unsigned char v[], int n); void mpmul(unsigned char w[], unsigned char u[], unsigned char v[], int n, int m); void mpsdv(unsigned char w[], unsigned char u[], int n, int iv, int *ir); void mpsqrt(unsigned char w[], unsigned char u[], unsigned char v[], int n, int m); int ir,j,m; unsigned char mm,*x,*y,*sx,*sxi,*t,*s,*pi; x=cvector(1,n+1); y=cvector(1,n
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Numerical Recipes in C The Art of S
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Contents Preface to the Second Edit
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Contents vii 7.2 Transformation Met
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Contents ix 15.6 Confidence Limits
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Preface to the Second Edition Our a
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Preface to the Second Edition xiii
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Preface to the First Edition xv lin
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Types of License Offered License In
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Computer Programs by Chapter and Se
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Chapter 1. Preliminaries 1.0 Introd
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1.0 Introduction 3 Tested Machines
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1.1 Program Organization and Contro
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true 1.1 Program Organization and C
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} 1.2 Some C Conventions for Scient
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1.2 Some C Conventions for Scientif
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(a) **m (b) **m 1.2 Some C Conventi
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= 0 3 = 0 = 0 = 0 = 0 = 0 1 ⁄ 2 1
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1.3 Error, Accuracy, and Stability
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2.0 Introduction 33 Accumulated rou
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2.0 Introduction 35 between singula
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2.1 Gauss-Jordan Elimination 37 Eli
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2.1 Gauss-Jordan Elimination 39 pre
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2.2 Gaussian Elimination with Backs
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2.3 LU Decomposition and Its Applic
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a b c d 2.3 LU Decomposition and It
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} 2.3 LU Decomposition and Its Appl
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2.3 LU Decomposition and Its Applic
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2.4 Tridiagonal and Band Diagonal S
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2.4 Tridiagonal and Band Diagonal S
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δx 2.5 Iterative Improvement of a
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2.5 Iterative Improvement of a Solu
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2.6 Singular Value Decomposition 59
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SVD of a Square Matrix 2.6 Singular
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solutions of A ⋅ x = d null space
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} 2.6 Singular Value Decomposition
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2.7 Sparse Linear Systems 2.7 Spars
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2.7 Sparse Linear Systems 73 prescr
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2.7 Sparse Linear Systems 75 Here
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2.7 Sparse Linear Systems 77 Finall
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2.7 Sparse Linear Systems 79 In row
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} 2.7 Sparse Linear Systems 81 jp=i
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2.7 Sparse Linear Systems 83 Matrix
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2.7 Sparse Linear Systems 85 while
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p=dvector(1,n); pp=dvector(1,n); r=
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} 2.7 Sparse Linear Systems 89 void
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2.8 Vandermonde Matrices and Toepli
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2.9 Cholesky Decomposition 97 If yo
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2.10 QR Decomposition 99 The standa
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#include #include "nrutil.h" 2.10
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in terms of which 2.11 Is Matrix In
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Chapter 3. Interpolation and Extrap
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(a) (b) 3.0 Introduction 107 Figure
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3.1 Polynomial Interpolation and Ex
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3.2 Rational Function Interpolation
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3.3 Cubic Spline Interpolation 113
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3.3 Cubic Spline Interpolation 115
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3.4 How to Search an Ordered Table
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} 3.4 How to Search an Ordered Tabl
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#include "nrutil.h" 3.5 Coefficient
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3.6 Interpolation in Two or More Di
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} int j; float *ymtmp; 3.6 Interpol
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3.6 Interpolation in Two or More Di
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Chapter 4. Integration of Functions
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4.1 Classical Formulas for Equally
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#define FUNC(x) ((*func)(x)) 4.2 El
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4.2 Elementary Algorithms 139 trapz
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4.4 Improper Integrals 141 which co
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4.4 Improper Integrals 143 The rout
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4.4 Improper Integrals 145 If you n
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4.5 Gaussian Quadratures and Orthog
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Gauss-Legendre: Gauss-Chebyshev: Ga
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} 4.5 Gaussian Quadratures and Orth
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4.6 Multidimensional Integrals 161
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y 4.6 Multidimensional Integrals 16
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Chapter 5. Evaluation of Functions
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5.1 Series and Their Convergence 16
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5.2 Evaluation of Continued Fractio
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5.2 Evaluation of Continued Fractio
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5.3 Polynomials and Rational Functi
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5.3 Polynomials and Rational Functi
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5.4 Complex Arithmetic 177 Actually
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5.5 Recurrence Relations and Clensh
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5.5 Recurrence Relations and Clensh
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5.6 Quadratic and Cubic Equations 1
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5.6 Quadratic and Cubic Equations 1
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5.7 Numerical Derivatives 187 With
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} 5.7 Numerical Derivatives 189 The
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Chebyshev polynomials 1 .5 0 −.5
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} 5.8 Chebyshev Approximation 193 f
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5.9 Derivatives or Integrals of a C
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5.10 Polynomial Approximation from
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5.11 Economization of Power Series
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5.12 Padé Approximants 201 then R(
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5.12 Padé Approximants 203 cof[2*n
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5.13 Rational Chebyshev Approximati
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5.13 Rational Chebyshev Approximati
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5.14 Evaluation of Functions by Pat
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5.14 Evaluation of Functions by Pat
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6.1 Gamma, Beta, and Related Functi
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6.1 Gamma, Beta, and Related Functi
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incomplete gamma function P(a,x) 1.
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} void nrerror(char error_text[]);
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#include 6.2 Incomplete Gamma Func
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6.3 Exponential Integrals 223 where
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6.3 Exponential Integrals 225 #incl
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6.4 Incomplete Beta Function 227 Th
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6.4 Incomplete Beta Function 229 So
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Bessel functions 1 .5 0 −.5 −1
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} 6.5 Bessel Functions of Integer O
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6.5 Bessel Functions of Integer Ord
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modified Bessel functions 4 3 2 1 6
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6.6 Modified Bessel Functions of In
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6.7 Bessel Functions of Fractional
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} 6.7 Bessel Functions of Fractiona
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6.8 Spherical Harmonics 253 The fir
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6.9 Fresnel Integrals, Cosine and S
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} 6.9 Fresnel Integrals, Cosine and
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} 6.10 Dawson’s Integral 259 } if
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6.11 Elliptic Integrals and Jacobia
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where 6.11 Elliptic Integrals and J
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#define TINY 1.0e-25 #define BIG 4.
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6.11 Elliptic Integrals and Jacobia
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6.12 Hypergeometric Functions 271 C
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#include "complex.h" #define ONE Co
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7.1 Uniform Deviates 275 As for ref
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7.1 Uniform Deviates 277 This corre
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#define IA 16807 #define IM 2147483
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RAN 3 1 7.1 Uniform Deviates 281 iy
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7.1 Uniform Deviates 283 from the w
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7.1 Uniform Deviates 285 Constants
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7.2 Transformation Method: Exponent
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7.2 Transformation Method: Exponent
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first random deviate in 7.3 Rejecti
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} 7.3 Rejection Method: Gamma, Pois
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} 7.3 Rejection Method: Gamma, Pois
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7.4 Generation of Random Bits 297 s
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} } *iseed
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7.5 Random Sequences Based on Data
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} } 7.5 Random Sequences Based on D
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7.6 Simple Monte Carlo Integration
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7.6 Simple Monte Carlo Integration
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7.7 Quasi- (that is, Sub-) Random S
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} 7.7 Quasi- (that is, Sub-) Random
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7.8 Adaptive and Recursive Monte Ca
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} } 7.8 Adaptive and Recursive Mont
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} 7.8 Adaptive and Recursive Monte
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Chapter 8. Sorting 8.0 Introduction
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8.1 Straight Insertion and Shell’
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8.2 Quicksort 333 question of the b
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8.2 Quicksort 335 istack=lvector(1,
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a 4 a 2 8.3 Heapsort 337 a 5 a 1 a
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original array 1 2 3 4 5 6 14 8 32
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8.5 Selecting the Mth Largest 341 w
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8.5 Selecting the Mth Largest 343 r
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} } } } 8.6 Determination of Equiva
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Chapter 9. Root Finding and Nonline
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9.0 Introduction 349 good first gue
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(a) (b) (c) (d) 9.1 Bracketing and
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Bisection Method 9.1 Bracketing and
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9.2 Secant Method, False Position M
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} 9.2 Secant Method, False Position
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} 9.3 Van Wijngaarden-Dekker-Brent
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9.3 Van Wijngaarden-Dekker-Brent Me
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9.4 Newton-Raphson Method Using Der
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9.4 Newton-Raphson Method Using Der
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} 9.4 Newton-Raphson Method Using D
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9.5 Roots of Polynomials 9.5 Roots
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9.5 Roots of Polynomials 371 tentat
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9.5 Roots of Polynomials 373 The me
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Eigenvalue Methods 9.5 Roots of Pol
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9.5 Roots of Polynomials 377 of goi
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9.6 Newton-Raphson Method for Nonli
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9.6 Newton-Raphson Method for Nonli
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9.7 Globally Convergent Methods for
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X1 A B C 10.0 Introduction 395 D Fi
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10.1 Golden Section Search in One D
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10.1 Golden Section Search in One D
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} } 10.1 Golden Section Search in O
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10.2 Parabolic Interpolation and Br
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} 10.3 One-Dimensional Search with
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10.3 One-Dimensional Search with Fi
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(a) (b) (c) (d) 10.4 Downhill Simpl
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10.4 Downhill Simplex Method in Mul
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10.5 Direction Set (Powell’s) Met
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(a) 10.6 Conjugate Gradient Methods
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10.6 Conjugate Gradient Methods in
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} 10.7 Variable Metric Methods in M
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10.7 Variable Metric Methods in Mul
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} 10.7 Variable Metric Methods in M
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10.8 Linear Programming and the Sim
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} 10.8 Linear Programming and the S
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10.9 Simulated Annealing Methods 44
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(a) (b) (c) 1 .5 0 1 .5 0 1 .5 0 10
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} } 10.9 Simulated Annealing Method
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} free_ivector(jorder,1,ncity); #in
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Definitions and Basic Facts 11.0 In
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11.0 Introduction 459 In both the d
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11.0 Introduction 461 “Eigenpacka
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11.1 Jacobi Transformations of a Sy
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} } 11.2 Reduction of a Symmetric M
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11.2 Reduction of a Symmetric Matri
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11.2 Reduction of a Symmetric Matri
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11.3 Eigenvalues and Eigenvectors o
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} } 11.4 Hermitian Matrices 481 for
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11.5 Reduction of a General Matrix
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11.5 Reduction of a General Matrix
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11.6 The QR Algorithm for Real Hess
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11.7 Eigenvalues or Eigenvectors by
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11.7 Eigenvalues or Eigenvectors by
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and equation (12.0.1) looks like th
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⏐h(t)⏐ 2 (a) Ph( f ) (one-sided
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12.1 Fourier Transform of Discretel
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12.1 Fourier Transform of Discretel
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12.2 Fast Fourier Transform (FFT) 5
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12.2 Fast Fourier Transform (FFT) 5
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Other FFT Algorithms 12.2 Fast Four
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12.3 FFT of Real Functions, Sine an
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12.3 FFT of Real Functions, Sine an
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(a) (b) (c) +1 0 −1 +1 0 −1 +1
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#include 12.3 FFT of Real Function
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} 12.3 FFT of Real Functions, Sine
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} } } 12.4 FFT in Two or More Dimen
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data [1] Re Im 12.4 FFT in Two or M
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12.5 Fourier Transforms of Real Dat
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12.6 External Storage or Memory-Loc
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} } 12.6 External Storage or Memory
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Chapter 13. Fourier and Spectral Ap
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(t) r*s(t) 13.1 Convolution and Dec
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13.1 Convolution and Deconvolution
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13.1 Convolution and Deconvolution
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13.2 Correlation and Autocorrelatio
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13.3 Optimal (Wiener) Filtering wit
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log scale 13.4 Power Spectrum Estim
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13.4 Power Spectrum Estimation Usin
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amplitude 1 .8 .6 .4 .2 0 13.4 Powe
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13.5 Digital Filtering in the Time
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IIR (Recursive) Filters 13.5 Digita
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(a) (b) 13.5 Digital Filtering in t
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13.6 Linear Prediction and Linear P
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} 13.6 Linear Prediction and Linear
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13.7 Maximum Entropy (All Poles) Me
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13.8 Spectral Analysis of Unevenly
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} 13.8 Spectral Analysis of Unevenl
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} 13.8 Spectral Analysis of Unevenl
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13.9 Computing Fourier Integrals Us
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Cubic order: 13.9 Computing Fourier
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13.9 Computing Fourier Integrals Us
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13.10 Wavelet Transforms 591 The sp
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13.10 Wavelet Transforms 593 p.”
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13.10 Wavelet Transforms 595 comple
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} 13.10 Wavelet Transforms 597 wfil
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.2 0 −.2 .2 0 −.2 DAUB4 e10 + e
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wavelet amplitude 1.5 1 .5 .0001 10
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} nprev=nnew; } free_vector(wksp,1,
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13.10 Wavelet Transforms 605 Figure
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13.11 Numerical Use of the Sampling
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Chapter 14. Statistical Description
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14.1 Moments of a Distribution: Mea
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14.1 Moments of a Distribution: Mea
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14.2 Do Two Distributions Have the
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14.3 Are Two Distributions Differen
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1. male 2. female . 14.4 Contingenc
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14.4 Contingency Table Analysis of
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14.5 Linear Correlation 637 two dis
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14.6 Nonparametric or Rank Correlat
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} } 14.7 Do Two-Dimensional Distrib
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14.7 Do Two-Dimensional Distributio
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14.8 Savitzky-Golay Smoothing Filte
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8 6 4 2 0 8 6 4 2 0 8 6 4 2 0 befor
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14.8 Savitzky-Golay Smoothing Filte
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15.1 Least Squares as a Maximum Lik
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15.1 Least Squares as a Maximum Lik
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15.2 Fitting Data to a Straight Lin
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15.2 Fitting Data to a Straight Lin
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15.3 Straight-Line Data with Errors
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15.3 Straight-Line Data with Errors
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15.4 General Linear Least Squares 6
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} 15.4 General Linear Least Squares
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15.5 Nonlinear Models 681 Lawson, C
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15.5 Nonlinear Models 683 Note that
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15.5 Nonlinear Models 685 pivots, b
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#include "nrutil.h" 15.5 Nonlinear
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15.6 Confidence Limits on Estimated
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actual data set 15.6 Confidence Lim
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15.7 Robust Estimation 699 M� 1 C
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15.7 Robust Estimation 701 where th
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15.7 Robust Estimation 703 algorith
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} *a=aa; *b=bb; *abdev=abdevt/ndata
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Chapter 16. Integration of Ordinary
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16.0 Introduction 709 problem where
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y(x) 16.1 Runge-Kutta Method 711 1
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} 16.1 Runge-Kutta Method 713 yt=ve
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16.2 Adaptive Stepsize Control for
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16.3 Modified Midpoint Method 723 H
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16.4 Richardson Extrapolation and t
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16.5 Second-Order Conservative Equa
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y 16.6 Stiff Sets of Equations 735
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16.6 Stiff Sets of Equations 737 If
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16.6 Stiff Sets of Equations 739 od
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} 16.6 Stiff Sets of Equations 741
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16.6 Stiff Sets of Equations 743 Co
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16.6 Stiff Sets of Equations 745 Se
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} 16.7 Multistep, Multivalue, and P
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16.7 Multistep, Multivalue, and Pre
Page 775 and 776:
16.7 Multistep, Multivalue, and Pre
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Chapter 17. Two Point Boundary Valu
Page 779 and 780:
equired boundary value 17.0 Introdu
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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
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17.3 Relaxation Methods 767 Next co
Page 793 and 794:
17.3 Relaxation Methods 769 calcula
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} 17.3 Relaxation Methods 771 for (
Page 797 and 798:
17.4 A Worked Example: Spheroidal H
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Page 801 and 802:
Page 803 and 804:
} 17.4 A Worked Example: Spheroidal
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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
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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
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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 899 and 900: S S S E γ = 1 19.6 Multigrid Metho
Page 905 and 906: } 19.6 Multigrid Methods for Bounda
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 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: } 20.6 Arithmetic at Arbitrary Prec
Page 949: 20.6 Arithmetic at Arbitrary Precis
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