vini Contents 11.6 The OR Algorithm for Real Hessenberg Matrices 486 11.7 Improving Eigenvalues and/or Finding Eigenvectors by Inverse Iteration 493 12 Fast Fourier Transform 496 12.0 Introduction 496 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 537 13.0 Introduction 537 13.1 Convolution and Deconvolution Using the FFT 538 13.2 Correlation and Autocorrelation Using the FFT 545 547 (North America from NUMERICAL RECIPES IN C:THE 19881992 13.3 Optimal (Wiener)Filtering with the FFT 13.4 Power Spectrum Estimation Using the FFT 549 13.5 Digital Filtering in the Time Domain 558 tusers to make one paper 13.6 Linear Prediction and Linear Predictive Coding 564 13.7 Power Spectrum Estimation by the Maximum Entropy 9 (All Poles)Method 572 Programs 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 609 14.0 Introduction 609 14.1 Moments of a Distribution:Mean.Variance.Skewness 610 ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) and So Forth 14.2 Do Two Distributions Have the Same Means or Variances? 615 1988-1992 by Numerical Recipes 14.3 Are Two Distributions Different? 620 14.4 Contingency Table Analysis of Two Distributions 628 14.5 Linear Correlation 636 (outside 14.6 Nonparametric or Rank Correlation 639 Software. 14.7 Do Two-Dimensional Distributions Differ? 645 North 14.8 Savitzky-Golay Smoothing Filters 650 15 Modeling of Data 656 15.0 Introduction 656 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 681viii Contents Permission is granted for internet users to make one paper copy for their own personal use. Further reproduction, or any copyin Copyright (C) 1988-1992 by Cambridge University Press. Programs Copyright (C) 1988-1992 by Numerical Recipes Software. Sample page from NUMERICAL RECIPES IN C: THE ART OF SCIENTIFIC COMPUTING (ISBN 0-521-43108-5) g of machine￾readable 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). 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 496 12.0 Introduction 496 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 537 13.0 Introduction 537 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 609 14.0 Introduction 609 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 656 15.0 Introduction 656 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