Index 979 Leakage width 554f. Cholesky decomposition 96ff.,430,462, Leapfrog method 842f. 674 Least squares filters see Savitzky-Golay filters complex 49f. Least squares fitting 650f,657ff,661ff., computing A-1.B 48 666f狂，671任 conjugate gradient method 83ff.,606 contrasted to general minimization prob- cyclic tridiagonal 74f. lems 689 direct methods 35,71 degeneracies in 677,679 Gauss-Jordan elimination 36ff. Fourier components 577 Gaussian elimination 41f. freezing parameters in 674,705 Hilbert matrix 90 general linear case 671ff. Hotelling's method 57,606 Levenberg-Marquardt method 683ff,825 and integral equations 788ff.,792 Lomb periodogram 577 iterative improvement 55ff..201 as M-estimate for normal errors 701 iterative methods 35,83ff. as maximum likelihood estimator 658 large sets of 33 as method for smoothing data 650f. least squares solution 62,65f.,205,676 multidimensional 680 LU decomposition 43ff.,201,393,739, nonlinear 393,681ff.,825 792.795.810 nonlinear,advanced methods 688 nonsingular 33 normal equations 651,672ff.,809f. overdetermined 34f..205.676.806 normal equations often singular 676,679 partitioned 77f. optimal (Wiener)filtering 547 OR decomposition 98f..389,393,674 QR method in 100,674 row vs.column elimination 40f. for rational Chebyshev approximation 205 Schultz's method 57.606 relation to linear correlation 636.664 Sherman-Morrison formula 73ff.,90 Savitzky-Golay filter as 650f. singular32,61,66,205,676 singular value decomposition(SVD)34f., singular value decomposition(SVD)59ff., 59E,205,676ff 205,676f任，806 skewed by outliers 659 sparse33,51ff.,71ff,739,813 for spectral analysis 577 summary of tasks 34 standard (probable)errors on fitted parame- Toeplitz 90.92ff..201 tes673,677 Vandermonde 90ff.,120 weighted 658 wavelet solution 603ff.,791 Woodbury formula 75ff.,90 see also Fitting L'Ecuyer's long period random generator 280ff. see also Eigensystems Linear congruential random number generator Left eigenvalues or eigenvectors 458 276f Legal matters xvi Legendre elliptic integral see Elliptic integrals choice of constants for 284f. Linear constraints 431 Legendre polynomials 252f. fitting data to 680 Linear convergence 353.400 Linear correlation (statistics)636ff. recurrence relation 178 Linear dependency shifted monic 159 constructing orthonormal basis 66,100 see also Associated Legendre polynomials. of directions in N-dimensional space 415 Spherical harmonics in linear algebraic equations 32f. Lehmer-Schur algorithm 376 Linear equations see Differential equations;In- Lemarie's wavelet 600 tegral equations;Linear algebraic equa- Lentz's method for continued fraction 171. tions 219 Linear inversion method,constrained 808ff. Lepage,P.319 Linear prediction 564ff. Leptokurtic distribution 612 characteristic polynomial 567 Levenberg-Marquardt algorithm 393,683ff., coefficients 564ff. 825 compared with regularization 810 advanced implementation 688 contrasted to polynomial extrapolation Levinson's method 92f 567 Lewis,H.W.284 related to optimal filtering 565f. License information xvi removal of bias in 570 Limbo 362 stability 567 Limit cycle,in Laguerre's method 372 Linear predictive coding (LPC)571f Line minimization see Minimization,along a Linear programming 394,430ff. ray artificial variables 437 Line search see Minimization,along a ray auxiliary objective function 437 Linear algebraic equations 32ff. basic variables 434 band diagonal 51ff. composite simplex algorithm 443 biconjugate gradient method 84f. constraints 431Index 979 Leakage width 554f. Leapfrog method 842f. Least squares filters see Savitzky-Golay filters Least squares fitting 650f., 657ff., 661ff., 666ff., 671ff. contrasted to general minimization prob￾lems 689 degeneracies in 677, 679 Fourier components 577 freezing parameters in 674, 705 general linear case 671ff. Levenberg-Marquardt method 683ff., 825 Lomb periodogram 577 as M-estimate for normal errors 701 as maximum likelihood estimator 658 as method for smoothing data 650f. multidimensional 680 nonlinear 393, 681ff., 825 nonlinear, advanced methods 688 normal equations 651, 672ff., 809f. normal equations often singular 676, 679 optimal (Wiener) filtering 547 QR method in 100, 674 for rational Chebyshev approximation 205 relation to linear correlation 636, 664 Savitzky-Golay filter as 650f. singular value decomposition (SVD) 34f., 59ff., 205, 676ff. skewed by outliers 659 for spectral analysis 577 standard (probable) errors on fitted parame￾ters 673, 677 weighted 658 see also Fitting L’Ecuyer’s long period random generator 280ff. Left eigenvalues or eigenvectors 458 Legal matters xvi Legendre elliptic integral see Elliptic integrals Legendre polynomials 252f. fitting data to 680 recurrence relation 178 shifted monic 159 see also Associated Legendre polynomials; Spherical harmonics Lehmer-Schur algorithm 376 Lemarie’s wavelet 600 Lentz’s method for continued fraction 171, 219 Lepage, P. 319 Leptokurtic distribution 612 Levenberg-Marquardt algorithm 393, 683ff., 825 advanced implementation 688 Levinson’s method 92f. Lewis, H.W. 284 License information xvi Limbo 362 Limit cycle, in Laguerre’s method 372 Line minimization see Minimization, along a ray Line search see Minimization, along a ray Linear algebraic equations 32ff. band diagonal 51ff. biconjugate gradient method 84f. Cholesky decomposition 96ff., 430, 462, 674 complex 49f. computing A−1 · B 48 conjugate gradient method 83ff., 606 cyclic tridiagonal 74f. direct methods 35, 71 Gauss-Jordan elimination 36ff. Gaussian elimination 41f. Hilbert matrix 90 Hotelling’s method 57, 606 and integral equations 788ff., 792 iterative improvement 55ff., 201 iterative methods 35, 83ff. large sets of 33 least squares solution 62, 65f., 205, 676 LU decomposition 43ff., 201, 393, 739, 792, 795, 810 nonsingular 33 overdetermined 34f., 205, 676, 806 partitioned 77f. QR decomposition 98f., 389, 393, 674 row vs. column elimination 40f. Schultz’s method 57, 606 Sherman-Morrison formula 73ff., 90 singular 32, 61, 66, 205, 676 singular value decomposition (SVD) 59ff., 205, 676ff., 806 sparse 33, 51ff., 71ff., 739, 813 summary of tasks 34 Toeplitz 90, 92ff., 201 Vandermonde 90ff., 120 wavelet solution 603ff., 791 Woodbury formula 75ff., 90 see also Eigensystems Linear congruential random number generator 276f. choice of constants for 284f. Linear constraints 431 Linear convergence 353, 400 Linear correlation (statistics) 636ff. Linear dependency constructing orthonormal basis 66, 100 of directions in N-dimensional space 415 in linear algebraic equations 32f. Linear equations see Differential equations; In￾tegral equations; Linear algebraic equa￾tions Linear inversion method, constrained 808ff. Linear prediction 564ff. characteristic polynomial 567 coefficients 564ff. compared with regularization 810 contrasted to polynomial extrapolation 567 related to optimal filtering 565f. removal of bias in 570 stability 567 Linear predictive coding (LPC) 571f. Linear programming 394, 430ff. artificial variables 437 auxiliary objective function 437 basic variables 434 composite simplex algorithm 443 constraints 431