Index 971 semi-implicit midpoint rule 743 Downhill simplex method see Simplex,method shooting method 754,757E of Nelder and Mead shooting method,example 779f.,781 Driver programs 3 similarity to Volterra integral equations Dual viewpoint,in multigrid method 883 794f Duplication theorem,elliptic integrals 262f. singular points 724f,760,784ff. dvector()utility 943 step doubling 715 DWT (discrete wavelet transform)see Wavelet stepsize control 709,714ff.,724,733f., transform 738.744.749f.751 stiff 709.734 stiff methods compared 747 Eardley,D.M.346 Stoermer's rule 732f. EBCDIC 898 see also Partial differential equations:Two- Economization of power series 198ff.,201 point boundary value problems Eigensystems 456ff. Diffusion equation 827,847ff.,864 balancing matrix 483 Crank-Nicolson method 848,853,855 bounds on eigenvalues 58 Forward Time Centered Space (FTCS) calculation of few eigenvectors or eigenval- 847E.850f.864 ues461,494 implicit differencing 848 canned routines 461 multidimensional 855f. characteristic polynomial 456,475f Digamma function 222 completeness 457 defective 457.482.494 Digital filtering see Filter deflation 478 Dihedral group Ds 902 degenerate eigenvalues 456,458 Dimensions (units)683f. elimination method 460,485 Diminishing increment sort 331 factorization method 460 Dirac delta function 293,789 fast Givens reduction 470 Direct method see Periodogram Direct methods for linear algebraic equations generalized eigenproblem 462 35 Givens reduction 469f. Hermitian matrix 481f. Direct product see Outer product of matrices Hessenberg matrix 460,477,482ff.,494 Direction of largest decrease 416f. Householder transformation 460,469ff., Direction numbers,Sobol's sequence 311 476,480,481.484f Direction-set methods for minimization 396, ill-conditioned eigenvalues 483 412ff Dirichlet boundary conditions 829,849,859, implicit shifts 478f and integral equations 788ff.,794 865.867 invariance under similarity transform 459 Disclaimer of warranty xvi inverse iteration 462,476,483,493ff. Discordant pair for Kendall's tau 643 Jacobi transformation 460,463ff.,469 Discrete convolution theorem 538ff. 481,495 Discrete Fourier transform (DFT)500ff. left eigenvalues 458 as approximate continuous transform 503 list of tasks 461 see also Fast Fourier transform (FFT) Discrete optimization 444ff. multiple eigenvalues 495 nonlinear 462 Discriminant 184.464 nonsymmetric matrix 482ff. Diskettes,how to order xvi,996f. operation count of balancing 483 Dispersion 840 operation count of Givens reduction 470 DISPO see Savitzky-Golay filters operation count of Householder reduction Dissipation.numerical 839 474 Divergent series 167 operation count of inverse iteration 494 Division operation count of Jacobi method 467 complex 177 operation count of QL method 477,480 multiple precision 919f. operation count of QR method for Hessen- of polynomials 175,369,377 berg matrices 490 dmatrix()utility 944 operation count of reduction to Hessenberg dn function 269 form 485 Do-while iteration 12 orthogonality 457 Dogleg step methods 393 polynomial roots and 375 Domain of integration 161f. QL method 476ff,481,494f. Dominant solution of recurrence relation 179 QL method with implicit shifts 478ff. Dot (denotes matrix multiplication)33 QR method60,460,463,476f. Double exponential error distribution 701 QR method for Hessenberg matrices 486ff. Double precision real,symmetric matrix 156,474,794 as refuge of scoundrels 890 reduction to Hessenberg form 484f. use in iterative improvement 56 right eigenvalues 458 Double root 348 shifting eigenvalues 456,477f.,486fIndex 971 semi-implicit midpoint rule 743 shooting method 754, 757ff. shooting method, example 779f., 781 similarity to Volterra integral equations 794f. singular points 724f., 760, 784ff. step doubling 715 stepsize control 709, 714ff., 724, 733f., 738, 744, 749f., 751 stiff 709, 734ff. stiff methods compared 747 Stoermer’s rule 732f. see also Partial differential equations; Two￾point boundary value problems Diffusion equation 827, 847ff., 864 Crank-Nicolson method 848, 853, 855 Forward Time Centered Space (FTCS) 847ff., 850f., 864 implicit differencing 848 multidimensional 855f. Digamma function 222 Digital filtering see Filter Dihedral group D5 902 Dimensions (units) 683f. Diminishing increment sort 331 Dirac delta function 293, 789 Direct method see Periodogram Direct methods for linear algebraic equations 35 Direct product see Outer product of matrices Direction of largest decrease 416f. Direction numbers, Sobol’s sequence 311 Direction-set methods for minimization 396, 412ff. Dirichlet boundary conditions 829, 849, 859, 865, 867 Disclaimer of warranty xvi Discordant pair for Kendall’s tau 643 Discrete convolution theorem 538ff. Discrete Fourier transform (DFT) 500ff. as approximate continuous transform 503 see also Fast Fourier transform (FFT) Discrete optimization 444ff. Discriminant 184, 464 Diskettes, how to order xvi, 996f. Dispersion 840 DISPO see Savitzky-Golay filters Dissipation, numerical 839 Divergent series 167 Division complex 177 multiple precision 919f. of polynomials 175, 369, 377 dmatrix() utility 944 dn function 269 Do-while iteration 12 Dogleg step methods 393 Domain of integration 161f. Dominant solution of recurrence relation 179 Dot (denotes matrix multiplication) 33 Double exponential error distribution 701 Double precision as refuge of scoundrels 890 use in iterative improvement 56 Double root 348 Downhill simplex method see Simplex, method of Nelder and Mead Driver programs 3 Dual viewpoint, in multigrid method 883 Duplication theorem, elliptic integrals 262f. dvector() utility 943 DWT (discrete wavelet transform) see Wavelet transform Eardley, D.M. 346 EBCDIC 898 Economization of power series 198ff., 201 Eigensystems 456ff. balancing matrix 483 bounds on eigenvalues 58 calculation of few eigenvectors or eigenval￾ues 461, 494 canned routines 461 characteristic polynomial 456, 475f. completeness 457 defective 457, 482, 494 deflation 478 degenerate eigenvalues 456, 458 elimination method 460, 485 factorization method 460 fast Givens reduction 470 generalized eigenproblem 462 Givens reduction 469f. Hermitian matrix 481f. Hessenberg matrix 460, 477, 482ff., 494 Householder transformation 460, 469ff., 476, 480, 481, 484f. ill-conditioned eigenvalues 483 implicit shifts 478ff. and integral equations 788ff., 794 invariance under similarity transform 459 inverse iteration 462, 476, 483, 493ff. Jacobi transformation 460, 463ff., 469, 481, 495 left eigenvalues 458 list of tasks 461 multiple eigenvalues 495 nonlinear 462 nonsymmetric matrix 482ff. operation count of balancing 483 operation count of Givens reduction 470 operation count of Householder reduction 474 operation count of inverse iteration 494 operation count of Jacobi method 467 operation count of QL method 477, 480 operation count of QR method for Hessen￾berg matrices 490 operation count of reduction to Hessenberg form 485 orthogonality 457 polynomial roots and 375 QL method 476ff., 481, 494f. QL method with implicit shifts 478ff. QR method 60, 460, 463, 476ff. QR method for Hessenberg matrices 486ff. real, symmetric matrix 156, 474, 794 reduction to Hessenberg form 484f. right eigenvalues 458 shifting eigenvalues 456, 477f., 486f