970 Index unevenly or irregularly sampled 576, Determinant 34,49 581f,654 Deviates,random see Random deviates use of CRCs in manipulating 897 DFP algorithm see Davidon-Fletcher-Powell windowing 553ff. algorithm see also Statistical tests Diagonal dominance 51,684,789,865 Data compression 603,889 Difference equations,finite see Finite differ- arithmetic coding 910ff ence equations (FDEs) cosine transform 519 Difference operator 167 Huffman coding 903ff.,910 Differential equations 707ff. linear predictive coding (LPC)571f. accuracy vs.stability 710,736 lossless 903 Adams-Bashforth-Moulton schemes 749 Data Encryption Standard(DES)300ff. adaptive stepsize control 709.714ff..725. Data type 28 733f,738.744.749f,751 DAUB4592ff.595,598f.601 algebraically difficult sets 772 DAUB6 593 backward Euler's method 735 DAUB12 605 Bader-Deuflhard method for stiff 737, DAUB20 598f. 742f Daubechies wavelet coefficients 592ff.,596, boundary conditions 707f.,753ff.,757, 598f.,601,605 760.779f Davidon-Fletcher-Powell algorithm 397,426f. Bulirsch-Stoer method 209,272,708f., Dawson's integral 259f.,606 712,722.724f.747 approximation for 259 Bulirsch-Stoer method for conservative routine for 260 equations 733 D.C.(direct current)498 comparison of methods 708f.,747,751 Debugging 7 conservative 732f. DEC (Digital Equipment Corp.)xvii,3,894 danger of too small stepsize 720 Declarations 930ff. eigenvalue problem 756,772ff.,779f., Decomposition see Cholesky decomposition; 781 LU decomposition;QR decomposition; embedded Runge-Kutta method 715f.,738 Singular value decomposition(SVD) equivalence of multistep and multivalue Deconvolution 542,549 methods 751 see also Convolution;Fast Fourier trans- Euler's method 708.710.735 form (FFT);Fourier transform forward Euler's method 735 Defect,in multigrid method 872 free boundary problem 756,785 Deferred approach to the limit see Richard- high-order implicit methods 737ff. son's deferred approach to the limit implicit differencing 735f.,749 Deflation initial value problems 708 of matrix 478 internal boundary conditions 784ff of polynomials 369ff.,377,378 internal singular points 784ff. Degeneracy of linear algebraic equations 32, interpolation on right-hand sides 117 61,65,676 Kaps-Rentrop method for stiff 737 Degenerate kernel 794 local extrapolation 715 Degenerate minimization principle 804 modified midpoint method 722f.,726 Degrees of freedom 621f.660.695f. multistep methods 747ff. Demonstration programs 3 multivalue methods 747 Derivatives order of method 710f.,725 computation via Chebyshev approximation path integration for function evaluation 189,195 208f,271 computation via Savitzky-Golay filters predictor-corrector methods 708,737, 189,651 747 matrix of first partial see Jacobian determi- reduction to first-order sets 707,753 nant relaxation method 754f,762ff. matrix of second partial see Hessian ma- relaxation method,example of 772ff. trix r.h.s.independent of c 735 numerical computation 186ff,386,651, Rosenbrock methods for stiff 737 738.758.779 Runge-Kutta method 708f.,710ff.,714ff., of polynomial 173f. 738.747 use in optimization 395f.,406 Runge-Kutta method,high-order 711 DES see Data Encryption Standard Runge-Kutta-Fehlberg method 715f. Descending transformation,elliptic integrals scaling stepsize to required accuracy 716f. 262 second order 732f. Descent direction 383,390,427 semi-implicit differencing 737 Descriptive statistics 609ff. semi-implicit Euler method 737,743 see also Statistical tests semi-implicit extrapolation method 737, Design matrix 651,671,804,809 743970 Index unevenly or irregularly sampled 576, 581f., 654 use of CRCs in manipulating 897 windowing 553ff. see also Statistical tests Data compression 603, 889 arithmetic coding 910ff. cosine transform 519 Huffman coding 903ff., 910 linear predictive coding (LPC) 571f. lossless 903 Data Encryption Standard (DES) 300ff. Data type 28 DAUB4 592ff., 595, 598f., 601 DAUB6 593 DAUB12 605 DAUB20 598f. Daubechies wavelet coefficients 592ff., 596, 598f., 601, 605 Davidon-Fletcher-Powell algorithm 397, 426f. Dawson’s integral 259f., 606 approximation for 259 routine for 260 D.C. (direct current) 498 Debugging 7 DEC (Digital Equipment Corp.) xvii, 3, 894 Declarations 930ff. Decomposition see Cholesky decomposition; LU decomposition; QR decomposition; Singular value decomposition (SVD) Deconvolution 542, 549 see also Convolution; Fast Fourier trans￾form (FFT); Fourier transform Defect, in multigrid method 872 Deferred approach to the limit see Richard￾son’s deferred approach to the limit Deflation of matrix 478 of polynomials 369ff., 377, 378 Degeneracy of linear algebraic equations 32, 61, 65, 676 Degenerate kernel 794 Degenerate minimization principle 804 Degrees of freedom 621f., 660, 695f. Demonstration programs 3 Derivatives computation via Chebyshev approximation 189, 195 computation via Savitzky-Golay filters 189, 651 matrix of first partial see Jacobian determi￾nant matrix of second partial see Hessian ma￾trix numerical computation 186ff., 386, 651, 738, 758, 779 of polynomial 173f. use in optimization 395f., 406 DES see Data Encryption Standard Descending transformation, elliptic integrals 262 Descent direction 383, 390, 427 Descriptive statistics 609ff. see also Statistical tests Design matrix 651, 671, 804, 809 Determinant 34, 49 Deviates, random see Random deviates DFP algorithm see Davidon-Fletcher-Powell algorithm Diagonal dominance 51, 684, 789, 865 Difference equations, finite see Finite differ￾ence equations (FDEs) Difference operator 167 Differential equations 707ff. accuracy vs. stability 710, 736 Adams-Bashforth-Moulton schemes 749 adaptive stepsize control 709, 714ff., 725, 733f., 738, 744, 749f., 751 algebraically difficult sets 772 backward Euler’s method 735 Bader-Deuflhard method for stiff 737, 742f. boundary conditions 707f., 753ff., 757, 760, 779f. Bulirsch-Stoer method 209, 272, 708f., 712, 722, 724ff., 747 Bulirsch-Stoer method for conservative equations 733 comparison of methods 708f., 747, 751 conservative 732f. danger of too small stepsize 720 eigenvalue problem 756, 772ff., 779f., 781 embedded Runge-Kutta method 715f., 738 equivalence of multistep and multivalue methods 751 Euler’s method 708, 710, 735 forward Euler’s method 735 free boundary problem 756, 785 high-order implicit methods 737ff. implicit differencing 735f., 749 initial value problems 708 internal boundary conditions 784ff. internal singular points 784ff. interpolation on right-hand sides 117 Kaps-Rentrop method for stiff 737 local extrapolation 715 modified midpoint method 722f., 726 multistep methods 747ff. multivalue methods 747 order of method 710f., 725 path integration for function evaluation 208ff., 271 predictor-corrector methods 708, 737, 747ff. reduction to first-order sets 707, 753 relaxation method 754f., 762ff. relaxation method, example of 772ff. r.h.s. independent of x 735 Rosenbrock methods for stiff 737 Runge-Kutta method 708f., 710ff., 714ff., 738, 747 Runge-Kutta method, high-order 711 Runge-Kutta-Fehlberg method 715f. scaling stepsize to required accuracy 716f. second order 732f. semi-implicit differencing 737 semi-implicit Euler method 737, 743 semi-implicit extrapolation method 737, 743