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Acton F.S. — Numerical Methods That Work
Acton F.S. — Numerical Methods That Work

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Название: Numerical Methods That Work

Автор: Acton F.S.


A commonsense approach to numerical algorithms for the solution of equations.

Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

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Год издания: 1990

Количество страниц: 550

Добавлена в каталог: 13.12.2009

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Предметный указатель
"Stiff" differential equations      148
$e^{x}$ calculated symmetrically from $tanh\frac{x}{2}$      27
Accelerating convergence of Gauss — Seidel iterations      480 483
Accelerating convergence, by Aitken extrapolation      216
Accelerating convergence, by shifting eigenvalues of a matrix      215
Activities      see "PERT"
Activities, activities represented as edges of a graph      508
Activities, events represented as nodes of a graph      508
Adams — Bashforth      146
Adaptive quadrature      105
Aitken extrapolation      216
Aitken interpolation      93
Alabama      516
Alternating directions, method of      483
Alternating series, loss of significant figures in      14 69
Approximating by a series of exponentials      252
Approximating error curves by matching, at extrema      301
Approximating error curves by matching, at zeros      300
Approximating quadratic factors of polynomials      195
Approximation for x      26
arctangent function      6
Arctangent function, continued fraction for      7
Arctangent function, Gauss iteration for      9
Arctangent function, series for      6
Arctangent function, series for large arguments      30
Arctangent function, series of Chebyshev polynomials for      36
Asymptotic series      16 262
Asymptotic series for erfc x      16
Asymptotic series, derivation of, from an integral      17 262
Asymptotic series, derivation of, from differential equations      18
Asymptotic series, minimum term of      17 263
Backward difference in parabolic partial differential equations      442
Backward polynomial division      191
Baltimore traffic      504
Banded symmetric matrices, reduction of      329
Bessel functions, recursive evaluation of $J_{n}(x)$      23
Bessel functions, series for $J_{0}(x)$      70
Bessel's interpolation formula      90
Binary chop      179
Biorthognality of the eigenvectors of an unsymmetric matrix with those of its transpose      220
Bisection method      see "Binary chop"
Block tridiagonal systems      482
Bottleneck problems in a network      524
Boundary conditions for ordinary differential equations      Chapter 6
Boundary conditions, at infinity      163
Boundary-value problems of physics, orthogonal functions defined by      495
Bounds for integrals on infinite regions      113
Breaking an infinite range of integration      112
Bulirsch and Stoer      136
Characteristic polynomial of a matrix, evaluation from Hessenberg form      344
Characteristic polynomial of a matrix, evaluation from tridiagonal form      332
Characteristic roots      see "Eigenvalues (Chapters 8 and 13)"
Characteristic value problem      see "Eigenvalue problems (Chapters 8 and 13)"
Chebyshev polynomials      292
Chebyshev polynomials, expansion of $tan^{-1}kx$ in      36
Chebyshev polynomials, matching approximations to functions at extrema of      301
Chebyshev polynomials, minimax property of      290 293
Chebyshev polynomials, used in economizing power series      293
Cholesky algorithm for factoring matrices      348
Circle of indeterminacy      76
Clustered roots      363
Complex polynomial roots      189f.
Complex polynomial roots by forward-backward division      195
Complex polynomial roots by Newton's method for quadratic factors      194
Complex polynomial roots by Newton's method in the complex plane      189
Computational singularities      65 410
Condenser, exact map for      497
Condenser, integral equation for      420 493
Condenser, parallel-plate      491
Condenser, Rosser iteration for      495
Conformal map for the parallel-plate condenser      497
Continued fraction      7
Continued fraction for arctan x      7
Continued fraction for tanh x      27
Continued fraction from a power series      285
Continued fraction, computational forms of      280
Continued fraction, evaluation of as a rational function      287
Continued fraction, tail of, expresion for      312
Continued fraction, truncation of      312
Convergence of an iteration      54 216
Convergence, acceleration of by Aitken extrapolation      216
Convergence, analytic discussion      394f.
Convergent iteration, construction of      398
Corrector step      see "Predictor-corrector methods"
Cosh z, roots of cosh z = w, by iteration      390f.
Cosines, evaluation of series of      11
Cosines, series for, economized      293 305
Crank — Nicholson      447
Critical paths in a network      508
Curve crawlers in several dimensions      379f. 382
Curved boundaries, Laplace's equation      485
Deferred approach to the limit      see "Richardson extrapolation"
Deflating a matrix      219
Derivative of a polynomial, evaluation by synthetic division      183
Derivative of a polynomial, evaluation from a tridiagonal form      334
Derivative, first, estimated from two points      158
Derivative, second, estimated from three points      159
Descent to a nearby subsurface in n dimensions      384
Difference equation, solution of linear      144
Difference table      91
Direct fitting of rational approximations      310
Direct fitting of the discrepancy, of a continued fraction      311
Direct fitting of the discrepancy, of a power series      304
Dirichlet problem      477 488
Discontinuities, removal of to speed convergence of series      233
Discovery of loops in ordered networks      313 315
Discrepancy caused by truncating a continued fraction      311
Divergent iterations      44 145 440
Division of one polynomial by another      190
Double length accumulation of vector innerproducts      343
Double root strategy in several dimensions      401
Double roots of polynomials      186
Economization, of $sin\theta/\theta$      426
Economization, of a power series      291
Economization, of rational functions      308
Eigenvectors      355
Eigenvectors, by inverse iteration      359
Eigenvectors, from specialized matrices      356
Ellipsoids, contours of      210 462
Ellipsoids, second derivative geometries of      461 469f.
Ellipsoids, seeking the center of      458
Ellipsoids, strategy for minimization via      474
Elliptic functions (Jacobian)      415
Elliptic integral K(k)      416
Elliptic integral K(k), computation of      417
Equal-ripple error      291f.
Equal-ripple error, direct production of      299
Error amplifier      440
Error caused by truncating a continued fraction      311
Error function, erf (x), continued fraction for      313
Error function, series representations for      15f.
Error growth, exponential      440
Error growth, linear      216
Error growth, quadratic      216
Error terms for quadrature formulas      111
Estimation of the tail of an integral      113 117
Euler's constant      268
Evaluating $e^{x}$ on a computer      27 28
Evaluating a determinant      354
Evaluating a determinant in Hessenberg form      344
Evaluating a determinant in tridiagonal form      332
Evaluating a polynomial for complex arguments      192—193
Evaluating a polynomial from Hessenberg form      344
Evaluating a polynomial from tridiagonal form      332
Evaluating a polynomial, by synthetic division      59 181
Evaluating finite Fourier series      11
Evaluating series of orthogonal functions      11
Evaluating the characteristic polynomial derivative from tridiagonal form      334
Evaluating the characteristic polynomial from Hessenberg form      344
Evaluating the characteristic polynomial from tridiagonal form      332
Evaluating the discrepancy caused by truncating a continued fraction      311
Everett's interpolation formula      92
Everett's relation of coefficients to Lagrange's      98
Exponential approximation      252
Exponential equations, the integration of boundary-value problems leading to      169f.
Exponential error      440
Exponential fitting      252
Exponential fitting of data      252
Exponential form of Fourier series      240
Extraneous solution to a difference equation      144
Extrapolation to the limit, Richardsonian      106
Factorization of the second order exponential equation      170
False Position in 2 dimensions      374
False Position in 2 dimensions, computational details      378
False Position in 2 dimensions, speed of convergence      379
False position, method of      52
False Position, rate of convergence of      56
Finding non-extremal eigenvalues by the power method      218
Finding the smallest eigenvalue by shifting      217
Finite Fourier series      228f.
Finite Fourier series for unequally spaced points      238
Finite Fourier series, discontinuity removal in      225 233
Finite Fourier series, improving the convergence of      225
Finite Fourier series, in exponential form      239—242
Finite Fourier series, number of terms possible      222 230
Finite Fourier series, recursive calculation of the coefficients      231
Finite Fourier series, recursive evaluation      11
Finite orthogonal polynomials on unequally spaced points      235
Fitting a cubic to two functional values and two derivatives      454f.
Fitting the discrepancy      304
Fletcher      467
Flow through a network of pipes      517f.
Flow through a network of pipes, circuit equations for      519
Flow through a network of pipes, nodal equations for      519
Ford      524
Forsythe      483
Forward differences in parabolic partial differential equations      438
Forward-backward division algorithm for polynomial roots      195
Fourier series, limit on number of coefficients for finite      222
Fourier series, optimality of      224
Fourier series, orthogonality in      223
Fourier series, rate of convergence of      225
Fourier series, recursive evaluation of      11
Fringed tridiagonal equation systems      480
Fulkerson      524
Functional iteration, convergence of      52 399
Functional iteration, starting values for      47
Functional iteration, stopping      48
Gauss — Seidel      480 483
Gaussian elimination      342 358
Gaussian quadrature      103
Gaussian triangularization      see "Gaussian elimination"
Gibbs phenomenon      227
Gibbs phenomenon, Lanczos's $\sigma$ factors for      228
Gill      156
Givens reduction of a matrix      322 330
Goldfeld      476
Gradient methods      see "Functional minimization (Chapter 17)"
Gradient vector      210 368 381
Graeffe's method      198
Gram — Schmidt orthogonalization      219
Graphical extrapolation of a differential equation solution      130
Hamming's method, (o.d.e.)      146
Hartree      170
Hessenberg form      317
Hessenberg form, direct reduction of unsymmetric matrices to      341
Hessenberg form, evaluation of the characteristic polynomial from      344
Hessenberg form, reduction to tridiagonal form      342
Highly oscillatory ordinary differential equations      152
Homogeneous algebraic equations      206
Hotelling's deflation method for eigenvalues      218
Householder's reduction for symmetric matrices      324f.
Ill-conditioned systems      253
Improving the convergence of series by recasting as continued fractions      296
Improving the convergence of series by recasting as rational functions      263
Infinite integrals      413
Infinite integrands      66 120 412
Infinite product for sin x      20
Infinite regions, quadrature over      112
Infinite regions, transformation of dummy variable      112
Initial-value treatment of boundary-value problems      173
Innerproduct doublelength accumulation, its necessity in large linear systems      343
Instabilities, in initial-value problems      434
Instabilities, in ordinary differential equations      143 434
Instabilities, in parabolic partial differential equations      436f.
Instabilities, in recurrence relations      21 432
Integral equation for the parallel plate condenser      420 493
Integral equations, singular      420 493
Integration of "stiff" differential equations      148
Integration of highly oscillatory differential equations      152
Interchanging, example of, with a tridiagonal matrix      358
Interior eigenvalues by orthogonalization      218
Interpolation formulas      Chapter 3
Interpolation formulas, Aitken      93
Interpolation formulas, Bessel      90
Interpolation formulas, by the method of undetermined coefficients      108
Interpolation formulas, Everett      92
Interpolation formulas, higher order      90
Interpolation formulas, Lagrange      96
Interpolation formulas, quadratic      90
Interpolation formulas, unequally spaced values      95
Interpolation, special methods for      98
Inverse iteration for eigenvectors      357
Isoclines      see "Graphical extrapolation of a differential equation solution"
Isolated roots      81
Iterated multiplication      211
Iterated vector, multiplication by a matrix      214
Iteration method for the largest eigenvalue      213
Iterative solution of a quadratic      58
Jacobi plane rotation      319
Jacobi plane rotation, comparative inefficiency of for eigenvalues      321
Jacobian elliptic functions, properties      415
Jacobian elliptic functions: sn, cn, dn, computation of      417
Kahn      512
Lagrange's interpolation formula      96 108
Lagrange's interpolation formula, disadvantages of      96
Lagrangian interpolation coefficients      96
Lagrangian interpolation coefficients, their relation to Everett's      98
Laguerre's method for polynomial roots      187 335
Laplace's equation      Chapter 18
Laplace's equation, bordered tridiagonal form from      480
Laplace's equation, closed boundary      411f.
Laplace's equation, curved boundary conditions      485
Laplace's equation, normal derivative boundary conditions      488
Laplace's equation, open boundary      491f.
Latent roots      see "Eigenvalues (Chapters 8 and 13)"
Latent vectors      see "Eigenvectors"
Least-squares fitting      25 253
Least-squares property of orthogonal expansions      223—225
Lehmer method for polynomials      196
Lin's method for quartics with complex roots      198
Linear convergence      216
Linear error      216
Linearization of nonlinear differential equations      171—173
List representation of a network      501 503
ln x, integral representation for      268
Lobaschevski      see "Graeffe"
Loop detection in an ordered network      513 515
Loss of significant figures by subtraction      72
Loss of significant figures in alternating series      14 69
LR algorithms      346
LR algorithms for tridiagonal forms      350
LR algorithms, convergence of      348
Madelung transformation      154
Maehly      296
Maximal network flows, Ford and Fulkerson's algorithm for      524
Method of undetermined coefficients      108
Milne's method      132 see
Milne's method, instability of      143
Minimum methods      Chapter 17
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