"Stiff" differential equations 148
 calculated symmetrically from  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  23
Bessel functions, series for  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  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  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 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  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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