Index C n (a, b), 453 O(h n ), 456 1-norm, 85 ", 7 1-norm, 85 2-norm, 85 A-stable, 288 adaptive quadrature, 258 adjacency matrix, 156, 178, 179 benzene, 179 Hückel Hamiltonian matrices, 179 naphthalene, 179 airfoil cross-section, 223 Akima algorithm, 218 allometric function, 382 amplification factor, 288, 293, 297, 319 annoying ± problem, 130, 141 Armijo s method, 362 asymptotically stable, 276 augmented matrix, 78 backtracking sampling method, 363 backward Euler method, 292 barycentric interpolation, 228 barycentric weights, 189, 228 Bernoulli equation, 321, 451 bisection method, 35 blind source separation problem, 418 brachistochrone problem, 375 causality, 312 chapeau function, 192 characteristic equation, 121 Chebyshev interpolation, 210 exponential convergence, 212, 216 interpolation error, 211 used for integration, 255 used for solving nonlinear equations, 55 Chebyshev points, 211 Chebyshev polynomial, 213 Cholesky factorization, 98 banded, 359 sparse, 102 Clenshaw-Curtis quadrature, 255 Colebrook equation, 64 compensated summation, 14 integration, 263 complex Taylor series expansion, 311 composite Hermite rule, 245 condition number, 89, 90, 357 conjugate gradient method, 355, 360 finite termination property, 356 consistent approximation, 285 contrast function, 424 corrected trapezoidal rule, 245 covariance matrix, 422, 450 crime rates, 410 Crout factorization, 74, 80 cubic B-splines, 197, 200, 244 least squares, 396 cubic spline interpolation, 194 integration rule, 245 interpolation error, 208 minimum curvature, 202 Deming regression, 333 descent direction requirement, 349 steepest, 349, 360 digamma function, 14 dominant eigenvalue, 125 Doolittle factorization, 74, 80 483
484 Index dot product, 122 double precision, 5, 8 downsampling, 428 drag on sphere, 31, 64 Du ng equation, 295 Eckart-Young theorem, 169 eigenvalue problem, 121 elastic bar minimum potential energy, 391 elastic beam minimum potential energy, 393 elastic string elastic foundation, 176 minimum potential energy, 372 elliptic integral first kind, 59 second kind, 59 error, 18 asymptotic form, 237, 458 for IVP solver, 290 iterative, 19 relative, 18, 90 vector, 89 error function, 59, 267 Euclidean norm, 85 Euler s constant, 14 exascale computers, 110 explicit method, 286 Fermi-Pasta-Ulam (FPU) chain, 124 Fibonacci sequence, 53 finite termination property, 47, 54, 356 FitzHugh-Nagumo equations, 442 Fletcher-Reeves method, 360 floating-point numbers, 5 flop, 10, 80, 356 Freudenstein equation, 61 Fritsch-Butland procedure, 218 Frobenius norm, 416 Fundamental Theorem of Calculus, 296 Gauss-Kronrod rules, 263 Gaussian elimination, 78 Gaussian quadrature `-point, 252 1-point, 249 2-point, 250 exponential convergence, 254 Givens rotations, 147 Golub-Reinsch algorithm, 165 Golub-Welsch algorithm, 254 Google Flu Trends, 434 Gram matrix, 423, 433 Gram-Schmidt, 140, 141, 143 modified, 144 Great Internet Mersenne Prime Search, 12 Halley s method, 67 Hamiltonian, 310 hat function, 192 Hermite interpolation, 217 Hessian, 361 Homer Simpson, 24 Horner s method, 15 Householder transformations, 147 ideal gas law, 66 IEEE-754, 5, 12 ill-conditioned, 90 image compression, 166 implicit method, 292 independent component analysis contrast function, 424 downsampling, 428 EEG rule, 428 kurtosis, 425 mixing matrix, 420 unmixing matrix, 423 whitening source data, 421 Inf, 10 interpolation Chebyshev, 464 cubic splines, 194, 441, 464 global polynomial, 185, 204 Lagrange, 187, 464 piecewise linear, 190, 206, 464 piecewise quadratic, 218, 229 inverse iteration, 134 inverse orthogonal iteration, 154 iteration error, 41, 352 iterative error, 19, 48, 127, 131 IVP solvers A-stable, 288 backward Euler, 292 Euler s method, 286 Heun s method, 300 leapfrog method, 294 RK2, 299 RK4, 302, 307, 312, 313, 440, 443 that conserve energy, 324 trapezoidal method, 297, 307, 308 velocity Verlet method, 309 Jacobian, 103, 361 Kermack-McKendrick model, 305
Index 485 kurtosis, 425 Lagrange interpolation, 187 barycentric weights, 228 interpolation error, 203 Lagrange multipliers, 405 law of mass action, 32, 305 leapfrog method, 294, 298 least squares cubic splines, 395 ill-conditioned, 338 nonlinear, 344 normal equation, 335, 337 QR approach, 339 Legendre polynomial, 254 Leslie matrix, 180 line search problem, 349, 361 Lobatto quadrature, 273, 315 IVP solver, 315 logistic equation, 276, 286, 297, 301, 332, 439 Lorentzian function, 259 LU factorization, 72, 79 Crout, 74, 80 Doolittle, 74, 77, 80 faster than, 109 flops, 80 pivoting, 76 tri-diagonal matrix, 100 machine epsilon, 6, 7 magic matrix, 115 mantissa, 5 Mars, 322 matrix bandwidth, 359 condition number, 90 defective, 123, 125 dense, 102 diagonally dominant, 97 ill-conditioned, 90 inverse of 2 2, 91 lower triangular, 72, 79 negative definite, 385 normal, 112 orthogonal, 146 penta-diagonal, 394 positive definite, 95, 98, 117, 351, 356 rotation, 423 sparse, 102 strictly diagonal dominant, 97 trace, 151 tri-diagonal, 82, 99, 154, 374 unit lower triangular, 80 unit upper lower triangular, 80 upper triangular, 72, 78, 79 matrix factorization LU, 79 LDL T,114 QP, 181 QR, 146 QDQ T,159 UL, 113 U V T,162 U T U,98 ZZ T,448 matrix norm, 87, 169 maximum error, 291 Mersenne prime, 12 Michaelis-Menten function, 332 Michaelis-Menten model, 32, 63 numerical solution, 317, 321 midpoint rule, 296 minimum potential energy, 373 mixing matrix, 420, 423 model function allometric, 382 asymptotic regression, 331 logistic, 331 Michaelis-Menten, 331, 344 power law, 382 Mooney-Rivlin law, 269 Moore-Penrose pseudoinverse, 434 multifrontal method, 102 mutually orthogonal, 356 NaN, 10 National Vegetable Research Station, 367 natural matrix norms, 88 Nelder-Mead algorithm, 55, 367, 442, 445 Newton s method, 40, 103 discoverer, 55 finding implicit functions, 44 finding inverse functions, 58 minimization, 361 nonlinear system, 102 order of convergence, 45, 49 Newton s second law, 305 ngram program, i, 407 norm 1-norm, 85, 87 1-norm, 85, 87 2-norm, 85 dimensionally consistent, 85, 409 Euclidean, 85
486 Index Frobenius, 416 matrix, 87, 169 vector, 85 normal equation, 335, 337 ill-conditioned, 338 normal modes, 124 numerical di erentiation, 282, 441 complex Taylor series expansion, 311 optimal step size, 281 objective function, 327 one-step method, 294 order of convergence, 43, 54, 138 orthogonal iteration, 141 orthogonal matrix, 146, 159 proper, 181 orthogonal regression, 333, 399 oscillators chain, 123 coupled, 177, 323 outer product, 166, 420, 436 overflow negative, 10 positive, 10 partial pivoting, 77 Pascal matrix, 116 pendulum equation, 307, 309 persistence forecast, 434 perturbation matrix, 106 piecewise linear interpolation, 190 integration rule, 238 interpolation error, 206 pivoting, 75, 76 Planck s law of blackbody radiation, 269 Polak-Ribière method, 360, 363 polar decomposition, 181 population integration, 268 interpolation, 223 power law fluid, 317 power law function, 382 power method, 126, 127 precision, 247 arbitrary, 12 double, 5, 8 of integration rule, 247 quadruple, 8 single, 8 solving matrix equations, 93 prime numbers, 13 principal component analysis autoscaling, 409 Cattell s scree test, 416 centering data, 399, 402 error function, 400, 403 Guttman-Kaiser criterion, 416 how many components, 416 how many points, 415 range scaling, 409 relative error, 416 relative residual variance, 416 scaling data, 400, 402 QR factorization, 147 flops, 340 least squares, 339 QR method, 148 divide and conquer, 150 implicitly shifted, 150 Radau quadrature, 273 radioactive decay, 275, 287 rank, 165 Rayleigh s quotient, 121 Rayleigh s quotient iteration, 137 regression error function, 327, 332, 340 fitting circle, 343, 371 fitting ellipse, 387 model function, 331 nonlinear, 344 objective function, 327 relative error, 346 relative error, 18 relative iterative error, 19 residual, 89, 351, 358 Reynolds number, 32, 64 robustification, 432 Rolle s theorem, 204 Romberg integration, 256, 272 Rosenbrock function, 366 Rosser matrix, 114, 173 round to nearest, 9 Runge s function, 190, 215 Runge Kutta methods, 299 Heun, 300 Lobatto quadrature, 315 order conditions, 302, 304, 313 RK2, 299, 300 RK4, 312, 441 Simpson s rule, 302 secant method, 51 golden ratio, 53 order of convergence, 53, 54 significant digits, 18, 90
Index 487 Simpson s rule, 241, 296 adaptive, 259 error, 242 singular value decomposition derivation, 160 flops, 165 ICA, 421 image compression, 166 PCA, 400, 406 summary, 162 SIR model, 305 sparse matrix, 102 spline clamped, 195, 208, 312, 441 natural, 195, 200 202, 209, 216 not-a-knot, 196, 201 square-and-multiply algorithm, 28 stability IVPs, 287 steady-state solution, 276 steepest descent direction, 349, 352, 360 method, 352, 360 Strassen s method, 109 strictly convex, 330 subnormal floats, 9 superlinear convergence, 54 symplectic method, 310 synthetic data, 441, 442, 451 tangent linear model, 438 taxi-cabs, 110 Taylor s theorem, iii, 453 terminal velocity, 31, 64 Thomas algorithm, 100 trace, 151 transpose notation, 73 trapezoidal method, 297 trapezoidal rule, 238, 296 error, 239 periodic function, 264, 267 traveling salesman problem, 328 tri-diagonal matrix, 82, 99, 154 eigenvalues, 118 eigenvectors, 181 invertibility, 101 LU factorization, 100 positive definite, 118 truncation error IVPs, 285, 297 quadrature approach, 297 two-step method, 294 Tyrannosaurus rex, 382, 447 unimodular matrix, 112 unit lower triangular matrix, 80 unit upper triangular matrix, 80 unmixing matrix, 423 van der Waals equation of state, 66 Van Wijngaarden - Dekker - Brent method, 55 Vandermonde matrix, 84, 94, 145, 186, 338 vector norm, 85 velocity Verlet, 309 whitening source data, 421, 447 Wolfe conditions, 363 Yogi Berra, 106 Young s modulus, 316