sign in sign up
<<
Home , Adjacency matrix, Circulant matrix, Degree matrix, Laplacian matrix

Index

A beat, 128 absolute stability, 189 bell-shaped curve, 16, 190, 458 absolute value, 83, 86 Bernoulli equation, 61 acceleration, 73, 478 Bessel function, 367, 460, 478 accuracy, 184, 185, 190, 191 better notation, 113, 124, 125 Adams method, 192, 193 big picture, 300, 303, 306, 400 add exponents, 9 Black-Scholes, 457 addition formula, 87 block , 231, 237, 420 , 318, 320, 427 block multiplication, 226, 227 boundary conditions, 406, 411, 431, 457 Airy’s equation, 130 boundary value problem, 406, 457, 470 albedo, 49 box, 176 amplitude, 75, 82, 111 box function, 407, 439, 445, 469, 478, 488 amplitude response, 34, 77 Brauer, 180 antisymmetric, 245, 323, 352, 409 applied mathematics, 316, 423, 487 C arrows, 156, 318 capacitance, 119 associative law, 220 carbon, 46 attractor, 170, 181 carrying capacity, 53, 55, 61 , 231, 259, 273, 280 Castillo-Chavez, 180 autocorrelation, 480 catalyst, 180 autonomous, 57, 71, 157, 158, 160 Cayley-Hamilton theorem, 348 average, 436, 440 cell phone, 44, 176 center, 161, 163, 174 B centered difference, 6, 190 back substitution, 213, 264 chain rule, 3, 4, 368, 371 backslash, 221 change of variables, 365 backward difference, 6, 12, 246, 415 chaos, 155, 181 backward Euler, 188, 189 characteristic equation, 90, 103, 108, 164 bad news, 329 chebfun, 405 balance equation, 48, 118, 316, 424 chemical engineering, 457 balance of forces, 118 chess matrix, 311 bank, 12, 40, 485 Cholesky factorization, 403 bar, 406, 408, 412, 455, 457 , 205, 449, 486, 488 basis, 285, 289-291, 296, 338, 446, 447 circular motion, 76, 351 beam, 469 closed-, 64

493 494 Index closest line, 387, 393 Rule, 476, 480, 484, 485 coefficient matrix, 199 Cooley-Tukey, 451 cofactor, 331 cooling (Newton’s Law), 46 column picture, 198, 206 cosine series, 436 column , 275, 322 Counting Theorem, 267, 304, 314 column space, 254, 259, 278 Cramer’s Rule, 331 column-times-row, 222, 226, 429 critical damping, 96, 100, 115 combination of columns, 199, 202 critical point, 170, 171, 182 combination of eigenvectors, 329, 349, cubic spline, 139 356, 371, 374 Current Law, 123, 317, 318 commute, 221, 224 cyclic convolution, 485–487 , 164, 165, 167, 335, D 354–356, 360, 369 d’Alembert, 464, 467 competition, 53, 174 damped frequency, 99, 105, 113 , 427, 428 damped gain, 113 complete solution, 1, 17, 18, 105, 106, damping, 96, 112, 118, 122 203, 211, 265, 274, 276 damping ratio, 99, 113, 114 complex conjugate, 32, 87, 94, 379 dashpot, 118 complex eigenvalues, 166 data, 401, 431 complex exponential, 13, 432 decay rate, 46, 437, 444, 456, 467 complex Fourier series, 440 deconvolution, 485, 487 complex gain, 111 matrix, 318, 427, 429 complex impedance, 120 delta function, 23, 28, 78, 97, 98, 407, complex matrix, 376 438, 439, 442, 458, 471 complex numbers, 31-33, 82-89 delta vector, 415, 447, 482 complex roots, 90, 163 dependent, 288 complex solution, 36, 38, 39, 89 dependent columns, 209 complex vector, 433 rule, 141, 441, 476 compound interest, 12, 185 , 175, 228, 232, 326, 330, computational mechanics, 372 332, 336, 347, 353, 402, 492 computational science, 419, 447 DFT, 432, 446, 449, 454, 485 concentration, 47, 180 , 229, 398 condition number, 401 diagonalizable, 363, 382 matrix, 124, 385, 425, 426 difference equation, 45, 52, 184, 188, 338 conjugate , 377 difference matrix, 240, 314, 405, 423 constant coefficients, 1, 98, 117, 432, differential equation, 1, 40, 349 470, 487 diffusion, 358, 456, 457 constant diagonals, 482, 486, 487 diagonalization, 337, 400 constant source, 20 , 44, 52, 267, 285, 291–293, continuous, 154, 358 304, 322 continuous interest, 44 dimensionless, 34, 99, 113, 124 convergence, 10, 196 direction field, 157 convex, 73 Discrete Cosine Transform (DCT), 454 convolution, 117, 136, 479-489 Discrete Fourier Transform, (see DFT) Index 495 discrete sines, 405, 432, 454 F displacements, 124 factorization, 382, 490 distributive law, 220 farad, 122 divergence, 417 , (see FFT) dot product, 201, 214, 248, 377 feedback, 64 double angle, 84 FFT, 88, 432, 446, 447, 450, 451 double pole, 145, 472 fftw, 452 double root, 91, 92, 101 Fibonacci, 340, 345, 405 doublet, 151 filter, 480 doubling time, 46, 47 finite elements, 124, 373, 419, 430 driving function, 77, 112, 476 finite speed, 463 dropoff curve, 57, 62, 157 first order, 164 E flow graph, 452 echelon matrix, 263, 266, 267 football, 176, 178 edge, 313, 423 force balance, 426 eigenfunction, 408, 421, 455, 459, 467 forced oscillation, 80, 105, 110 eigenvalue, 164, 325, 326, 382 forward differences, 240 eigenvalue matrix, 337 Four Fundamental Subspaces, 300, 303 eigenvector, 167, 325, 326, 382 Fourier coefficients, 435–437, 440 eigenvector matrix, 337, 363 Fourier cosine series, 457 Einstein, 464 Fourier , 449 elapsed time, 98 Fourier matrix, 85, 243, 446-448, 450 elimination, 210, 212, 334 Fourier series, 419, 436, 439, 443, 455 elimination matrix, 224, 229, 303 Fourier sine series, 410, 434, 467 empty set, 293 fourth order, 80, 93, 469 energy, 396, 397, 409, 411, 424, 443 foxes, 172, 174 energy balance, 48 free column, 262 energy identity, 440, 444 free variable, 262, 266, 269, 270, 274 enzyme, 180 free-free boundary conditions, 412 epidemic, 179, 180 frequency, 31, 76, 79, 373, 466 equal roots, 90, 92, 100 frequency domain, 120, 145, 449, 480 equilibrium, 417 frequency response, 36, 77, 432 error, 185, 186, 191, 193 frisbee, 176 error function, 458 full rank, 275-277, 281, 287, 385 error vector, 386, 394 function space, 293, 298, 433, 440, 480 Euler, 317 fundamental matrix, 366, 371, 384 Euler equations, 176, 183 fundamental solution, 78, 81, 97, 117, 458 Euler’s Formula, 13, 82, 83, 450 Fundamental Theorem, 5, 8, 42, 244, Euler’s method, 185, 186, 189, 384 304, 307, 400 even permutation, 246 exact equations, 65 G existence, 154, 196 gain, 30, 33, 84, 104, 111 exponential, 2, 7, 10, 25, 131, 362, 369 Gauss-Jordan, 230–232, 236, 283, 331 exponential response, 104, 108, 117 gene, 431 496 Index general solution, 280 impulse response, 23, 24, 78, 97, 102, generalized eigenvalues, 372 117, 121, 136, 140, 150, 482 geometric series, 7 , 124, 313, 317, 320, 423 Gibbs phenomenon, 435, 436 independence, 204 gold, 153 independent columns, 273, 276, 290, Gompertz equation, 63 322, 385, 391 Google, 328 independent eigenvectors, 362 GPS, 464 independent rows, 273 gradient, 417, 421 inductance, 119 graph, 313, 317, 318, 320, 416, 423 infection rate, 179 graph Laplacian, 316, 318, 423 infinite series, 10, 13, 329, 369, 434, 455 Green’s function, 136, 482, 483 inflection point, 54, 55 greenhouse effect, 49 initial conditions, 2, 40, 73, 349, 457 grid, 416, 419, 429 initial values, 470, 483 ground a node, 424, 426 inner product, 226, 323, 377, 409, 433 growth factor, 24, 40–42, 51, 97, 135, 482 instability, 193 growth rate, 2, 40, 364 integrating factor, 19, 26, 41, 482 integration by parts, 248, 323, 409, 413, 431 H interest rate, 12, 43, 485 Henon map, 181 intersection, 201, 258, 299 HadamardK matrix, 243, 344 inverse matrix, 31, 228, 231, 482 half-life, 46 inverse transform, 140, 446, 473, 477 harmonic motion, 75, 76, 79 invertible, 205, 213, 228, 290 harvesting, 59, 60, 62 isocline, 156, 159, 160 hat function, 467 J , 410, 455, 456 Jacobian matrix, 171, 177 heat , 457, 458, 460 Jordan form, 357, 382, 383 Heaviside, 21, 477 Julia, 330 Henry, 122 jump, 21, 474, 475 , 377 Hertz, 76 K higher order, 93, 102, 105, 107, 117, 355 key formula, 8, 19, 78, 112, 117, 135, 482 Hilbert space, 433 kinetic energy, 79 homogeneous, 17, 103 Kirchhoff’s Current Law, 316, 424 Hooke’s Law, 74, 374, 424 Kirchhoff’s Laws, 123, 272 hyperplane, 207 Kirchhoff’s Voltage Law, 315 KKT matrix, 428 I kron .A;B/, 420 , 201, 219 image, 484 L imaginary eigenvalues, 331, 351 l’Hopital’s Rule, 43, 109 O impedance, 39, 120, 121, 127 LAPACK, 242, 332 implicit, 67, 188 Laplace convolution, 481, 483 impulse, 23, 78 Laplace equation, 416, 417 Index 497

Laplace transform, 121, 141-151, 470-478 companion, 164, 355, 360 Laplace’s equation, 418, 442, 443 complex, 376 , 318, 320, 424 difference, 240, 314, 405, 422 law of mass action, 180 echelon, 266 least squares, 385–387 eigenvalue, 337 left eigenvectors, 348 eigenvector, 337, 363 left nullspace, 300, 302 elimination, 224, 229, 303 left-inverse, 228, 232, 242 exponential, 14, 362, 368 length, 242 factorizations, 382, 490 Lienard, 182 K Fourier, 85, 243, 446, 447, 450 linear combination, 199, 201, 254, 288 fundamental, 366 , 4, 17, 105, 134, 177, 349 Hadamard, 243, 344 linear shift-invariant, 459 Hermitian, 377 linear time-invariant (LTI), 71, 349 identity, 201, 219 linear transformation, 209 incidence, 124, 313, 314, 317, 423 linearity, 221, 471 inverse, 228, 231 linearization, 172-179 invertible, 204, 213, 231, 290 linearly independent, 277, 287, 289 Jacobian, 171, 177 lobster trap, 159 KKT, 428 logistic equation, 47, 53, 62, 157, 190 Laplacian, 318, 320, 424 loop, 315–317 Markov, 327, 333 loop equation, 119, 123, 127 Lorenz equation, ix, 154, 181 orthogonal, 238, 247, 376 Lotka-Volterra, 173 permutation, 241, 246, 299, 450 positive definite, 372, 385, 396 M projection orthogonal, 238, 242, 247, magic matrix, 209 334, 376, 378, 382, 390, 394 magnitude, 112 rank one, 305, 382, 404 magnitude response, 34, 77 rectangular, 385 Markov matrix, 327, 329, 333, 382 reflection, 247 mass action, 180 rotation, 331 mass matrix, 372, 381 saddle-point, 428, 430 Mathematica, 194, 467 second difference, 414 mathematical finance, 457 semidefinite, 398, 412, 413 MATLAB, 191, 332, 372, 447, 451, 486 similar, 365, 370, 383 singular, 202, 326, 328, 492 The single heading “Matrix” indexes skew-symmetric, 382 the active life of . sparse, 223 Matrix stable, 352 . 1 2 1/, 246, 415, 454 stiffness, 124, 372, 385 adjacency, 318 symmetric, 238, 375, 409 antisymmetric, 352, 376 Toeplitz, 480, 482 augmented, 230, 271, 278 tridiagonal, 382, 454 circulant, 486, 488 unitary, 377 498 Index , 219-223, 249 orthogonal eigenvectors, 239, 375 mean, 392, 395 orthogonal functions, 323, 405, 434 mechanics, 74 , 238, 242, 376, 381 mesh, 420 orthogonal subspace, 306 Michaelis-Menten, 180 orthonormal basis, 398, 400, 440 minimum, 404 orthonormal columns, 242, 397 model problem, 40, 115, 374, 423 oscillation, 74, 75 modulus, 32, 83 oscillation equation, 372 multiplication, 202, 219, 479 overdamping, 96, 100, 102 multiplicity, 93, 343 overshoot (Gibbs), 435, 436 multiplier, 210, 214, 225 P multistep method, 192 PF2, 62, 142, 149, 472 N PF3, 143, 149, 472 natural frequency, 77, 99, 102, 466 parabolas, 91, 96 network, 313-323, 416, 425, 426 parallel, 122, 127 neutral stability, 166, 339, 352 partial differential equation, (see PDE) Newton’s Law, 46, 73, 239, 370 partial fractions, 56, 62, 142-149, 474 Newton’s method, 6, 181 partial sums, 438 nodal analysis, 123 particular solution, 17, 18, 41, 106, 203, node, 313, 423 274, 276, 278 nondiagonalizable, 339, 342, 346, 383 PDE, 416, 455, 466 nonlinear equation, 1, 53, 172 peak time, 113, 128 nonlinear oscillation, 71 pendulum, 71, 81, 182 norm, 400, 401 period, 76, 163, 444 normal distribution, 458 periodic, 173 normal equations, 387, 389 , 241, 246, 299, 450 normal modes, 373 perpendicular, 201, 243, 389, 433, 434 Nth order equation, 107, 117 perpendicular eigenvectors, 383 null solution, 17, 18, 78, 92, 103, 106, perpendicular subspaces, 312 113, 203 phase angle, 32, 80 , 267 phase lag, 30, 33, 75, 81, 112 nullspace, 261 phase line, 170 number of solutions, 282 phase plane, 59, 351 phase response, 77 O pictures, 153, 162 ODE 45, 191, 193 pivot, 210, 212, 225, 233, 402 off-diagonal ratios, 227 pivot column, 262, 264, 290, 294 Ohm’s Law, 39, 122, 424, 425, 427 pivot variable, 264, 270 one-way wave, 463, 468 plane, 201, 207, 258 open-loop, 64 Pluto, 155 operation count, 452 point source, 23, 457, 458 optimal control, 478 point-spread function, 484 order of accuracy, 186, 190, 192 Poisson’s equation, 417 orthogonal basis, 399, 433, 447, 448 polar angle, 38, 83 Index 499 polar form, 30, 32, 84, 110, 112, 121, rectangular form, 110, 111 244, 418, 431, 448 rectangular matrix, 385 poles, 100, 129, 140, 471–473 recursion, 452, 453 , 131 red lights, 478 Pontryagin, 478 reflection matrix, 247, 382 population, 47, 55, 61, 63 relativity, 464 positive definite, 372, 385, 396, 403-411 relaxation time, 46 positive definite matrix, 372, 382, 396 repeated eigenvalues, 338, 339, 355, 383 positive semidefinite, 412, 413 repeated roots, 90, 92, 101, 355 potential energy, 79 repeating ramp, 436 powers, 221, 328, 341 resistance, 119, 426 practical resonance, 126 resonance, 26, 27, 29, 79, 82, 108, 109, 114, predator-prey, 172, 174, 180 116, 132, 137, 364 prediction-correction, 191 response, 77 present value, 51 reverse order, 229, 238, 248 principal axis, 376 right triangle, 129, 386 Principal Component Analysis, 401, 431 right-inverse, 228, 232, 233 probability, 458 RLC loop, 39, 118, 119, 122 product integral, 384 roots, 101, 108, 129 product of pivots, 330, 492 roots of zN 1, 448 D product rule, 8 , 331 projection, 387, 389–391, 394 row exchange, 212, 216, 242 , 247, 334, 382, 389, 394 row picture, 198, 199, 214 pulse, 392, 393 row space, 289, 323 Python, 330 rref .A/, 263, 265, 267, 268, 284 Runge-Kutta, 16, 191–193 Q quadratic formula, 90 S quiver, 155 S-curve, 54, 64, 157 saddle, 162, 169, 173, 177, 402, 428 R saddle-point matrix, 428, 430 rabbits, 172, 174 SciPy, 194 radians, 76 second difference, 240, 246, 410, 414, 415 radioactive decay, 45 semidefinite, 398, 412 ramp function, 23, 98, 407, 408, 477 separable, 56, 65 ramp response, 129 separation of variables, 421, 422, 456, rank, 267, 273, 277, 301 459, 460, 466 rank of AB, 311 shift, 441 rank one matrix, 305, 382, 401 shift invariance, 98, 459, 480, 482, 487 rank theorem, 322 shift rule for transform, 475 Rayleigh quotient, 431 sign reversal, 492 reactance, 121 similar matrix, 365, 370, 383 real eigenvalues, 166, 239, 375 Simpson’s Rule, 195 real roots, 90, 162 sines and cosines, 439 real solution, 31, 111 singular matrix, 202, 205, 218, 326, 492 500 Index singular value, 398, 400, 405 stock prices, 457 Singular Value Decomposition, (see SVD) straight line, 386 singular vector, 385 subspace, 251–254, 256, 258, 296 sink, 17, 162 Sudoku matrix, 209 sinusoid, 19, 30, 34 sum of spaces, 260 sinusoidal identity, 35, 37, 112 sum of squares, 386, 388 SIR model, 179 superposition, 8, 349, 460 six pictures, 162, 171 SVD, 244, 398, 382, 385, 399-405, 431 skew-symmetric, 381 switch, 22 smoothness, 437 symmetric and orthogonal, 244, 378 solution curve, 154 , 238, 239, 292, 375, 409 Solution Page, 117 symmetry, 468 solvable, 255, 257, 277, 311 system, 164, 197, 325 source, 17, 19, 40, 162 T span, 256, 260, 285, 288, 296 Table of Eigenvalues, 382 sparse matrices, 223 Table of Rules, 476 special inputs, 131, 139 Table of Transforms, 146, 471 special solution, 261, 265, 302 tangent, 75, 80, 156 , 376, 383 tangent line, 6, 184 speed of light, 464 tangent parabola, 7, 191 spike, 23, 407, 437, 438 Taylor series, 7, 10, 14, 16, 185 spiral, 33, 86, 88, 95, 161 temperature, 46, 442, 455, 459 spiral sink, 163 test grades, 395 spring, 74, 119 three steps, 341, 349, 369 square root, 397 time constant, 100 square wave, 435, 437, 443, 456 time domain, 120, 127 stability, 49, 58–60, 187, 188 time lag, 81 stability limit, 190, 195 time-varying, 367, 371, 384 stability line, 58, 170 , 480, 482 stability test, 165-170, 175, 188, 339, 353 Toomre, 178 stable, 161, 169, 352, 472 , 175, 331, 332, 336, 347, 353, 384 standing wave, 465 transfer function, 104, 121, 432, 477, 481 starting value (initial condition), 2, 9 transient, 27, 103 state space, 127 tree, 317 , 401, 458 , 213, 238, 293, 490, 492 steady state, 21, 49, 53, 58, 155, 328, 357 , 232, 246, 382, 410, 454 Stefan-Boltzmann Law, 49, 63 tumbling box, 176, 178, 183 step function, 21, 23, 474, 475, 478, 489 step response, 22, 81, 97, 102, 124–128 U stepsize, 184 underdamping, 96, 100, 102, 117 stiff equation, 187 undetermined coefficients, 117, 130-137 stiff system, 193 uniqueness, 154, 289 stiffness, 118, 468 unit circle, 33, 84, 85, 94, 448 stiffness matrix, 124, 372, 385 unit vector, 334 Index 501 , 377 Index of Symbols units, 44, 52, 456 A LU , 414, 490 unstable, 49, 53, 166 D A QR, 490 upper triangular, 210, 213 D A QS, 431 V D A U†V T, 382, 398, 401 variable coefficient, 1, 42, 130 D A VƒV 1, 337, 341 variance, 392, 395, 401, 431 D variation of parameters, 41, 43, 130, ATA, 239, 276, 312, 385, 395, 417, 423 133–135, 138, 482 ATCA, 392, 404, 416, 425, 427 vector, 164, 199, 200, 251, 252 A A T, 413 , 251, 252, 298, 321  D N very particular, 26, 27, 117, 144 K2D, 419, 420 violin, 465, 469 K ATCA, 410, 423, 424 D Voltage Law, 123, 317, 318 P.D/, 108, 117 voltage source, 425 Q, 238 W S LDLT, 403 D wave equation, 463–466, 469 S QƒQT, 376 weighted Laplacian, 424 D S ?, 307 weighted least squares, 390, 392 C .A/ and N .A/, 255, 261 Wikipedia, 243, 431 n n Wolfram Alpha, 194 R and C , 251 , 134, 135, 366, 384 Z zerocline, 157 zeta, 99, 113