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Differential Equations and Linear Algebra

Differential Equations and Linear Algebra

Index

A beat, 128 absolute stability, 188 bell-shaped curve, 16, 189, 458 , 83, 86 Bernoulli equation, 61 acceleration, 73, 478 Bessel function, 364, 460, 478 accuracy, 183, 185, 190, 191 better notation, 113, 124, 125 Adams method, 191, 192 big picture, 298, 301, 304, 397 addition formula, 87 Black-Scholes, 457 add exponents, 9 block , 230, 236, 418 , 316, 318, 425 block multiplication, 225, 226 boundaryconditions, 417, 403, 409, 429, 457 Airy’s equation, 130 boundary value problem, 403, 457, 470 albedo, 49 box, 175 amplitude, 75, 82, 111 box function, 404, 437, 443, 469, 478, 488 amplitude response, 34, 77 Brauer, 179 antisymmetric, 244, 321, 349, 406 applied , 314, 421, 487 C arrows, 155, 316 capacitance, 119 associative law, 219 carbon, 46 attractor, 169, 180 carrying capacity, 53, 55, 61 , 230, 257, 271, 278 Castillo-Chavez, 179 , 480 catalyst, 179 autonomous, 57, 71, 156, 157, 159 Cayley-Hamilton theorem, 345 average, 434, 438 cell phone, 44, 175 center, 160, 162, 173 B centered difference, 6, 189 back substitution, 212, 262 chain rule, 3, 4, 365, 368 backslash, 220 change of variables, 362 backward difference, 6, 12, 245, 413 chaos, 154, 180 backward Euler, 187, 188 characteristic equation, 90, 103, 108, 163 bad news, 326 chebfun, 402 balance equation, 48, 118, 314, 424 chemical engineering, 457 balance of forces, 118 chess matrix, 309 bank, 12, 40, 485 Cholesky factorization, 400 bar, 403, 405, 409, 455, 457 , 204, 448, 486, 488 , 283, 287, 289, 291, 295, 335, 444, 445 circular motion, 76, 348 beam, 469 closed-, 64

493 494 Index closest line, 384, 390 Rule, 476, 480, 484, 485 coefficient matrix, 198 Cooley-Tukey, 451 cofactor, 328 cooling (Newton’s Law), 46 column picture, 197, 205 cosine series, 434 column , 273, 320 Counting Theorem, 265, 302, 312 column space, 252, 257, 276 Cramer’s Rule, 328 column-times-row, 221, 225, 427 critical damping, 96, 100, 115 combination of columns, 198, 201 critical point, 169, 170, 181 combination of eigenvectors, 326, 346, cubic spline, 139 353, 368, 371 Current Law, 123, 315, 316 commute, 219, 223 cyclic convolution, 485–487 , 163, 164, 166, 332, D 351–353, 357, 366 d’Alembert, 464, 467 competition, 53, 173 damped frequency, 99, 105, 113 , 425, 426 damped gain, 113 complete solution, 1, 17, 18, 105, 106, damping, 96, 112, 118, 122 202, 210, 263, 272, 274 damping ratio, 99, 113, 114 complex conjugate, 32, 87, 94, 376 dashpot, 118 complex eigenvalues, 165 data, 398, 429 complex exponential, 13, 430 DCT, 454 complex , 438 decay rate, 46, 435, 442, 456, 467 complex gain, 111 deconvolution, 485, 487 complex impedance, 120 matrix, 316, 421, 425 complex matrix, 373 delta function, 23, 28, 78, 97, 98, 404, complex numbers, 32, 82 436, 437, 440, 458, 471 complex roots, 90, 162 delta vector, 413, 445, 482 complex solution, 36, 38, 39, 89 dependent, 286 complex vector, 431 dependent columns, 208 compound interest, 12, 184 rule, 141, 439, 476 computational mechanics, 369 , 174, 227, 231, 323, 327, computational science, 417, 445 329, 333, 344, 350, 399, 492 concentration, 47, 179 DFT, 430, 444, 448, 454, 485 , 398 , 228, 395 matrix, 124, 382, 423, 424 diagonalizable, 354, 379 conjugate , 374 difference equation, 45, 52, 183, 187, 335 constant coefficients, 1, 98, 117, 430, difference matrix, 239, 312, 402, 421 470, 487 differential equation, 1, 40 constant diagonals, 482, 486, 487 diffusion, 355, 456, 457 constant source, 20 diagonalization, 334, 397 continuous, 153, 314 , 44, 52, 265, 283, 289–291, continuous interest, 44 302, 320 convergence, 10, 195 dimensionless, 34, 99, 113, 124 convex, 73 direction field, 156 convolution, 117, 136, 479-489 Discrete Fourier Transform, (see DFT) Index 495 discrete sines, 402, 430, 454 F displacements, 124 factorization, 379, 490 distributive law, 219 farad, 122 divergence, 415 , 88, 449 dot product, 200, 213, 247, 374 feedback, 64 double angle, 84 FFT, 430, 444, 445, 449, 451 double pole, 145, 472 fftw, 452 double root, 90, 92, 101 Fibonacci, 337, 342, 402 doublet, 151 filter, 480 doubling time, 46, 47 finite elements, 124, 370, 417, 428 driving function, 77, 112, 476 finite speed, 463 dropoff curve, 57, 62, 156 first order, 163 E flow graph, 452 echelon matrix, 261, 264, 265 football, 175, 177 edge, 311, 421 force balance, 424 eigenfunction, 405, 419, 455, 459, 467 forced oscillation, 80, 105, 110 eigenvalue, 163, 322, 323, 379 forward differences, 239 eigenvalue matrix, 334 Four Fundamental Subspaces, 298, 301 eigenvector, 166, 322, 323, 379 Fourier coefficients, 433–435, 438 eigenvector matrix, 334, 360 Fourier cosine series, 457 Einstein, 464 Fourier , 448 elapsed time, 98 Fourier matrix, 85, 242, 444-447, 449 elimination, 209, 211, 331 Fourier series, 419, 434, 437, 441, 455 elimination matrix, 223, 228, 301 Fourier sine series, 407, 432, 467 empty set, 291 fourth order, 80, 93, 469 energy, 393, 394, 406, 408, 422, 441 foxes, 171, 173 energy balance, 48 free column, 260 energy identity, 438, 442 free variable, 262, 264, 267, 268, 272 enzyme, 179 free-free boundary conditions, 409 epidemic, 178, 179 frequency, 31, 76, 79, 370, 466 equal roots, 90, 92, 101 frequency domain, 120, 145, 448, 480 equilibrium, 415 frequency response, 36, 77, 430 error, 184, 185, 190, 193 frisbee, 175 error function, 458 full rank, 273-275, 279, 285, 382 error vector, 383, 391 function space, 291, 296, 431, 438, 480 Euler, 315 fundamental matrix, 363, 368, 381 Euler equations, 175, 182 fundamental solution, 78, 81, 97, 117, 458 Euler’s Formula, 13, 82, 83, 449 Fundamental Theorem, 5, 8, 42, 243, Euler’s method, 184, 185, 188, 381 302, 305, 397 even permutation, 245 exact equations, 65 G existence, 153, 195 gain, 30, 33, 84, 104, 111 exponential, 2, 7, 10, 25, 131, 359, 366 Gauss-Jordan, 229–231, 235, 281, 328 exponential response, 104, 108, 117 gene, 429 496 Index general solution, 278 impulse response, 23, 24, 78, 97, 102, generalized eigenvalues, 369 117, 121, 136, 140, 150, 482 geometric series, 7 matrix, 124, 311, 315, 318, 421 Gibbs phenomenon, 433, 434 independence, 203 gold, 152 independent columns, 271, 274, 288, Gompertz equation, 63 320, 382, 388 Google, 325 independent eigenvectors, 359 GPS, 464 independent rows, 271 gradient, 415, 419 inductance, 119 graph, 311, 315, 316, 318, 414, 421 infection rate, 178 graph Laplacian, 316, 318, 422 infinite series, 10, 13, 326, 366, 432, 455 Green’s function, 136, 482, 483 inflection point, 54, 55 greenhouse effect, 49 initial conditions, 2, 40, 73, 346, 457 grid, 414, 417, 427 initial values, 470, 483 ground a node, 422, 424 inner product, 225, 321, 374, 406, 431 growth factor, 24, 40–42, 51, 97, 135, 482 instability, 192 growth rate, 2, 40, 362 integrating factor, 19, 26, 41, 482 integration by parts, 247, 321, 406, 411, 429 H interest rate, 12, 43, 485 Henon map, 180 intersection, 200, 256, 297 HadamardK matrix, 242, 341 inverse matrix, 31, 227, 230, 482 half-life, 46 inverse transform, 140, 444, 473, 477 harmonic motion, 75, 76, 79 invertible, 204, 212, 227, 288 harvesting, 59, 60, 62 isocline, 155, 158, 159 hat function, 467 J , 407, 455, 456 Jacobian matrix, 170, 176 heat , 457, 458, 460 Jordan form, 354, 379, 380 Heaviside, 21, 477 Julia, 327 Henry, 122 jump, 21, 474, 475 , 374 Hertz, 76 K higher order, 93, 102, 105, 107, 117, 352 key formula, 8, 19, 78, 112, 117, 135, 482 Hilbert space, 431 kinetic energy, 79 homogeneous, 17, 103 Kirchhoff’s Current Law, 314, 422 Hooke’s Law, 74, 371, 422 Kirchhoff’s Laws, 123, 270 hyperplane, 206 Kirchhoff’s Voltage Law, 313 KKT matrix, 426 I kron .A;B/, 418 , 200, 218 image, 484 L imaginary eigenvalues, 329, 348 l’Hopital’s Rule, 43, 109 O impedance, 39, 120, 121, 127 LAPACK, 241, 329 implicit, 67, 187 Laplace convolution, 481, 483 impulse, 23, 78 Laplace equation, 414, 415 Index 497

Laplace transform, 121, 140-150, 470-478 companion, 163, 352, 357 Laplace’s equation, 416, 440, 441 complex, 373 , 316, 318, 422 difference, 239, 312, 402, 421, law of mass action, 179 echelon, 264 least squares, 382–384 eigenvalue, 334 left eigenvectors, 345 eigenvector, 334, 360 left nullspace, 298, 300 elimination, 223, 228, 301 left-inverse, 227, 231, 241 exponential, 14, 359, 367 length, 241 factorizations, 379, 490 Lienard, 181 K Fourier, 85, 242, 444, 447, 449 linear combination, 198, 200, 252, 286 fundamental, 363 , 4, 17, 105, 130, 176, 346 Hadamard, 242, 341 linear shift-invariant, 459 Hermitian, 374 linear time-invariant (LTI), 71, 346 identity, 200, 218 linear transformation, 208 incidence, 124, 311, 312, 315, 421 linearity, 220, 471 inverse, 227, 230 linearization, 171-178 invertible, 204, 212, 227, 288 linearly independent, 275, 285, 287 Jacobian, 170, 176 lobster trap, 158 KKT, 426, 428 logistic equation, 47, 53, 62, 156, 189 Laplacian, 316, 318, 422 loop, 313–315 Markov, 324, 330 loop equation, 119, 120, 123 Lorenz equation, ix, 154, 180 orthogonal, 237, 246, 373 Lotka-Volterra, 172 permutation, 240, 245, 297, 449 positive definite, 369, 382, 393 M projection orthogonal, 237, 241, 246, magic matrix, 208 331, 373, 378, 379, 386, 391 magnitude, 112 rank one, 303, 379, 398 magnitude response, 34, 77 rectangular, 382 Markov matrix, 324, 326, 330, 379 reflection, 246 mass action, 179 rotation, 328 mass matrix, 369, 378 saddle-point, 426, 428 Mathematica, 193, 467 second difference, 412 mathematical finance, 457 semidefinite, 395, 409, 411 MATLAB, 190, 327, 369, 444, 451, 486 similar, 362, 367, 380 singular, 201, 323, 325, 500 The single heading “Matrix” indexes skew-symmetric, 379 the active life of . sparse, 222 Matrix stable, 349 1;2; 1, 245, 413, 454 stiffness, 124, 369, 382 adjacency, 316 symmetric, 165, 237, 372, 406 antisymmetric, 349, 379 Toeplitz, 480, 482 augmented, 230, 271, 278 tridiagonal, 379, 454 circulant, 486, 488 unitary, 374 498 Index , 218-222, 248 orthogonal eigenvectors, 238, 372 mean, 389, 392 orthogonal functions, 321, 402, 432 mechanics, 74 , 237, 246, 373, 378 mesh, 418 orthogonal subspace, 304 Michaelis-Menten, 179 orthonormal basis, 395, 397, 438 minimum, 401 orthonormal columns, 241, 395 model problem, 40, 115, 371, 421 oscillation, 74, 75 modulus, 32, 83 oscillation equation, 369 multiplication, 201, 218, 479 overdamping, 96, 100, 102 multiplicity, 93, 340 overshoot (Gibbs), 433, 434 multiplier, 209, 213, 224 P multistep method, 191 PF2, 62, 142, 149, 472 N PF3, 143, 149, 472 natural frequency, 77, 99, 102, 466 parabolas, 91, 96 network, 311-321, 414, 423, 424 parallel, 122, 127 neutral stability, 165, 336, 350 partial differential equation, (see PDE) Newton’s Law, 46, 73, 239, 370 partial fractions, 56, 62, 142-149, 474 Newton’s method, 6, 180 partial sums, 436 nodal analysis, 123 particular solution, 17, 18, 41, 106, 202, node, 311, 421 272, 274, 276 nondiagonalizable, 336, 339, 343, 380 PDE, 414, 455, 463 nonlinear equation, 1, 53, 171 peak time, 113, 128 nonlinear oscillation, 71 pendulum, 71, 81, 181 norm, 397, 398 period, 76, 162, 442 normal distribution, 458 periodic, 172 normal equations, 384, 386 , 240, 245, 297, 449 normal modes, 370 perpendicular, 200, 242, 389, 433, 434 Nth order equation, 107, 117 perpendicular eigenvectors, 380 null solution, 17, 18, 78, 92, 103, 106, perpendicular subspaces, 310 113, 202 phase angle, 32, 80 , 265 phase lag, 30, 33, 75, 81, 104, 112 nullspace, 259 phase line, 169 number of solutions, 280 phase plane, 59, 348 phase response, 77 O pictures, 152, 161 ODE 45, 190, 192 pivot, 209, 211, 224, 232, 399 off-diagonal ratios, 226 pivot column, 260, 262, 288, 292 Ohm’s Law, 39, 122, 422, 423, 425 pivot variable, 262, 268 one-way wave, 463, 468 plane, 200, 206, 256 open-loop, 64 Pluto, 154 operation count, 452 point source, 23, 457, 458 optimal control, 478 point-spread function, 484 order of accuracy, 185, 190, 191 Poisson’s equation, 415 orthogonal basis, 396, 431, 446, 447 polar angle, 38, 83 Index 499 polar form, 30, 32, 84, 110, 112, 121, rectangular form, 110, 111 243, 416, 429, 447 rectangular matrix, 382 poles, 100, 129, 140, 471–473 recursion, 452, 453 , 131 red lights, 478 Pontryagin, 478 reflection matrix, 246, 379 population, 47, 55, 61, 63 relativity, 464 positive definite, 369, 382, 394, 400-408 relaxation time, 46 positive definite matrix, 369, 382, 393 repeated eigenvalues, 335, 336, 352, 380 positive semidefinite, 409, 411 repeated roots, 90, 92, 101, 352 potential energy, 79 repeating ramp, 434 powers, 220, 325, 338 resistance, 119, 424 practical resonance, 126 resonance, 26, 27, 29, 79, 82, 108, 109, 114, predator-prey, 171, 173, 179 116, 132, 137, 361 prediction-correction, 190 response, 77 present value, 51 reverse order, 228, 237, 247 principal axis, 373 right triangle, 129, 383 Principal Component Analysis, 398, 429 right-inverse, 227, 231, 232 probability, 458 RLC loop, 39, 118, 119, 122 product integral, 381 roots, 101, 108, 129 product of pivots, 327, 500 roots of zN 1, 447 D product rule, 8 , 328 projection, 384, 386–388, 391 row exchange, 211, 215, 241 , 246, 331, 379, 386, 391 row picture, 196, 197, 213 pulse, 389, 390 row space, 287, 321 Python, 327 rref .A/, 261, 263, 265, 266, 282 Runge-Kutta, 16, 190–192 Q quadratic formula, 90 S quiver, 154 S-curve, 54, 64, 156 saddle, 161, 168, 172, 176, 399, 426 R saddle-point matrix, 428 rabbits, 171, 173 SciPy, 193 radians, 76 second difference, 239, 245, 407, 412, 413 radioactive decay, 45 semidefinite, 395, 409 ramp function, 23, 98, 404, 405, 477 separable, 56, 65 ramp response, 129 separation of variables, 419, 420, 456, rank, 265, 271, 275, 299 459, 460, 466 rank of AB, 309 shift, 439 rank one matrix, 303, 379, 398 shift invariance, 98, 459, 480, 482, 487 rank theorem, 320 shift rule for transform, 475 Rayleigh quotient, 429 sign reversal, 500 reactance, 121 similar matrix, 362, 367, 380 real eigenvalues, 165, 238, 372 Simpson’s Rule, 194 real roots, 90, 161 sines and cosines, 437 real solution, 31, 111 singular matrix, 201, 204, 214, 323, 500 500 Index singular value, 395, 397, 402 stock prices, 457 Singular Value Decomposition, 395 straight line, 383 singular vector, 382 subspace, 249–252, 254, 256, 294 sink, 17, 161 sudoku matrix, 208 sinusoid, 19, 30, 34 sum of spaces, 258 sinusoidal identity, 35, 37, 112 sum of squares, 383, 385 SIR model, 178 superposition, 8, 346, 460 six pictures, 161, 170 SVD, 243, 379, 382, 395-402, 429 skew-symmetric, 378 switch, 22 smoothness, 435 symmetric and orthogonal, 243, 375 solution curve, 153 , 237, 238, 290, 372, 406 Solution Page, 117 symmetry, 468 solvable, 253, 255, 276, 309 system, 163, 196, 322 source, 17, 19, 40, 161 T span, 254, 258, 283, 286, 294 Table of eigenvalues, 379 sparse matrices, 222 Table of Rules, 476 special inputs, 131, 139 Table of Transforms, 146, 471 special solution, 259, 263, 300 tangent, 75, 80, 155 , 373, 380 tangent line, 6, 183 speed of light, 464 tangent parabola, 7, 190 spike, 23, 404, 435, 436 Taylor series, 7, 10, 14, 16, 184 spiral, 33, 86, 88, 95, 160 temperature, 46, 440, 455, 459 spiral sink, 162 test grades, 392 spring, 74, 119 three steps, 338, 346, 366 square root, 394 time constant, 100 square wave, 433, 435, 441, 456 time domain, 120, 127 stability, 49, 58–60, 186, 187 time lag, 81 stability limit, 187, 194 time-varying, 364, 368, 381 stability line, 58, 169 , 480, 482 stability test, 164-169, 174, 187, 336, 350 Toomre, 177 stable, 160, 168, 349, 372 , 174, 328, 329, 333, 344, 350, 381 standing wave, 465 transfer function, 104, 121, 430, 477, 481 starting value (initial condition), 2, 9 transient, 27, 103 state space, 127 tree, 315 , 398, 458 , 213, 236, 293, 490, 492 steady state, 21, 49, 53, 58, 154, 325, 354 , 231, 245, 379, 407, 454 Stefan-Boltzmann Law, 49, 63 tumbling box, 175, 177, 182 step function, 21, 23, 474, 475, 478, 489 step response, 22, 81, 97, 102, 124–128 U stepsize, 183 underdamping, 96, 100, 102, 117 stiff equation, 186 undetermined coefficients, 117, 130-137 stiff system, 192 uniqueness, 153, 287 stiffness, 118, 468 unit circle, 33, 84, 85, 94, 447 stiffness matrix, 124, 369, 382 unit vector, 331 Index 501 , 374 Index of Symbols units, 44, 52, 456 A LU , 412, 490 unstable, 49, 53, 165 D A QR, 490 upper triangular, 209, 213 D A QS, 429 V D A U†V T, 379, 395, 398 variable coefficient, 1, 42, 130 D A VƒV 1, 334, 338 , 389, 392, 398, 429 D variation of , 41, 43, 130, ATA, 238, 274, 310, 382, 392, 415, 421 133–135, 138, 482 ATCA, 389, 401, 414, 423, 425 vector, 163, 198, 199, 249, 250 A A T, 411 , 249, 250, 296, 319  D very particular, 26, 27, 117, 144 K2D, 417, 418 violin, 465, 469 K ATCA, 408, 421, 422 D Voltage Law, 123, 315, 316 P.D/, 108, 117 voltage source, 423 Q, 237 W S LDLT, 400 D wave equation, 463–466, 469 S QƒQT, 373 weighted Laplacian, 422, 425 D S ?, 305 , 388, 389 C .A/ and N.A/, 253, 259 Wikipedia, 242, 429 n n Wolfram Alpha, 193 R and C , 249 , 134, 135, 363, 381 Z zerocline, 156 zeta, 99, 113

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