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Mathews J.H., Fink K.D. — Numerical Methods Using MATLAB
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Название: Numerical Methods Using MATLABАвторы: Mathews J.H., Fink K.D. Аннотация: Provides an introduction to numerical analysis for undergraduates in mathematics, computer science, physical sciences, and engineering. Emphasis is on why numerical methods work and their limitations, with material balanced between theory, error analysis, and readability. MATLAB programs are the vehicle for presenting underlying numerical algorithms. Includes exercises and computer assignments, with extensive use of MATLAB, new to this third edition. Assumes previous courses in calculus and structured programming. Book News, Inc.®, Portland, OR
Язык: Рубрика: Математика /Численные методы /Численный анализ /Статус предметного указателя: Готов указатель с номерами страниц ed2k: ed2k stats Издание: third editionГод издания: 1999Количество страниц: 662Добавлена в каталог: 25.02.2005Операции: Положить на полку |
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29, 32, 214, 313, 314, 329, 333, 373, 377, 437, 445, 452, 461, 475.477,505,517,527,538 Accelerating convergence, Aitken’s process 90, 99 (#10—#14) Accelerating convergence, Newton — Raphson 71, 82, 88 (#23), 176 Accelerating convergence, Steffensen’s method 90, 95 Adam — Bashforth — Moulton method 474, 482 Adaptive quadrature 382, 387 Aitken’s process 90, 99 (#10—#14) Approximate significant digils 25 Approximation of data, least-squares curves 211, 257 Approximation of data, least-squares line 255, 258 Approximation of data, least-squares polynomial 271, 274 Approximation of functions, Chebyshev polynomial 230, 233, 238, 240 Approximation of functions, Lagrange polynomial 207, 211, 213, 217, 238 Approximation of functions, least squares 255, 257, 271 Approximation of functions, Newton polynomial 220, 224, 227 Approximation of functions, Pade approximation 243, 246 Approximation of functions, rational functions 243 Approximation of functions, splines 280, 281, 285, 293 Approximation of functions, Taylor polymomials 8, 26, 31, 189 Augmented matrix 126, 129 Back substitution 121, 123, 136 Backward difference 334 Basis 557 Binary numbers 13, 17, 19 Binomial series 197 (#14) Bisection method 53, 54, 59 Bolzano’s method 53 Boole’s rule 344, 372, 375, 380 (#3, #4), 389 (#3) Boundary value problems 497, 503, 505, 510 Bracketing methods 51, 53 Central difference 313, 314, 329, 340 (#7, #8) Characteristic polynomial 559 Chebyshev nodes 232 Chebyshev polynomial, interpolation 230, 233, 238, 240 Chebyshev polynomial, minimization 233 Chebyshev polynomial, nodes 234 Chopped number 27 Composite Simpson’s rule 350, 354, 359, 363 Composite trapezoidal rule 350, 354, 358, 363 Computer accuracy 21 Continuous function 3 Convergence, acceleration 82, 87 (#2I—#23), 90, 92, 95 Convergence, criteria 62, 66 Convergence, global (local) 62 Convergence, linear 76, 77, 90 Convergence, Newton — Raphson 77, 82, 87 (#21, #23) Convergence, order of 32, 75 Convergence, quadratic 76, 77, 82, 87 (#21, #23) Convergence, sequence 3 Convergence, series 8, 99 (#10—#14) Convergence, speed 75 Corrector formula 475, 477 Crank — Nicholson method 531, 535 Cube-root algorithm 86 (#11) Cubic spline, clamped 284, 285, 293 Cubic spline, natural 284, 285 Deflation of eigenvalues 578 Derivative, definition 5, 311 Derivative, formulas 204, 313, 322, 329, 333, 505,517,527,538 Derivative, higher 329, 333, 505 Derivative, partial 325 (#7), 517, 527, 538 Derivative, polynomials 204, 334, 336 Determinant 113, 114, 123, 151 Difference equation 505, 517, 527, 531, 539 Difference, backward 334 Difference, central 313, 314, 329, 340 (#7, #8) Difference, divided 223 Difference, finite-difference method 505, 510, 514,517,527,539 Difference, forward 334, 341 (#13) Difference, table 224 Differential equation, Adams — Bashforth — Moulton method 474, 482 Differential equation, boundary value problems 497, 503, 505, 510 Differential equation, Crank — Nicholson method 531, 535 Differential equation, Dirichlet method for Laplace’s equation 549 Differential equation, Euler’s method 433, 437, 440 Differential equation, existence-uniqueness 430 Differential equation, finite-difference method 505, 510, 514, 517, 527, 539 Differential equation, forward-difference method 528, 533 Differential equation, Hamming’s method 484 Differential equation, Heun’s method 443, 445, 448, 465 Differential equation, higher-order equations 490 Differential equation, initial value problem 428, 430, 487, 498 Differential equation, Milne — Simpson method 477,483 Differential equation, modified Euler method 465 Differential equation, partial differential equations 514, 516, 526, 538 Differential equation, predictor 474, 477 Differential equation, Runge — Ktitta — Fehlberg method 466, 469 Differential equation, Runge — Kutta method 458, 461, 466, 468, 488, 502 Differential equation, shooting method 498, 503 Differential equation, stability of solutions 478, 481 Differential equation, Taylor methods 451, 452, 455 Digit, binary 14, 17, 19 Digit, decimal 14, 19, 22 Dirichlet method for Laplace’s equation 549 Distance between points 103, 162 Divided differences 223 Division by zero 74, 77 Division, synthetic 10, 200 Dot product 103 DOUBLE PRECISION 22 Double root 75, 77, 87 (#21) d’Alembert’s solution 519 Eigenvalues, characteristic polynomial 559 Eigenvalues, definition 559 Eigenvalues, dominant 568 Eigenvalues, Householder’s method 594 Eigenvalues, inverse power method 573, 575, 576 Eigenvalues, Jacobi’s. method 581 Eigenvalues, power method 568, 570, 573, 576 Eigenvalues, QR method 601,606 Eigenvectors, definition 559 Eigenvectors, dominant 568 Elementary row operations 126 Elementary transformations 125 Elliptic equations 538 Endpoint constraints for splines 284 Epidemic model 442 (#9) Equivalent linear systems 125 Error, absolute 24 Error, bound 189, 194,213 Error, computer 21, 27, 135 Error, data 36, 203, 316 Error, differential equations 437, 445, 452, 462, 475, 477, 519 Error, differentiation 313, 314, 316, 318 Error, integration 344, 358, 359, 377 Error, interpolating polynomial 189, 213, 238 Error, loss of significance 28 Error, propagation 32 Error, relative 24, 66 Error, root mean square 253 Error, round-off 27 Error, sequence 3 Error, stable (unstable) 33 Error, subtractive cancellation 28 Error, truncation 26, 313, 314 Euclidean norm 103, 162, 163 Euler formulas 299 Euler’s method 433, 437, 440 Euler’s method, global error 437 Euler’s method, modified 465 Euler’s method, systems 488 Even function 300 Exponential fit 263 Extrapolated value 199 extrema 400, 404 Extreme Value Theorem 4 False position method 56, 60 Final global error 437, 445, 452, 462 Finite difference method! 505, 510, 514, 517, 527, 539 Fixed-point iteration 42, 49, 173 Fixed-point iteration, error bound 46 Floating-point number 21, 22 Floating-point number, accuracy 21 Forward difference 334, 341 (#13) Forward difference method 527, 528, 533 Forward substitution 125 (#2) Fourier series 299 Fourier series, discrete 304 Fractions, binary 17 Fundamental theorem of calculus 6 Gauss — Legendre integration 389, 392, 394 Gauss — Seidel iteration 159, 161, 164 Gaussian elimination 125, 128, 143, 150 Gaussian elimination, back substitution 121, 123 Gaussian elimination, computational complexity 147 Gaussian elimination, LU factorization 141, 143, 150 Gaussian elimination, multipliers 127, 129 Gaussian elimination, pivoting 127, 131 Gaussian elimination, tridiagonal systems 140 (#1), 166 (#3), 284, 506, 599 Generalized Rolle’s theorem 6,198 (#20) Geometric series 16, 51 Gerschgorin’s circle theorem 566 Golden ratio search 401, 412 Gradient 412, 420 Graphical analysis, fixed-point iteration 47 Graphical analysis, Newton’s method 70, 78, 79 Graphical analysis, secant method 80 Halley’s method 87 (#22) Hamming’s method 484 Heat equation 515 Helmholtz’s equation 538, 548 Heun’s method 443, 445, 448, 465 Higher derivatives 329, 333 Hilbert matrix 139 (#15) Hooke’s law 262 (#1) Horner’s method 10, 200 Householder’s method 594 Hyperbolic equations 516 Ill-conditioning, least-squares data fitting 134 Ill-conditioning, matrices 133, 139 (#15) Initial value problem 428, 430, 487, 498 Integration, adaptive quadrature 382, 387 Integration, Boole’s rule 344, 372, 375, 380 (#3, #4), 389 (#3) Integration, composite rules 350, 354, 358, 363 Integration, cubic splines 296 (#12) Integration, Gauss — Legendre integration 389, 392, 394 Integration, midpoint rule 366 (#12), 381 (#11) Integration, Newton — Cotes 344 Integration, Romberg integration 373, 375, 377, 378, 381 (#11) Integration, Simpson’s rule 344, 353 (#9), 354, 359, 363, 370, 380 (#6), 387 Integration, trapezoidal rule 344, 354, 358, 363, 368, 377 Intermediate Value Theorem 3 Interpolation, Chebyshev polynomials 230, 233, 238, 240 Interpolation, cubic splines 281, 285—287, 293 Interpolation, error, polynomials 8, 31, 189, 211, 213, 238 Interpolation, extrapolation 199 Interpolation, integration 296 (#12), 344 Interpolation, Lagrange polynomials 207, 211, 213, 217, 238 Interpolation, least squares 255, 271 Interpolation, linear 207, 219 (#12), 255, 277 (#17), 280 Interpolation, Newton polynomials 220, 224, 227 Interpolation, Pade approximations 243, 246 Interpolation, piecewise linear 280 Interpolation, polynomial wiggle 273 Interpolation, rational functions 243 Interpolation, Runge phenomenon 236 Interpolation, Taylor polynomials 8, 26, 31, 189, 313, 329 Interpolation, trigonometric polynomials 297, 303, 306 Iteration methods, bisection 53, 54, 59 Iteration methods, fixed point 42,49, 173, 544 Iteration methods, Gauss — Seidel 159, 161, 164 Iteration methods, Jacobi iteration 156, 161, 163 Iteration methods, Muller 92, 97 Iteration methods, Newton 70, 82, 84, 88 (#23), 176, 179 Iteration methods, partial differential equations 546 Iteration methods, regula falsi 56, 60 Iteration methods, secant 80, 84, 87 (#20) Iteration methods, Steffensen 92, 95 Jacobi iteration for linear systems 156, 161, 163 Jacobian matrix 170, 176 Jacobi’s method for eigenvalues 581, 590 Lagrange polynomials 207, 211, 213, 236 Laplace’s equation 538, 549 Least-squares data fitting, data linearization 266 Least-squares data fitting, linear fit 255, 258, 260 (#7), 277 (#17) Least-squares data fitting, nonlinear fit 257, 266, 271 Least-squares data fitting, plane 277 (#17, #18) Least-squares data fitting, polynomial fit 271,274 Least-squares data fitting, root-mean-square error 253 Least-squares data fitting, trigonometric polynomials 297, 303, 306 Length of a curve 364 (#2) Length of a vector 103, 162, 163 Limit, function 2 Limit, sequence 3 Limit, series 8 Linear approximation 219 (#12), 255, 258, 277 (#17), 280 linear combination 103, 499 Linear convergence 76, 77,90 Linear independence 557 Linear least-squares fit 255, 258, 260 (#7), 277 (#17) Linear system 114, 121, 128, 143, 152, 156, 163 Linear systems of equations, back substitution 121, 123, 136 Linear systems of equations, forward substitution 125 (#2) Linear systems of equations, Gaussian elimination 125, 128, 143, 150 Linear systems of equations, LU factorization 141, 143, 150 Linear systems of equations, tridiagonal systems 140 (#1), 166 (#3), 284, 506, 599 Linear systems, theory, matrix form 111, 114, 127, 141 Linear systems, theory, nonsingular 114 Lipschitz condition 430 Location of roots 68 Logistic rule of population growth 276 (#6, #7) Loss of significance 28 Lower triangular determinant 123 LU factorization 141, 143, 150 Machine numbers 20 Maclaurin series 243 Mantissa 20, 22 Markov process 579 (#5) Matrix, addition 107 Matrix, augmented 126, 129 Matrix, determinant 113, 114, 123, 151 Matrix, diagonalization 563 Matrix, eigenvalue 559 Matrix, eigenvector 559 Matrix, equality 106 Matrix, Hilbert 139 (#15) Matrix, identity 112 Matrix, ill-conditioned 133, 139 (#15) Matrix, inverse 112, 114 Matrix, lower triangular 120, 125 (#2), 143 Matrix, LU factorization 141, 143, 150 Matrix, multiplication 110, 112, 143, 150 Matrix, nonsingular 112 Matrix, norm 566 Matrix, orthogonal 565, 594 Matrix, permutation 148, 150 Matrix, singular 113 Matrix, strictly diagonally dominant 160, 162, 163 Matrix, symmetric 109 (#6), 565, 581, 590, 594 Matrix, transpose 104, 108 (#5), 270 Matrix, triangular 120, 125 (#2) Matrix, tridiagonal 140 (#1), 166 (#3), 284, 506, 599 Mean of data 260 (#4, #5, #6) Mean value theorems, derivative 5, 45 Mean value theorems, integrals 6 Mean value theorems, intermediate 3 Mean value theorems, weighted integral 7 Midpoint rule 366 (#12), 381 (#11) Milne — Simpson method 477, 483 Minimax approximation, Chebyshev 231, 233, 238
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