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Kincaid D., Cheney W. — Numerical analysis: mathematics of scientific computing
Kincaid D., Cheney W. — Numerical analysis: mathematics of scientific computing

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Название: Numerical analysis: mathematics of scientific computing

Авторы: Kincaid D., Cheney W.

Аннотация:

This book has evolved over many years from lecture notes that accompany cer-
certain upper-division courses in mathematics and computer sciences at our university. These courses introduce students to the algorithms and methods that are commonly needed in scientific computing. The mathematical underpinnings of these methods
are emphasized as much as their algorithmic aspects. The students have been diverse: mathematics, engineering, science, and computer science undergraduates, as well as
graduate students from various disciplines. Portions of the book also have been used to lay the groundwork in several graduate courses devoted to special topics in numerical analysis, such as the numerical solution of differential equations, numerical
linear algebra, and approximation theory. Our approach has always been to treat the subject from a mathematical point of view, with attention given to its rich offering of
theorems, proofs, and interesting ideas. From these arise many computational procedures and intriguing questions of computer science. Of course, our motivation comes from the practical world of scientific computing, which dictates the choice of topics and the manner of treating each. For example, with some topics it is more instructive to discuss the theoretical foundations of the subject and not attempt to analyze algorithms in detail. In other cases, the reverse is true, and the students learn much from programming simple algorithms themselves and experimenting with them—although
we offer a blanket admonishment to use well-tested software, such as from program libraries, on problems arising from applications.


Язык: en

Рубрика: Математика/Численные методы/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1991

Количество страниц: 690

Добавлена в каталог: 15.07.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$fl(x \bigodot y)$      34 219
$fl(x)$      33—35
$l_{1}$-norm      162 168
$L_{2}$-norm      162 192
$l_{\infty}$-norm      162 168
$o$-notation      10 12
$V$-cycle algorithm      628
$w$-orthogonal      457
$w$-orthogonality      459
a posteriori bounds      219
A-orthogonal      209
A-orthonormality      208
A-stability      569
Absolute error      41
Adams — Bashforth formula, fifth-order      509
Adams — Bashforth formula, fourth-order      514
Adams — Bashforth — Moulton, systems      529
Adams — Moulton formula, fifth- order      510
Adams — Moulton formula, fourth-order      514
Adaptive approximation      424
Adaptive quadrature      471
Adaptive quadrature, algorithm      475
Adaptive Runge — Kutta — Fehlberg, algorithm      505
Adaptive Runge — Kutta — Fehlberg, method      503
Aitken acceleration      231
Algorithm, see Pseudocode Aliasing      421 422
Antidifferentiation      443
Applications of B-splines      343
Approximate inverse      175
Approximating functions      278
Approximation formula, $f’(x)$      433
Approximation formula, $f’’(x)$      433
Approximation sense of Sard      477
Associated matrix norm      163
Attraction, basin of      105
Autonomous      528 555 570
B-splines applications      343
B-splines, degree 0, $B_{i}^{0}(x)$      333
B-splines, degree 1, $B_{i}^{1}(x)$      334
Back substitution      127 145
Back substitution, permuted system      128
Backward SOR      194
Badly conditioned problems      48
Bairstow’s method      96
Bairstow’s method, algorithm      98
Barycentric interpolation formula      295
Basic feasible point      654
Basic functions      552
Basic Gaussian elimination      139
Basic vectors      551
Basin of attraction      105
Bernoulli numbers      358 468
Bernstein polynomials      289
Bessel functions $J_{n}$      25 27 53 55
Bessel’s inequality      364
Best approximation      359
Big $O$, functions and sequences      10
Binary system      28
Binomial coefficient      289
Birkhoff interpolation      307
Bisection method      57
Bisection method, algorithm      59
Bisection theorem      61
Boolean sum $\bar{P}\bigoplus\bar{Q}$      387
Boundary-value problems      531 540 547 551
Bounded sequence      25
Boundedness      14
Branches      16
Canonical form      614
Carath$\acute{e}$odory’s Theorem      375
Cardinal functions      282 435
Cardinal property      386
Cartesian grid      385
Cartesian product      385
Cauchy criterion      82 172
Cauchy — Schwarz inequality      103 478
Chain rule      492
Change of intervals      449
Characteristic curves      598 599 606 614
Characteristic equation      187 228
Characteristic polynomial      21 228 557
Characteristics      606
Characteristics, alternative approach      613
Characterizing best approximations      372
Chebyshev acceleration      197
Chebyshev algorithm      201
Chebyshev Alternation Theorem      288 380
Chebyshev polynomial of the second kind      451
Chebyshev polynomials      285 366
Chebyshev solution of linear equations      376
Chebyshev theory      370
Chebyshev’s quadrature formulas      456
Cholesky factorization      130 133 154
Cholesky factorization, algorithm      134
Choosing nodes      287
Chopped to $n$-digit approximation      6
Chopping      31
Classification, partial differential equations      607
Coarse grid      623
Coarse grid, correction scheme      628
Collocation      551 592
Column equilibration      177
Companion matrix      276
Complex Fourier series      410
Complex Newton’s method      105
Composite rule      445
Computer arithmetic      28
Condition number      51 52 165
Conditioning      51
conjugate      96 191 226
Conjugate direction methods      208
Conjugate gradient method      204 210
Conjugate gradient method, algorithm      210
Conjugate transpose      192 227 237
Consistent      517 636
Consistent systems      644
Continuation methods      108
Continued fractions      403
continuous      2
Continuous functions      13
contours      207
Contractive function/mapping      81
Contractive mapping theorem      82
Convergence      549
Convergence of interpolating polynomials      287
Convergence, linear      12
Convergence, order of      85
Convergence, quadratic      12
Convergence, sequences      9—11
Convergence, superlinear      12
Convergent method      517
Conversion of series to continued fractions      405
Convex      374
Convex combinations      374
Convex hull      374 638
Convexity      374 636
Cosine integral      357
Cramer’s Rule      303
Crank — Nicolson method      582
Critical set, crit ($f$)      372
Crout’s factorization      130
Cube roots of unity      105
Cubic $B$-splines      552
Cubic splines      317
Curvature      322
Damping of errors      625
Decimal system      28
Defective matrix      235
Deflation      92 239 274
Derivative      2
Derivatives, $B$-splines      337
Deviation      359
Deviation array      425
Diagonal structure      126
Diagonalizable matrices      558
Diagonally dominant      152 319
Difference equations      20 22 50
Differentiable      2
Differentiation via polynomial interpolation      434
Diffusion equation      572
Dilogarithm function      357 497
Direct method for computing eigenvalues      228
Direct methods      181
Direction vector      207
Dirichlet problem      586 592
Discrete problem      587
Displacement operator      21
Distance from a function to a spline space      352
Divided differences      296 298
Divided differences with repetitions      311
Divided-difference properties      301
Doolittle’s factorization      130
Doolittle’s factorization, algorithm      131
Double-precision      30
Dual problem      649
Duality theory      649
Eigenvalue      187 227 555
Eigenvalue problem      226
Eigenvector      227 555
Elementary matrix      120
Elementary operations      118
Elementary row and column operations      275
Elliptic integral      497
Elliptic integral, second kind      496
Elliptic partial differential equations      607
Embedded Runge — Kutta procedures      505
Entire function      288
Equilibration      176
Equivalent systems      118 140
Error analysis      460 619
Error analysis, bisection method      61
Error analysis, functional iteration      84
Error analysis, Gaussian quadrature      460
Error analysis, Newton’s method      66
Error analysis, secant method      76
Error function, erf(x)      15 356 495
Error in polynomial interpolation      284
Error vector      166 174
Errors      494 502 583
Euclidean norm      162 192 227
Euler — Maclaurin formula      468 484
Euler’s formula      410
Euler’s method      113 495 566
Evaluation of functions      45 51
Excess in row $i$      160
Exchange      383
Exchange method      382
Explicit functions      15
Explicit method      511 517 572 575
EXPONENT      29
Exponential integrals      54
Exponential polynomial      413 422
Extended-precision      30
Extrapolation      194
Extreme point      639
Extremum problem      359
Factor Theorem      89
Factorization phase      145
Factorization phase, algorithm      148
Factorizations $LU$      129
Factorizations $PA = LU$      149
Farkas Theorem      643
Fast Fourier sine transformation      632
Fast Fourier Transform      409 414
Fast Fourier transform, algorithm      419
Fast methods for Poisson’s equation      631
Feasible point      113 648
Feasible set      113 648 652
Fibonacci sequence      19 54
Finite-difference method      547 574 586
Finite-element method      597
First-degree spline      424
First-order partial differential equations      598
First-variational equation      543
Fixed point      81
Floating-point arithmetic      31
Floating-point error analysis      35
floating-point numbers      28
Forward elimination      145
Forward SOR      194
Forward substitution      127
Forward substitution, permuted system      128
Fourier method      578
Fourier series      410
Fractal      106
Fresnel integral      357 496
Frobenius norm      169
Full pivoting      156
Fully implicit method      581
Functional iteration      80 511
Fundamental matrix      562
Fundamental polynomials for interpolation      444
Fundamental theorem of algebra      89 226
Galerkin method      591 620
Gauss — Jordan method      161
Gauss — Seidel method      182 189 589 624 629
Gaussian elimination, scaled row pivoting      145
Gaussian quadrature      456
Generalized eigenvalue problem      276
Generalized Pythagorean law      364
Gerschgorin’s theorem      240
Global errors      516
Global roundoff error      494
Global truncation error      494 520
Gram matrix      368
Gram — Schmidt process      246 365
Gram — Schmidt process, algorithm      247
Greatest lower bound, glb      14—15
Green’s function      536 537
Grid correction      627
Haar subspace      378
Hailey’s method      74
Half-space      642
Harmonic      592
Harmonic series      47
Heat equation      572 607
Hermite interpolation      305
Hermite’s quadrature formula      456
Hermitian      192 238
Hessenberg matrix      270
Heun’s method      500
Higher transcendental functions      355
Higher-degree natural spline      324
Higher-order divided differences      298
Higher-order ordinary differential equations      524
Hilbert matrix      53 369
Homotopy      108
Horner’s algorithm, complete      93
Horner’s method      14
Horner’s method, algorithm      41 91 280
Householder factorization      251 253
Householder transformations      251
Hyperbolic partial differential equations      607
Hyperbolic problems      616
Idempotent      267
Identity matrix      119
Ill-conditioned      51 52 167 283
Implicit function theorem      16
Implicit functions      15 68
Implicit method      512 571 580
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