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9780198508526

Numerical Analysis A Mathematical Introduction

by
  • ISBN13:

    9780198508526

  • ISBN10:

    0198508522

  • Format: Paperback
  • Copyright: 2002-12-26
  • Publisher: Oxford University Press

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Summary

This book provides professionals and students with a thorough understanding of the interface between mathematics and scientific computation. Ranging from classical questions to modern techniques, it explains why numerical computations succeed or fail. The book is divided into four sections, with an emphasis on the use of mathematics as a tool in determining the success rate of numerical methods. The text requires only a modest level of mathematical training, and is ideally suited for scientists and students in mathematics, physics and engineering.

Table of Contents

I The entrance fee 1(46)
Floating numbers
3(8)
Counting in base β
3(2)
Expansion of the rational numbers in base β
5(1)
The machine representation of numbers
5(2)
Summation of series in floating-point numbers
7(1)
Even the obvious problems are rotten
8(1)
Even the easy problems are hard
9(1)
A floating conclusion
10(1)
A flavour of numerical analysis
11(14)
Comparison of exponentials and powers
12(1)
Convergence and divergence of classic series
13(1)
Discrete approximation of the logarithm
13(3)
Comparison of means
16(1)
Elementary construction of the exponential
17(3)
Exponentials of matrices
20(5)
Algebraic preliminaries
25(22)
Linear algebra refresher
25(9)
The matrix of a linear mapping
25(2)
The determinant
27(2)
The fundamental theorem of linear algebra and its consequences
29(1)
Eigenvalues and eigenvectors
30(1)
Scalar products, adjoints, and company
31(3)
Triangular matrices
34(1)
Block matrices
34(3)
Block decomposition of a linear mapping or matrix
34(2)
Block multiplication
36(1)
Exercises from Chapter 3
37(10)
Elementary algebra
37(1)
Block decomposition
38(1)
Graphs and matrices
39(1)
Functions of matrices
40(1)
Square roots, cosines, and sines of matrices
41(1)
Companion matrices and bounds of matrix powers
42(2)
The Kantorovich inequality
44(3)
II Polynomial and trigonometric approximation 47(158)
Interpolation and divided differences
51(26)
Lagrange interpolation
51(3)
The Lagrange interpolation problem
51(3)
Newton's interpolation
54(5)
Newton's basis is better than Lagrange's basis
54(3)
Integral representation of divided differences
57(2)
Interpolation error
59(5)
Hermite and osculating interpolation
64(3)
Divided differences as operators
67(5)
Finite differences on uniform grids
70(2)
Exercises from Chapter 4
72(5)
More on divided differences
72(2)
Boundary value problem for an ODE
74(1)
Extrapolation to the limit
75(2)
Least-squares approximation for polynomials
77(29)
Posing the problem
77(4)
Least-squares is Pythagoras in many dimensions
77(3)
Is it really calculable?
80(1)
Orthogonal polynomials
81(7)
Construction of orthogonal polynomials
82(1)
Examples of orthogonal polynomials
83(4)
Revival of special functions
87(1)
Orthogonal polynomials and least-squares
88(1)
Polynomial density: Bernstein polynomials
88(8)
Modulus of continuity
89(1)
Bernstein polynomials and Bernstein approximation
90(4)
Application of Bernstein polynomials to graphics software: the Bezier curves
94(2)
Least-squares convergence
96(1)
Qualitative properties
97(2)
Exercises from Chapter 5
99(7)
Laguerre polynomials
99(2)
Pade type and Pade approximations
101(5)
Splines
106(27)
Natural splines: the functional approach
107(9)
Weak equality of functions
107(1)
Weak integrals of functions
108(2)
The space of natural splines
110(6)
Numerics for cubic natural splines
116(3)
Spaces of splines, B-splines
119(11)
Splines with distinct knots
119(1)
The beautiful properties of B-splines
120(5)
Numerics with B-splines
125(2)
Using B-splines to understand natural splines
127(1)
B-splines in CAGD
128(2)
Exercises from Chapter 6
130(3)
Varied exercises on splines
130(1)
Approximation by splines
131(1)
Coincident knots
132(1)
Fourier's world
133(32)
Trigonometric approximation and Fourier series
133(13)
Trigonometric polynomials
134(1)
Integration of periodic functions
134(1)
Least-squares approximation for trigonometric polynomials
135(1)
Density of trigonometric polynomials in the space of continuous periodic functions
136(3)
Convergence in the mean square of trigonometric approximation to continuous functions
139(1)
Asymptotic behaviour of Fourier coefficients
140(2)
Convergence of trigonometric approximation to L2# functions
142(2)
Uniform convergence of Fourier series
144(2)
Convolution and pointwise convergence
146(13)
Convolution
146(2)
Regularization
148(2)
Constructive density results
150(1)
Convolution and Fourier series
151(1)
Convergence of Fourier series as a local phenomenon
152(1)
Pointwise convergence of partial Fourier sums of absolutely continuous functions
153(3)
Pointwise convergence of partial Fourier sums of piecewise absolutely continuous functions
156(1)
Gibbs phenomenon
157(2)
Exercises from Chapter 7
159(6)
Elementary exercises on Fourier series
159(1)
Fejer, La Vallee Poussin, and Poisson kernels
160(2)
There exists an integrable function whose Fourier coefficients decrease arbitrarily slowly to 0
162(1)
The existence of sequences of numbers ak tending to 0 as |k| tends to infinity which are not the Fourier coefficients of any integrable function
162(1)
Discrete least-squares approximation by trigonometric polynomials
163(2)
Quadrature
165(40)
Numerical integration
166(3)
Numerical integration for dummies
166(3)
The analysis of quadrature formulae
169(5)
Order of a quadrature formula
169(2)
On the practical interest of weighted formulae
171(1)
Examples of simple formulae
172(1)
Composite formulae
173(1)
The Peano kernel and error estimates
174(6)
Definition of the Peano kernel
174(5)
Quadrature error in composite formulae
179(1)
Gaussian quadrature
180(3)
Periodic numerical integration
183(1)
Bernoulli, Euler--MacLaurin
184(5)
Detailed analysis of the trapezium formula
185(2)
The Bernoulli polynomials
187(2)
The Euler--MacLaurin formula
189(1)
Discrete Fourier and fast Fourier transforms
189(8)
Discrete Fourier transforms
190(1)
Principle of the fast Fourier transform algorithm
191(2)
FFT algorithm: decimation-in-frequency
193(4)
Exercises from Chapter 8
197(8)
Summation of series with Bernoulli numbers and polynomials
197(3)
The Fredholm integral equation of the first kind
200(2)
Towards Franklin's periodic wavelets
202(3)
III Numerical linear algebra 205(100)
Gauss's world
207(33)
Elimination without pivoting
207(5)
Just elimination
207(2)
Matrix interpretation of Gaussian elimination
209(3)
Putting it into practice: operation counts
212(5)
The madness of Cramer's rule
212(2)
Putting elimination into practice
214(1)
Operation counts for elimination
214(1)
Inverting a matrix: putting it into practice and the operation count
215(1)
Do we need to invert matrices?
216(1)
Elimination with pivoting
217(10)
The effect of a small pivot
217(2)
Partial pivoting and total pivoting: general description and cost
219(2)
Aside: permutation matrices
221(1)
Matrix interpretation of partial and total pivoting
222(2)
The return of the determinant
224(1)
Banded matrices
224(3)
Other decompositions: LDU and Cholesky
227(4)
The LDU decomposition
227(1)
The Cholesky method
228(3)
Putting the Cholesky method into practice and operation counts
231(1)
Exercises from Chapter 9
231(9)
Exercises on the rank of systems of vectors
231(1)
Echelon matrices and least-squares
232(3)
The conditioning of a linear system
235(2)
Inverting persymmetric matrices
237(3)
Theoretical interlude
240(17)
The Rayleigh quotient
240(2)
Spectral radius and norms
242(2)
Spectral radius
242(1)
Norms of vectors, operators, and matrices
243(1)
Topology and norms
244(10)
Topology refresher
244(1)
Equivalence of norms
244(3)
Linear mappings: continuity, norm
247(1)
Subordinate norms
248(2)
Examples of subordinate norms
250(2)
The Frobenius norm is not subordinate
252(2)
Exercises from Chapter 10
254(3)
Continuity of the eigenvalues of a matrix with respect to itself
254(1)
Various questions on norms
255(2)
Iterations and recurrence
257(33)
Iterative solution of linear systems
258(12)
Linear recurrence and powers of matrices
270(6)
The spectrum of a finite difference matrix
273(3)
Exercises from Chapter 11
276(14)
Finite difference matrix of the Laplacian in a rectangle
276(1)
Richardson's and pre-conditioned Richardson's methods
277(3)
Convergence rate of the gradient method
280(2)
The conjugate gradient
282(3)
Introduction to multigrid methods
285(5)
Pythagoras' world
290(15)
About orthogonalization
290(13)
The Gram--Schmidt orthonormalization revisited
290(3)
Paths of inertia
293(2)
Topology for QR and Cholesky
295(1)
Operation counts and numeric strategies
296(1)
Hessenberg form
297(1)
Householder transformations
298(1)
QR decomposition by Householder transformations
299(3)
Hessenberg form by Householder transformations
302(1)
Exercises from Chapter 12
303(2)
The square root of a Hermitian positive definite matrix
303(2)
IV Nonlinear problems 305(174)
Spectra
307(24)
Eigenvalues: the naive approach
311(2)
Seeking eigenvalues and polynomial equations
311(1)
The bisection method
312(1)
Resonance and vibration
313(3)
Galloping Gertie
313(1)
Small vibrations
314(2)
Power method
316(6)
The straightforward case
316(3)
Modification of the power method
319(2)
Inverse power method
321(1)
QR method
322(6)
The algorithm and its basic properties
322(2)
Convergence in a special case
324(3)
Effectiveness of QR
327(1)
Exercises from Chapter 13
328(3)
Spectral pathology
328(1)
QR flow and Lax pairs
328(3)
Nonlinear equations and systems
331(31)
From existence to construction
331(5)
Existence and non-existence of solutions
331(2)
Existence proofs translate into algorithms
333(1)
A long and exciting history
333(1)
An overview of existence proofs
334(2)
Construction of several methods
336(17)
The strictly contracting fixed point theorem
336(2)
Newton's method: geometric interpretation and examples
338(3)
Convergence of Newton's method
341(3)
The secant method
344(4)
The golden ratio and Fibonacci's rabbits
348(3)
Order of an iterative method
351(1)
Ideas on the solution of vector problems
352(1)
Exercises from Chapter 14
353(9)
The Cardano formulae
353(1)
Brouwer's fixed point theorem in dimension 2
354(1)
Comparison of two methods for calculating square roots
355(2)
Newton's method for finding the square roots of matrices
357(5)
Solving differential systems
362(23)
Cauchy--Lipschitz theory
362(5)
Idea of the proof of existence for ODEs
362(1)
Cauchy--Lipschitz existence theorem
363(3)
Systems of order 1 and of order p
366(1)
Autonomous and non-autonomous systems
367(1)
Linear differential equations
367(15)
Constant coefficient linear systems
367(2)
Matrix exponentials
369(2)
Duhamel's formula
371(1)
Linear equations and systems with variable coefficients
372(5)
Gronwall's lemma
377(2)
Applications of Gronwall's lemma
379(1)
Smoother solutions
380(2)
Exercises from Chapter 15
382(3)
Lyapunov function for a 2 x 2 linear system
382(1)
A delay differential equation
382(2)
A second-order ordinary differential equation
384(1)
Single-step schemes
385(29)
Single-step schemes: the basics
386(6)
Convergence, stability, consistency
387(2)
Necessary and sufficient condition of consistency
389(1)
Sufficient condition for stability
390(2)
Order of a one-step scheme
392(3)
Explicit and implicit Euler schemes
395(4)
The forward Euler scheme
395(1)
Backwards Euler scheme
396(2)
θ-method
398(1)
Runge--Kutta formulae
399(4)
Examples of Runge--Kutta schemes
400(3)
Exercises from Chapter 16
403(11)
Detailed study of the θ-scheme
403(1)
Euler scheme with variable step size and asymptotic error estimates
404(3)
Numerical schemes for a delay differential equation
407(2)
Alternate directions
409(3)
Numerical analysis of a second-order differential equation
412(2)
Linear multistep schemes
414(25)
Constructing multistep methods
414(7)
Adams methods
415(1)
The multistep methods of Adams
416(3)
Backward differentiation
419(1)
Other multistep methods
420(1)
Order of multistep methods
421(3)
The order is nice and easy for multistep methods
421(2)
Order of some multistep methods
423(1)
Stability of multistep methods
424(8)
Multistep methods can be very unstable
424(2)
The stability theory for multistep methods
426(6)
Stability of some multistep schemes
432(1)
Convergence of multistep schemes
432(1)
Initializing multistep methods
432(1)
Solving in the implicit case
433(1)
Exercises from Chapter 17
433(6)
Short exercises
433(1)
An alternative formulation of the order condition
434(1)
Weak instability
434(1)
Predictor-corrector methods
435(2)
One-leg methods
437(2)
Towards partial differential equations
439(40)
The advection equation
439(7)
The advection equation and its physical origin
439(3)
Solving the advection equation
442(2)
More general advection equations and systems
444(2)
Numerics for the advection equation
446(9)
Definition of some good and some bad schemes
446(4)
Convergence of the scheme (18.2.6)
450(5)
The wave equation in one dimension
455(6)
Masses and springs
455(3)
Elementary facts about the wave equation
458(1)
A numerical scheme for the wave equation
459(2)
The heat equation and separation of variables
461(10)
Derivation of the heat equation
461(4)
Seeking a particular solution by separation of variables
465(2)
Solution by Fourier series
467(2)
Relation between the heat equation and the discrete model
469(2)
Exercises from Chapter 18
471(8)
The eigenvectors of a strictly hyperbolic matrix
471(1)
More on the upwind scheme
472(2)
Fourier analysis of difference schemes for the advection equation
474(1)
The Lax--Friedrichs scheme
474(1)
The Lax--Wendroff scheme
475(1)
Stability of the leap-frog scheme
475(1)
Elementary questions on the wave equation
476(1)
Generalized solutions for the advection equation
476(2)
Advection-diffusion equation
478(1)
References 479(6)
Index 485

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