Numerical Mathematics / Edition 2

Numerical Mathematics / Edition 2

by Alfio Quarteroni, Riccardo Sacco, Fausto Saleri

ISBN-10: 3642071015

ISBN-13: 9783642071010

Pub. Date: 11/25/2010

Publisher: Springer Berlin Heidelberg

Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and…  See more details below


Numerical mathematics is the branch of mathematics that proposes, develops, analyzes and applies methods from scientific computing to several fields including analysis, linear algebra, geometry, approximation theory, functional equations, optimization and differential equations. Other disciplines, such as physics, the natural and biological sciences, engineering, and economics and the financial sciences frequently give rise to problems that need scientific computing for their solutions. As such, numerical mathematics is the crossroad of several disciplines of great relevance in modern applied sciences, and can become a crucial tool for their qualitative and quantitative analysis.

One of the purposes of this book is to provide the mathematical foundations of numerical methods, to analyze their basic theoretical properties (stability, accuracy, computational complexity) and demonstrate their performance on examples and counterexamples which outline their pros and cons. This is done using the MATLAB[Trademark] software environment which is user-friendly and widely adopted. Within any specific class of problems, the most appropriate scientific computing algorithms are reviewed, their theoretical analyses are carried out and the expected results are verified on a MATLAB[Trademark] computer implementation. Every chapter is supplied with examples, exercises and applications of the discussed theory to the solution of real-life problems.

This book is addressed to senior undergraduate and graduate students with particular focus on degree courses in engineering, mathematics, physics and computer sciences. The attention which is paid to the applications and the related development ofsoftware makes it valuable also for researchers and users of scientific computing in a large variety of professional fields. In this second edition, the readability of pictures, tables and program headings has been improved. Several changes in the chapters on iterative methods and on polynomial approximation have also been added.

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Product Details

Springer Berlin Heidelberg
Publication date:
Texts in Applied Mathematics Series, #37
Edition description:
Softcover reprint of hardcover 2nd ed. 2007
Product dimensions:
6.14(w) x 9.21(h) x 1.36(d)

Table of Contents

Getting Started
Foundations of Matrix Analysis     3
Vector Spaces     3
Matrices     5
Operations with Matrices     6
Inverse of a Matrix     7
Matrices and Linear Mappings     8
Operations with Block-Partitioned Matrices     9
Trace and Determinant of a Matrix     10
Rank and Kernel of a Matrix     11
Special Matrices     12
Block Diagonal Matrices     12
Trapezoidal and Triangular Matrices     12
Banded Matrices     13
Eigenvalues and Eigenvectors     13
Similarity Transformations     15
The Singular Value Decomposition (SVD)     17
Scalar Product and Norms in Vector Spaces     18
Matrix Norms     22
Relation between Norms and the Spectral Radius of a Matrix     25
Sequences and Series of Matrices     26
Positive Definite, Diagonally Dominant and M-matrices     27
Exercises     30
Principles of Numerical Mathematics     33
Well-posedness and Condition Number of a Problem     33
Stability of Numerical Methods     37
Relations between Stability and Convergence     40
Apriori and a posteriori Analysis     42
Sources of Error in Computational Models     43
Machine Representation of Numbers     45
The Positional System     45
The Floating-point Number System     46
Distribution of Floating-point Numbers     49
IEC/IEEE Arithmetic     49
Rounding of a Real Number in its Machine Representation     50
Machine Floating-point Operations     52
Exercises     54
Numerical Linear Algebra
Direct Methods for the Solution of Linear Systems     59
Stability Analysis of Linear Systems     60
The Condition Number of a Matrix     60
Forward a priori Analysis     62
Backward a priori Analysis     65
A posteriori Analysis     65
Solution of Triangular Systems     66
Implementation of Substitution Methods     67
Rounding Error Analysis     69
Inverse of a Triangular Matrix     70
The Gaussian Elimination Method (GEM) and LU Factorization     70
GEM as a Factorization Method     73
The Effect of Rounding Errors     78
Implementation of LU Factorization     78
Compact Forms of Factorization      80
Other Types of Factorization     81
LDM[superscript T] Factorization     81
Symmetric and Positive Definite Matrices: The Cholesky Factorization     82
Rectangular Matrices: The QR Factorization     84
Pivoting     87
Computing the Inverse of a Matrix     91
Banded Systems     92
Tridiagonal Matrices     93
Implementation Issues     94
Block Systems     96
Block LU Factorization     97
Inverse of a Block-partitioned Matrix     97
Block Tridiagonal Systems     98
Sparse Matrices     99
The Cuthill-McKee Algorithm     102
Decomposition into Substructures     103
Nested Dissection     105
Accuracy of the Solution Achieved Using GEM     106
An Approximate Computation of K(A)     108
Improving the Accuracy of GEM     112
Scaling     112
Iterative Refinement     113
Undetermined Systems     114
Applications     117
Nodal Analysis of a Structured Frame     117
Regularization of a Triangular Grid     120
Exercises     123
Iterative Methods for Solving Linear Systems     125
On the Convergence of Iterative Methods     125
Linear Iterative Methods     128
Jacobi, Gauss-Seidel and Relaxation Methods     128
Convergence Results for Jacobi and Gauss-Seidel Methods     130
Convergence Results for the Relaxation Method     132
A priori Forward Analysis     133
Block Matrices     134
Symmetric Form of the Gauss-Seidel and SOR Methods     135
Implementation Issues     137
Stationary and Nonstationary Iterative Methods     138
Convergence Analysis of the Richardson Method     139
Preconditioning Matrices     141
The Gradient Method     148
The Conjugate Gradient Method     152
The Preconditioned Conjugate Gradient Method     158
The Alternating-Direction Method     160
Methods Based on Krylov Subspace Iterations     160
The Arnoldi Method for Linear Systems     164
The GMRES Method     167
The Lanczos Method for Symmetric Systems     168
The Lanczos Method for Unsymmetric Systems     170
Stopping Criteria     173
A Stopping Test Based on the Increment     174
A Stopping Test Based on the Residual      175
Applications     175
Analysis of an Electric Network     176
Finite Difference Analysis of Beam Bending     178
Exercises     180
Approximation of Eigenvalues and Eigenvectors     183
Geometrical Location of the Eigenvalues     183
Stability and Conditioning Analysis     186
A priori Estimates     187
A posteriori Estimates     190
The Power Method     192
Approximation of the Eigenvalue of Largest Module     192
Inverse Iteration     195
Implementation Issues     196
The QR Iteration     199
The Basic QR Iteration     201
The QR Method for Matrices in Hessenberg Form     203
Householder and Givens Transformation Matrices     203
Reducing a Matrix in Hessenberg Form     207
QR Factorization of a Matrix in Hessenberg Form     209
The Basic QR Iteration Starting from Upper Hessenberg Form     209
Implementation of Transformation Matrices     212
The QR Iteration with Shifting Techniques     214
The QR Method with Single Shift     215
The QR Method with Double Shift     217
Computing the Eigenvectors and the SVD of a Matrix     220
The Hessenberg Inverse Iteration     220
Computing the Eigenvectors from the Schur Form of a Matrix     221
Approximate Computation of the SVD of a Matrix     222
The Generalized Eigenvalue Problem     223
Computing the Generalized Real Schur Form     224
Generalized Real Schur Form of Symmetric-Definite Pencils     225
Methods for Eigenvalues of Symmetric Matrices     226
The Jacobi Method     226
The Method of Sturm Sequences     229
The Lanczos Method     233
Applications     236
Analysis of the Buckling of a Beam     236
Free Dynamic Vibration of a Bridge     238
Exercises     240
Around Functions and Functionals
Rootfinding for Nonlinear Equations     247
Conditioning of a Nonlinear Equation     248
A Geometric Approach to Rootfinding     250
The Bisection Method     250
The Methods of Chord, Secant and Regula Falsi and Newton's Method     253
The Dekker-Brent Method     259
Fixed-point Iterations for Nonlinear Equations     260
Convergence Results for Some Fixed-point Methods     263
Zeros of Algebraic Equations      264
The Horner Method and Deflation     265
The Newton-Horner Method     266
The Muller Method     269
Stopping Criteria     273
Post-processing Techniques for Iterative Methods     275
Aitken's Acceleration     275
Techniques for Multiple Roots     278
Applications     280
Analysis of the State Equation for a Real Gas     280
Analysis of a Nonlinear Electrical Circuit     281
Exercises     283
Nonlinear Systems and Numerical Optimization     285
Solution of Systems of Nonlinear Equations     286
Newton's Method and Its Variants     286
Modified Newton's Methods     288
Quasi-Newton Methods     292
Secant-like Methods     292
Fixed-point Methods     295
Unconstrained Optimization     298
Direct Search Methods     300
Descent Methods     305
Line Search Techniques     307
Descent Methods for Quadratic Functions     309
Newton-like Methods for Function Minimization     311
Quasi-Newton Methods     312
Secant-like methods     313
Constrained Optimization      315
Kuhn-Tucker Necessary Conditions for Nonlinear Programming     318
The Penalty Method     319
The Method of Lagrange Multipliers     321
Applications     325
Solution of a Nonlinear System Arising from Semiconductor Device Simulation     325
Nonlinear Regularization of a Discretization Grid     328
Exercises     330
Polynomial Interpolation     333
Polynomial Interpolation     333
The Interpolation Error     335
Drawbacks of Polynomial Interpolation on Equally Spaced Nodes and Runge's Counterexample     336
Stability of Polynomial Interpolation     337
Newton Form of the Interpolating Polynomial     339
Some Properties of Newton Divided Differences     341
The Interpolation Error Using Divided Differences     343
Barycentric Lagrange Interpolation     344
Piecewise Lagrange Interpolation     346
Hermite-Birkoff Interpolation     349
Extension to the Two-Dimensional Case     351
Polynomial Interpolation     351
Piecewise Polynomial Interpolation     352
Approximation by Splines     355
Interpolatory Cubic Splines     357
B-splines      361
Splines in Parametric Form     365
Bezier Curves and Parametric B-splines     367
Applications     370
Finite Element Analysis of a Clamped Beam     370
Geometric Reconstruction Based on Computer Tomographies     374
Exercises     375
Numerical Integration     379
Quadrature Formulae     379
Interpolatory Quadratures     381
The Midpoint or Rectangle Formula     381
The Trapezoidal Formula     383
The Cavalieri-Simpson Formula     385
Newton-Cotes Formulae     386
Composite Newton-Cotes Formulae     392
Hermite Quadrature Formulae     394
Richardson Extrapolation     396
Romberg Integration     397
Automatic Integration     400
Nonadaptive Integration Algorithms     400
Adaptive Integration Algorithms     402
Singular Integrals     406
Integrals of Functions with Finite Jump Discontinuities     406
Integrals of Infinite Functions     407
Integrals over Unbounded Intervals     409
Multidimensional Numerical Integration     411
The Method of Reduction Formula      411
Two-Dimensional Composite Quadratures     413
Monte Carlo Methods for Numerical Integration     416
Applications     417
Computation of an Ellipsoid Surface     417
Computation of the Wind Action on a Sailboat Mast     418
Exercises     421
Transforms, Differentiation and Problem Discretization
Orthogonal Polynomials in Approximation Theory     425
Approximation of Functions by Generalized Fourier Series     425
The Chebyshev Polynomials     427
The Legendre Polynomials     428
Gaussian Integration and Interpolation     429
Chebyshev Integration and Interpolation     433
Legendre Integration and Interpolation     436
Gaussian Integration over Unbounded Intervals     438
Programs for the Implementation of Gaussian Quadratures     439
Approximation of a Function in the Least-Squares Sense     441
Discrete Least-Squares Approximation     442
The Polynomial of Best Approximation     443
Fourier Trigonometric Polynomials     445
The Gibbs Phenomenon     449
The Fast Fourier Transform     450
Approximation of Function Derivatives     452
Classical Finite Difference Methods      452
Compact Finite Differences     454
Pseudo-Spectral Derivative     458
Transforms and Their Applications     460
The Fourier Transform     460
(Physical) Linear Systems and Fourier Transform     463
The Laplace Transform     465
The Z-Transform     467
The Wavelet Transform     468
The Continuous Wavelet Transform     468
Discrete and Orthonormal Wavelets     471
Applications     472
Numerical Computation of Blackbody Radiation     472
Numerical Solution of Schrodinger Equation     474
Exercises     476
Numerical Solution of Ordinary Differential Equations     479
The Cauchy Problem     479
One-Step Numerical Methods     482
Analysis of One-Step Methods     483
The Zero-Stability     484
Convergence Analysis     486
The Absolute Stability     489
Difference Equations     492
Multistep Methods     497
Adams Methods     500
BDF Methods     502
Analysis of Multistep Methods     502
Consistency     502
The Root Conditions      504
Stability and Convergence Analysis for Multistep Methods     505
Absolute Stability of Multistep Methods     509
Predictor-Corrector Methods     511
Runge-Kutta (RK) Methods     518
Derivation of an Explicit RK Method     521
Stepsize Adaptivity for RK Methods     521
Implicit RK Methods     523
Regions of Absolute Stability for RK Methods     525
Systems of ODEs     526
Stiff Problems     528
Applications     530
Analysis of the Motion of a Frictionless Pendulum     531
Compliance of Arterial Walls     532
Exercises     536
Two-Point Boundary Value Problems     539
A Model Problem     539
Finite Difference Approximation     541
Stability Analysis by the Energy Method     542
Convergence Analysis     546
Finite Differences for Two-Point Boundary Value Problems with Variable Coefficients     548
The Spectral Collocation Method     550
The Galerkin Method     552
Integral Formulation of Boundary Value Problems     552
A Quick Introduction to Distributions     554
Formulation and Properties of the Galerkin Method      555
Analysis of the Galerkin Method     556
The Finite Element Method     558
Implementation Issues     564
Spectral Methods     566
Advection-Diffusion Equations     568
Galerkin Finite Element Approximation     569
The Relationship between Finite Elements and Finite Differences; the Numerical Viscosity     572
Stabilized Finite Element Methods     574
A Quick Glance at the Two-Dimensional Case     580
Applications     583
Lubrication of a Slider     583
Vertical Distribution of Spore Concentration over Wide Regions     584
Exercises     586
Parabolic and Hyperbolic Initial Boundary Value Problems     589
The Heat Equation     589
Finite Difference Approximation of the Heat Equation     591
Finite Element Approximation of the Heat Equation     593
Stability Analysis of the [theta]-Method     595
Space-Time Finite Element Methods for the Heat Equation     601
Hyperbolic Equations: A Scalar Transport Problem     604
Systems of Linear Hyperbolic Equations     607
The Wave Equation     608
The Finite Difference Method for Hyperbolic Equations      609
Discretization of the Scalar Equation     610
Analysis of Finite Difference Methods     611
Consistency     612
Stability     612
The CFL Condition     613
Von Neumann Stability Analysis     615
Dissipation and Dispersion     618
Equivalent Equations     619
Finite Element Approximation of Hyperbolic Equations     624
Space Discretization with Continuous and Discontinuous Finite Elements     625
Time Discretization     627
Applications     630
Heat Conduction in a Bar     630
A Hyperbolic Model for Blood Flow Interaction with Arterial Walls     630
Exercises     632
References     635
Index of MATLAB Programs     645
Index     649

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