Numerical Computation 2: "Methods, Software, And Analysis" - Brossura

Ueberhuber, Christoph W.

Brossura

ISBN 10: 3540620575 ISBN 13: 9783540620570

Casa editrice: Springer, 2013

Riassunto

This book is the second part of a modern, two-volume introduction to numerical computation, which strongly emphasizes software aspects. It can serve as a textbook for courses on numerical analysis, particularly for engineers. The book can also be used as a reference book and it includes an extensive bibliography. The author is a well-known specialist in numerical analysis who was involved in the creation of the software package QUADPACK.

Su questo libro

Recensione

"The two volumes can be highly recommended for newcomers in the area as well as for people working for a long time in or with computer numerics." 

EUROSIM - Simulation News Europe

 

"... This book is highly recommended to students, scientists, anad engineers interested in the numerical solution of mathematical problems. It is very useful as a handbook for both newcomers and experts. Every science/engineering/mathematics/computer science library should have a copy of this book."

Matti Vuorinen, Mathematical Reviews 2003

Contenuti

10 Best Approximation.- 10.1 Mathematical Foundations.- 10.2 Continuous Least Squares Approximation.- 10.2.1 Mathematical Foundations.- 10.2.2 Best Approximation with Orthogonality.- 10.2.3 The Normal Equations.- 10.2.4 The Approximation Error.- 10.2.5 Orthogonal Polynomials.- 10.3 Discrete Least Squares Approximation.- 10.3.1 Linear Least Squares Approximation.- 10.3.2 Nonlinear Least Squares Approximation.- 10.4 Uniform Best Approximation.- 10.4.1 Uniformly Best Approximating Polynomials.- 10.5 Approximation Algorithms.- 10.6 Approximation Software for Special Functions.- 10.6.1 Elementary Functions.- 10.6.2 Funpack.- 10.6.3 IMSL Libraries.- 10.6.4 NAG Libraries.- 11 The Fourier Transform.- 11.1 Background.- 11.2 Mathematical Foundations.- 11.2.1 Trigonometric Approximation.- 11.2.2 The Spectrum.- 11.3 Trigonometric Interpolation.- 11.4 Convolution.- 11.5 Manipulation of the Signal Spectrum.- 11.5.1 Case Study: The Filtering of a Noisy Signal.- 11.6 DFT Algorithms.- 11.6.1 The Fast Fourier Transform (FFT).- 11.6.2 FFT of Real Functions.- 11.6.3 FFT in Two or More Dimensions.- 11.7 FFT Software Packages.- 11.7.1 FFTPACK.- 11.7.2 VFFTPK.- 11.8 FFT Routines in Software Libraries.- 11.8.1 IMSL Software Libraries.- 11.8.2 NAG Software Libraries.- 11.9 Other FFT Programs.- 11.9.1 TOMS Collection.- 11.9.2 Various NETLIB Software.- 12 Numerical Integration.- 12.1 Fundamentals of Integration.- 12.1.1 Integration Regions.- 12.1.2 Weight Functions.- 12.1.3 Integration Methods.- 12.1.4 Sensitivity of Integration Problems.- 12.1.5 Inherent Uncertainty of Numerical Integration.- 12.2 Preprocessing of Integration Problems.- 12.2.1 Transformation of Integrals.- 12.2.2 Decomposition of Integration Regions.- 12.2.3 Iteration of Integrals.- 12.3 Univariate Integration Formulas.- 12.3.1 Construction of Integration Formulas.- 12.3.2 Simple Interpolatory Quadrature Formulas.- 12.3.3 Compound Quadrature Formulas.- 12.3.4 Romberg Formulas.- 12.3.5 Nonlinear Extrapolation.- 12.3.6 Special Methods.- 12.4 Multivariate Integration Formulas.- 12.4.1 Construction Principles.- 12.4.2 Polynomial Integration Formulas.- 12.4.3 Number-Theoretic Integration Formulas.- 12.4.4 Monte Carlo Techniques.- 12..4.5 Lattice Rules.- 12.4.6 Special Methods.- 12.5 Integration Algorithms.- 12.5.1 Error Estimation.- 12.5.2 Sampling Strategy.- 12.5.3 Adaptive Integration Algorithms and Programs.- 12.5.4 Software for Univariate Problems: Globally Adaptive Integration Programs.- 12.5.5 Software for Multivariate Problems: Globally Adaptive Integration Programs.- 12.5.6 Reliability Enhancement.- 12.5.7 Multiple Integrands.- 13 Systems of Linear Equations.- 13.1 Design Stage.- 13.1.1 Type of Problem.- 13.1.2 Structural Properties of System Matrices.- 13.1.3 Types of Solutions.- 13.1.4 Algorithms and Software Requirements.- 13.2 Realization Stage.- 13.3 Verification Stage.- 13.4 Mathematical Foundations.- 13.4.1 Linear Spaces.- 13.4.2 Vector Norms.- 13.4.3 Orthogonality.- 13.4.4 Linear Functions.- 13.4.5 Matrices.- 13.4.6 Inverse of a Matrix.- 13.4.7 Eigenvalues of a Matrix.- 13.4.8 Matrix Norms.- 13.4.9 Determinant of a Matrix.- 13.5 Special Properties of Matrices.- 13.5.1 Symmetric and Hermitian Matrices.- 13.5.2 Orthogonal and Unitary Matrices.- 13.5.3 Positive Definite Matrices.- 13.6 Special Types of Matrices.- 13.6.1 Diagonal Matrices.- 13.6.2 Triangular Matrices.- 13.6.3 Block Matrices.- 13.6.4 Hessenberg Matrices.- 13.6.5 Tridiagonal Matrices.- 13.6.6 Band Matrices.- 13.6.7 Permutation Matrices.- 13.7 Singular Value Decomposition.- 13.7.1 Geometry of Linear Transformations.- 13.7.2 Structure of Linear Transformations.- 13.7.3 Generalized Inverse Mappings.- 13.7.4 General Solution of Linear Systems.- 13.7.5 The Solution of Homogeneous Systems.- 13.7.6 Linear Data Fitting.- 13.8 The Condition of Linear Systems.- 13.8.1 Condition of Regular Systems.- 13.8.2 Effects of a Perturbed Right-Hand Side.- 13.8.3 Effects of a Perturbed Matrix.- 13.9 Condition of Least Squares Problems.- 13.10 Condition Analysis Using the SVD.- 13.10.1 Case Study: Condition Analysis.- 13.11 Direct Methods.- 13.11.1 The Elimination Principle.- 13.11.2 LU Factorization.- 13.11.3 Pivot Strategies.- 13.12 Special Types of Linear Systems.- 13.12.1 Symmetric, Positive Definite Matrices.- 13.12.2 Band Matrices.- 13.13 Assessment of the Accuracy Achieved.- 13.13.1 Condition Number Estimates.- 13.13.2 Backward Error Analysis.- 13.13.3 Iterative Refinement.- 13.13.4 Experimental Condition Analysis.- 13.14 Methods for Least Squares Problems.- 13.14.1 Normal Equations.- 13.14.2 QR Method.- 13.15 LAPACK—The Fundamental Linear Algebra Package.- 13.15.1 The History of LAPACK.- 13.15.2 LAPACK and BLAS.- 13.15.3 Block Algorithms.- 13.15.4 Structure of LAPACK.- 13.16 LAPACK Black Box Programs.- 13.16.1 Linear Equations.- 13.16.2 Linear Least Squares Problems.- 13.17 LAPACK Computational Routines.- 13.17.1 Error Bounds.- 13.17.2 Orthogonal Factorizations.- 13.17.3 Singular Value Decomposition (SVD).- 13.18 LAPACK Documentation.- 13.18.1 Parameters.- 13.18.2 Error Handling.- 13.19 LAPACK Storage Schemes.- 13.19.1 Conventional Storage.- 13.19.2 Packed Storage.- 13.19.3 Storage of Band Matrices.- 13.19.4 Tridiagonal and Bidiagonal Matrices.- 13.19.5 Orthogonal or Unitary Matrices.- 13.20 Block Size for Block Algorithms.- 13.21 LAPACK Variants and Extensions.- 14 Nonlinear Equations.- 14.1 Iterative Methods.- 14.1.1 Fixed-Point Iteration.- 14.1.2 Convergence of Iterative Methods.- 14.1.3 Rate of Convergence.- 14.1.4 Determination of the Initial Values.- 14.1.5 The Termination of an Iteration.- 14.2 Nonlinear Scalar Equations.- 14.2.1 The Multiplicity of a Zero.- 14.2.2 The Condition of a Nonlinear Equation.- 14.2.3 The Bisection Method.- 14.2.4 Newton’s Method.- 14.2.5 The Secant Method.- 14.2.6 Muller’s Method.- 14.2.7 Efficiency Assessment.- 14.2.8 The Acceleration of Convergence.- 14.2.9 Polyalgorithms.- 14.2.10 Roots of Polynomials.- 14.3 Systems of Nonlinear Equations.- 14.3.1 Generalized Linear Methods.- 14.3.2 Newton’s Method.- 14.3.3 The Secant Method.- 14.3.4 Modification Methods.- 14.3.5 Large Nonlinear Systems.- 14.4 Nonlinear Data Fitting.- 14.4.1 Minimization Methods.- 14.4.2 The Levenberg-Marquardt Method.- 14.4.3 The Powell Method.- 14.4.4 Special Functions.- 15 Eigenvalues and Eigenvectors.- 15.1 Mathematical Foundations.- 15.1.1 The Characteristic Polynomial.- 15.1.2 Similarity.- 15.1.3 Eigenvectors.- 15.1.4 Unitary Similarity.- 15.1.5 Similarity to (Quasi) Diagonal Matrices.- 15.1.6 Bounds for the Eigenvalues.- 15.2 Condition of Eigenvalue Problems.- 15.3 The Power Method.- 15.3.1 The Inverse Power Method.- 15.3.2 The Inverse Power Method with Spectral Shifts.- 15.4 The QR Algorithm.- 15.4.1 The QR Algorithm with Spectral Shifts.- 15.4.2 Efficiency Improvement of the QR Algorithm.- 15.5 The Diagonal Reduction.- 15.5.1 ‘The Jacobi Method.- 15.6 The Hessenberg Reduction.- 15.6.1 The Givens Algorithm.- 15.6.2 The Householder Algorithm.- 15.7 LAPACK Programs.- 15.7.1 Symmetric Eigenproblems.- 15.7.2 Nonsymmetric Eigenproblems.- 15.7.3 Singular Value Decomposition (SVD).- 15.7.4 Generalized, Symmetric Eigenproblems.- 15.7.5 Generalized, Nonsymmetric Eigenproblems.- 16 Large, Sparse Linear Systems.- 16.1 Storage Schemes for Iterative Methods.- 16.1.1 Coordinate (COO) Format.- 16.1.2 Compressed Row Storage (CRS) Format.- 16.1.3 Modified CRS (MRS) Format.- 16.1.4 Compressed Column Storage (CCS Format).- 16.1.5 Block Compressed Row Storage (BCRS) Format.- 16.1.6 Compressed Diagonal Storage (CDS) Format.- 16.1.7 LAPACK (BND) Format for Band Matrices.- 16.1.8 Jagged Diagonal Storage (JDS) Format.- 16.1.9 Skyline Storage (SKS) Format.- 16.2 Storage Schemes for Symmetric Matrices.- 16.3 Storage Schemes for Direct Methods.- 16.3.1 Band Format.- 16.3.2 General Storage Schemes.- 16.4 Comparison of Storage Schemes.- 16.5 Direct Methods.- 16.5.1 Gaussian Elimination for Sparse Linear Systems.- 16.5.2 Band Matrices.- 16.5.3 Poisson Matrices.- 16.5.4 Matrices with General Sparsity Structure.- 16.6 Iterative Methods.- 16.7 Minimization Methods.- 16.7.1 The Gauss-Seidel Method.- 16.7.2 Gradient Methods.- 16.7.3 The Jacobi Method.- 16.7.4 The Conjugate Gradient Method.- 16.7.5 The Krylov Method.- 16.8 Stationary Iterative Methods.- 16.8.1 The Jacobi Method.- 16.8.2 The Gauss-Seidel Method.- 16.8.3 The Successive Over-Relaxation (SOR) Method.- 16.8.4 The Symmetric SOR (SSOR) Method.- 16.9 Non-Stationary Iterative Methods.- 16.9.1 The Conjugate Gradient Method (CG Method).- 16.9.2 The CG Method on the Normal Equations.- 16.9.3 The MINRES and the SYMMLQ Method.- 16.9.4 The Generalized Minimal Residual (GMRES) Method.- 16.9.5 The Bi-Conjugate Gradient (BiCG) Method.- 16.9.6 The Quasi-Minimal Residual (QMR) Method.- 16.9.7 The Squared CG (CGS) Method.- 16.9.8 The Stabilized BiCG (BiCGSTAB) Method.- 16.9.9 Chebyshev Iteration.- 16.10 Preconditioning.- 16.10.1 Jacobi Preconditioning.- 16.10.2 SSOR Preconditioning.- 16.10.3 Incomplete Factorization.- 16.10.4 Incomplete Block Factorization.- 16.10.5 Incomplete LQ Factorization.- 16.10.6 Polynomial Preconditioning.- 16.11 Matrix-Vector Products.- 16.11.1 Matrix-Vector Product Using the CRS Format.- 16.11.2 Matrix-Vector Product Using the CDS Format.- 16.12 Parallelism.- 16.13 Selecting an Iterative Method.- 16.13.1 Properties of Iterative Methods.- 16.13.2 Case Study: Comparison of Iterative Methods.- 16.14 Software for Sparse Systems.- 16.15 Basic Software.- 16.15.1 The Harwell-Boeing Collection.- 16.15.2 Sparse-Blas.- 16.15.3 Sparskit.- 16.16 Dedicated Software Packages.- 16.16.1 Itpack.- 16.16.2 Templates.- 16.16.3 Slap.- 16.16.4 Y12M.- 16.16.5 Umfpack.- 16.16.6 Pim.- 16.17 Routines from Software Libraries.- 16.17.1 IMSL Software Libraries.- 16.17.2 NAG Software Libraries.- 16.17.3 Harwell Library.- 17 Random Numbers.- 17.1 Random Number Generators.- 17.2 The Generation of Uniform Random Numbers.- 17.2.1 Congruential Generators.- 17.2.2 Uniform Random Vectors.- 17.2.3 Improving Random Number Generators.- 17.3 The Generation of Non-uniform Random Numbers.- 17.3.1 The Inversion Method for Univariate Distributions.- 17.3.2 The Rejection Method.- 17.3.3 The Composition Method.- 17.4 Testing Random Number Generators.- 17.5 Software for Generating Random Numbers.- 17.5.1 Programming Languages.- 17.5.2 IMSL Library.- 17.5.3 NAG Library.- Glossary of Notation.- Author Index.

Dettagli dell'edizione

• Editore: Springer
• Data di pubblicazione: 2013
• Lingua: Inglese
• ISBN 10: 3540620575
• ISBN 13: 9783540620570
• Rilegatura: Copertina flessibile
• Numero di pagine: 516
• Contatto del produttore: non disponibile
• Persona responsabile: CMN Printpool Inh. Mathis Neumann; https://www.also.com/ec/cms5/en_6000/6000/index.jsp; ; Friedrich-Karl-Str. 9; Hamburg; Germania