Numerical Linear Algebra: Theory and Applications

This book combines a solid theoretical background in linear algebra with practical algorithms for numerical solution of linear algebra problems. Developed from a number of courses taught repeatedly by the authors, the material covers topics like matrix algebra, theory for linear systems of equations, spectral theory, vector and matrix norms combined with main direct and iterative numerical methods, least squares problems, and eigenproblems. Numerical algorithms illustrated by computer programs written in MATLAB® are also provided as supplementary material on SpringerLink to give the reader a better understanding of professional numerical software for the solution of real-life problems. Perfect for a one- or two-semester course on numerical linear algebra, matrix computation, and large sparse matrices, this text will interest students at the advanced undergraduate or graduate level.

[1]  M. Marcus,et al.  A Survey of Matrix Theory and Matrix Inequalities , 1965 .

[2]  A. I. Malʹt︠s︡ev.  Foundations of linear algebra , 1963 .

[3]  J. Cullum,et al.  A generalized nonsymmetric Lanczos procedure , 1989 .

[4]  Robert D. Skeel,et al.  Scaling for Numerical Stability in Gaussian Elimination , 1979, JACM.

[5]  James Demmel,et al.  Accurate Singular Values of Bidiagonal Matrices , 1990, SIAM J. Sci. Comput..

[6]  W. Hackbusch.  Iterative Solution of Large Sparse Systems of Equations , 1993 .

[7]  Fuzhen Zhang.  Matrix Theory: Basic Results and Techniques , 1999 .

[8]  L. Mirsky,et al.  Introduction to Linear Algebra , 1965, The Mathematical Gazette.

[9]  V. Prasolov.  Problems and theorems in linear algebra , 1994 .

[10]  Mei Han An,et al.  accuracy and stability of numerical algorithms , 1991 .

[11]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[12]  W. Gragg,et al.  On computing accurate singular values and eigenvalues of matrices with acyclic graphs , 1992 .

[13]  G. Stewart,et al.  Matrix Perturbation Theory , 1990 .

[14]  Ė. B. Vinberg,et al.  A course in algebra , 2003 .

[15]  Kenneth Kuttler,et al.  Elementary Linear Algebra , 2012 .

[16]  K. Ikramov.  Matrix pencils: Theory, applications, and numerical methods , 1993 .

[17]  S. Godunov.  Modern Aspects of Linear Algebra , 1998 .

[18]  Zhaojun Bai,et al.  Progress in the numerical solution of the nonsymmetric eigenvalue problem , 1995, Numer. Linear Algebra Appl..

[19]  S. F. Karpenko.  Approximate solution of a class of operator equations , 1982 .

[20]  Charles R. Johnson,et al.  Topics in Matrix Analysis , 1991 .

[21]  Xian-He Sun,et al.  Parallel Homotopy Algorithm for the Symmetric Tridiagonal Eigenvalue Problem , 1991, SIAM J. Sci. Comput..

[22]  L. Mirsky,et al.  An introduction to linear algebra , 1957, Mathematical Gazette.

[23]  J. G. Lewis,et al.  A Shifted Block Lanczos Algorithm for Solving Sparse Symmetric Generalized Eigenproblems , 1994, SIAM J. Matrix Anal. Appl..

[24]  J. Demmel,et al.  On the correctness of some bisection-like parallel eigenvalue algorithms in floating point arithmetic. , 1995 .

[25]  T. Y. Li,et al.  Solving eigenvalue problems of real nonsymmetric matrices with real homotopies , 1992 .

[26]  R. Skeel.  Iterative refinement implies numerical stability for Gaussian elimination , 1980 .

[27]  Beresford N. Parlett.  The Construction of Orthogonal Eigenvectors for Tight Clusters by Use of Submatrices , 1996 .

[28]  S. Batterson.  Convergence of the shifted QR algorithm on 3×3 normal matrices , 1990 .

[29]  Richard H. Byrd,et al.  A Trust Region Algorithm for Nonlinearly Constrained Optimization , 1987 .

[30]  Nicholas J. Highham.  A survey of condition number estimation for triangular matrices , 1987 .

[32]  H. Engl,et al.  Regularization of Inverse Problems , 1996 .

[33]  J. Bunch,et al.  Some stable methods for calculating inertia and solving symmetric linear systems , 1977 .

[34]  Stanley C. Eisenstat,et al.  A Divide-and-Conquer Algorithm for the Symmetric Tridiagonal Eigenproblem , 1995, SIAM J. Matrix Anal. Appl..

[35]  J. Demmel,et al.  Solving Sparse Linear Systems with Sparse Backward Error , 2015 .

[36]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[37]  James Demmel,et al.  Jacobi's Method is More Accurate than QR , 1989, SIAM J. Matrix Anal. Appl..

[38]  I. Gel'fand,et al.  Lectures on Linear Algebra , 1961 .

[39]  C. Bischof.  Incremental condition estimation , 1990 .

[40]  O. Axelsson.  Iterative solution methods , 1995 .

[41]  J. Cuppen.  A divide and conquer method for the symmetric tridiagonal eigenproblem , 1980 .

[42]  Y. Saad,et al.  Numerical solution of large nonsymmetric eigenvalue problems , 1989 .

[43]  Tien-Yien Li,et al.  Homotopy-determinant algorithm for solving nonsymmetric eigenvalue problems , 1992 .

[44]  A. Sluis.  Condition numbers and equilibration of matrices , 1969 .

[45]  Nicholas J. Higham,et al.  FORTRAN codes for estimating the one-norm of a real or complex matrix, with applications to condition estimation , 1988, TOMS.

[46]  Nicholas J. Higham,et al.  Experience with a Matrix Norm Estimator , 1990, SIAM J. Sci. Comput..

[47]  Inderjit S. Dhillon,et al.  Fernando's solution to Wilkinson's problem: An application of double factorization , 1997 .

[48]  B. Parlett,et al.  Accurate singular values and differential qd algorithms , 1994 .

[49]  Inderjit S. Dhillon,et al.  A Way to Find the Most Redundant Equation in a Tridiagonal System , 1995 .

[50]  Beresford N. Parlett,et al.  The New qd Algorithms , 1995, Acta Numerica.

[51]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[52]  Nicholas J. Higham,et al.  Functions of matrices - theory and computation , 2008 .

[53]  Robert D. Skeel,et al.  EFFECT OF EQUILIBRATION ON RESIDUAL SIZE FOR PARTIAL PIVOTING , 1981 .

[54]  I. Dhillon.  Algorithm for the Symmetric Tridiagonal Eigenvalue/Eigenvector Problem , 1998 .

[55]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[56]  H. T. Lau,et al.  A Numerical Library in C for Scientists and Engineers , 1995 .

[57]  P. G. Ciarlet,et al.  Introduction to Numerical Linear Algebra and Optimisation , 1989 .

[58]  Paul R. Halmos.  Linear algebra problem book , 1995 .

[59]  Michael A. Malcolm,et al.  Computer methods for mathematical computations , 1977 .

[60]  I. M. Glazman,et al.  Finite-Dimensional Linear Analysis: A Systematic Presentation in Problem Form , 1974 .

[61]  A. Tikhonov,et al.  Numerical Methods for the Solution of Ill-Posed Problems , 1995 .

[62]  B. Parlett.  The Symmetric Eigenvalue Problem , 1981 .

[63]  M. H. Wright,et al.  FORTRAN Subroutines to Solve the Linear Least-Squares Problem and Compute the Complete Orthogonal Factorization. , 1978 .

[64]  H. Simon.  The Lanczos algorithm with partial reorthogonalization , 1984 .

[65]  P. Halmos.  Finite-Dimensional Vector Spaces , 1960 .

[66]  Carl D. Meyer,et al.  Matrix Analysis and Applied Linear Algebra , 2000 .

[67]  D. Day.  How the QR algorithm fails to converge and how to fix it , 1996 .

[68]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[69]  C. Kelley.  Iterative Methods for Linear and Nonlinear Equations , 1987 .

[70]  F. Gantmacher,et al.  Applications of the theory of matrices , 1960 .

[71]  M. Saunders,et al.  Solution of Sparse Indefinite Systems of Linear Equations , 1975 .