Today’s Computational Methods of Linear Algebra

Abstract : This is a survey of selected computational aspects of linear algebra, addressed to the nonspecialist in numerical analysis. Some current methods of solving systems of linear equations, and computing eigenvalues of symmetric and unsymmetric matrices are outlined. A bibliography containing 62 titles is included.

[1]  J. Neumann,et al.  Numerical inverting of matrices of high order , 1947 .

[2]  A. Turing.  ROUNDING-OFF ERRORS IN MATRIX PROCESSES , 1948 .

[3]  C. Lanczos.  An iteration method for the solution of the eigenvalue problem of linear differential and integral operators , 1950 .

[4]  J. Todd,et al.  The condition of a certain matrix , 1950, Mathematical Proceedings of the Cambridge Philosophical Society.

[5]  J. Neumann,et al.  Numerical inverting of matrices of high order. II , 1951 .

[6]  M. Hestenes,et al.  Methods of conjugate gradients for solving linear systems , 1952 .

[7]  George E. Forsythe,et al.  Solving linear algebraic equations can be interesting , 1953 .

[8]  W. Givens.  Numerical Computation of the Characteristic Values of a Real Symmetric Matrix , 1954 .

[9]  J. Greenstadt.  A method for finding roots of arbitrary matrices , 1955 .

[10]  M. Lotkin.  Characteristic values of arbitrary matrices , 1956, ACM '56.

[11]  David A. Pope,et al.  Maximizing Functions of Rotations—Experiments Concerning Speed of Diagonalization of Symmetric Matrices Using Jacobi's Method , 1957, JACM.

[12]  P. Henrici.  On the Speed of Convergence of Cyclic and Quasicyclic Jacobi Methods for Computing Eigenvalues of Hermitian Matrices , 1958 .

[13]  Friedrich L. Bauer,et al.  On certain methods for expanding the characteristic polynomial , 1959, Numerische Mathematik.

[14]  G. Forsythe,et al.  The cyclic Jacobi method for computing the principal values of a complex matrix , 1960 .

[15]  Richard Bellman,et al.  Introduction to Matrix Analysis , 1972 .

[16]  George E. Forsythe,et al.  Finite-Difference Methods for Partial Differential Equations , 1961 .

[17]  D. Faddeev,et al.  Computational methods of linear algebra , 1959 .

[18]  James Hardy Wilkinson,et al.  Error Analysis of Direct Methods of Matrix Inversion , 1961, JACM.

[19]  J. G. F. Francis,et al.  The QR Transformation A Unitary Analogue to the LR Transformation - Part 1 , 1961, Comput. J..

[20]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[21]  W. Wasow,et al.  Finite-Difference Methods for Partial Differential Equations , 1961 .

[22]  P. J. Ebertein,et al.  A Jacobi-Like Method for the Automatic Computation of Eigenvalues and Eigenvectors of an Arbitrary Matrix , 1962 .

[23]  James M. Ortega,et al.  The LLT and QR methods for symmetric tridiagonal matrices , 1963, Comput. J..

[24]  Eldon R. Hansen,et al.  On Cyclic Jacobi Methods , 1963 .

[25]  Alston S. Householder,et al.  The Theory of Matrices in Numerical Analysis , 1964 .

[26]  L. Fox.  An introduction to numerical linear algebra , 1964 .

[27]  George B. Dantzig,et al.  Linear programming and extensions , 1965 .

[28]  B. Parlett.  Laguerre's Method Applied to the Matrix Eigenvalue Problem , 1964 .

[29]  Arnold Schönhage,et al.  Zur quadratischen Konvergenz des Jacobi-Verfahrens , 1964 .

[30]  V. V. Klyuev,et al.  Minimization of the number of arithmetic operations in the solution of linear algebraic systems of equations , 1965 .

[31]  W. Kahan.  Numerical Linear Algebra , 1966, Canadian Mathematical Bulletin.

[32]  William Kahan,et al.  Two working algorithms for the eigenvalues of a symmetric tridiagonal matrix , 1966 .

[33]  Cleve B. Moler,et al.  Iterative Refinement in Floating Point , 1967, JACM.

[34]  C. Moler,et al.  APPROXIMATIONS AND BOUNDS FOR EIGENVALUES OF ELLIPTIC OPERATORS , 1967 .