Computational methods of linear algebra

The authors' survey paper is devoted to the present state of computational methods in linear algebra. Questions discussed are the means and methods of estimating the quality of numerical solution of computational problems, the generalized inverse of a matrix, the solution of systems with rectangular and poorly conditioned matrices, the inverse eigenvalue problem, and more traditional questions such as algebraic eigenvalue problems and the solution of systems with a square matrix (by direct and iterative methods).

[1]  S. Falk.  Klassifikation gedämpfter Schwingungssysteme und Eingrenzung ihrer Eigenwerte , 1960 .

[2]  H. Keller.  Special Block Iterations with Applications to Laplace and Biharmonic Difference Equations , 1960 .

[3]  R. Duffin.  The Rayleigh-Ritz method for dissipative or gyroscopic systems , 1960 .

[4]  L. Mirsky.  SYMMETRIC GAUGE FUNCTIONS AND UNITARILY INVARIANT NORMS , 1960 .

[5]  James Hardy Wilkinson,et al.  Rounding errors in algebraic processes , 1964, IFIP Congress.

[6]  E. Stiefel,et al.  Über diskrete und lineare Tschebyscheff-Approximationen , 1959, Numerische Mathematik.

[7]  H. Rutishauser.  Deflation by elementary rotations for the solution of algebraic eigenvalue problems , 1959, ACM '59.

[8]  Robert L. Ashenhurst,et al.  Unnormalized Floating Point Arithmetic , 1959, JACM.

[9]  J. Roth.  An application of algebraic topology: Kron’s method of tearing , 1959 .

[10]  G. Golub.  The Use of Chebyshev Matrix Polynomials in The Iterative Solution of Linear Equations Compared to The Method of Successive Overrelaxation , 1959 .

[11]  M. Pearl.  On normal and $EP_r$ matrices. , 1959 .

[12]  J. Schröder.  Fehlerabschätzung bei linearen Gleichungssystemen mit dem Brouwerschen Fixpunktsatz , 1959 .

[13]  E. Stiefel,et al.  Refined Iterative Methods for Computation of the Solution and the Eigenvalues of Self-Adjoint Boundary Value Problems , 1959 .

[14]  K. G. Guderley.  On Nonlinear Eiegenvalue Problems for Matrices , 1958 .

[15]  Robert L. Ashenhurst,et al.  Significant Digit Computer Arithmetic , 1958, IRE Trans. Electron. Comput..

[16]  Elliott Ward Cheney,et al.  A finite algorithm for the solution of consistent linear equations and inequalities and for the Tchebycheff approximation of inconsistent linear equations. , 1958 .

[17]  A. Bjerhammar,et al.  A generalized matrix algebra , 1958 .

[18]  H. Markowitz.  The Elimination form of the Inverse and its Application to Linear Programming , 1957 .

[19]  P. Wynn,et al.  On a Device for Computing the e m (S n ) Transformation , 1956 .

[20]  R. Penrose.  A Generalized inverse for matrices , 1955 .

[21]  R. Duffin.  A Minimax Theory for Overdamped Networks , 1955 .

[22]  A. Hoffman,et al.  Some metric inequalities in the space of matrices , 1955 .

[23]  G. Azumaya.  STRONGLY $\pi$-REGULAR RINGS , 1954 .

[24]  E. Lukács,et al.  Tables of inverses of finite segments of the Hilbert matrix , 1953 .

[25]  M. Hestenes,et al.  Solutions of Ax = gamma Bx1 , 1951 .

[26]  Lothar Collatz,et al.  Eigenwertaufgaben mit technischen Anwendungen , 1949 .

[27]  L. Kantorovitch.  The method of successive approximation for functional equations , 1939 .

[28]  G. Schulz.  Iterative Berechung der reziproken Matrix , 1933 .