Journal ArticleDOI
On the Numerical Solution of Ill-Conditioned Linear Systems with Applications to Ill-Posed Problems
TLDR
In this paper, the singular value decomposition (SVDC) is used to solve ill-conditioned linear systems using the singular values decomposition. But the SVDC can improve the accuracy of the computed solution for certain kinds of right-hand sides.Abstract:
We consider the solution of ill-conditioned linear systems using the singular value decomposition, and show how this can improve the accuracy of the computed solution for certain kinds of right-hand sides Then we indicate how this technique is especially appropriate for some classical ill-posed problems of mathematical physicsread more
Citations
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Journal ArticleDOI
Generalized Cross-Validation as a Method for Choosing a Good Ridge Parameter
TL;DR: The generalized cross-validation (GCV) method as discussed by the authors is a generalized version of Allen's PRESS, which can be used in subset selection and singular value truncation, and even to choose from among mixtures of these methods.
Journal ArticleDOI
REGULARIZATION TOOLS: A Matlab package for analysis and solution of discrete ill-posed problems
TL;DR: The package REGULARIZATION TOOLS consists of 54 Matlab routines for analysis and solution of discrete ill-posed problems, i.e., systems of linear equations whose coefficient matrix has the properties that its condition number is very large, and its singular values decay gradually to zero.
Journal ArticleDOI
Tikhonov regularization and total least squares
TL;DR: It is shown how Tikhonov's regularization method can be recast in a total least squares formulation suited for problems in which both the coefficient matrix and the right-hand side are known only approximately.
Journal ArticleDOI
The truncated SVD as a method for regularization
TL;DR: In this article, the truncated singular value decomposition (SVD) is considered as a method for regularization of ill-posed linear least squares problems and compared with the usual regularized solution.
Journal ArticleDOI
Computational methods of linear algebra
D. K. Faddeev,V. N. Faddeeva +1 more
TL;DR: A survey of computational methods in linear algebra can be found in this article, where the authors discuss 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, and more traditional questions such as algebraic eigenvalue problems and systems with a square matrix.
References
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Book
The Theory of Matrices
TL;DR: In this article, the Routh-Hurwitz problem of singular pencils of matrices has been studied in the context of systems of linear differential equations with variable coefficients, and its applications to the analysis of complex matrices have been discussed.
Book
The algebraic eigenvalue problem
TL;DR: Theoretical background Perturbation theory Error analysis Solution of linear algebraic equations Hermitian matrices Reduction of a general matrix to condensed form Eigenvalues of matrices of condensed forms The LR and QR algorithms Iterative methods Bibliography.
Journal ArticleDOI
Singular value decomposition and least squares solutions
Gene H. Golub,C. Reinsch +1 more
TL;DR: The decomposition of A is called the singular value decomposition (SVD) and the diagonal elements of ∑ are the non-negative square roots of the eigenvalues of A T A; they are called singular values.
Journal ArticleDOI
A Technique for the Numerical Solution of Certain Integral Equations of the First Kind
TL;DR: Here the authors will consider only nonsingular linear integral equations of the first kind, where the known functions h(x), K(x, y) and g(x) are assumed to be bounded and usually to be continuous.
Journal ArticleDOI
Calculating the Singular Values and Pseudo-Inverse of a Matrix
Gene H. Golub,William Kahan +1 more
TL;DR: In this article, a numerically stable and fairly fast scheme is described to compute the unitary matrices U and V which transform a given matrix A into a diagonal form π = U^ * AV, thus exhibiting A's singular values on π's diagonal.