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Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems. (English) Zbl 0789.65029

The underlying theory on regularization methods for analysis and solution of discrete ill-posed problems is dealt with and an overview of the package of 54 Matlab routines for this analysis is presented. Such problems involve arbitrarily large perturbations caused by small perturbations. A Fredholm integral equation of the first kind is an example.

Systems of linear equations and linear least squares problems ${min}_{x}{\parallel Ax-b\parallel }_{2}$, $A\in {ℝ}^{m×n}$, $m>n$ arising from discretization of ill-posed problems (i.e. the singular values of $A$ decay gradually to zero and the ratio between the largest and the smallest nonzero singular value is large) are analyzed. The purpose of the regularization is to stabilize the problem and to single out a stable solution.

An analysis of the singular value decomposition (SVD) and the generalized SVD, the Picard condition, the discrete Picard condition, filter factors, a graphical analysis by the $L$-curve, the transformation of regularization problems by direct and iterative methods to standard form, methods for choosing the regularization parameter $\lambda$ are discussed. 54 regularization routines are given and characterized. A complete manual of these routines can be obtained.

Reviewer: V.Burjan (Praha)

##### MSC:
 65F20 Overdetermined systems, pseudoinverses (numerical linear algebra) 65R30 Improperly posed problems (integral equations, numerical methods) 65F15 Eigenvalues, eigenvectors (numerical linear algebra) 65R20 Integral equations (numerical methods) 65F30 Other matrix algorithms 45B05 Fredholm integral equations
##### Software:
LSQR; Matlab; Regularization tools