This paper is a comprehensive report on test matrices for the generalized inversion of matrices. Two principles are described how to construct singular square or arbitrary rectangular test matrices and their Moore-Penrose inverses. By prescribing the singular values of the matrices or by suitably choosing the free parameters test matrices with condition numbers of any size can be obtained. We also deal with test matrices which are equal to their Moore-Penrose inverse. In addition to many advices how to construct test matrices the paper presents many test matrices explicitly, in particular singular square matrices of ordern, sets of 7×6 and 7×5 matrices of different rank, a set of 5×5 matrices which are equal to their Moore-Penrose inverse and some special test matrices known from literature. For the set of 7×6 parameter matrices also the singular values corresponding to six values of the parameter are listed. For three simple parameter matrices of order 5×4 and 6×5 even test results obtained by eight different algorithms are quoted.

Report on test matrices for generalized inverses

It is well known that if Gaussian elimination with iterative refinement (IR) is used in the solution of systems of linear algebraic equations $Ax = b$ whose matrices are dense, then the accuracy of the results will usually be greater than the accuracy obtained by the use of Gaussian elimination without iterative refinement (DS). However, both more storage (about $100\% $, because a copy of matrix A is needed) and more computing time (some extra time is needed to perform the iterative process) must be used with IR. Normally, when the matrix is sparse the accuracy of the solution computed by some sparse matrix technique and IR will still be greater. In this paper it is verified (by many numerical experiments) that the use of sparse matrix techniques with IR may also result in a reduction of both the computing time and the storage requirements (this will never happen when IR is applied for dense matrices). Two parameters, a drop-tolerance $T \geqq 0$ and a stability factor $u > 1$, are introduced in the effo...

Use of Iterative Refinement in the Solution of Sparse Linear Systems

The experimental evidence indicates that the implementation of Newton's method in the numerical solution of systems of ordinary differential equations (ODE's) $y' = f(t,y)$, $y(a) = y_0 $, $t \in [a,b]$ by implicit computational schemes may cause difficulties. This is especially true if (i) $f(t,y)$ and/or $f'_y (t,y)$ are quickly varying in t and/or y and (ii) a low degree of accuracy is required. Such difficulties may also arise when diagonally implicit Runge–Kutta methods (DIRKM's) are used in the situation described by (i) and (ii). In this paper some modified DIRKM's (MDIRKM's) are derived. The use of MDIRKM's is an attempt to improve the performance of Newton's method in the case where f and $f'_y $ are quickly varying only in t. The stability properties of the MDIRKM's are studied. An error estimation technique for the new methods is proposed. Some numerical examples are presented.

Modified Diagonally Implicit Runge–Kutta Methods

Encapsulation, controlled release, and antitumor efficacy of cisplatin delivered in liposomes composed of sterol-modified phospholipids.

Stability Restrictions on Time-Stepsize for Numerical Integration of First-Order Partial Differential Equations*

The use of sparse matrix technique in the numerical integration of stiff systems of linear ordinary differential equations

A testing scheme for subroutines solving large linear problems

Use of a semiexplicit Runge-Kutta integration algorithm in a spectroscopic problem

Classification of the systems of ordinary differential equations and practical aspects in the numerical integration of large systems

Solution of ordinary differential equations with time dependent coefficients. Development of a semiexplicit Runge Kutta algorithm and application to a spectroscopic problem

Jerzy Wasniewski

Papers

The use of sparse matrix technique in the numerical integration of stiff systems of linear ordinary differential equations

A testing scheme for subroutines solving large linear problems

Use of a semiexplicit Runge-Kutta integration algorithm in a spectroscopic problem

Classification of the systems of ordinary differential equations and practical aspects in the numerical integration of large systems

Solution of ordinary differential equations with time dependent coefficients. Development of a semiexplicit Runge Kutta algorithm and application to a spectroscopic problem