Interval Gaussian Elimination with Pivot Tightening

We present a method by which the breakdown of the interval Gaussian elimination caused by division of an interval containing zero can be avoided for some classes of matrices. These include the inverse nonnegative matrices, the totally nonnegative matrices, and the inverse $M$-matrices—all classes with identically signed inverses. The approach consists of a tightening of the interval pivot by determining the exact range of the pivot over the matrix interval.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Ronald L. Smith,et al.  Intervals of Inverse M-Matrices , 2002, Reliab. Comput..

[3]  Juan Manuel Peña,et al.  Total positivity and Neville elimination , 1992 .

[4]  G. Alefeld,et al.  Introduction to Interval Computation , 1983 .

[5]  Charles R. Johnson,et al.  Almost principal minors of inverse M-matrices , 2001 .

[6]  R. Willoughby.  The inverse M-matrix problem , 1977 .

[7]  Jürgen Garloff,et al.  Vertex Implications for Totally Nonnegative Matrices , 1996 .

[8]  W. F. Trench.  An Algorithm for the Inversion of Finite Toeplitz Matrices , 1964 .

[9]  Jürgen Garloff,et al.  Criteria for sign regularity of sets of matrices , 1982 .

[10]  Jan Mayer.  An Approach to Overcome Division by Zero in the Interval Gauss Algorithm , 2002, Reliab. Comput..

[11]  A. Neumaier.  Interval methods for systems of equations , 1990 .

[12]  Jürgen Garloff,et al.  TOTALLY NONNEGATIVE INTERVAL MATRICES , 1980 .

[13]  Günter Mayer,et al.  A Contribution to the Feasibility of the Interval Gaussian Algorithm , 2006, Reliab. Comput..

[14]  R. B. Kearfott,et al.  A Comparison of some Methods for Solving Linear Interval Equations , 1997 .

[15]  Charles R. Johnson.  Inverse M-matrices☆ , 1982 .

[16]  Jürgen Garloff,et al.  Intervals of almost totally positive matrices , 2003 .

[17]  Shalhav Zohar,et al.  The Solution of a Toeplitz Set of Linear Equations , 1974, JACM.

[18]  E. Bareiss.  Numerical solution of linear equations with Toeplitz and Vector Toeplitz matrices , 1969 .

[19]  Götz Alefeld,et al.  The Cholesky method for interval data , 1993 .

[20]  J. Kuttler,et al.  A fourth-order finite-difference approximation for the fixed membrane eigenproblem , 1971 .

[21]  T. Andô.  Totally positive matrices , 1987 .

[22]  Kurt Metelmann.  Inverspositive Bandmatrizen und totalnichtnegative Green'sche Matrizen , 1972 .

[23]  F. R. Gantmakher.  The Theory of Matrices , 1984 .