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Current Inverse Iteration Software Can Fail
 BIT
, 1998
"... Inverse Iteration is widely used to compute the eigenvectors of a matrix once accurate eigenvalues are known. We discuss various issues involved in any implementation of inverse iteration for real, symmetric matrices. Current implementations resort to reorthogonalization when eigenvalues agree to mo ..."
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Cited by 13 (2 self)
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Inverse Iteration is widely used to compute the eigenvectors of a matrix once accurate eigenvalues are known. We discuss various issues involved in any implementation of inverse iteration for real, symmetric matrices. Current implementations resort to reorthogonalization when eigenvalues agree to more than three digits relative to the norm. Such reorthogonalization can have unexpected consequences. Indeed, as we show in this paper, the implementations in EISPACK [18] and LAPACK [1] may fail. We illustrate with both theoretical and empirical failures. Keywords : Inverse iteration, symmetric, tridiagonal matrix, eigenvalues, eigenvectors. AMS subject classification : 15A18, 65F15, 65F25. 1 Introduction Given an eigenvalue of the matrix A, a corresponding eigenvector is defined as a nonzero solution of the homogeneous system (A \Gamma I)v = 0: However in a computer implementation we can only expect, in general, to have an approximation oe to . In such a case, we may attempt to compu...
Reliable Computation of the Condition Number of a Tridiagonal Matrix in O(n) Time
, 1997
"... We present one more algorithm to compute the condition number (for inversion) of a n \Theta n tridiagonal matrix J in O(n) time. Previous O(n) algorithms for this task given by Higham in [17] are based on the tempting compact representation of the upper (lower) triangle of J \Gamma1 as the upper (lo ..."
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Cited by 13 (1 self)
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We present one more algorithm to compute the condition number (for inversion) of a n \Theta n tridiagonal matrix J in O(n) time. Previous O(n) algorithms for this task given by Higham in [17] are based on the tempting compact representation of the upper (lower) triangle of J \Gamma1 as the upper (lower) triangle of a rankone matrix. However, they suffer from severe overflow and underflow problems, especially on diagonally dominant matrices. Our new algorithm avoids these problems and is as efficient as the earlier algorithms. Keywords. Tridiagonal matrix, inverse, condition number, norm, overflow, underflow. AMS subject classifications. 15A12, 15A60, 65F35. 1 Introduction When solving a linear system Bx = r we are interested in knowing how accurate the solution is. This question is often answered by showing that the solution computed in finite precision is exact for a matrix "close" to B, and then measuring how sensitive the solution is to a small perturbation. The condition numb...
A TESTING INFRASTRUCTURE FOR LAPACK’S SYMMETRIC EIGENSOLVERS
"... LAPACK is often mentioned as a positive example of a software library that encapsulates complex, robust, and widely used numerical algorithms for a wide range of applications. At installation time, the user has the option of running a (limited) number of test cases to verify the integrity of the ins ..."
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Cited by 3 (3 self)
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LAPACK is often mentioned as a positive example of a software library that encapsulates complex, robust, and widely used numerical algorithms for a wide range of applications. At installation time, the user has the option of running a (limited) number of test cases to verify the integrity of the installation process. On the algorithm developer’s side, however, more exhaustive tests are usually performed to study algorithm behavior on a variety of problem settings and also computer architectures. In this process, difficult test cases need to be found that reflect particular challenges of an application or push algorithms to extreme behavior. These tests are then assembled into a comprehensive collection, therefore making it possible for any new or competing algorithm to be stressed in a similar way. This note describes such an infrastructure for exhaustively testing the symmetric tridiagonal eigensolvers implemented in LAPACK. It consists of two parts: a selection of carefully chosen test matrices with particular idiosyncrasies and a portable testing framework that allows easy testing and data processing. The tester facilitates experiments with algorithmic choices, parameter and threshold studies, and performance comparisons on different architectures.