TY - JOUR

T1 - Generation of test matrices with specified eigenvalues using floating-point arithmetic

AU - Ozaki, Katsuhisa

AU - Ogita, Takeshi

N1 - Funding Information:
This research is partially supported by JST / CREST and JSPS Kakenhi (20H04195). We sincerely express our thanks to the reviewer for the fruitful comments and Dr. Takeshi Terao for his kind advice of simplified codes of MATLAB.
Publisher Copyright:
© 2021, The Author(s).

PY - 2022/5

Y1 - 2022/5

N2 - This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by S′ to compute YS′X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.

AB - This paper concerns test matrices for numerical linear algebra using an error-free transformation of floating-point arithmetic. For specified eigenvalues given by a user, we propose methods of generating a matrix whose eigenvalues are exactly known based on, for example, Schur or Jordan normal form and a block diagonal form. It is also possible to produce a real matrix with specified complex eigenvalues. Such test matrices with exactly known eigenvalues are useful for numerical algorithms in checking the accuracy of computed results. In particular, exact errors of eigenvalues can be monitored. To generate test matrices, we first propose an error-free transformation for the product of three matrices YSX. We approximate S by S′ to compute YS′X without a rounding error. Next, the error-free transformation is applied to the generation of test matrices with exactly known eigenvalues. Note that the exactly known eigenvalues of the constructed matrix may differ from the anticipated given eigenvalues. Finally, numerical examples are introduced in checking the accuracy of numerical computations for symmetric and unsymmetric eigenvalue problems.

KW - Eigenvalue problems

KW - Floating-point arithmetic

KW - Numerical linear algebra

KW - Test matrices

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U2 - 10.1007/s11075-021-01186-7

DO - 10.1007/s11075-021-01186-7

M3 - Article

AN - SCOPUS:85115063252

SN - 1017-1398

VL - 90

SP - 241

EP - 262

JO - Numerical Algorithms

JF - Numerical Algorithms

IS - 1

ER -