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Comments on “EIGN1M: A matrix diagonalization subroutine with minimal storage requirements”
Cmpwem B Cknimy Vol. 2. p. 153 Pergamm Press Ltd. 1578. Printed in Coral Britaie
LETTERS COMMENTS
TO THE EDITOR
ON “EIGNlM: A MATRIX DIAGONALIZATION WITH MINIMAL STORAGE REQUIREMENTS”
The communication by Paul A. Dobosh, “EIGNIM: A matrix diagonalization subroutine with minimal storage requirements”, Comput. CIurm. l(4) (1977) contains some misleading statements which should be corrected. EIGNlh4 is based on the Householder QR method of Dr. J. H. Wilkinson. Whereas Dobosh states “Available computer subroutines implementing this method require storage space for the input matrix... and the output eigenvector matrix,” several routines usiug the same method and requiring only one N by N array have been widely available for over five years. (IV is the Order of the matrix for which eigensolutions are to be found). Wilkinson (197 1) points out that the ALGOL procedures tred 2 .and tq12 (pp. 212-240) can be called so that the eigenvectors overwrite the original matrix (p. 213). The Numerical Afgorithms Group Library has incorporated these routines as the ALGOL/FORTRAN programs FU2ABNF since 1971. Very similar programs have heen available in EJSPACK since at least 1973 when the *‘User’s Information” [later published by Springer, (Smith et al. 1974)) from the Argonne National Laboratory &ted, in Section 2.1.11, that the storage requirement could be reduced to N* + 2N words of array storage, the same as required by F02ABF. Readers may wish tc emulate the very high standard of FQRTRAN coding found in EJSPACK. EIGNIM, on the other hand, apparently requires N*+ 1ON words if EQUIVALENCE is ignored and N2 +5 N if it is in effect. This is quite far from the stated goal of “minimum storage requirements”. Storage requirements should, in any case, take into account code length. A one-sided transformation methodf2.31 probably
offers the smallest overall storage and code if the segments are chained, while the very short code of s Jacobi method, requiring 2N* words of storage, often causes the latter to be the most compact approach when chaining is not possible. The Jacobi method, by the way, is the only “diagonalization” among the methods mentioned here for the algebraic eigenproblem. This work has yet to be published, but tests run on a vsriety of problems and computing systems suggest that the HouseIrolderlQL based programs are 4-6 times faster than the Jacobi based ones. Agriculfure Canada J. C. NASH Policy and Economics Branch Research Division Sir John Catfing Building Gtfdwa Ontorio, K 1A OC5 Canada -cFs Nash, 1. C., (197% Cornput. J! I8 (I), 7476. Nash J. C., Compact Numerical Methods for Comprrters: Linear Algebm andFunction MinimizattOn, Adam Hilger Bristol. To be published. Smith, 3. T., Boyle, 1. M., Garbow, B. S., Ikebe, Y., Klema, V. C. & Moler, C. 8. (1974), L..xt~re Notes in Compufer Science, Eigensystun Routines-EISPACK Guide Vol. 6, Marti Springer, New York. Wilkinson I. H. & Reinsch C. (1971), Linear Algebra, Vol. 2, Handbook for Automatic Compufution, Springer, Berlin.
REJOINDER h. Nash is entirely correct. After procrastinating for years and martment
of Chemistry Mount Hofyoke Collage South Had&y MA 01075 U.S.A.
finally deciding EIGNlM was wad publishing, it was embarrassing Lo discover the EISPACK routines just as my article was being published. The EISPACK mutines are indeed superb. While I would not quibble over 3N, the EISPACK routines are reported to be faster, they are well-commented, and various combinations of routines can handle Hermitian matrices as well as the general eigenvalue problem for real matrices, (FE. S)C = 0. Anyone running a molecular orbital problem should consider these routines immediately.
SUBROUTINE
PAULA. J3o8osH
Editor’s note: Dr. Dobosh was not alone in overlooking the value of EISPACK for treating these problems.
153
LETTERS COMMENTS
TO THE EDITOR
ON “EIGNlM: A MATRIX DIAGONALIZATION WITH MINIMAL STORAGE REQUIREMENTS”
The communication by Paul A. Dobosh, “EIGNIM: A matrix diagonalization subroutine with minimal storage requirements”, Comput. CIurm. l(4) (1977) contains some misleading statements which should be corrected. EIGNlh4 is based on the Householder QR method of Dr. J. H. Wilkinson. Whereas Dobosh states “Available computer subroutines implementing this method require storage space for the input matrix... and the output eigenvector matrix,” several routines usiug the same method and requiring only one N by N array have been widely available for over five years. (IV is the Order of the matrix for which eigensolutions are to be found). Wilkinson (197 1) points out that the ALGOL procedures tred 2 .and tq12 (pp. 212-240) can be called so that the eigenvectors overwrite the original matrix (p. 213). The Numerical Afgorithms Group Library has incorporated these routines as the ALGOL/FORTRAN programs FU2ABNF since 1971. Very similar programs have heen available in EJSPACK since at least 1973 when the *‘User’s Information” [later published by Springer, (Smith et al. 1974)) from the Argonne National Laboratory &ted, in Section 2.1.11, that the storage requirement could be reduced to N* + 2N words of array storage, the same as required by F02ABF. Readers may wish tc emulate the very high standard of FQRTRAN coding found in EJSPACK. EIGNIM, on the other hand, apparently requires N*+ 1ON words if EQUIVALENCE is ignored and N2 +5 N if it is in effect. This is quite far from the stated goal of “minimum storage requirements”. Storage requirements should, in any case, take into account code length. A one-sided transformation methodf2.31 probably
offers the smallest overall storage and code if the segments are chained, while the very short code of s Jacobi method, requiring 2N* words of storage, often causes the latter to be the most compact approach when chaining is not possible. The Jacobi method, by the way, is the only “diagonalization” among the methods mentioned here for the algebraic eigenproblem. This work has yet to be published, but tests run on a vsriety of problems and computing systems suggest that the HouseIrolderlQL based programs are 4-6 times faster than the Jacobi based ones. Agriculfure Canada J. C. NASH Policy and Economics Branch Research Division Sir John Catfing Building Gtfdwa Ontorio, K 1A OC5 Canada -cFs Nash, 1. C., (197% Cornput. J! I8 (I), 7476. Nash J. C., Compact Numerical Methods for Comprrters: Linear Algebm andFunction MinimizattOn, Adam Hilger Bristol. To be published. Smith, 3. T., Boyle, 1. M., Garbow, B. S., Ikebe, Y., Klema, V. C. & Moler, C. 8. (1974), L..xt~re Notes in Compufer Science, Eigensystun Routines-EISPACK Guide Vol. 6, Marti Springer, New York. Wilkinson I. H. & Reinsch C. (1971), Linear Algebra, Vol. 2, Handbook for Automatic Compufution, Springer, Berlin.
REJOINDER h. Nash is entirely correct. After procrastinating for years and martment
of Chemistry Mount Hofyoke Collage South Had&y MA 01075 U.S.A.
finally deciding EIGNlM was wad publishing, it was embarrassing Lo discover the EISPACK routines just as my article was being published. The EISPACK mutines are indeed superb. While I would not quibble over 3N, the EISPACK routines are reported to be faster, they are well-commented, and various combinations of routines can handle Hermitian matrices as well as the general eigenvalue problem for real matrices, (FE. S)C = 0. Anyone running a molecular orbital problem should consider these routines immediately.
SUBROUTINE
PAULA. J3o8osH
Editor’s note: Dr. Dobosh was not alone in overlooking the value of EISPACK for treating these problems.
153