# Matrix Eigensystem Routines — EISPACK Guide Extension by Burton S. Garbow, J. M. Boyle, J. J. Dongarra, C. B. Moler

By Burton S. Garbow, J. M. Boyle, J. J. Dongarra, C. B. Moler (auth.)

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Additional info for Matrix Eigensystem Routines — EISPACK Guide Extension

Sample text

The algorithm used in EISPACK determines two unitary matrices Q and Z such that QAZ and QBZ are both upper triangular. j and Bj are simply the diagonal elements of the two triangular matrices, and it is these quantities which are returned by the subroutine. /B.. JJ there is more in the individual ~. and B. than there is in the ratios ] 3 The errors in aj and \$. are all about the same absolute size, J namely the size of the errors in the input data (or of roundoff errors in the computation). 3-3 the same size as the original matrix elements, determined by the data and the computation.

1-16 The array storage required to execute this path is 3N 2 + 3N working precision words. However, for this particular path the eigenvectors can overwrite the A matrix if the same array parameter is used for both A and Z, thereby reducing the storage required to 2N 2 + 3N working precision words. 0 seconds for sample systems of order I0, 20, 40, and 80 respectively. 3 of [I0]. 2 of this volume. 1~6 ALL EIGENVALUES OF A GENEP~IIZED REAL SYMMETRIC MATRIX SYSTEM The real generalized eigenproblem Az = hBz is further characterized as symmetric if both A and B are symmetric and B is positive definite (all positive eigenvalues).

NE. 0) GO TO 99999 CALL TRBAKI(NM,N,A,fv2,M,Z) CALL REBAK(NM,N,B,fv9,M,Z) or, using EISPAC: CALL EISPAC(NM,N,MATA('REAL',A,'SYMMETRIC'),MATB('REAL',B,'SYMMETRIC', 'POSITIVE DEFINITE'),VALUES(W,MM,M,RLB,RUB),VECTOR(Z)) The parameter computation. EPSI is used to control the accuracy of the eigenvalue Setting it to zero or calling EISPAC without supplying causes the use of a default value suitable for most matrices. 3 and in the BISECT and EISPAC documents. 1-20 number of eigenvalues determined to lie in the interval defined by RLB and RUB and, provided M !