
NAMEzggevx.f SYNOPSISFunctions/Subroutinessubroutine zggevx (BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, ILO, IHI, LSCALE, RSCALE, ABNRM, BBNRM, RCONDE, RCONDV, WORK, LWORK, RWORK, IWORK, BWORK, INFO) Function/Subroutine Documentationsubroutine zggevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN, complex*16, dimension( lda, * )A, integerLDA, complex*16, dimension( ldb, * )B, integerLDB, complex*16, dimension( * )ALPHA, complex*16, dimension( * )BETA, complex*16, dimension( ldvl, * )VL, integerLDVL, complex*16, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, double precision, dimension( * )LSCALE, double precision, dimension( * )RSCALE, double precisionABNRM, double precisionBBNRM, double precision, dimension( * )RCONDE, double precision, dimension( * )RCONDV, complex*16, dimension( * )WORK, integerLWORK, double precision, dimension( * )RWORK, integer, dimension( * )IWORK, logical, dimension( * )BWORK, integerINFO)ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices Purpose:ZGGEVX computes for a pair of NbyN complex nonsymmetric matrices (A,B) the generalized eigenvalues, and optionally, the left and/or right generalized eigenvectors. Optionally, it also computes a balancing transformation to improve the conditioning of the eigenvalues and eigenvectors (ILO, IHI, LSCALE, RSCALE, ABNRM, and BBNRM), reciprocal condition numbers for the eigenvalues (RCONDE), and reciprocal condition numbers for the right eigenvectors (RCONDV). A generalized eigenvalue for a pair of matrices (A,B) is a scalar lambda or a ratio alpha/beta = lambda, such that A  lambda*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for beta=0, and even for both being zero. The right eigenvector v(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies A * v(j) = lambda(j) * B * v(j) . The left eigenvector u(j) corresponding to the eigenvalue lambda(j) of (A,B) satisfies u(j)**H * A = lambda(j) * u(j)**H * B. where u(j)**H is the conjugatetranspose of u(j). BALANC
Author:
BALANC is CHARACTER*1 Specifies the balance option to be performed: = 'N': do not diagonally scale or permute; = 'P': permute only; = 'S': scale only; = 'B': both permute and scale. Computed reciprocal condition numbers will be for the matrices after permuting and/or balancing. Permuting does not change condition numbers (in exact arithmetic), but balancing does.JOBVL JOBVL is CHARACTER*1 = 'N': do not compute the left generalized eigenvectors; = 'V': compute the left generalized eigenvectors.JOBVR JOBVR is CHARACTER*1 = 'N': do not compute the right generalized eigenvectors; = 'V': compute the right generalized eigenvectors.SENSE SENSE is CHARACTER*1 Determines which reciprocal condition numbers are computed. = 'N': none are computed; = 'E': computed for eigenvalues only; = 'V': computed for eigenvectors only; = 'B': computed for eigenvalues and eigenvectors.N N is INTEGER The order of the matrices A, B, VL, and VR. N >= 0.A A is COMPLEX*16 array, dimension (LDA, N) On entry, the matrix A in the pair (A,B). On exit, A has been overwritten. If JOBVL='V' or JOBVR='V' or both, then A contains the first part of the complex Schur form of the "balanced" versions of the input A and B.LDA LDA is INTEGER The leading dimension of A. LDA >= max(1,N).B B is COMPLEX*16 array, dimension (LDB, N) On entry, the matrix B in the pair (A,B). On exit, B has been overwritten. If JOBVL='V' or JOBVR='V' or both, then B contains the second part of the complex Schur form of the "balanced" versions of the input A and B.LDB LDB is INTEGER The leading dimension of B. LDB >= max(1,N).ALPHA ALPHA is COMPLEX*16 array, dimension (N)BETA BETA is COMPLEX*16 array, dimension (N) On exit, ALPHA(j)/BETA(j), j=1,...,N, will be the generalized eigenvalues. Note: the quotient ALPHA(j)/BETA(j) ) may easily over or underflow, and BETA(j) may even be zero. Thus, the user should avoid naively computing the ratio ALPHA/BETA. However, ALPHA will be always less than and usually comparable with norm(A) in magnitude, and BETA always less than and usually comparable with norm(B).VL VL is COMPLEX*16 array, dimension (LDVL,N) If JOBVL = 'V', the left generalized eigenvectors u(j) are stored one after another in the columns of VL, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVL = 'N'.LDVL LDVL is INTEGER The leading dimension of the matrix VL. LDVL >= 1, and if JOBVL = 'V', LDVL >= N.VR VR is COMPLEX*16 array, dimension (LDVR,N) If JOBVR = 'V', the right generalized eigenvectors v(j) are stored one after another in the columns of VR, in the same order as their eigenvalues. Each eigenvector will be scaled so the largest component will have abs(real part) + abs(imag. part) = 1. Not referenced if JOBVR = 'N'.LDVR LDVR is INTEGER The leading dimension of the matrix VR. LDVR >= 1, and if JOBVR = 'V', LDVR >= N.ILO ILO is INTEGERIHI IHI is INTEGER ILO and IHI are integer values such that on exit A(i,j) = 0 and B(i,j) = 0 if i > j and j = 1,...,ILO1 or i = IHI+1,...,N. If BALANC = 'N' or 'S', ILO = 1 and IHI = N.LSCALE LSCALE is DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the left side of A and B. If PL(j) is the index of the row interchanged with row j, and DL(j) is the scaling factor applied to row j, then LSCALE(j) = PL(j) for j = 1,...,ILO1 = DL(j) for j = ILO,...,IHI = PL(j) for j = IHI+1,...,N. The order in which the interchanges are made is N to IHI+1, then 1 to ILO1.RSCALE RSCALE is DOUBLE PRECISION array, dimension (N) Details of the permutations and scaling factors applied to the right side of A and B. If PR(j) is the index of the column interchanged with column j, and DR(j) is the scaling factor applied to column j, then RSCALE(j) = PR(j) for j = 1,...,ILO1 = DR(j) for j = ILO,...,IHI = PR(j) for j = IHI+1,...,N The order in which the interchanges are made is N to IHI+1, then 1 to ILO1.ABNRM ABNRM is DOUBLE PRECISION The onenorm of the balanced matrix A.BBNRM BBNRM is DOUBLE PRECISION The onenorm of the balanced matrix B.RCONDE RCONDE is DOUBLE PRECISION array, dimension (N) If SENSE = 'E' or 'B', the reciprocal condition numbers of the eigenvalues, stored in consecutive elements of the array. If SENSE = 'N' or 'V', RCONDE is not referenced.RCONDV RCONDV is DOUBLE PRECISION array, dimension (N) If JOB = 'V' or 'B', the estimated reciprocal condition numbers of the eigenvectors, stored in consecutive elements of the array. If the eigenvalues cannot be reordered to compute RCONDV(j), RCONDV(j) is set to 0; this can only occur when the true value would be very small anyway. If SENSE = 'N' or 'E', RCONDV is not referenced.WORK WORK is COMPLEX*16 array, dimension (MAX(1,LWORK)) On exit, if INFO = 0, WORK(1) returns the optimal LWORK.LWORK LWORK is INTEGER The dimension of the array WORK. LWORK >= max(1,2*N). If SENSE = 'E', LWORK >= max(1,4*N). If SENSE = 'V' or 'B', LWORK >= max(1,2*N*N+2*N). If LWORK = 1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.RWORK RWORK is DOUBLE PRECISION array, dimension (lrwork) lrwork must be at least max(1,6*N) if BALANC = 'S' or 'B', and at least max(1,2*N) otherwise. Real workspace.IWORK IWORK is INTEGER array, dimension (N+2) If SENSE = 'E', IWORK is not referenced.BWORK BWORK is LOGICAL array, dimension (N) If SENSE = 'N', BWORK is not referenced.INFO INFO is INTEGER = 0: successful exit < 0: if INFO = i, the ith argument had an illegal value. = 1,...,N: The QZ iteration failed. No eigenvectors have been calculated, but ALPHA(j) and BETA(j) should be correct for j=INFO+1,...,N. > N: =N+1: other than QZ iteration failed in ZHGEQZ. =N+2: error return from ZTGEVC. Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
April 2012
Further Details:
Balancing a matrix pair (A,B) includes, first, permuting rows and columns to isolate eigenvalues, second, applying diagonal similarity transformation to the rows and columns to make the rows and columns as close in norm as possible. The computed reciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not change the condition numbers (in exact arithmetic) but diagonal scaling will. For further explanation of balancing, see section 4.11.1.2 of LAPACK Users' Guide. An approximate error bound on the chordal distance between the ith computed generalized eigenvalue w and the corresponding exact eigenvalue lambda is chord(w, lambda) <= EPS * norm(ABNRM, BBNRM) / RCONDE(I) An approximate error bound for the angle between the ith computed eigenvector VL(i) or VR(i) is given by EPS * norm(ABNRM, BBNRM) / DIF(i). For further explanation of the reciprocal condition numbers RCONDE and RCONDV, see section 4.11 of LAPACK User's Guide. AuthorGenerated automatically by Doxygen for LAPACK from the source code.
Visit the GSP FreeBSD Man Page Interface. 