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cggevx-f(3)

cggevx.f –

Synopsis

Functions/Subroutines

subroutine cggevx (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)
CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Function/Subroutine Documentation

subroutine cggevx (characterBALANC, characterJOBVL, characterJOBVR, characterSENSE, integerN, complex, dimension( lda, * )A, integerLDA, complex, dimension( ldb, * )B, integerLDB, complex, dimension( * )ALPHA, complex, dimension( * )BETA, complex, dimension( ldvl, * )VL, integerLDVL, complex, dimension( ldvr, * )VR, integerLDVR, integerILO, integerIHI, real, dimension( * )LSCALE, real, dimension( * )RSCALE, realABNRM, realBBNRM, real, dimension( * )RCONDE, real, dimension( * )RCONDV, complex, dimension( * )WORK, integerLWORK, real, dimension( * )RWORK, integer, dimension( * )IWORK, logical, dimension( * )BWORK, integerINFO)

CGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Purpose:

 CGGEVX computes for a pair of N-by-N 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 conjugate-transpose of u(j).

 

Parameters:

BALANC

 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 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 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 array, dimension (N)

BETA

 BETA is COMPLEX 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 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 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 INTEGER

IHI

 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,...,ILO-1 or i = IHI+1,...,N.
 If BALANC = 'N' or 'S', ILO = 1 and IHI = N.

LSCALE

 LSCALE is REAL 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,...,ILO-1
 = 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 ILO-1.

RSCALE

 RSCALE is REAL 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,...,ILO-1
 = 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 ILO-1.

ABNRM

 ABNRM is REAL
 The one-norm of the balanced matrix A.

BBNRM

 BBNRM is REAL
 The one-norm of the balanced matrix B.

RCONDE

 RCONDE is REAL 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 REAL array, dimension (N)
 If SENSE = '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 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 REAL 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 i-th 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 CHGEQZ.
 =N+2: error return from CTGEVC.

 

Author:

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 i-th
 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 i-th 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.

 

Definition at line 372 of file cggevx.f.

Author

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