cggevx.f (3) Linux Manual Page

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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