Man page - ggev(3)

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Manual

ggev

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cggev (character jobvl, character jobvr, integer n, complex,dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,integer ldb, complex, dimension( * ) alpha, complex, dimension( * )beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex,dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work,integer lwork, real, dimension( * ) rwork, integer info)
subroutine dggev (character jobvl, character jobvr, integer n, doubleprecision, dimension( lda, * ) a, integer lda, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )alphar, double precision, dimension( * ) alphai, double precision,dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integerldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, doubleprecision, dimension( * ) work, integer lwork, integer info)
subroutine sggev (character jobvl, character jobvr, integer n, real,dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai,real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl,real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work,integer lwork, integer info)
subroutine zggev (character jobvl, character jobvr, integer n, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( *) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16,dimension( ldvr, * ) vr, integer ldvr, complex*16, dimension( * ) work,integer lwork, double precision, dimension( * ) rwork, integer info)
Author

NAME

ggev - ggev: eig, unblocked

SYNOPSIS

Functions

subroutine cggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
CGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
DGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sggev (jobvl, jobvr, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, work, lwork, info)
SGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zggev (jobvl, jobvr, n, a, lda, b, ldb, alpha, beta, vl, ldvl, vr, ldvr, work, lwork, rwork, info)
ZGGEV computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

Function Documentation

subroutine cggev (character jobvl, character jobvr, integer n, complex,dimension( lda, * ) a, integer lda, complex, dimension( ldb, * ) b,integer ldb, complex, dimension( * ) alpha, complex, dimension( * )beta, complex, dimension( ldvl, * ) vl, integer ldvl, complex,dimension( ldvr, * ) vr, integer ldvr, complex, dimension( * ) work,integer lwork, real, dimension( * ) rwork, integer info)

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

Purpose:

CGGEV 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.

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 generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues 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

JOBVL

JOBVL is CHARACTER*1
= ā€™Nā€™: do not compute the left generalized eigenvectors;
= ā€™Vā€™: compute the left generalized eigenvectors.

JOBVR

JOBVR is CHARACTER*1
= ā€™Nā€™: do not compute the right generalized eigenvectors;
= ā€™Vā€™: compute the right generalized 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.

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.

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 quotients 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 = ā€™Vā€™, 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 is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVL = ā€™Nā€™.

LDVL

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = ā€™Vā€™, LDVL >= N.

VR

VR is COMPLEX array, dimension (LDVR,N)
If JOBVR = ā€™Vā€™, 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 is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVR = ā€™Nā€™.

LDVR

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = ā€™Vā€™, LDVR >= N.

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).
For good performance, LWORK must generally be larger.

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 (8*N)

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 then 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.

subroutine dggev (character jobvl, character jobvr, integer n, doubleprecision, dimension( lda, * ) a, integer lda, double precision,dimension( ldb, * ) b, integer ldb, double precision, dimension( * )alphar, double precision, dimension( * ) alphai, double precision,dimension( * ) beta, double precision, dimension( ldvl, * ) vl, integerldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, doubleprecision, dimension( * ) work, integer lwork, integer info)

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

Purpose:

DGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right
generalized eigenvectors.

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

JOBVL

JOBVL is CHARACTER*1
= ā€™Nā€™: do not compute the left generalized eigenvectors;
= ā€™Vā€™: compute the left generalized eigenvectors.

JOBVR

JOBVR is CHARACTER*1
= ā€™Nā€™: do not compute the right generalized eigenvectors;
= ā€™Vā€™: compute the right generalized eigenvectors.

N

N is INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.

LDA

LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).

B

B is DOUBLE PRECISION array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.

LDB

LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).

ALPHAR

ALPHAR is DOUBLE PRECISION array, dimension (N)

ALPHAI

ALPHAI is DOUBLE PRECISION array, dimension (N)

BETA

BETA is DOUBLE PRECISION array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. If ALPHAI(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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 DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = ā€™Vā€™, the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVL = ā€™Nā€™.

LDVL

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = ā€™Vā€™, LDVL >= N.

VR

VR is DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = ā€™Vā€™, the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVR = ā€™Nā€™.

LDVR

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = ā€™Vā€™, LDVR >= N.

WORK

WORK is DOUBLE PRECISION 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(1,8*N).
For good performance, LWORK must generally be larger.

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.

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 ALPHAR(j), ALPHAI(j), and BETA(j)
should be correct for j=INFO+1,...,N.
> N: =N+1: other than QZ iteration failed in DHGEQZ.
=N+2: error return from DTGEVC.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sggev (character jobvl, character jobvr, integer n, real,dimension( lda, * ) a, integer lda, real, dimension( ldb, * ) b,integer ldb, real, dimension( * ) alphar, real, dimension( * ) alphai,real, dimension( * ) beta, real, dimension( ldvl, * ) vl, integer ldvl,real, dimension( ldvr, * ) vr, integer ldvr, real, dimension( * ) work,integer lwork, integer info)

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

Purpose:

SGGEV computes for a pair of N-by-N real nonsymmetric matrices (A,B)
the generalized eigenvalues, and optionally, the left and/or right
generalized eigenvectors.

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

JOBVL

JOBVL is CHARACTER*1
= ā€™Nā€™: do not compute the left generalized eigenvectors;
= ā€™Vā€™: compute the left generalized eigenvectors.

JOBVR

JOBVR is CHARACTER*1
= ā€™Nā€™: do not compute the right generalized eigenvectors;
= ā€™Vā€™: compute the right generalized eigenvectors.

N

N is INTEGER
The order of the matrices A, B, VL, and VR. N >= 0.

A

A is REAL array, dimension (LDA, N)
On entry, the matrix A in the pair (A,B).
On exit, A has been overwritten.

LDA

LDA is INTEGER
The leading dimension of A. LDA >= max(1,N).

B

B is REAL array, dimension (LDB, N)
On entry, the matrix B in the pair (A,B).
On exit, B has been overwritten.

LDB

LDB is INTEGER
The leading dimension of B. LDB >= max(1,N).

ALPHAR

ALPHAR is REAL array, dimension (N)

ALPHAI

ALPHAI is REAL array, dimension (N)

BETA

BETA is REAL array, dimension (N)
On exit, (ALPHAR(j) + ALPHAI(j)*i)/BETA(j), j=1,...,N, will
be the generalized eigenvalues. If ALPHAI(j) is zero, then
the j-th eigenvalue is real; if positive, then the j-th and
(j+1)-st eigenvalues are a complex conjugate pair, with
ALPHAI(j+1) negative.

Note: the quotients ALPHAR(j)/BETA(j) and ALPHAI(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, ALPHAR and ALPHAI 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 REAL array, dimension (LDVL,N)
If JOBVL = ā€™Vā€™, the left eigenvectors u(j) are stored one
after another in the columns of VL, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
u(j) = VL(:,j), the j-th column of VL. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
u(j) = VL(:,j)+i*VL(:,j+1) and u(j+1) = VL(:,j)-i*VL(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVL = ā€™Nā€™.

LDVL

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = ā€™Vā€™, LDVL >= N.

VR

VR is REAL array, dimension (LDVR,N)
If JOBVR = ā€™Vā€™, the right eigenvectors v(j) are stored one
after another in the columns of VR, in the same order as
their eigenvalues. If the j-th eigenvalue is real, then
v(j) = VR(:,j), the j-th column of VR. If the j-th and
(j+1)-th eigenvalues form a complex conjugate pair, then
v(j) = VR(:,j)+i*VR(:,j+1) and v(j+1) = VR(:,j)-i*VR(:,j+1).
Each eigenvector is scaled so the largest component has
abs(real part)+abs(imag. part)=1.
Not referenced if JOBVR = ā€™Nā€™.

LDVR

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = ā€™Vā€™, LDVR >= N.

WORK

WORK is REAL 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(1,8*N).
For good performance, LWORK must generally be larger.

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.

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 ALPHAR(j), ALPHAI(j), and BETA(j)
should be correct for j=INFO+1,...,N.
> N: =N+1: other than QZ iteration failed in SHGEQZ.
=N+2: error return from STGEVC.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zggev (character jobvl, character jobvr, integer n, complex*16,dimension( lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b,integer ldb, complex*16, dimension( * ) alpha, complex*16, dimension( *) beta, complex*16, dimension( ldvl, * ) vl, integer ldvl, complex*16,dimension( ldvr, * ) vr, integer ldvr, complex*16, dimension( * ) work,integer lwork, double precision, dimension( * ) rwork, integer info)

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

Purpose:

ZGGEV 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.

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 generalized eigenvector v(j) corresponding to the
generalized eigenvalue lambda(j) of (A,B) satisfies

A * v(j) = lambda(j) * B * v(j).

The left generalized eigenvector u(j) corresponding to the
generalized eigenvalues 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

JOBVL

JOBVL is CHARACTER*1
= ā€™Nā€™: do not compute the left generalized eigenvectors;
= ā€™Vā€™: compute the left generalized eigenvectors.

JOBVR

JOBVR is CHARACTER*1
= ā€™Nā€™: do not compute the right generalized eigenvectors;
= ā€™Vā€™: compute the right generalized 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.

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.

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 quotients 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 = ā€™Vā€™, 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 is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVL = ā€™Nā€™.

LDVL

LDVL is INTEGER
The leading dimension of the matrix VL. LDVL >= 1, and
if JOBVL = ā€™Vā€™, LDVL >= N.

VR

VR is COMPLEX*16 array, dimension (LDVR,N)
If JOBVR = ā€™Vā€™, 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 is scaled so the largest component has
abs(real part) + abs(imag. part) = 1.
Not referenced if JOBVR = ā€™Nā€™.

LDVR

LDVR is INTEGER
The leading dimension of the matrix VR. LDVR >= 1, and
if JOBVR = ā€™Vā€™, LDVR >= N.

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).
For good performance, LWORK must generally be larger.

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 (8*N)

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 then QZ iteration failed in ZHGEQZ,
=N+2: error return from ZTGEVC.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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