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Manual

ggevx

NAME

ggevx - ggevx: eig, expert

SYNOPSIS

Functions

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
subroutine dggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)
DGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sggevx (balanc, jobvl, jobvr, sense, n, a, lda, b, ldb, alphar, alphai, beta, vl, ldvl, vr, ldvr, ilo, ihi, lscale, rscale, abnrm, bbnrm, rconde, rcondv, work, lwork, iwork, bwork, info)
SGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine 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)
ZGGEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

Function Documentation

subroutine cggevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex, dimension( lda, * ) a, integerlda, complex, dimension( ldb, * ) b, integer ldb, complex, dimension( ) alpha, complex, dimension( ) beta, complex, dimension( ldvl, * )vl, integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr,integer ilo, integer ihi, real, dimension( * ) lscale, real, dimension(* ) rscale, real abnrm, real bbnrm, real, dimension( * ) rconde, real,dimension( * ) rcondv, complex, dimension( * ) work, integer lwork,real, dimension( * ) rwork, integer, dimension( * ) iwork, logical,dimension( * ) bwork, integer info)

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

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 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 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 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.

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.

subroutine dggevx (character balanc, character jobvl, character jobvr,character sense, integer n, double precision, 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, integer ldvl, double precision, dimension(ldvr, * ) vr, integer ldvr, integer ilo, integer ihi, double precision,dimension( * ) lscale, double precision, dimension( * ) rscale, doubleprecision abnrm, double precision bbnrm, double precision, dimension( ) rconde, double precision, dimension( ) rcondv, double precision,dimension( * ) work, integer lwork, integer, dimension( * ) iwork,logical, dimension( * ) bwork, integer info)

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

Purpose:

DGGEVX 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.

Optionally also, it 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).

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

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

A is DOUBLE PRECISION 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 real 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 is DOUBLE PRECISION 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 real 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).

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 is DOUBLE PRECISION array, dimension (LDVL,N)
If JOBVL = ’V’, 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 will be scaled so the largest component 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 is DOUBLE PRECISION array, dimension (LDVR,N)
If JOBVR = ’V’, 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 will be scaled so the largest component 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 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,...,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 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,...,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 DOUBLE PRECISION
The one-norm of the balanced matrix A.

BBNRM

BBNRM is DOUBLE PRECISION
The one-norm 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.
For a complex conjugate pair of eigenvalues two consecutive
elements of RCONDE are set to the same value. Thus RCONDE(j),
RCONDV(j), and the j-th columns of VL and VR all correspond
to the j-th eigenpair.
If SENSE = ’N or ’V’, RCONDE is not referenced.

RCONDV

RCONDV is DOUBLE PRECISION array, dimension (N)
If SENSE = ’V’ or ’B’, the estimated reciprocal condition
numbers of the eigenvectors, stored in consecutive elements
of the array. For a complex eigenvector two consecutive
elements of RCONDV are set to the same value. 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 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(1,2*N).
If BALANC = ’S’ or ’B’, or JOBVL = ’V’, or JOBVR = ’V’,
LWORK >= max(1,6*N).
If SENSE = ’E’ or ’B’, LWORK >= max(1,10*N).
If SENSE = ’V’ or ’B’, LWORK >= 2*N*N+8*N+16.

IWORK

IWORK is INTEGER array, dimension (N+6)
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 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.

Further Details:

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.

subroutine sggevx (character balanc, character jobvl, character jobvr,character sense, 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, integer ilo, integer ihi, real, dimension( * ) lscale,real, dimension( * ) rscale, real abnrm, real bbnrm, real, dimension( ) rconde, real, dimension( ) rcondv, real, dimension( * ) work,integer lwork, integer, dimension( * ) iwork, logical, dimension( * )bwork, integer info)

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

Purpose:

SGGEVX 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.

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

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

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

A is REAL 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 real 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 is REAL 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 real 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).

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.

VL is REAL array, dimension (LDVL,N)
If JOBVL = ’V’, 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 will be scaled so the largest component 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 is REAL array, dimension (LDVR,N)
If JOBVR = ’V’, 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 will be scaled so the largest component 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

RSCALE

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.
For a complex conjugate pair of eigenvalues two consecutive
elements of RCONDE are set to the same value. Thus RCONDE(j),
RCONDV(j), and the j-th columns of VL and VR all correspond
to the j-th eigenpair.
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. For a complex eigenvector two consecutive
elements of RCONDV are set to the same value. 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 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(1,2*N).
If BALANC = ’S’ or ’B’, or JOBVL = ’V’, or JOBVR = ’V’,
LWORK >= max(1,6*N).
If SENSE = ’E’, LWORK >= max(1,10*N).
If SENSE = ’V’ or ’B’, LWORK >= 2*N*N+8*N+16.

IWORK

IWORK is INTEGER array, dimension (N+6)
If SENSE = ’E’, IWORK is not referenced.

BWORK

BWORK is LOGICAL array, dimension (N)
If SENSE = ’N’, BWORK is not referenced.

INFO

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

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.

subroutine zggevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex16, dimension( lda, ) a, integerlda, complex16, dimension( ldb, ) b, integer ldb, complex16,dimension( ) alpha, complex16, dimension( ) beta, complex16,dimension( ldvl, ) vl, integer ldvl, complex16, dimension( ldvr, )vr, integer ldvr, integer ilo, integer ihi, double precision,dimension( * ) lscale, double precision, dimension( * ) rscale, doubleprecision abnrm, double precision bbnrm, double precision, dimension( ) rconde, double precision, dimension( ) rcondv, complex16,dimension( ) work, integer lwork, double precision, dimension( * )rwork, integer, dimension( * ) iwork, logical, dimension( * ) bwork,integer info)

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

Purpose:

ZGGEVX 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.

Parameters

BALANC

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

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

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 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.

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 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 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

RSCALE

ABNRM

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

BBNRM

BBNRM is DOUBLE PRECISION
The one-norm 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).

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

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Further Details:

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.

Author

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