Man page - geevx(3)

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Manual

geevx

NAME
SYNOPSIS
Functions
Detailed Description
Function Documentation
subroutine cgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex, dimension( lda, * ) a, integerlda, complex, dimension( * ) w, complex, dimension( ldvl, * ) vl,integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integerilo, integer ihi, real, dimension( * ) scale, real abnrm, real,dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension(* ) work, integer lwork, real, dimension( * ) rwork, integer info)
subroutine dgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, double precision, dimension( lda, * ) a,integer lda, double precision, dimension( * ) wr, double precision,dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integerldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integerilo, integer ihi, double precision, dimension( * ) scale, doubleprecision abnrm, double precision, dimension( * ) rconde, doubleprecision, dimension( * ) rcondv, double precision, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer info)
subroutine sgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, real, dimension( lda, * ) a, integer lda,real, dimension( * ) wr, real, dimension( * ) wi, real, dimension(ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integerldvr, integer ilo, integer ihi, real, dimension( * ) scale, real abnrm,real, dimension( * ) rconde, real, dimension( * ) rcondv, real,dimension( * ) work, integer lwork, integer, dimension( * ) iwork,integer info)
subroutine zgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex*16, dimension( lda, * ) a, integerlda, complex*16, dimension( * ) w, complex*16, dimension( ldvl, * ) vl,integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr,integer ilo, integer ihi, double precision, dimension( * ) scale,double precision abnrm, double precision, dimension( * ) rconde, doubleprecision, dimension( * ) rcondv, complex*16, dimension( * ) work,integer lwork, double precision, dimension( * ) rwork, integer info)
Author

NAME

geevx - geevx: eig, expert

SYNOPSIS

Functions

subroutine cgeevx (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info)
CGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine dgeevx (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info)
DGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine sgeevx (balanc, jobvl, jobvr, sense, n, a, lda, wr, wi, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, iwork, info)
SGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices
subroutine zgeevx (balanc, jobvl, jobvr, sense, n, a, lda, w, vl, ldvl, vr, ldvr, ilo, ihi, scale, abnrm, rconde, rcondv, work, lwork, rwork, info)
ZGEEVX computes the eigenvalues and, optionally, the left and/or right eigenvectors for GE matrices

Detailed Description

Function Documentation

subroutine cgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex, dimension( lda, * ) a, integerlda, complex, dimension( * ) w, complex, dimension( ldvl, * ) vl,integer ldvl, complex, dimension( ldvr, * ) vr, integer ldvr, integerilo, integer ihi, real, dimension( * ) scale, real abnrm, real,dimension( * ) rconde, real, dimension( * ) rcondv, complex, dimension(* ) work, integer lwork, real, dimension( * ) rwork, integer info)

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

Purpose:

CGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK
Users’ Guide.

Parameters

BALANC

BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= ’N’: Do not diagonally scale or permute;
= ’P’: Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= ’S’: Diagonally scale the matrix, ie. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= ’B’: Both diagonally scale and permute A.

Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.

JOBVL

JOBVL is CHARACTER*1
= ’N’: left eigenvectors of A are not computed;
= ’V’: left eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVL must = ’V’.

JOBVR

JOBVR is CHARACTER*1
= ’N’: right eigenvectors of A are not computed;
= ’V’: right eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVR must = ’V’.

SENSE

SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= ’N’: None are computed;
= ’E’: Computed for eigenvalues only;
= ’V’: Computed for right eigenvectors only;
= ’B’: Computed for eigenvalues and right eigenvectors.

If SENSE = ’E’ or ’B’, both left and right eigenvectors
must also be computed (JOBVL = ’V’ and JOBVR = ’V’).

N

N is INTEGER
The order of the matrix A. N >= 0.

A

A is COMPLEX array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten. If JOBVL = ’V’ or
JOBVR = ’V’, A contains the Schur form of the balanced
version of the matrix A.

LDA

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

W

W is COMPLEX array, dimension (N)
W contains the computed eigenvalues.

VL

VL is COMPLEX 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 JOBVL = ’N’, VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = ’V’, LDVL >= N.

VR

VR is COMPLEX 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 JOBVR = ’N’, VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = ’V’, LDVR >= N.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

SCALE is REAL array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J), for J = 1,...,ILO-1
= D(J), for J = ILO,...,IHI
= P(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 (the maximum
of the sum of absolute values of elements of any column).

RCONDE

RCONDE is REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.

RCONDV

RCONDV is REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.

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. If SENSE = ’N’ or ’E’,
LWORK >= max(1,2*N), and if SENSE = ’V’ or ’B’,
LWORK >= N*N+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 (2*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of W
contain eigenvalues which have converged.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine dgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, double precision, dimension( lda, * ) a,integer lda, double precision, dimension( * ) wr, double precision,dimension( * ) wi, double precision, dimension( ldvl, * ) vl, integerldvl, double precision, dimension( ldvr, * ) vr, integer ldvr, integerilo, integer ihi, double precision, dimension( * ) scale, doubleprecision abnrm, double precision, dimension( * ) rconde, doubleprecision, dimension( * ) rcondv, double precision, dimension( * )work, integer lwork, integer, dimension( * ) iwork, integer info)

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

Purpose:

DGEEVX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate-transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK
Users’ Guide.

Parameters

BALANC

BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= ’N’: Do not diagonally scale or permute;
= ’P’: Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= ’S’: Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= ’B’: Both diagonally scale and permute A.

Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.

JOBVL

JOBVL is CHARACTER*1
= ’N’: left eigenvectors of A are not computed;
= ’V’: left eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVL must = ’V’.

JOBVR

JOBVR is CHARACTER*1
= ’N’: right eigenvectors of A are not computed;
= ’V’: right eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVR must = ’V’.

SENSE

SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= ’N’: None are computed;
= ’E’: Computed for eigenvalues only;
= ’V’: Computed for right eigenvectors only;
= ’B’: Computed for eigenvalues and right eigenvectors.

If SENSE = ’E’ or ’B’, both left and right eigenvectors
must also be computed (JOBVL = ’V’ and JOBVR = ’V’).

N

N is INTEGER
The order of the matrix A. N >= 0.

A

A is DOUBLE PRECISION array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten. If JOBVL = ’V’ or
JOBVR = ’V’, A contains the real Schur form of the balanced
version of the input matrix A.

LDA

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

WR

WR is DOUBLE PRECISION array, dimension (N)

WI

WI is DOUBLE PRECISION array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.

VL

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 JOBVL = ’N’, VL is not referenced.
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)-st 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).

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = ’V’, LDVL >= N.

VR

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 JOBVR = ’N’, VR is not referenced.
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)-st 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).

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
JOBVR = ’V’, LDVR >= N.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J), for J = 1,...,ILO-1
= D(J), for J = ILO,...,IHI
= P(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 (the maximum
of the sum of absolute values of elements of any column).

RCONDE

RCONDE is DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.

RCONDV

RCONDV is DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.

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. If SENSE = ’N’ or ’E’,
LWORK >= max(1,2*N), and if JOBVL = ’V’ or JOBVR = ’V’,
LWORK >= 3*N. If SENSE = ’V’ or ’B’, LWORK >= N*(N+6).
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.

IWORK

IWORK is INTEGER array, dimension (2*N-2)
If SENSE = ’N’ or ’E’, not referenced.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of WR
and WI contain eigenvalues which have converged.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine sgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, real, dimension( lda, * ) a, integer lda,real, dimension( * ) wr, real, dimension( * ) wi, real, dimension(ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integerldvr, integer ilo, integer ihi, real, dimension( * ) scale, real abnrm,real, dimension( * ) rconde, real, dimension( * ) rcondv, real,dimension( * ) work, integer lwork, integer, dimension( * ) iwork,integer info)

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

Purpose:

SGEEVX computes for an N-by-N real nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate-transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK
Users’ Guide.

Parameters

BALANC

BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= ’N’: Do not diagonally scale or permute;
= ’P’: Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= ’S’: Diagonally scale the matrix, i.e. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= ’B’: Both diagonally scale and permute A.

Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.

JOBVL

JOBVL is CHARACTER*1
= ’N’: left eigenvectors of A are not computed;
= ’V’: left eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVL must = ’V’.

JOBVR

JOBVR is CHARACTER*1
= ’N’: right eigenvectors of A are not computed;
= ’V’: right eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVR must = ’V’.

SENSE

SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= ’N’: None are computed;
= ’E’: Computed for eigenvalues only;
= ’V’: Computed for right eigenvectors only;
= ’B’: Computed for eigenvalues and right eigenvectors.

If SENSE = ’E’ or ’B’, both left and right eigenvectors
must also be computed (JOBVL = ’V’ and JOBVR = ’V’).

N

N is INTEGER
The order of the matrix A. N >= 0.

A

A is REAL array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten. If JOBVL = ’V’ or
JOBVR = ’V’, A contains the real Schur form of the balanced
version of the input matrix A.

LDA

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

WR

WR is REAL array, dimension (N)

WI

WI is REAL array, dimension (N)
WR and WI contain the real and imaginary parts,
respectively, of the computed eigenvalues. Complex
conjugate pairs of eigenvalues will appear consecutively
with the eigenvalue having the positive imaginary part
first.

VL

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 JOBVL = ’N’, VL is not referenced.
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)-st 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).

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = ’V’, LDVL >= N.

VR

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 JOBVR = ’N’, VR is not referenced.
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)-st 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).

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1, and if
JOBVR = ’V’, LDVR >= N.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

SCALE is REAL array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J), for J = 1,...,ILO-1
= D(J), for J = ILO,...,IHI
= P(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 (the maximum
of the sum of absolute values of elements of any column).

RCONDE

RCONDE is REAL array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.

RCONDV

RCONDV is REAL array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.

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. If SENSE = ’N’ or ’E’,
LWORK >= max(1,2*N), and if JOBVL = ’V’ or JOBVR = ’V’,
LWORK >= 3*N. If SENSE = ’V’ or ’B’, LWORK >= N*(N+6).
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.

IWORK

IWORK is INTEGER array, dimension (2*N-2)
If SENSE = ’N’ or ’E’, not referenced.

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of WR
and WI contain eigenvalues which have converged.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

subroutine zgeevx (character balanc, character jobvl, character jobvr,character sense, integer n, complex*16, dimension( lda, * ) a, integerlda, complex*16, dimension( * ) w, complex*16, dimension( ldvl, * ) vl,integer ldvl, complex*16, dimension( ldvr, * ) vr, integer ldvr,integer ilo, integer ihi, double precision, dimension( * ) scale,double precision abnrm, double precision, dimension( * ) rconde, doubleprecision, dimension( * ) rcondv, complex*16, dimension( * ) work,integer lwork, double precision, dimension( * ) rwork, integer info)

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

Purpose:

ZGEEVX computes for an N-by-N complex nonsymmetric matrix A, the
eigenvalues and, optionally, the left and/or right eigenvectors.

Optionally also, it computes a balancing transformation to improve
the conditioning of the eigenvalues and eigenvectors (ILO, IHI,
SCALE, and ABNRM), reciprocal condition numbers for the eigenvalues
(RCONDE), and reciprocal condition numbers for the right
eigenvectors (RCONDV).

The right eigenvector v(j) of A satisfies
A * v(j) = lambda(j) * v(j)
where lambda(j) is its eigenvalue.
The left eigenvector u(j) of A satisfies
u(j)**H * A = lambda(j) * u(j)**H
where u(j)**H denotes the conjugate transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm
equal to 1 and largest component real.

Balancing a matrix means permuting the rows and columns to make it
more nearly upper triangular, and applying a diagonal similarity
transformation D * A * D**(-1), where D is a diagonal matrix, to
make its rows and columns closer in norm and the condition numbers
of its eigenvalues and eigenvectors smaller. 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 4.10.2 of the LAPACK
Users’ Guide.

Parameters

BALANC

BALANC is CHARACTER*1
Indicates how the input matrix should be diagonally scaled
and/or permuted to improve the conditioning of its
eigenvalues.
= ’N’: Do not diagonally scale or permute;
= ’P’: Perform permutations to make the matrix more nearly
upper triangular. Do not diagonally scale;
= ’S’: Diagonally scale the matrix, ie. replace A by
D*A*D**(-1), where D is a diagonal matrix chosen
to make the rows and columns of A more equal in
norm. Do not permute;
= ’B’: Both diagonally scale and permute A.

Computed reciprocal condition numbers will be for the matrix
after balancing and/or permuting. Permuting does not change
condition numbers (in exact arithmetic), but balancing does.

JOBVL

JOBVL is CHARACTER*1
= ’N’: left eigenvectors of A are not computed;
= ’V’: left eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVL must = ’V’.

JOBVR

JOBVR is CHARACTER*1
= ’N’: right eigenvectors of A are not computed;
= ’V’: right eigenvectors of A are computed.
If SENSE = ’E’ or ’B’, JOBVR must = ’V’.

SENSE

SENSE is CHARACTER*1
Determines which reciprocal condition numbers are computed.
= ’N’: None are computed;
= ’E’: Computed for eigenvalues only;
= ’V’: Computed for right eigenvectors only;
= ’B’: Computed for eigenvalues and right eigenvectors.

If SENSE = ’E’ or ’B’, both left and right eigenvectors
must also be computed (JOBVL = ’V’ and JOBVR = ’V’).

N

N is INTEGER
The order of the matrix A. N >= 0.

A

A is COMPLEX*16 array, dimension (LDA,N)
On entry, the N-by-N matrix A.
On exit, A has been overwritten. If JOBVL = ’V’ or
JOBVR = ’V’, A contains the Schur form of the balanced
version of the matrix A.

LDA

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

W

W is COMPLEX*16 array, dimension (N)
W contains the computed eigenvalues.

VL

VL is COMPLEX*16 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 JOBVL = ’N’, VL is not referenced.
u(j) = VL(:,j), the j-th column of VL.

LDVL

LDVL is INTEGER
The leading dimension of the array VL. LDVL >= 1; if
JOBVL = ’V’, LDVL >= N.

VR

VR is COMPLEX*16 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 JOBVR = ’N’, VR is not referenced.
v(j) = VR(:,j), the j-th column of VR.

LDVR

LDVR is INTEGER
The leading dimension of the array VR. LDVR >= 1; if
JOBVR = ’V’, LDVR >= N.

ILO

ILO is INTEGER

IHI

IHI is INTEGER
ILO and IHI are integer values determined when A was
balanced. The balanced A(i,j) = 0 if I > J and
J = 1,...,ILO-1 or I = IHI+1,...,N.

SCALE

SCALE is DOUBLE PRECISION array, dimension (N)
Details of the permutations and scaling factors applied
when balancing A. If P(j) is the index of the row and column
interchanged with row and column j, and D(j) is the scaling
factor applied to row and column j, then
SCALE(J) = P(J), for J = 1,...,ILO-1
= D(J), for J = ILO,...,IHI
= P(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 (the maximum
of the sum of absolute values of elements of any column).

RCONDE

RCONDE is DOUBLE PRECISION array, dimension (N)
RCONDE(j) is the reciprocal condition number of the j-th
eigenvalue.

RCONDV

RCONDV is DOUBLE PRECISION array, dimension (N)
RCONDV(j) is the reciprocal condition number of the j-th
right eigenvector.

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. If SENSE = ’N’ or ’E’,
LWORK >= max(1,2*N), and if SENSE = ’V’ or ’B’,
LWORK >= N*N+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 (2*N)

INFO

INFO is INTEGER
= 0: successful exit
< 0: if INFO = -i, the i-th argument had an illegal value.
> 0: if INFO = i, the QR algorithm failed to compute all the
eigenvalues, and no eigenvectors or condition numbers
have been computed; elements 1:ILO-1 and i+1:N of W
contain eigenvalues which have converged.

Author

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

Author

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