## table of contents

complex16GEeigen(3) | LAPACK | complex16GEeigen(3) |

# NAME¶

complex16GEeigen

# SYNOPSIS¶

## Functions¶

subroutine **zgees** (JOBVS, SORT, SELECT, N, A, LDA, SDIM, W,
VS, LDVS, WORK, LWORK, RWORK, BWORK, INFO)

** ZGEES computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices** subroutine **zgeesx** (JOBVS,
SORT, SELECT, SENSE, N, A, LDA, SDIM, W, VS, LDVS, RCONDE, RCONDV, WORK,
LWORK, RWORK, BWORK, INFO)

** ZGEESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices** subroutine **zgeev** (JOBVL,
JOBVR, N, A, LDA, W, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

** ZGEEV 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** subroutine **zgges** (JOBVSL, JOBVSR,
SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL, LDVSL, VSR, LDVSR,
WORK, LWORK, RWORK, BWORK, INFO)

** ZGGES computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices** subroutine **zgges3**
(JOBVSL, JOBVSR, SORT, SELCTG, N, A, LDA, B, LDB, SDIM, ALPHA, BETA, VSL,
LDVSL, VSR, LDVSR, WORK, LWORK, RWORK, BWORK, INFO)

** ZGGES3 computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors for GE matrices (blocked algorithm)** subroutine
**zggesx** (JOBVSL, JOBVSR, SORT, SELCTG, SENSE, N, A, LDA, B, LDB, SDIM,
ALPHA, BETA, VSL, LDVSL, VSR, LDVSR, RCONDE, RCONDV, WORK, LWORK, RWORK,
IWORK, LIWORK, BWORK, INFO)

** ZGGESX computes the eigenvalues, the Schur form, and, optionally, the
matrix of Schur vectors 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** subroutine **zggev3** (JOBVL, JOBVR, N,
A, LDA, B, LDB, ALPHA, BETA, VL, LDVL, VR, LDVR, WORK, LWORK, RWORK, INFO)

** ZGGEV3 computes the eigenvalues and, optionally, the left and/or right
eigenvectors for GE matrices (blocked algorithm)** 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¶

This is the group of complex16 eigenvalue driver functions for GE matrices

# Function Documentation¶

## subroutine zgees (character JOBVS, character SORT, external SELECT, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS, integer LDVS, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

** ZGEES computes the eigenvalues, the Schur form, and,
optionally, the matrix of Schur vectors for GE matrices**

**Purpose:**

ZGEES computes for an N-by-N complex nonsymmetric matrix A, the

eigenvalues, the Schur form T, and, optionally, the matrix of Schur

vectors Z. This gives the Schur factorization A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal of the

Schur form so that selected eigenvalues are at the top left.

The leading columns of Z then form an orthonormal basis for the

invariant subspace corresponding to the selected eigenvalues.

A complex matrix is in Schur form if it is upper triangular.

**Parameters**

*JOBVS*

JOBVS is CHARACTER*1

= 'N': Schur vectors are not computed;

= 'V': Schur vectors are computed.

*SORT*

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the

diagonal of the Schur form.

= 'N': Eigenvalues are not ordered:

= 'S': Eigenvalues are ordered (see SELECT).

*SELECT*

SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument

SELECT must be declared EXTERNAL in the calling subroutine.

If SORT = 'S', SELECT is used to select eigenvalues to order

to the top left of the Schur form.

IF SORT = 'N', SELECT is not referenced.

The eigenvalue W(j) is selected if SELECT(W(j)) is true.

*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 by its Schur form T.

*LDA*

LDA is INTEGER

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

*SDIM*

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues for which

SELECT is true.

*W*

W is COMPLEX*16 array, dimension (N)

W contains the computed eigenvalues, in the same order that

they appear on the diagonal of the output Schur form T.

*VS*

VS is COMPLEX*16 array, dimension (LDVS,N)

If JOBVS = 'V', VS contains the unitary matrix Z of Schur

vectors.

If JOBVS = 'N', VS is not referenced.

*LDVS*

LDVS is INTEGER

The leading dimension of the array VS. LDVS >= 1; if

JOBVS = 'V', LDVS >= 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 (N)

*BWORK*

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, and i is

<= N: the QR algorithm failed to compute all the

eigenvalues; elements 1:ILO-1 and i+1:N of W

contain those eigenvalues which have converged;

if JOBVS = 'V', VS contains the matrix which

reduces A to its partially converged Schur form.

= N+1: the eigenvalues could not be reordered because

some eigenvalues were too close to separate (the

problem is very ill-conditioned);

= N+2: after reordering, roundoff changed values of

some complex eigenvalues so that leading

eigenvalues in the Schur form no longer satisfy

SELECT = .TRUE.. This could also be caused by

underflow due to scaling.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgeesx (character JOBVS, character SORT, external SELECT, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, integer SDIM, complex*16, dimension( * ) W, complex*16, dimension( ldvs, * ) VS, integer LDVS, double precision RCONDE, double precision RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

** ZGEESX computes the eigenvalues, the Schur form, and,
optionally, the matrix of Schur vectors for GE matrices**

**Purpose:**

ZGEESX computes for an N-by-N complex nonsymmetric matrix A, the

eigenvalues, the Schur form T, and, optionally, the matrix of Schur

vectors Z. This gives the Schur factorization A = Z*T*(Z**H).

Optionally, it also orders the eigenvalues on the diagonal of the

Schur form so that selected eigenvalues are at the top left;

computes a reciprocal condition number for the average of the

selected eigenvalues (RCONDE); and computes a reciprocal condition

number for the right invariant subspace corresponding to the

selected eigenvalues (RCONDV). The leading columns of Z form an

orthonormal basis for this invariant subspace.

For further explanation of the reciprocal condition numbers RCONDE

and RCONDV, see Section 4.10 of the LAPACK Users' Guide (where

these quantities are called s and sep respectively).

A complex matrix is in Schur form if it is upper triangular.

**Parameters**

*JOBVS*

JOBVS is CHARACTER*1

= 'N': Schur vectors are not computed;

= 'V': Schur vectors are computed.

*SORT*

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the

diagonal of the Schur form.

= 'N': Eigenvalues are not ordered;

= 'S': Eigenvalues are ordered (see SELECT).

*SELECT*

SELECT is a LOGICAL FUNCTION of one COMPLEX*16 argument

SELECT must be declared EXTERNAL in the calling subroutine.

If SORT = 'S', SELECT is used to select eigenvalues to order

to the top left of the Schur form.

If SORT = 'N', SELECT is not referenced.

An eigenvalue W(j) is selected if SELECT(W(j)) is true.

*SENSE*

SENSE is CHARACTER*1

Determines which reciprocal condition numbers are computed.

= 'N': None are computed;

= 'E': Computed for average of selected eigenvalues only;

= 'V': Computed for selected right invariant subspace only;

= 'B': Computed for both.

If SENSE = 'E', 'V' or 'B', SORT must equal 'S'.

*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 is overwritten by its Schur form T.

*LDA*

LDA is INTEGER

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

*SDIM*

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues for which

SELECT is true.

*W*

W is COMPLEX*16 array, dimension (N)

W contains the computed eigenvalues, in the same order

that they appear on the diagonal of the output Schur form T.

*VS*

VS is COMPLEX*16 array, dimension (LDVS,N)

If JOBVS = 'V', VS contains the unitary matrix Z of Schur

vectors.

If JOBVS = 'N', VS is not referenced.

*LDVS*

LDVS is INTEGER

The leading dimension of the array VS. LDVS >= 1, and if

JOBVS = 'V', LDVS >= N.

*RCONDE*

RCONDE is DOUBLE PRECISION

If SENSE = 'E' or 'B', RCONDE contains the reciprocal

condition number for the average of the selected eigenvalues.

Not referenced if SENSE = 'N' or 'V'.

*RCONDV*

RCONDV is DOUBLE PRECISION

If SENSE = 'V' or 'B', RCONDV contains the reciprocal

condition number for the selected right invariant subspace.

Not referenced if SENSE = 'N' or 'E'.

*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).

Also, if SENSE = 'E' or 'V' or 'B', LWORK >= 2*SDIM*(N-SDIM),

where SDIM is the number of selected eigenvalues computed by

this routine. Note that 2*SDIM*(N-SDIM) <= N*N/2. Note also

that an error is only returned if LWORK < max(1,2*N), but if

SENSE = 'E' or 'V' or 'B' this may not be large enough.

For good performance, LWORK must generally be larger.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates upper bound on the optimal size of the

array WORK, 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 (N)

*BWORK*

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = 'N'.

*INFO*

INFO is INTEGER

= 0: successful exit

< 0: if INFO = -i, the i-th argument had an illegal value.

> 0: if INFO = i, and i is

<= N: the QR algorithm failed to compute all the

eigenvalues; elements 1:ILO-1 and i+1:N of W

contain those eigenvalues which have converged; if

JOBVS = 'V', VS contains the transformation which

reduces A to its partially converged Schur form.

= N+1: the eigenvalues could not be reordered because some

eigenvalues were too close to separate (the problem

is very ill-conditioned);

= N+2: after reordering, roundoff changed values of some

complex eigenvalues so that leading eigenvalues in

the Schur form no longer satisfy SELECT=.TRUE. This

could also be caused by underflow due to scaling.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgeev (character JOBVL, character JOBVR, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( * ) W, 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)¶

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

**Purpose:**

ZGEEV computes for an N-by-N complex nonsymmetric matrix A, the

eigenvalues and, optionally, the left and/or right eigenvectors.

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.

**Parameters**

*JOBVL*

JOBVL is CHARACTER*1

= 'N': left eigenvectors of A are not computed;

= 'V': left eigenvectors of are computed.

*JOBVR*

JOBVR is CHARACTER*1

= 'N': right eigenvectors of A are not computed;

= 'V': right eigenvectors of A are computed.

*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.

*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.

*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 (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 have been computed;

elements i+1:N of W contain eigenvalues which have

converged.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgeevx (character BALANC, character JOBVL, character JOBVR, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, 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, double precision, 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 California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgges (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

** ZGGES computes the eigenvalues, the Schur form, and,
optionally, the matrix of Schur vectors for GE matrices**

**Purpose:**

ZGGES computes for a pair of N-by-N complex nonsymmetric matrices

(A,B), the generalized eigenvalues, the generalized complex Schur

form (S, T), and optionally left and/or right Schur vectors (VSL

and VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster

of eigenvalues appears in the leading diagonal blocks of the upper

triangular matrix S and the upper triangular matrix T. The leading

columns of VSL and VSR then form an unitary basis for the

corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver

ZGGEV instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w

or a ratio alpha/beta = w, such that A - w*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.

A pair of matrices (S,T) is in generalized complex Schur form if S

and T are upper triangular and, in addition, the diagonal elements

of T are non-negative real numbers.

**Parameters**

*JOBVSL*

JOBVSL is CHARACTER*1

= 'N': do not compute the left Schur vectors;

= 'V': compute the left Schur vectors.

*JOBVSR*

JOBVSR is CHARACTER*1

= 'N': do not compute the right Schur vectors;

= 'V': compute the right Schur vectors.

*SORT*

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the

diagonal of the generalized Schur form.

= 'N': Eigenvalues are not ordered;

= 'S': Eigenvalues are ordered (see SELCTG).

*SELCTG*

SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments

SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.

If SORT = 'S', SELCTG is used to select eigenvalues to sort

to the top left of the Schur form.

An eigenvalue ALPHA(j)/BETA(j) is selected if

SELCTG(ALPHA(j),BETA(j)) is true.

Note that a selected complex eigenvalue may no longer satisfy

SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since

ordering may change the value of complex eigenvalues

(especially if the eigenvalue is ill-conditioned), in this

case INFO is set to N+2 (See INFO below).

*N*

N is INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA, N)

On entry, the first of the pair of matrices.

On exit, A has been overwritten by its generalized Schur

form S.

*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 second of the pair of matrices.

On exit, B has been overwritten by its generalized Schur

form T.

*LDB*

LDB is INTEGER

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

*SDIM*

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues (after sorting)

for which SELCTG is true.

*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. ALPHA(j), j=1,...,N and BETA(j),

j=1,...,N are the diagonals of the complex Schur form (A,B)

output by ZGGES. The BETA(j) will be non-negative real.

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).

*VSL*

VSL is COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors.

Not referenced if JOBVSL = 'N'.

*LDVSL*

LDVSL is INTEGER

The leading dimension of the matrix VSL. LDVSL >= 1, and

if JOBVSL = 'V', LDVSL >= N.

*VSR*

VSR is COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors.

Not referenced if JOBVSR = 'N'.

*LDVSR*

LDVSR is INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and

if JOBVSR = 'V', LDVSR >= 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)

*BWORK*

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = '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. (A,B) are not in Schur

form, but ALPHA(j) and BETA(j) should be correct for

j=INFO+1,...,N.

> N: =N+1: other than QZ iteration failed in ZHGEQZ

=N+2: after reordering, roundoff changed values of

some complex eigenvalues so that leading

eigenvalues in the Generalized Schur form no

longer satisfy SELCTG=.TRUE. This could also

be caused due to scaling.

=N+3: reordering failed in ZTGSEN.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zgges3 (character JOBVSL, character JOBVSR, character SORT, external SELCTG, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, logical, dimension( * ) BWORK, integer INFO)¶

** ZGGES3 computes the eigenvalues, the Schur form, and,
optionally, the matrix of Schur vectors for GE matrices (blocked
algorithm)**

**Purpose:**

ZGGES3 computes for a pair of N-by-N complex nonsymmetric matrices

(A,B), the generalized eigenvalues, the generalized complex Schur

form (S, T), and optionally left and/or right Schur vectors (VSL

and VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL)*S*(VSR)**H, (VSL)*T*(VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster

of eigenvalues appears in the leading diagonal blocks of the upper

triangular matrix S and the upper triangular matrix T. The leading

columns of VSL and VSR then form an unitary basis for the

corresponding left and right eigenspaces (deflating subspaces).

(If only the generalized eigenvalues are needed, use the driver

ZGGEV instead, which is faster.)

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w

or a ratio alpha/beta = w, such that A - w*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.

A pair of matrices (S,T) is in generalized complex Schur form if S

and T are upper triangular and, in addition, the diagonal elements

of T are non-negative real numbers.

**Parameters**

*JOBVSL*

JOBVSL is CHARACTER*1

= 'N': do not compute the left Schur vectors;

= 'V': compute the left Schur vectors.

*JOBVSR*

JOBVSR is CHARACTER*1

= 'N': do not compute the right Schur vectors;

= 'V': compute the right Schur vectors.

*SORT*

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the

diagonal of the generalized Schur form.

= 'N': Eigenvalues are not ordered;

= 'S': Eigenvalues are ordered (see SELCTG).

*SELCTG*

SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments

SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.

If SORT = 'S', SELCTG is used to select eigenvalues to sort

to the top left of the Schur form.

An eigenvalue ALPHA(j)/BETA(j) is selected if

SELCTG(ALPHA(j),BETA(j)) is true.

Note that a selected complex eigenvalue may no longer satisfy

SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since

ordering may change the value of complex eigenvalues

(especially if the eigenvalue is ill-conditioned), in this

case INFO is set to N+2 (See INFO below).

*N*

N is INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA, N)

On entry, the first of the pair of matrices.

On exit, A has been overwritten by its generalized Schur

form S.

*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 second of the pair of matrices.

On exit, B has been overwritten by its generalized Schur

form T.

*LDB*

LDB is INTEGER

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

*SDIM*

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues (after sorting)

for which SELCTG is true.

*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. ALPHA(j), j=1,...,N and BETA(j),

j=1,...,N are the diagonals of the complex Schur form (A,B)

output by ZGGES3. The BETA(j) will be non-negative real.

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).

*VSL*

VSL is COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors.

Not referenced if JOBVSL = 'N'.

*LDVSL*

LDVSL is INTEGER

The leading dimension of the matrix VSL. LDVSL >= 1, and

if JOBVSL = 'V', LDVSL >= N.

*VSR*

VSR is COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors.

Not referenced if JOBVSR = 'N'.

*LDVSR*

LDVSR is INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and

if JOBVSR = 'V', LDVSR >= 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.

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)

*BWORK*

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = '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. (A,B) are not in Schur

form, but ALPHA(j) and BETA(j) should be correct for

j=INFO+1,...,N.

> N: =N+1: other than QZ iteration failed in ZHGEQZ

=N+2: after reordering, roundoff changed values of

some complex eigenvalues so that leading

eigenvalues in the Generalized Schur form no

longer satisfy SELCTG=.TRUE. This could also

be caused due to scaling.

=N+3: reordering failed in ZTGSEN.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zggesx (character JOBVSL, character JOBVSR, character SORT, external SELCTG, character SENSE, integer N, complex*16, dimension( lda, * ) A, integer LDA, complex*16, dimension( ldb, * ) B, integer LDB, integer SDIM, complex*16, dimension( * ) ALPHA, complex*16, dimension( * ) BETA, complex*16, dimension( ldvsl, * ) VSL, integer LDVSL, complex*16, dimension( ldvsr, * ) VSR, integer LDVSR, double precision, dimension( 2 ) RCONDE, double precision, dimension( 2 ) RCONDV, complex*16, dimension( * ) WORK, integer LWORK, double precision, dimension( * ) RWORK, integer, dimension( * ) IWORK, integer LIWORK, logical, dimension( * ) BWORK, integer INFO)¶

** ZGGESX computes the eigenvalues, the Schur form, and,
optionally, the matrix of Schur vectors for GE matrices**

**Purpose:**

ZGGESX computes for a pair of N-by-N complex nonsymmetric matrices

(A,B), the generalized eigenvalues, the complex Schur form (S,T),

and, optionally, the left and/or right matrices of Schur vectors (VSL

and VSR). This gives the generalized Schur factorization

(A,B) = ( (VSL) S (VSR)**H, (VSL) T (VSR)**H )

where (VSR)**H is the conjugate-transpose of VSR.

Optionally, it also orders the eigenvalues so that a selected cluster

of eigenvalues appears in the leading diagonal blocks of the upper

triangular matrix S and the upper triangular matrix T; computes

a reciprocal condition number for the average of the selected

eigenvalues (RCONDE); and computes a reciprocal condition number for

the right and left deflating subspaces corresponding to the selected

eigenvalues (RCONDV). The leading columns of VSL and VSR then form

an orthonormal basis for the corresponding left and right eigenspaces

(deflating subspaces).

A generalized eigenvalue for a pair of matrices (A,B) is a scalar w

or a ratio alpha/beta = w, such that A - w*B is singular. It is

usually represented as the pair (alpha,beta), as there is a

reasonable interpretation for beta=0 or for both being zero.

A pair of matrices (S,T) is in generalized complex Schur form if T is

upper triangular with non-negative diagonal and S is upper

triangular.

**Parameters**

*JOBVSL*

JOBVSL is CHARACTER*1

= 'N': do not compute the left Schur vectors;

= 'V': compute the left Schur vectors.

*JOBVSR*

JOBVSR is CHARACTER*1

= 'N': do not compute the right Schur vectors;

= 'V': compute the right Schur vectors.

*SORT*

SORT is CHARACTER*1

Specifies whether or not to order the eigenvalues on the

diagonal of the generalized Schur form.

= 'N': Eigenvalues are not ordered;

= 'S': Eigenvalues are ordered (see SELCTG).

*SELCTG*

SELCTG is a LOGICAL FUNCTION of two COMPLEX*16 arguments

SELCTG must be declared EXTERNAL in the calling subroutine.

If SORT = 'N', SELCTG is not referenced.

If SORT = 'S', SELCTG is used to select eigenvalues to sort

to the top left of the Schur form.

Note that a selected complex eigenvalue may no longer satisfy

SELCTG(ALPHA(j),BETA(j)) = .TRUE. after ordering, since

ordering may change the value of complex eigenvalues

(especially if the eigenvalue is ill-conditioned), in this

case INFO is set to N+3 see INFO below).

*SENSE*

SENSE is CHARACTER*1

Determines which reciprocal condition numbers are computed.

= 'N': None are computed;

= 'E': Computed for average of selected eigenvalues only;

= 'V': Computed for selected deflating subspaces only;

= 'B': Computed for both.

If SENSE = 'E', 'V', or 'B', SORT must equal 'S'.

*N*

N is INTEGER

The order of the matrices A, B, VSL, and VSR. N >= 0.

*A*

A is COMPLEX*16 array, dimension (LDA, N)

On entry, the first of the pair of matrices.

On exit, A has been overwritten by its generalized Schur

form S.

*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 second of the pair of matrices.

On exit, B has been overwritten by its generalized Schur

form T.

*LDB*

LDB is INTEGER

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

*SDIM*

SDIM is INTEGER

If SORT = 'N', SDIM = 0.

If SORT = 'S', SDIM = number of eigenvalues (after sorting)

for which SELCTG is true.

*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. ALPHA(j) and BETA(j),j=1,...,N are

the diagonals of the complex Schur form (S,T). BETA(j) will

be non-negative real.

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).

*VSL*

VSL is COMPLEX*16 array, dimension (LDVSL,N)

If JOBVSL = 'V', VSL will contain the left Schur vectors.

Not referenced if JOBVSL = 'N'.

*LDVSL*

LDVSL is INTEGER

The leading dimension of the matrix VSL. LDVSL >=1, and

if JOBVSL = 'V', LDVSL >= N.

*VSR*

VSR is COMPLEX*16 array, dimension (LDVSR,N)

If JOBVSR = 'V', VSR will contain the right Schur vectors.

Not referenced if JOBVSR = 'N'.

*LDVSR*

LDVSR is INTEGER

The leading dimension of the matrix VSR. LDVSR >= 1, and

if JOBVSR = 'V', LDVSR >= N.

*RCONDE*

RCONDE is DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'E' or 'B', RCONDE(1) and RCONDE(2) contain the

reciprocal condition numbers for the average of the selected

eigenvalues.

Not referenced if SENSE = 'N' or 'V'.

*RCONDV*

RCONDV is DOUBLE PRECISION array, dimension ( 2 )

If SENSE = 'V' or 'B', RCONDV(1) and RCONDV(2) contain the

reciprocal condition number for the selected deflating

subspaces.

Not referenced if SENSE = 'N' or 'E'.

*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 N = 0, LWORK >= 1, else if SENSE = 'E', 'V', or 'B',

LWORK >= MAX(1,2*N,2*SDIM*(N-SDIM)), else

LWORK >= MAX(1,2*N). Note that 2*SDIM*(N-SDIM) <= N*N/2.

Note also that an error is only returned if

LWORK < MAX(1,2*N), but if SENSE = 'E' or 'V' or 'B' this may

not be large enough.

If LWORK = -1, then a workspace query is assumed; the routine

only calculates the bound on the optimal size of the WORK

array and the minimum size of the IWORK array, returns these

values as the first entries of the WORK and IWORK arrays, and

no error message related to LWORK or LIWORK is issued by

XERBLA.

*RWORK*

RWORK is DOUBLE PRECISION array, dimension ( 8*N )

Real workspace.

*IWORK*

IWORK is INTEGER array, dimension (MAX(1,LIWORK))

On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.

*LIWORK*

LIWORK is INTEGER

The dimension of the array IWORK.

If SENSE = 'N' or N = 0, LIWORK >= 1, otherwise

LIWORK >= N+2.

If LIWORK = -1, then a workspace query is assumed; the

routine only calculates the bound on the optimal size of the

WORK array and the minimum size of the IWORK array, returns

these values as the first entries of the WORK and IWORK

arrays, and no error message related to LWORK or LIWORK is

issued by XERBLA.

*BWORK*

BWORK is LOGICAL array, dimension (N)

Not referenced if SORT = '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. (A,B) are not in Schur

form, but ALPHA(j) and BETA(j) should be correct for

j=INFO+1,...,N.

> N: =N+1: other than QZ iteration failed in ZHGEQZ

=N+2: after reordering, roundoff changed values of

some complex eigenvalues so that leading

eigenvalues in the Generalized Schur form no

longer satisfy SELCTG=.TRUE. This could also

be caused due to scaling.

=N+3: reordering failed in ZTGSEN.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## 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

= '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.

*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 = '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 is scaled so the largest component has

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*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 is scaled so the largest component has

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.

*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 DHGEQZ,

=N+2: error return from DTGEVC.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zggev3 (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)¶

** ZGGEV3 computes the eigenvalues and, optionally, the left
and/or right eigenvectors for GE matrices (blocked algorithm)**

**Purpose:**

ZGGEV3 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

= '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.

*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 = '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 is scaled so the largest component has

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*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 is scaled so the largest component has

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.

*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 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 DHGEQZ,

=N+2: error return from DTGEVC.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

## subroutine zggevx (character BALANC, character JOBVL, character JOBVR, character SENSE, 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, integer ILO, integer IHI, double precision, dimension( * ) LSCALE, double precision, dimension( * ) RSCALE, double precision ABNRM, double precision BBNRM, double precision, dimension( * ) RCONDE, double precision, dimension( * ) RCONDV, complex*16, 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.

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*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*

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.

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*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*

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*

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.

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).

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 (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 ZHGEQZ.

=N+2: error return from ZTGEVC.

**Author**

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date**

**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.

# Author¶

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