NAME

zgeevx - compute for an N-by-N complex nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors

SYNOPSIS

  SUBROUTINE ZGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 
 *      LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONE, RCONV, WORK, 
 *      LDWORK, WORK2, INFO)
  CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
  DOUBLE COMPLEX A(LDA,*), W(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
  INTEGER N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  DOUBLE PRECISION ABNRM
  DOUBLE PRECISION SCALE(*), RCONE(*), RCONV(*), WORK2(*)

  SUBROUTINE ZGEEVX_64( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, W, VL, 
 *      LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONE, RCONV, WORK, 
 *      LDWORK, WORK2, INFO)
  CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
  DOUBLE COMPLEX A(LDA,*), W(*), VL(LDVL,*), VR(LDVR,*), WORK(*)
  INTEGER*8 N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  DOUBLE PRECISION ABNRM
  DOUBLE PRECISION SCALE(*), RCONE(*), RCONV(*), WORK2(*)

F95 INTERFACE

  SUBROUTINE GEEVX( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], W, VL, 
 *       [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV, [WORK], 
 *       [LDWORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
  COMPLEX(8), DIMENSION(:) :: W, WORK
  COMPLEX(8), DIMENSION(:,:) :: A, VL, VR
  INTEGER :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  REAL(8) :: ABNRM
  REAL(8), DIMENSION(:) :: SCALE, RCONE, RCONV, WORK2

  SUBROUTINE GEEVX_64( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], W, 
 *       VL, [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV, 
 *       [WORK], [LDWORK], [WORK2], [INFO])
  CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
  COMPLEX(8), DIMENSION(:) :: W, WORK
  COMPLEX(8), DIMENSION(:,:) :: A, VL, VR
  INTEGER(8) :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  REAL(8) :: ABNRM
  REAL(8), DIMENSION(:) :: SCALE, RCONE, RCONV, WORK2

void zgeevx(char balanc, char jobvl, char jobvr, char sense, int n, doublecomplex *a, int lda, doublecomplex *w, doublecomplex *vl, int ldvl, doublecomplex *vr, int ldvr, int *ilo, int *ihi, double *scale, double *abnrm, double *rcone, double *rconv, int *info);

void zgeevx_64(char balanc, char jobvl, char jobvr, char sense, long n, doublecomplex *a, long lda, doublecomplex *w, doublecomplex *vl, long ldvl, doublecomplex *vr, long ldvr, long *ilo, long *ihi, double *scale, double *abnrm, double *rcone, double *rconv, long *info);

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.

ARGUMENTS

BALANC (input)
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 (input)

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

 = 'V': left eigenvectors of A are computed.
If SENSE  = 'E' or 'B', JOBVL must  = 'V'.

JOBVR (input)

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

 = 'V': right eigenvectors of A are computed.
If SENSE  = 'E' or 'B', JOBVR must  = 'V'.

SENSE (input)
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 (input)
The order of the matrix A. N > = 0.
A (input/output)
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 (input)
The leading dimension of the array A. LDA > = max(1,N).
W (output)
W contains the computed eigenvalues.
VL (output)
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 (input)
The leading dimension of the array VL. LDVL > = 1; if JOBVL = 'V', LDVL > = N.
VR (output)
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 (input)
The leading dimension of the array VR. LDVR > = 1; if JOBVR = 'V', LDVR > = N.
ILO (output)
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.
IHI (output)
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 (output)
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 (output)
The one-norm of the balanced matrix (the maximum of the sum of absolute values of elements of any column).
RCONE (output)
RCONE(j) is the reciprocal condition number of the j-th eigenvalue.
RCONV (output)
RCONV(j) is the reciprocal condition number of the j-th right eigenvector.
WORK (workspace)
On exit, if INFO = 0, WORK(1) returns the optimal LDWORK.
LDWORK (input)
The dimension of the array WORK. If SENSE = 'N' or 'E', LDWORK > = max(1,2*N), and if SENSE = 'V' or 'B', LDWORK > = N*N+2*N. For good performance, LDWORK must generally be larger.
If LDWORK = -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 LDWORK is issued by XERBLA.
WORK2 (workspace)
dimension(2*N)

INFO (output)

 = 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.

NAME

SYNOPSIS

F95 INTERFACE

C INTERFACE

PURPOSE

ARGUMENTS