NAME

SYNOPSIS

  SUBROUTINE DGEEVX( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, WI, 
 *      VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONE, RCONV, WORK, 
 *      LDWORK, IWORK2, INFO)
  CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
  INTEGER N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  INTEGER IWORK2(*)
  DOUBLE PRECISION ABNRM
  DOUBLE PRECISION A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), SCALE(*), RCONE(*), RCONV(*), WORK(*)
 
  SUBROUTINE DGEEVX_64( BALANC, JOBVL, JOBVR, SENSE, N, A, LDA, WR, 
 *      WI, VL, LDVL, VR, LDVR, ILO, IHI, SCALE, ABNRM, RCONE, RCONV, 
 *      WORK, LDWORK, IWORK2, INFO)
  CHARACTER * 1 BALANC, JOBVL, JOBVR, SENSE
  INTEGER*8 N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  INTEGER*8 IWORK2(*)
  DOUBLE PRECISION ABNRM
  DOUBLE PRECISION A(LDA,*), WR(*), WI(*), VL(LDVL,*), VR(LDVR,*), SCALE(*), RCONE(*), RCONV(*), WORK(*)
 

F95 INTERFACE

  SUBROUTINE GEEVX( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], WR, 
 *       WI, VL, [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV, 
 *       [WORK], [LDWORK], [IWORK2], [INFO])
  CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
  INTEGER :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  INTEGER, DIMENSION(:) :: IWORK2
  REAL(8) :: ABNRM
  REAL(8), DIMENSION(:) :: WR, WI, SCALE, RCONE, RCONV, WORK
  REAL(8), DIMENSION(:,:) :: A, VL, VR
 
  SUBROUTINE GEEVX_64( BALANC, JOBVL, JOBVR, SENSE, [N], A, [LDA], WR, 
 *       WI, VL, [LDVL], VR, [LDVR], ILO, IHI, SCALE, ABNRM, RCONE, RCONV, 
 *       [WORK], [LDWORK], [IWORK2], [INFO])
  CHARACTER(LEN=1) :: BALANC, JOBVL, JOBVR, SENSE
  INTEGER(8) :: N, LDA, LDVL, LDVR, ILO, IHI, LDWORK, INFO
  INTEGER(8), DIMENSION(:) :: IWORK2
  REAL(8) :: ABNRM
  REAL(8), DIMENSION(:) :: WR, WI, SCALE, RCONE, RCONV, WORK
  REAL(8), DIMENSION(:,:) :: A, VL, VR
 

C INTERFACE

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

void dgeevx_64(char balanc, char jobvl, char jobvr, char sense, long n, double *a, long lda, double *wr, double *wi, double *vl, long ldvl, double *vr, long ldvr, long *ilo, long *ihi, double *scale, double *abnrm, double *rcone, double *rconv, long *info);

PURPOSE

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.

* JOBVL (input)

* JOBVR (input)

* SENSE (input)

Determines which reciprocal condition numbers are computed.

* 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 real Schur form of the balanced version of the input matrix A.

* LDA (input)

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

* WR (output)

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.

* WI (output)

See the description for WR.

* 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. 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 (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. 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 (input)

The leading dimension of the array VR. LDVR >= 1, and 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)

See the description of ILO.

* 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

* 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 JOBVL = 'V' or JOBVR = 'V', LDWORK >= 3*N. If SENSE = 'V' or 'B', LDWORK >= N*(N+6). 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.

* IWORK2 (workspace)

If SENSE = 'N' or 'E', not referenced.

* INFO (output)