TREVC - 5.2 English - 68552

TREVC computes some or all of the right and/or left eigenvectors of

a real upper quasi-triangular matrix T.

Purpose:

  TREVC computes some or all of the right and/or left eigenvectors of
  a real upper quasi-triangular matrix T.
  Matrices of this type are produced by the Schur factorization of
  a real general matrix:  A = Q*T*Q**T, as computed by SHSEQR.

  The right eigenvector x and the left eigenvector y of T corresponding
  to an eigenvalue w are defined by:

     T*x = w*x,     (y**H)*T = w*(y**H)

  where y**H denotes the conjugate transpose of y.
  The eigenvalues are not input to this routine, but are read directly
  from the diagonal blocks of T.

  This routine returns the matrices X and/or Y of right and left
  eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an
  input matrix.  If Q is the orthogonal factor that reduces a matrix
  A to Schur form T, then Q*X and Q*Y are the matrices of right and
  left eigenvectors of A.

Parameters:

• SIDE – [in]

  SIDE is CHARACTER*1

  = ‘R’: compute right eigenvectors only;

  = ‘L’: compute left eigenvectors only;

  = ‘B’: compute both right and left eigenvectors.

• HOWMNY – [in]

  HOWMNY is CHARACTER*1

  = ‘A’: compute all right and/or left eigenvectors;

  = ‘B’: compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL;

  = ‘S’: compute selected right and/or left eigenvectors, as indicated by the logical array SELECT.

• SELECT – [inout]

  SELECT is LOGICAL array, dimension (N)

  If HOWMNY = ‘S’, SELECT specifies the eigenvectors to be computed.

  If w(j) is a real eigenvalue, the corresponding real eigenvector is computed if SELECT(j) is .TRUE..

  If w(j) and w(j+1) are the real and imaginary parts of a complex eigenvalue, the corresponding complex eigenvector is computed if either SELECT(j) or SELECT(j+1) is .TRUE., and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is set to .FALSE..

  Not referenced if HOWMNY = ‘A’ or ‘B’.

• N – [in]

  N is INTEGER

  The order of the matrix T. N >= 0.

• T – [in]

  T is REAL array, dimension (LDT,N)

  The upper quasi-triangular matrix T in Schur canonical form.

• LDT – [in]

  LDT is INTEGER

  The leading dimension of the array T. LDT >= fla_max(1,N).

• VL – [inout]

  VL is REAL array, dimension (LDVL,MM)

  On entry, if SIDE = ‘L’ or ‘B’ and HOWMNY = ‘B’, VL must contain an N-by-N matrix Q (usually the orthogonal matrix Q of Schur vectors returned by SHSEQR).

  On exit, if SIDE = ‘L’ or ‘B’, VL contains:

  if HOWMNY = ‘A’, the matrix Y of left eigenvectors of T;

  if HOWMNY = ‘B’, the matrix Q*Y;

  if HOWMNY = ‘S’, the left eigenvectors of T specified by SELECT, stored consecutively in the columns of VL, in the same order as their eigenvalues.

  A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part, and the second the imaginary part. Not referenced if SIDE = ‘R’.

• LDVL – [in]

  LDVL is INTEGER

  The leading dimension of the array VL. LDVL >= 1, and if SIDE = ‘L’ or ‘B’, LDVL >= N.

• VR – [inout]

  VR is REAL array, dimension (LDVR,MM)

  On entry, if SIDE = ‘R’ or ‘B’ and HOWMNY = ‘B’, VR must contain an N-by-N matrix Q (usually the orthogonal matrix Q of Schur vectors returned by SHSEQR).

  On exit, if SIDE = ‘R’ or ‘B’, VR contains:

  if HOWMNY = ‘A’, the matrix X of right eigenvectors of T;

  if HOWMNY = ‘B’, the matrix Q*X;

  if HOWMNY = ‘S’, the right eigenvectors of T specified by SELECT, stored consecutively in the columns of VR, in the same order as their eigenvalues.

  A complex eigenvector corresponding to a complex eigenvalue is stored in two consecutive columns, the first holding the real part and the second the imaginary part.

  Not referenced if SIDE = ‘L’.

• LDVR – [in]

  LDVR is INTEGER

  The leading dimension of the array VR. LDVR >= 1, and if SIDE = ‘R’ or ‘B’, LDVR >= N.

• MM – [in]

  MM is INTEGER

  The number of columns in the arrays VL and/or VR. MM >= M.

• M – [out]

  M is INTEGER

  The number of columns in the arrays VL and/or VR actually used to store the eigenvectors.

  If HOWMNY = ‘A’ or ‘B’, M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns.

• WORK – [out] WORK is REAL array, dimension (3*N)

• INFO – [out]

  INFO is INTEGER

  = 0: successful exit

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