ESMF_DTREVC3 Subroutine

subroutine ESMF_DTREVC3(SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, LWORK, INFO)

\brief \b ESMF_DTREVC3 \htmlonly Download ESMF_DTREVC3 + dependencies [TGZ] [ZIP] [TXT] \endhtmlonly \par Purpose:

\verbatim

ESMF_DTREVC3 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 = QTQ**T, as computed by ESMF_DHSEQR.

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

Tx = wx, (yH)T = w(yH)

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 QX and/or QY, where Q is an input matrix. If Q is the orthogonal factor that reduces a matrix A to Schur form T, then QX and QY are the matrices of right and left eigenvectors of A.

This uses a Level 3 BLAS version of the back transformation. \endverbatim \param[in] SIDE \verbatim SIDE is CHARACTER*1 = ‘R’: compute right eigenvectors only; = ‘L’: compute left eigenvectors only; = ‘B’: compute both right and left eigenvectors. \endverbatim

\param[in] HOWMNY \verbatim 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. \endverbatim

\param[in,out] SELECT \verbatim 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’. \endverbatim

\param[in] N \verbatim N is INTEGER The order of the matrix T. N >= 0. \endverbatim

\param[in] T \verbatim T is DOUBLE PRECISION array, dimension (LDT,N) The upper quasi-triangular matrix T in Schur canonical form. \endverbatim

\param[in] LDT \verbatim LDT is INTEGER The leading dimension of the array T. LDT >= max(1,N). \endverbatim

\param[in,out] VL \verbatim VL is DOUBLE PRECISION 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 ESMF_DHSEQR). 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’. \endverbatim

\param[in] LDVL \verbatim LDVL is INTEGER The leading dimension of the array VL. LDVL >= 1, and if SIDE = ‘L’ or ‘B’, LDVL >= N. \endverbatim

\param[in,out] VR \verbatim VR is DOUBLE PRECISION 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 ESMF_DHSEQR). 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’. \endverbatim

\param[in] LDVR \verbatim LDVR is INTEGER The leading dimension of the array VR. LDVR >= 1, and if SIDE = ‘R’ or ‘B’, LDVR >= N. \endverbatim

\param[in] MM \verbatim MM is INTEGER The number of columns in the arrays VL and/or VR. MM >= M. \endverbatim

\param[out] M \verbatim 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. \endverbatim

\param[out] WORK \verbatim WORK is DOUBLE PRECISION array, dimension (MAX(1,LWORK)) \endverbatim

\param[in] LWORK \verbatim LWORK is INTEGER The dimension of array WORK. LWORK >= max(1,3N). For optimum performance, LWORK >= N + 2N*NB, where NB is the optimal blocksize.

     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 ESMF_XERBLA.

\endverbatim

\param[out] INFO \verbatim INFO is INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value \endverbatim \author Univ. of Tennessee \author Univ. of California Berkeley \author Univ. of Colorado Denver \author NAG Ltd. \date November 2017 \ingroup doubleOTHERcomputational \par Further Details:

\verbatim

The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow.

Each eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the magnitude of a complex number (x,y) is taken to be |x| + |y|. \endverbatim

Arguments