.TH DTREVC 1 "November 2006" " LAPACK routine (version 3.1) " " LAPACK routine (version 3.1) " .SH NAME DTREVC - some or all of the right and/or left eigenvectors of a real upper quasi-triangular matrix T .SH SYNOPSIS .TP 19 SUBROUTINE DTREVC( SIDE, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO ) .TP 19 .ti +4 CHARACTER HOWMNY, SIDE .TP 19 .ti +4 INTEGER INFO, LDT, LDVL, LDVR, M, MM, N .TP 19 .ti +4 LOGICAL SELECT( * ) .TP 19 .ti +4 DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * ) .SH PURPOSE DTREVC 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 DHSEQR. .br The right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by: .br .br T*x = w*x, (y**H)*T = w*(y**H) .br .br where y**H denotes the conjugate transpose of y. .br The eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T. .br .br 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. .br .SH ARGUMENTS .TP 8 SIDE (input) CHARACTER*1 = \(aqR\(aq: compute right eigenvectors only; .br = \(aqL\(aq: compute left eigenvectors only; .br = \(aqB\(aq: compute both right and left eigenvectors. .TP 8 HOWMNY (input) CHARACTER*1 .br = \(aqA\(aq: compute all right and/or left eigenvectors; .br = \(aqB\(aq: compute all right and/or left eigenvectors, backtransformed by the matrices in VR and/or VL; = \(aqS\(aq: compute selected right and/or left eigenvectors, as indicated by the logical array SELECT. .TP 8 SELECT (input/output) LOGICAL array, dimension (N) If HOWMNY = \(aqS\(aq, 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 = \(aqA\(aq or \(aqB\(aq. .TP 8 N (input) INTEGER The order of the matrix T. N >= 0. .TP 8 T (input) DOUBLE PRECISION array, dimension (LDT,N) The upper quasi-triangular matrix T in Schur canonical form. .TP 8 LDT (input) INTEGER The leading dimension of the array T. LDT >= max(1,N). .TP 8 VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM) On entry, if SIDE = \(aqL\(aq or \(aqB\(aq and HOWMNY = \(aqB\(aq, VL must contain an N-by-N matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if SIDE = \(aqL\(aq or \(aqB\(aq, VL contains: if HOWMNY = \(aqA\(aq, the matrix Y of left eigenvectors of T; if HOWMNY = \(aqB\(aq, the matrix Q*Y; if HOWMNY = \(aqS\(aq, 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 = \(aqR\(aq. .TP 8 LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= 1, and if SIDE = \(aqL\(aq or \(aqB\(aq, LDVL >= N. .TP 8 VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM) On entry, if SIDE = \(aqR\(aq or \(aqB\(aq and HOWMNY = \(aqB\(aq, VR must contain an N-by-N matrix Q (usually the orthogonal matrix Q of Schur vectors returned by DHSEQR). On exit, if SIDE = \(aqR\(aq or \(aqB\(aq, VR contains: if HOWMNY = \(aqA\(aq, the matrix X of right eigenvectors of T; if HOWMNY = \(aqB\(aq, the matrix Q*X; if HOWMNY = \(aqS\(aq, 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 = \(aqL\(aq. .TP 8 LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= 1, and if SIDE = \(aqR\(aq or \(aqB\(aq, LDVR >= N. .TP 8 MM (input) INTEGER The number of columns in the arrays VL and/or VR. MM >= M. .TP 8 M (output) INTEGER The number of columns in the arrays VL and/or VR actually used to store the eigenvectors. If HOWMNY = \(aqA\(aq or \(aqB\(aq, M is set to N. Each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns. .TP 8 WORK (workspace) DOUBLE PRECISION array, dimension (3*N) .TP 8 INFO (output) INTEGER = 0: successful exit .br < 0: if INFO = -i, the i-th argument had an illegal value .SH FURTHER DETAILS The algorithm used in this program is basically backward (forward) substitution, with scaling to make the the code robust against possible overflow. .br 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|. .br