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NAME DTREVC - compute all or some right and/or left eigenvectors of a real upper quasi-triangular matrix T SYNOPSIS SUBROUTINE DTREVC( JOB, HOWMNY, SELECT, N, T, LDT, VL, LDVL, VR, LDVR, MM, M, WORK, INFO ) CHARACTER HOWMNY, JOB INTEGER INFO, LDT, LDVL, LDVR, M, MM, N LOGICAL SELECT( * ) DOUBLE PRECISION T( LDT, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( * ) PURPOSE DTREVC computes all or some right and/or left eigenvectors of a real upper quasi-triangular matrix T. 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. The routine may either return the matrices X and/or Y of right or left eigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input orthogonal matrix. If T was obtained from the real Schur factorization of an original matrix A = Q*T*Q**T, then Q*X and/or Q*Y are the matrices of right or left eigenvectors of A. T must be in Schur canonical form (as returned by DHSEQR), that is, block upper triangular with 1-by-1 and 2-by-2 diag- onal blocks; each 2-by-2 diagonal block has its diagonal elements equal and its off-diagonal elements of opposite sign. ARGUMENTS JOB (input) CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. HOWMNY (input) CHARACTER*1 = 'A': compute all right and/or left eigenvectors; = 'O': compute all right and/or left eigenvectors, multiplied on the left by an input (generally orthogonal) matrix; = 'S': compute some right and/or left eigenvectors, specified by the logical array SELECT. SELECT (input/output) LOGICAL array, dimension (N) If HOWMNY = 'S', SELECT specifies the eigenvectors to be computed. To select the real eigenvector corresponding to a real eigenvalue w(j), SELECT(j) must be set to .TRUE.. To select the complex eigen- vector corresponding to a complex conjugate pair w(j) and w(j+1), either SELECT(j) or SELECT(j+1) must be set to .TRUE.; then on exit SELECT(j) is .TRUE. and SELECT(j+1) is .FALSE.. If HOWMNY = 'A' or 'O', SELECT is not referenced. N (input) INTEGER The order of the matrix T. N >= 0. T (input) DOUBLE PRECISION array, dimension (LDT,N) The upper quasi-triangular matrix T in Schur canoni- cal form. LDT (input) INTEGER The leading dimension of the array T. LDT >= max(1,N). (LDVL,MM) VL (input/output) DOUBLE PRECISION array, dimension On entry, if JOB = 'L' or 'B' and HOWMNY = 'O', VL must contain an N-by-N matrix Q (usually the orthog- onal matrix Q of Schur vectors returned by DHSEQR). On exit, if JOB = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of T; if HOWMNY = 'O', 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. If JOB = 'R', VL is not referenced. LDVL (input) INTEGER The leading dimension of the array VL. LDVL >= max(1,N). (LDVR,MM) VR (input/output) DOUBLE PRECISION array, dimension On entry, if JOB = 'R' or 'B' and HOWMNY = 'O', VR must contain an N-by-N matrix Q (usually the orthog- onal matrix Q of Schur vectors returned by DHSEQR). On exit, if JOB = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of T; if HOWMNY = 'O', 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. If JOB = 'L', VR is not referenced. LDVR (input) INTEGER The leading dimension of the array VR. LDVR >= max(1,N). MM (input) INTEGER The number of columns in the arrays VL and/or VR. MM >= M. M (output) INTEGER The number of columns in the arrays VL and/or VR required to store the eigenvectors; each selected real eigenvector occupies one column and each selected complex eigenvector occupies two columns. If HOWMNY = 'A' or 'O', M is set to N. WORK (workspace) DOUBLE PRECISION array, dimension (3*N) INFO (output) INTEGER = 0: successful exit < 0: if INFO = -i, the i-th argument had an illegal value FURTHER DETAILS The algorithm used in this program is basically backward (forward) substitution, with scaling to make the code robust against possible overflow. Each eigenvector is normalized so that the element of larg- est magnitude has magnitude 1; here the magnitude of a com- plex number (x,y) is taken to be |x| + |y|.