.TH "SRC/stgevc.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*- .ad l .nh .SH NAME SRC/stgevc.f .SH SYNOPSIS .br .PP .SS "Functions/Subroutines" .in +1c .ti -1c .RI "subroutine \fBstgevc\fP (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info)" .br .RI "\fBSTGEVC\fP " .in -1c .SH "Function/Subroutine Documentation" .PP .SS "subroutine stgevc (character side, character howmny, logical, dimension( * ) select, integer n, real, dimension( lds, * ) s, integer lds, real, dimension( ldp, * ) p, integer ldp, real, dimension( ldvl, * ) vl, integer ldvl, real, dimension( ldvr, * ) vr, integer ldvr, integer mm, integer m, real, dimension( * ) work, integer info)" .PP \fBSTGEVC\fP .PP \fBPurpose:\fP .RS 4 .PP .nf STGEVC computes some or all of the right and/or left eigenvectors of a pair of real matrices (S,P), where S is a quasi-triangular matrix and P is upper triangular\&. Matrix pairs of this type are produced by the generalized Schur factorization of a matrix pair (A,B): A = Q*S*Z**T, B = Q*P*Z**T as computed by SGGHRD + SHGEQZ\&. The right eigenvector x and the left eigenvector y of (S,P) corresponding to an eigenvalue w are defined by: S*x = w*P*x, (y**H)*S = w*(y**H)*P, where y**H denotes the conjugate transpose of y\&. The eigenvalues are not input to this routine, but are computed directly from the diagonal blocks of S and P\&. This routine returns the matrices X and/or Y of right and left eigenvectors of (S,P), or the products Z*X and/or Q*Y, where Z and Q are input matrices\&. If Q and Z are the orthogonal factors from the generalized Schur factorization of a matrix pair (A,B), then Z*X and Q*Y are the matrices of right and left eigenvectors of (A,B)\&. .fi .PP .RE .PP \fBParameters\fP .RS 4 \fISIDE\fP .PP .nf SIDE is CHARACTER*1 = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors\&. .fi .PP .br \fIHOWMNY\fP .PP .nf 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, specified by the logical array SELECT\&. .fi .PP .br \fISELECT\fP .PP .nf 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'\&. .fi .PP .br \fIN\fP .PP .nf N is INTEGER The order of the matrices S and P\&. N >= 0\&. .fi .PP .br \fIS\fP .PP .nf S is REAL array, dimension (LDS,N) The upper quasi-triangular matrix S from a generalized Schur factorization, as computed by SHGEQZ\&. .fi .PP .br \fILDS\fP .PP .nf LDS is INTEGER The leading dimension of array S\&. LDS >= max(1,N)\&. .fi .PP .br \fIP\fP .PP .nf P is REAL array, dimension (LDP,N) The upper triangular matrix P from a generalized Schur factorization, as computed by SHGEQZ\&. 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S must be in positive diagonal form\&. .fi .PP .br \fILDP\fP .PP .nf LDP is INTEGER The leading dimension of array P\&. LDP >= max(1,N)\&. .fi .PP .br \fIVL\fP .PP .nf 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 left Schur vectors returned by SHGEQZ)\&. On exit, if SIDE = 'L' or 'B', VL contains: if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P); if HOWMNY = 'B', the matrix Q*Y; if HOWMNY = 'S', the left eigenvectors of (S,P) 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'\&. .fi .PP .br \fILDVL\fP .PP .nf LDVL is INTEGER The leading dimension of array VL\&. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N\&. .fi .PP .br \fIVR\fP .PP .nf 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 Z (usually the orthogonal matrix Z of right Schur vectors returned by SHGEQZ)\&. On exit, if SIDE = 'R' or 'B', VR contains: if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P); if HOWMNY = 'B' or 'b', the matrix Z*X; if HOWMNY = 'S' or 's', the right eigenvectors of (S,P) 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'\&. .fi .PP .br \fILDVR\fP .PP .nf LDVR is INTEGER The leading dimension of the array VR\&. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N\&. .fi .PP .br \fIMM\fP .PP .nf MM is INTEGER The number of columns in the arrays VL and/or VR\&. MM >= M\&. .fi .PP .br \fIM\fP .PP .nf 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\&. .fi .PP .br \fIWORK\fP .PP .nf WORK is REAL array, dimension (6*N) .fi .PP .br \fIINFO\fP .PP .nf INFO is INTEGER = 0: successful exit\&. < 0: if INFO = -i, the i-th argument had an illegal value\&. > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex eigenvalue\&. .fi .PP .RE .PP \fBAuthor\fP .RS 4 Univ\&. of Tennessee .PP Univ\&. of California Berkeley .PP Univ\&. of Colorado Denver .PP NAG Ltd\&. .RE .PP \fBFurther Details:\fP .RS 4 .PP .nf Allocation of workspace: ---------- -- --------- WORK( j ) = 1-norm of j-th column of A, above the diagonal WORK( N+j ) = 1-norm of j-th column of B, above the diagonal WORK( 2*N+1:3*N ) = real part of eigenvector WORK( 3*N+1:4*N ) = imaginary part of eigenvector WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector Rowwise vs\&. columnwise solution methods: ------- -- ---------- -------- ------- Finding a generalized eigenvector consists basically of solving the singular triangular system (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left) Consider finding the i-th right eigenvector (assume all eigenvalues are real)\&. The equation to be solved is: n i 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,\&. \&. \&.,1 k=j k=j where C = (A - w B) (The components v(i+1:n) are 0\&.) The 'rowwise' method is: (1) v(i) := 1 for j = i-1,\&. \&. \&.,1: i (2) compute s = - sum C(j,k) v(k) and k=j+1 (3) v(j) := s / C(j,j) Step 2 is sometimes called the 'dot product' step, since it is an inner product between the j-th row and the portion of the eigenvector that has been computed so far\&. The 'columnwise' method consists basically in doing the sums for all the rows in parallel\&. As each v(j) is computed, the contribution of v(j) times the j-th column of C is added to the partial sums\&. Since FORTRAN arrays are stored columnwise, this has the advantage that at each step, the elements of C that are accessed are adjacent to one another, whereas with the rowwise method, the elements accessed at a step are spaced LDS (and LDP) words apart\&. When finding left eigenvectors, the matrix in question is the transpose of the one in storage, so the rowwise method then actually accesses columns of A and B at each step, and so is the preferred method\&. .fi .PP .RE .PP .PP Definition at line \fB293\fP of file \fBstgevc\&.f\fP\&. .SH "Author" .PP Generated automatically by Doxygen for LAPACK from the source code\&.