SRC/stgevc.f(3) Library Functions Manual SRC/stgevc.f(3) NAME SRC/stgevc.f SYNOPSIS Functions/Subroutines subroutine stgevc (side, howmny, select, n, s, lds, p, ldp, vl, ldvl, vr, ldvr, mm, m, work, info) STGEVC Function/Subroutine Documentation 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) STGEVC Purpose: !> !> 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). !> !> Parameters SIDE !> SIDE is CHARACTER*1 !> = 'R': compute right eigenvectors only; !> = 'L': compute left eigenvectors only; !> = 'B': compute both right and left eigenvectors. !> HOWMNY !> 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. !> SELECT !> 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 !> N is INTEGER !> The order of the matrices S and P. N >= 0. !> S !> S is REAL array, dimension (LDS,N) !> The upper quasi-triangular matrix S from a generalized Schur !> factorization, as computed by SHGEQZ. !> LDS !> LDS is INTEGER !> The leading dimension of array S. LDS >= max(1,N). !> P !> 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. !> LDP !> LDP is INTEGER !> The leading dimension of array P. LDP >= max(1,N). !> VL !> 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'. !> LDVL !> LDVL is INTEGER !> The leading dimension of array VL. LDVL >= 1, and if !> SIDE = 'L' or 'B', LDVL >= N. !> VR !> 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'. !> LDVR !> LDVR is INTEGER !> The leading dimension of the array VR. LDVR >= 1, and if !> SIDE = 'R' or 'B', LDVR >= N. !> MM !> MM is INTEGER !> The number of columns in the arrays VL and/or VR. MM >= M. !> M !> 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 !> WORK is REAL array, dimension (6*N) !> INFO !> 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. !> Author Univ. of Tennessee Univ. of California Berkeley Univ. of Colorado Denver NAG Ltd. Further Details: !> !> 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 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 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 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. !> Definition at line 293 of file stgevc.f. Author Generated automatically by Doxygen for LAPACK from the source code. LAPACK Version 3.12.0 SRC/stgevc.f(3)