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NAME STGEVC - compute selected left and/or right generalized eigenvectors of a pair of real upper triangular matrices (A,B) SYNOPSIS SUBROUTINE STGEVC( JOB, SIDE, SELECT, N, A, LDA, B, LDB, VL, LDVL, VR, LDVR, MM, M, WORK, INFO ) CHARACTER JOB, SIDE INTEGER INFO, LDA, LDB, LDVL, LDVR, M, MM, N LOGICAL SELECT( * ) REAL A( LDA, * ), B( LDB, * ), VL( LDVL, * ), VR( LDVR, * ), WORK( N, * ) PURPOSE STGEVC computes selected left and/or right generalized eigenvectors of a pair of real upper triangular matrices (A,B). The j-th generalized left and right eigenvectors are y and x, resp., such that: H H y (A - wB) = 0 or (A - wB) y = 0 and (A - wB)x = 0 H Note: the left eigenvector is sometimes defined as the row vector y but STGEVC computes the column vector y. Reminder: the eigenvectors may be real or complex. If com- plex, the eigenvector for the eigenvalue w s.t. Im(w) > 0 is computed. ARGUMENTS JOB (input) CHARACTER*1 = 'A': compute All (left/right/left+right) general- ized eigenvectors of (A,B); = 'S': compute Selected (left/right/left+right) generalized eigenvectors of (A,B) -- see the description of the argument SELECT; = 'B' or 'T': compute all (left/right/left+right) generalized eigenvectors of (A,B), and Back Transform them using the initial contents of VL/VR -- see the descriptions of the arguments VL and VR. SIDE (input) CHARACTER*1 Specifies for which side eigenvectors are to be com- puted: = 'R': compute right eigenvectors only; = 'L': compute left eigenvectors only; = 'B': compute both right and left eigenvectors. SELECT (input) LOGICAL array, dimension (N) If JOB='S', then SELECT specifies the (generalized) eigenvectors to be computed. To get the eigenvector corresponding to the j-th eigenvalue, set SELECT(j) to .TRUE. If the j-th and (j+1)-st eigenvalues are conjugates, i.e., A(j+1,j) is nonzero, then only the eigenvector for the first may be selected (the second being just the conjugate of the first); this may be done by setting either SELECT(j) or SELECT(j+1) to .TRUE. If JOB='A', 'B', or 'T', SELECT is not referenced, and all eigenvectors are selected. N (input) INTEGER The order of the matrices A and B. N >= 0. A (input) REAL array, dimension (LDA,N) One of the pair of matrices whose generalized eigen- vectors are to be computed. It must be block upper triangular, with 1-by-1 or 2-by-2 blocks on the diagonal, the 1-by-1 blocks corresponding to real generalized eigenvalues and the 2-by-2 blocks corresponding to complex generalized eigenvalues. The eigenvalues are computed from the diagonal blocks of A and corresponding entries of B. LDA (input) INTEGER The leading dimension of array A. LDA >= max(1, N). B (input) REAL array, dimension (LDB,N) The other of the pair of matrices whose generalized eigenvectors are to be computed. It must be upper triangular, and if A has a 2-by-2 diagonal block in rows/columns j,j+1, then the corresponding 2-by-2 block of B must be diagonal with positive entries. LDB (input) INTEGER The leading dimension of array B. LDB >= max(1, N). VL (input/output) REAL array, dimension (LDVL,MM) On exit, the left eigenvectors (column vectors -- see the note in "Purpose".) Real eigenvectors take one column, complex take two columns, the first for the real part and the second for the imaginary part. If JOB='A', then all left eigenvectors of (A,B) will be computed and stored in VL. If JOB='S', then only the eigenvectors selected by SELECT will be com- puted, and they will be stored one right after another in VL; the first selected eigenvector will go in column 1 (and 2, if complex), the second in the next column(s), etc. If JOB='B' or 'T', then all left eigenvectors of (A,B) will be computed and multiplied (on the left) by the matrix found in VL on entry to STGEVC. Usually, this will be the Q matrix computed by SGGHRD and SHGEQZ, so that on exit, VL will contain the left eigenvectors of the original matrix pair. In any case, each eigenvector will be scaled so the largest component of each vec- tor has abs(real part) + abs(imag. part)=1, *unless* the diagonal blocks in A and B corresponding to the eigenvector are both zero (hence, 1-by-1), in which case the eigenvector will be zero. If SIDE = 'R', VL is not referenced. LDVL (input) INTEGER The leading dimension of array VL. LDVL >= 1; if SIDE = 'B' or 'L', LDVL >= N. VR (input/output) COMPLEX array, dimension (LDVR,MM) On exit, the right eigenvectors. Real eigenvectors take one column, complex take two columns, the first for the real part and the second for the imaginary part. If JOB='A', then all right eigenvectors of (A,B) will be computed and stored in VR. If JOB='S', then only the eigenvectors selected by SELECT will be computed, and they will be stored one right after another in VR; the first selected eigen- vector will go in column 1 (and 2, if complex), the second in the next column(s), etc. If JOB='B' or 'T', then all right eigenvectors of (A,B) will be computed and multiplied (on the left) by the matrix found in VR on entry to STGEVC. Usually, this will be the Z matrix computed by SGGHRD and SHGEQZ, so that on exit, VR will contain the right eigenvectors of the original matrix pair. In any case, each eigenvector will be scaled so the largest component of each vector has abs(real part) + abs(imag. part)=1, *unless* the diagonal blocks in A and B corresponding to the eigenvector are both zero (hence, 1-by-1), in which case the eigenvector will be zero. If SIDE = 'L', VR is not referenced. LDVR (input) INTEGER The leading dimension of array VR. LDVR >= 1; if SIDE = 'B' or 'R', LDVR >= N. MM (input) INTEGER The number of columns in VL and/or VR. If JOB='A', 'B', or 'T', then MM >= N. If JOB='S', then MM must be at least the number of columns required, as computed from SELECT. Each .TRUE. value in SELECT corresponding to a real eigenvalue (i.e., A(j+1,j) and A(j,j-1) are zero) counts for one column, and each .TRUE. value corresponding to the first of a complex conjugate pair (i.e., A(j+1,j) is not zero) counts for two columns. (.TRUE. values correspond- ing to the second of a pair -- A(j,j-1) is not zero -- are ignored.) M (output) INTEGER The number of columns in VL and/or VR actually used to store the eigenvectors. WORK (workspace) REAL array, dimension ( N, 6 ) INFO (output) 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. FURTHER DETAILS Allocation of workspace: ---------- -- --------- WORK( j, 1 ) = 1-norm of j-th column of A, above the diagonal WORK( j, 2 ) = 1-norm of j-th column of B, above the diagonal WORK( *, 3 ) = real part of eigenvector WORK( *, 4 ) = imaginary part of eigenvector WORK( *, 5 ) = real part of back-transformed eigenvector WORK( *, 6 ) = imaginary part of back-transformed eigen- vector Rowwise vs. columnwise solution methods: ------- -- ---------- -------- ------- Finding a generalized eigenvector consists basically of solving the singular triangular system H (A - w B) x = 0 (for right) or: (A - w B) 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 column- wise, this has the advantage that at each step, the entries of C that are accessed are adjacent to one another, whereas with the rowwise method, the entries accessed at a step are spaced LDA (and LDB) 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.