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---
:name: dtgevc
:md5sum: 05c0ef877f01207c516547bdc1379d50
:category: :subroutine
:arguments:
- side:
:type: char
:intent: input
- howmny:
:type: char
:intent: input
- select:
:type: logical
:intent: input
:dims:
- n
- n:
:type: integer
:intent: input
- s:
:type: doublereal
:intent: input
:dims:
- lds
- n
- lds:
:type: integer
:intent: input
- p:
:type: doublereal
:intent: input
:dims:
- ldp
- n
- ldp:
:type: integer
:intent: input
- vl:
:type: doublereal
:intent: input/output
:dims:
- ldvl
- mm
- ldvl:
:type: integer
:intent: input
- vr:
:type: doublereal
:intent: input/output
:dims:
- ldvr
- mm
- ldvr:
:type: integer
:intent: input
- mm:
:type: integer
:intent: input
- m:
:type: integer
:intent: output
- work:
:type: doublereal
:intent: workspace
:dims:
- 6*n
- info:
:type: integer
:intent: output
:substitutions: {}
:fortran_help: " SUBROUTINE DTGEVC( SIDE, HOWMNY, SELECT, N, S, LDS, P, LDP, VL, LDVL, VR, LDVR, MM, M, WORK, INFO )\n\n\
* Purpose\n\
* =======\n\
*\n\
* DTGEVC computes some or all of the right and/or left eigenvectors of\n\
* a pair of real matrices (S,P), where S is a quasi-triangular matrix\n\
* and P is upper triangular. Matrix pairs of this type are produced by\n\
* the generalized Schur factorization of a matrix pair (A,B):\n\
*\n\
* A = Q*S*Z**T, B = Q*P*Z**T\n\
*\n\
* as computed by DGGHRD + DHGEQZ.\n\
*\n\
* The right eigenvector x and the left eigenvector y of (S,P)\n\
* corresponding to an eigenvalue w are defined by:\n\
* \n\
* S*x = w*P*x, (y**H)*S = w*(y**H)*P,\n\
* \n\
* where y**H denotes the conjugate tranpose of y.\n\
* The eigenvalues are not input to this routine, but are computed\n\
* directly from the diagonal blocks of S and P.\n\
* \n\
* This routine returns the matrices X and/or Y of right and left\n\
* eigenvectors of (S,P), or the products Z*X and/or Q*Y,\n\
* where Z and Q are input matrices.\n\
* If Q and Z are the orthogonal factors from the generalized Schur\n\
* factorization of a matrix pair (A,B), then Z*X and Q*Y\n\
* are the matrices of right and left eigenvectors of (A,B).\n\
* \n\n\
* Arguments\n\
* =========\n\
*\n\
* SIDE (input) CHARACTER*1\n\
* = 'R': compute right eigenvectors only;\n\
* = 'L': compute left eigenvectors only;\n\
* = 'B': compute both right and left eigenvectors.\n\
*\n\
* HOWMNY (input) CHARACTER*1\n\
* = 'A': compute all right and/or left eigenvectors;\n\
* = 'B': compute all right and/or left eigenvectors,\n\
* backtransformed by the matrices in VR and/or VL;\n\
* = 'S': compute selected right and/or left eigenvectors,\n\
* specified by the logical array SELECT.\n\
*\n\
* SELECT (input) LOGICAL array, dimension (N)\n\
* If HOWMNY='S', SELECT specifies the eigenvectors to be\n\
* computed. If w(j) is a real eigenvalue, the corresponding\n\
* real eigenvector is computed if SELECT(j) is .TRUE..\n\
* If w(j) and w(j+1) are the real and imaginary parts of a\n\
* complex eigenvalue, the corresponding complex eigenvector\n\
* is computed if either SELECT(j) or SELECT(j+1) is .TRUE.,\n\
* and on exit SELECT(j) is set to .TRUE. and SELECT(j+1) is\n\
* set to .FALSE..\n\
* Not referenced if HOWMNY = 'A' or 'B'.\n\
*\n\
* N (input) INTEGER\n\
* The order of the matrices S and P. N >= 0.\n\
*\n\
* S (input) DOUBLE PRECISION array, dimension (LDS,N)\n\
* The upper quasi-triangular matrix S from a generalized Schur\n\
* factorization, as computed by DHGEQZ.\n\
*\n\
* LDS (input) INTEGER\n\
* The leading dimension of array S. LDS >= max(1,N).\n\
*\n\
* P (input) DOUBLE PRECISION array, dimension (LDP,N)\n\
* The upper triangular matrix P from a generalized Schur\n\
* factorization, as computed by DHGEQZ.\n\
* 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks\n\
* of S must be in positive diagonal form.\n\
*\n\
* LDP (input) INTEGER\n\
* The leading dimension of array P. LDP >= max(1,N).\n\
*\n\
* VL (input/output) DOUBLE PRECISION array, dimension (LDVL,MM)\n\
* On entry, if SIDE = 'L' or 'B' and HOWMNY = 'B', VL must\n\
* contain an N-by-N matrix Q (usually the orthogonal matrix Q\n\
* of left Schur vectors returned by DHGEQZ).\n\
* On exit, if SIDE = 'L' or 'B', VL contains:\n\
* if HOWMNY = 'A', the matrix Y of left eigenvectors of (S,P);\n\
* if HOWMNY = 'B', the matrix Q*Y;\n\
* if HOWMNY = 'S', the left eigenvectors of (S,P) specified by\n\
* SELECT, stored consecutively in the columns of\n\
* VL, in the same order as their eigenvalues.\n\
*\n\
* A complex eigenvector corresponding to a complex eigenvalue\n\
* is stored in two consecutive columns, the first holding the\n\
* real part, and the second the imaginary part.\n\
*\n\
* Not referenced if SIDE = 'R'.\n\
*\n\
* LDVL (input) INTEGER\n\
* The leading dimension of array VL. LDVL >= 1, and if\n\
* SIDE = 'L' or 'B', LDVL >= N.\n\
*\n\
* VR (input/output) DOUBLE PRECISION array, dimension (LDVR,MM)\n\
* On entry, if SIDE = 'R' or 'B' and HOWMNY = 'B', VR must\n\
* contain an N-by-N matrix Z (usually the orthogonal matrix Z\n\
* of right Schur vectors returned by DHGEQZ).\n\
*\n\
* On exit, if SIDE = 'R' or 'B', VR contains:\n\
* if HOWMNY = 'A', the matrix X of right eigenvectors of (S,P);\n\
* if HOWMNY = 'B' or 'b', the matrix Z*X;\n\
* if HOWMNY = 'S' or 's', the right eigenvectors of (S,P)\n\
* specified by SELECT, stored consecutively in the\n\
* columns of VR, in the same order as their\n\
* eigenvalues.\n\
*\n\
* A complex eigenvector corresponding to a complex eigenvalue\n\
* is stored in two consecutive columns, the first holding the\n\
* real part and the second the imaginary part.\n\
* \n\
* Not referenced if SIDE = 'L'.\n\
*\n\
* LDVR (input) INTEGER\n\
* The leading dimension of the array VR. LDVR >= 1, and if\n\
* SIDE = 'R' or 'B', LDVR >= N.\n\
*\n\
* MM (input) INTEGER\n\
* The number of columns in the arrays VL and/or VR. MM >= M.\n\
*\n\
* M (output) INTEGER\n\
* The number of columns in the arrays VL and/or VR actually\n\
* used to store the eigenvectors. If HOWMNY = 'A' or 'B', M\n\
* is set to N. Each selected real eigenvector occupies one\n\
* column and each selected complex eigenvector occupies two\n\
* columns.\n\
*\n\
* WORK (workspace) DOUBLE PRECISION array, dimension (6*N)\n\
*\n\
* INFO (output) INTEGER\n\
* = 0: successful exit.\n\
* < 0: if INFO = -i, the i-th argument had an illegal value.\n\
* > 0: the 2-by-2 block (INFO:INFO+1) does not have a complex\n\
* eigenvalue.\n\
*\n\n\
* Further Details\n\
* ===============\n\
*\n\
* Allocation of workspace:\n\
* ---------- -- ---------\n\
*\n\
* WORK( j ) = 1-norm of j-th column of A, above the diagonal\n\
* WORK( N+j ) = 1-norm of j-th column of B, above the diagonal\n\
* WORK( 2*N+1:3*N ) = real part of eigenvector\n\
* WORK( 3*N+1:4*N ) = imaginary part of eigenvector\n\
* WORK( 4*N+1:5*N ) = real part of back-transformed eigenvector\n\
* WORK( 5*N+1:6*N ) = imaginary part of back-transformed eigenvector\n\
*\n\
* Rowwise vs. columnwise solution methods:\n\
* ------- -- ---------- -------- -------\n\
*\n\
* Finding a generalized eigenvector consists basically of solving the\n\
* singular triangular system\n\
*\n\
* (A - w B) x = 0 (for right) or: (A - w B)**H y = 0 (for left)\n\
*\n\
* Consider finding the i-th right eigenvector (assume all eigenvalues\n\
* are real). The equation to be solved is:\n\
* n i\n\
* 0 = sum C(j,k) v(k) = sum C(j,k) v(k) for j = i,. . .,1\n\
* k=j k=j\n\
*\n\
* where C = (A - w B) (The components v(i+1:n) are 0.)\n\
*\n\
* The \"rowwise\" method is:\n\
*\n\
* (1) v(i) := 1\n\
* for j = i-1,. . .,1:\n\
* i\n\
* (2) compute s = - sum C(j,k) v(k) and\n\
* k=j+1\n\
*\n\
* (3) v(j) := s / C(j,j)\n\
*\n\
* Step 2 is sometimes called the \"dot product\" step, since it is an\n\
* inner product between the j-th row and the portion of the eigenvector\n\
* that has been computed so far.\n\
*\n\
* The \"columnwise\" method consists basically in doing the sums\n\
* for all the rows in parallel. As each v(j) is computed, the\n\
* contribution of v(j) times the j-th column of C is added to the\n\
* partial sums. Since FORTRAN arrays are stored columnwise, this has\n\
* the advantage that at each step, the elements of C that are accessed\n\
* are adjacent to one another, whereas with the rowwise method, the\n\
* elements accessed at a step are spaced LDS (and LDP) words apart.\n\
*\n\
* When finding left eigenvectors, the matrix in question is the\n\
* transpose of the one in storage, so the rowwise method then\n\
* actually accesses columns of A and B at each step, and so is the\n\
* preferred method.\n\
*\n\
* =====================================================================\n\
*\n"
|
