DTGEVC(3)

💖 DTGEVC(3) — Library Functions Manual 💖

okay bestie, welcome to the dtgevc.f(3) page from the Library Functions Manual — this is the whole lore, and I'm gonna walk you through it with zero gatekeeping. no cap. ✨

🌸 NAME 🌸

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dtgevc.f

This is literally the file this page is about, bestie. 🫶

✨ SYNOPSIS ✨

🎀 Functions/Subroutines 🎀

       subroutine DTGEVC (side, howmny, select, n, s, lds, p, ldp, vl, ldvl,
           vr, ldvr, mm, m, work, info)
           DTGEVC

The main character is DTGEVC, periodt. 👑

💅 Function/Subroutine Documentation 💅

   subroutine DTGEVC (character side, character howmny, logical, dimension( * )
       select, integer n, double precision, dimension( lds, * ) s, integer lds,
       double precision, dimension( ldp, * ) p, integer ldp, double precision,
       dimension( ldvl, * ) vl, integer ldvl, double precision, dimension(
       ldvr, * ) vr, integer ldvr, integer mm, integer m, double precision,
       dimension( * ) work, integer info)
       DTGEVC

🦋 Purpose: 🦋

okay so DTGEVC 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 DGGHRD + DHGEQZ. slay. 🔥

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. iconic behavior, honestly. 👑

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). no cap, that's the whole point. 💅

🍓 Parameters 🍓

SIDE

SIDE is CHARACTER*1.

• 'R' — compute right eigenvectors only;
• 'L' — compute left eigenvectors only;
• 'B' — compute both right and left eigenvectors.

bestie please pick one, the routine is not a mind reader. 🙈

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 DOUBLE PRECISION array, dimension (LDS,N). The upper quasi-triangular matrix S from a generalized Schur factorization, as computed by DHGEQZ.

LDS

LDS is INTEGER. The leading dimension of array S. LDS >= max(1,N).

P

P is DOUBLE PRECISION array, dimension (LDP,N). The upper triangular matrix P from a generalized Schur factorization, as computed by DHGEQZ. 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 DOUBLE PRECISION 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 DHGEQZ).

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 DOUBLE PRECISION 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 DHGEQZ).

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 DOUBLE PRECISION 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.

the academic icons behind this whole thing, honestly. 🎓✨

🔥 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 dtgevc.f.

🤖 Author 🤖

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xoxo, bestie — signing off with LAPACK Version 3.12.0 and the source page path /home/abui...RC/dtgevc.f(3). mwah 💋