SRC/stgevc.f(3)

📦
version 3.12.1 · 4 versions
, in arch rolling

🌸 NAME 🌸

okay so this page is SRC/stgevc.f — yes, bestie, from the Library Functions Manual, and we’re doing SRC/stgevc.f(3) energy today. 💅

✨ 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

This is STGEVC, and the vibe is “let’s get those eigenvectors.” 🌈

💖 Purpose 💖

OK so 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 — a CHARACTER*1, and it controls which side gets the eigenvector glow-up:
    • 'R': compute right eigenvectors only;
    • 'L': compute left eigenvectors only;
    • 'B': compute both right and left eigenvectors.
  • HOWMNY — a CHARACTER*1; it tells us how many eigenvectors to compute and whether to backtransform them:
    • '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 — 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 — INTEGER; the order of the matrices S and P. N >= 0.
  • S — REAL array, dimension (LDS,N); the upper quasi-triangular matrix S from a generalized Schur factorization, as computed by SHGEQZ.
  • LDS — INTEGER; the leading dimension of array S. LDS >= max(1,N).
  • P — 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 — INTEGER; the leading dimension of array P. LDP >= max(1,N).
  • VL — 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 — INTEGER; the leading dimension of array VL. LDVL >= 1, and if SIDE = 'L' or 'B', LDVL >= N.
  • VR — 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 — INTEGER; the leading dimension of the array VR. LDVR >= 1, and if SIDE = 'R' or 'B', LDVR >= N.
  • MM — INTEGER; the number of columns in the arrays VL and/or VR. MM >= M.
  • M — 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 — REAL array, dimension (6*N).
  • INFO — 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 💖

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okay bestie, that’s the whole page 🫶 This cutie is from LAPACK Version 3.12.0, page SRC/stgevc.f(3) — go solve those eigenvectors and slay. 💅✨

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