SRC/stgsja.f(3)

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💖 `SRC/stgsja.f` 💖

from the Library Functions Manual

okay bestie, sit down, hydrate, because this one is a whole ✨ journey ✨. <pink>stgsja</pink> is the Fortran routine that finishes the job SGGSVP started — it's the GSVD glow-up part two, and it's lowkey iconic. 🎀

✨ SYNOPSIS ✨

🦋 Functions/Subroutines

subroutine stgsja (jobu, jobv, jobq, m, p, n, k, l, a, lda, b, ldb,
    tola, tolb, alpha, beta, u, ldu, v, ldv, q, ldq, work, ncycle, info)
    STGSJA

💅 Function/Subroutine Documentation 💅

👑 subroutine `stgsja`

here's the full signature so you can clock every single argument, no cap:

subroutine stgsja (character jobu, character jobv, character jobq, integer
    m, integer p, integer n, integer k, integer l, real, dimension( lda, * )
    a, integer lda, real, dimension( ldb, * ) b, integer ldb, real tola,
    real tolb, real, dimension( * ) alpha, real, dimension( * ) beta, real,
    dimension( ldu, * ) u, integer ldu, real, dimension( ldv, * ) v, integer
    ldv, real, dimension( ldq, * ) q, integer ldq, real, dimension( * )
    work, integer ncycle, integer info)
    STGSJA

🌸 Purpose 🌸

okay so. STGSJA computes the generalized singular value decomposition, aka the GSVD, of two real upper triangular (or trapezoidal) matrices A and B. math serve 💅

On entry, it's assumed that the matrices A and B already have the following forms — which you get from the preprocessing subroutine SGGSVP working on a general M-by-N matrix A and a P-by-N matrix B:

             N-K-L  K    L
   A =    K ( 0    A12  A13 ) if M-K-L >= 0;
          L ( 0     0   A23 )
      M-K-L ( 0     0    0  )

           N-K-L  K    L
   A =  K ( 0    A12  A13 ) if M-K-L < 0;
      M-K ( 0     0   A23 )

           N-K-L  K    L
   B =  L ( 0     0   B13 )
      P-L ( 0     0    0  )

where the K-by-K matrix A12 and the L-by-L matrix B13 are nonsingular upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0, otherwise A23 is (M-K)-by-L upper trapezoidal. bestie that's a mouthful but we're locked in 🔒

On exit, you get:

       U**T *A*Q = D1*( 0 R ),    V**T *B*Q = D2*( 0 R ),

where U, V and Q are orthogonal matrices. R is a nonsingular upper triangular matrix, and D1 and D2 are "diagonal" matrices (yes, in air quotes — they're structured like this, keep reading) 🫶

If M-K-L >= 0:

                    K  L
       D1 =     K ( I  0 )
                L ( 0  C )
            M-K-L ( 0  0 )

                  K  L
       D2 = L   ( 0  S )
            P-L ( 0  0 )

               N-K-L  K    L
  ( 0 R ) = K (  0   R11  R12 ) K
            L (  0    0   R22 ) L

where

  C = diag( ALPHA(K+1), ... , ALPHA(K+L) ),
  S = diag( BETA(K+1),  ... , BETA(K+L) ),
  C**2 + S**2 = I.

  R is stored in A(1:K+L,N-K-L+1:N) on exit.

and if M-K-L < 0:

               K M-K K+L-M
    D1 =   K ( I  0    0   )
         M-K ( 0  C    0   )

                 K M-K K+L-M
    D2 =   M-K ( 0  S    0   )
         K+L-M ( 0  0    I   )
           P-L ( 0  0    0   )

               N-K-L  K   M-K  K+L-M
( 0 R ) =    K ( 0    R11  R12  R13  )
          M-K ( 0     0   R22  R23  )
        K+L-M ( 0     0    0   R33  )

where

C = diag( ALPHA(K+1), ... , ALPHA(M) ),
S = diag( BETA(K+1),  ... , BETA(M) ),
C**2 + S**2 = I.

R = ( R11 R12 R13 ) is stored in A(1:M, N-K-L+1:N) and R33 is stored
    (  0  R22 R23 )
in B(M-K+1:L,N+M-K-L+1:N) on exit.

the computation of the orthogonal transformation matrices U, V or Q is optional — genuinely your call, main character choice. these matrices may either be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1. slay, flexible ✨

🎀 Parameters 🎀

JOBU — is CHARACTER*1
- = 'U' — U must contain an orthogonal matrix U1 on entry, and the product U1*U is returned;
- = 'I' — U is initialized to the unit matrix, and the orthogonal matrix U is returned;
- = 'N' — U is not computed.
JOBV — is CHARACTER*1
- = 'V' — V must contain an orthogonal matrix V1 on entry, and the product V1*V is returned;
- = 'I' — V is initialized to the unit matrix, and the orthogonal matrix V is returned;
- = 'N' — V is not computed.
JOBQ — is CHARACTER*1
- = 'Q' — Q must contain an orthogonal matrix Q1 on entry, and the product Q1*Q is returned;
- = 'I' — Q is initialized to the unit matrix, and the orthogonal matrix Q is returned;
- = 'N' — Q is not computed.
M — is INTEGER. the number of rows of the matrix A. M >= 0.
P — is INTEGER. the number of rows of the matrix B. P >= 0.
N — is INTEGER. the number of columns of the matrices A and B. N >= 0.
K — is INTEGER.
L — is INTEGER.

K and L specify the subblocks in the input matrices A and B: A23 = A(K+1:MIN(K+L,M),N-L+1:N) and B13 = B(1:L,N-L+1:N) of A and B, whose GSVD is going to be computed by STGSJA. see Further Details (coming at you in the next part 👀).

A — is REAL array, dimension (LDA,N). on entry, the M-by-N matrix A. on exit, A(N-K+1:N,1:MIN(K+L,M) ) contains the triangular matrix R or part of R. see Purpose for details.
LDA — is INTEGER. the leading dimension of the array A. LDA >= max(1,M).
B — is REAL array, dimension (LDB,N). on entry, the P-by-N matrix B. on exit, if necessary, B(M-K+1:L,N+M-K-L+1:N) contains a part of R. see Purpose for details.
LDB — is INTEGER. the leading dimension of the array B. LDB >= max(1,P).
TOLA — is REAL.
TOLB — is REAL.

TOLA and TOLB are the convergence criteria for the Jacobi-Kogbetliantz iteration procedure. generally, they are the same as used in the preprocessing step, say

TOLA = max(M,N)*norm(A)*MACHEPS,
TOLB = max(P,N)*norm(B)*MACHEPS.

ALPHA — is REAL array, dimension (N).
BETA — is REAL array, dimension (N).

on exit, ALPHA and BETA contain the generalized singular value pairs of A and B:

ALPHA(1:K) = 1,
BETA(1:K)  = 0,

and if M-K-L >= 0,

ALPHA(K+1:K+L) = diag(C),
BETA(K+1:K+L)  = diag(S),

or if M-K-L < 0,

ALPHA(K+1:M)= C, ALPHA(M+1:K+L)= 0
BETA(K+1:M) = S, BETA(M+1:K+L) = 1.

furthermore, if K+L < N,

ALPHA(K+L+1:N) = 0 and
BETA(K+L+1:N)  = 0.

U — is REAL array, dimension (LDU,M). on entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix returned by SGGSVP). on exit:
- if JOBU = 'I', U contains the orthogonal matrix U;
- if JOBU = 'U', U contains the product U1*U.

if JOBU = 'N', U is not referenced. 💀

LDU — is INTEGER. the leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V — is REAL array, dimension (LDV,P). on entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix returned by SGGSVP). on exit:
- if JOBV = 'I', V contains the orthogonal matrix V;
- if JOBV = 'V', V contains the product V1*V.

if JOBV = 'N', V is not referenced.

LDV — is INTEGER. the leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q — is REAL array, dimension (LDQ,N). on entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix returned by SGGSVP). on exit:
- if JOBQ = 'I', Q contains the orthogonal matrix Q;
- if JOBQ = 'Q', Q contains the product Q1*Q.

if JOBQ = 'N', Q is not referenced.

LDQ — is INTEGER. the leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK — is REAL array, dimension (2*N).
NCYCLE — is INTEGER. the number of cycles required for convergence. how many laps did it take, bestie 🏃‍♀️
INFO — is INTEGER
- = 0 — successful exit 🎉
- < 0 — if INFO = -i, the i-th argument had an illegal value.
- = 1 — the procedure does not converge after MAXIT cycles.

🧚 Internal Parameters 🧚

MAXIT is INTEGER MAXIT specifies the total loops that the iterative procedure may take. if after MAXIT cycles, the routine fails to converge, we return INFO = 1. so yeah, it gave up — that's the vibe check failing, not you 💅

💜 Author 💜

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 Further Details

And the tea on STGSJA, bestie: it essentially uses a variant of the Kogbetliantz algorithm to reduce a min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and an L-by-L matrix B13 into this exact form:

U1**T *A13*Q1 = C1*R1; V1**T *B13*Q1 = S1*R1,

where U1, V1 and Q1 are orthogonal matrix (yes, the source says "matrix" — not me changing it), and Z**T is the transpose of Z. C1 and S1 are diagonal matrices satisfying

C1**2 + S1**2 = I,

and R1 is an L-by-L nonsingular upper triangular matrix. It's giving clean reduction, iconic diagonal energy. 💅✨

Definition at line 375 of file stgsja.f.

Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. Fr fr.

That's the little sign-off, bestie: this page is from LAPACK Version 3.12.0, for SRC/stgsja.f(3). Mwah. 🫶💖

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