STGSJA(3)

NAME 🎀

okay so this is stgsja.f(3) from the Library Functions Manual — the header has /home/abui...RC/stgsja.f(3) repeated like a chorus, and the NAME line is literally:

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

That's the file, bestie. Iconic path, no notes. ✨

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 (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 here's the tea: STGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B. no cap, that's the whole main-character job. 💖

On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine SGGSVP from a general M-by-N matrix A and 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 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. slay, the shapes are shaking. 🫶

On exit,

       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 so-called “diagonal” matrices, which are of the following structures:

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.

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. These matrices may either be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1. flexible AND iconic. 💅

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Parameters 🌸

• JOBU — 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 — 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 — 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 — M is INTEGER. The number of rows of the matrix A. M >= 0.
• P — P is INTEGER. The number of rows of the matrix B. P >= 0.
• N — N is INTEGER. The number of columns of the matrices A and B. N >= 0.
• K — K is INTEGER.
• L — 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. iykyk. ✨

• A — 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• B — 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 — LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,P).
• TOLA — TOLA is REAL.
• TOLB — 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 — ALPHA is REAL array, dimension (N).
• BETA — 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 — 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 — LDU is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
• V — 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 — LDV is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
• Q — 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 — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — WORK is REAL array, dimension (2*N).
• NCYCLE — NCYCLE is INTEGER. The number of cycles required for convergence.
• INFO — 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.

bestie, INFO = 1 is not a slay: it means the procedure does not converge after MAXIT cycles. 😭

Internal Parameters 🧚

• MAXIT — 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.

The source block wraps up with .fi. 💀

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Author 👑

• Univ. of Tennessee
• Univ. of California Berkeley
• Univ. of Colorado Denver
• NAG Ltd.

Further Details 🌈

okay bestie so here's the vibe of what this routine is actually doing 👑

STGSJA 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 to the form:

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

where U1, V1 and Q1 are orthogonal matrix, 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. ✨

ngl that's the whole glow-up right there — you're rotating A13 and B13 by orthogonal matrices so the result lands in that clean C1*R1 / S1*R1 shape, with the C1**2 + S1**2 = I condition holding down the diagonal like the main character it is 💅

Definition at line 375 of file stgsja.f. 📍

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🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. 🤖 no human wrote this, it's giving autopilot realness

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okay that's the whole page bestie, you ate that 🫶 this little number came straight out of /home/abui...RC/stgsja.f(3) and it's LAPACK Version 3.12.0 — iconic, versioned, documented, periodt 💖✨🎀