DTGSJA(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 dtgsja.f(3) — Library Functions Manual 💅

okay bestie, welcome to the Library Functions Manual page for dtgsja.f(3). This one is giving big-brain Fortran energy: DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B. No cap, it's a lot, but we're gonna get through it together. 🫶

🦋 NAME 🦋

Girl, the NAME section is literally just the source path. Here it is, exactly:

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

✨ SYNOPSIS ✨

🌸 Functions/Subroutines 🌸

Under Functions/Subroutines, we have this icon:

subroutine DTGSJA (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)

And yes, she's called DTGSJA. Main character energy, honestly. 💅

💖 Function/Subroutine Documentation 💖

Here's the full signature in all its glory:

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

🎀 Purpose 🎀

So, DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B. 💖

On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine DGGSVP 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. 🌸

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 ``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.

And this next bit is a whole vibe:

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. 💫

🌸 Parameters 🌸

  • JOBU — is CHARACTER*1, and here are her settings:
    • = '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. That's it, she's just 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 DTGSJA. See Further Details.
  • A — is DOUBLE PRECISION 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 DOUBLE PRECISION 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 DOUBLE PRECISION. Simple, iconic, no notes.
  • TOLB — is DOUBLE PRECISION. 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)*MAZHEPS,
TOLB = max(P,N)*norm(B)*MAZHEPS.
  • ALPHA — is DOUBLE PRECISION array, dimension (N).
  • BETA — is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDU,M). On entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix returned by DGGSVP). 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 DOUBLE PRECISION array, dimension (LDV,P). On entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix returned by DGGSVP). 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 DOUBLE PRECISION array, dimension (LDQ,N). On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix returned by DGGSVP). 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 DOUBLE PRECISION array, dimension (2*N).
  • NCYCLE — is INTEGER. The number of cycles required for convergence.
  • 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.

The original doc block also closes with .fi, so keeping that detail for the completionists: .fi 🫶

👑 Author 👑

These are the legends:

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

🌸 Further Details 🌸

okay bestie, here’s the math tea 👑 — DTGSJA essentially uses a variant of the Kogbetliantz algorithm to reduce the min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and the L-by-L matrix B13 into this 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. slay 💅✨

Definition at line 375 of file dtgsja.f. 🦋

🫶 Author 🫶

Generated automatically by Doxygen for LAPACK from the source code. iconic auto-gen behavior fr fr 🎀

💖 cute sign-off: the footer tea is LAPACK Version 3.12.0, source file /home/abui...RC/dtgsja.f(3) — and no date is listed in that footer line, so we’re not inventing one, periodt. bye bestie, go reduce those matrices 🌈✨

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