SRC/dtgsja.f(3)

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version 3.12.1 · 7 versions
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💖 SRC/dtgsja.f — DTGSJA 💖

okay bestie, this is the Library Functions Manual page for SRC/dtgsja.f(3), and the subroutine is DTGSJA. it's giving generalized singular value decomposition and we are seated. 🫶✨

🌸 NAME 🌸

SRC/dtgsja.f

🦋 SYNOPSIS 🦋

🧚 Functions/Subroutines 🧚

       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)

DTGSJA

💅 Function/Subroutine Documentation 💅

🫧 subroutine dtgsja 🫧

   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)

DTGSJA

🎯 Purpose 🎯

okay so DTGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B. yes bestie, two matrices. we're fancy. 💅

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. got it? good. 🫶

On exit, the tea is:

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 the so-called “diagonal” matrices, with 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. choose your fighter, bestie. ✨

🧩 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 DTGSJA. See Further Details.

A

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

LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).

B

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

LDB is INTEGER. The leading dimension of the array B. LDB >= max(1,P).

TOLA

TOLA is DOUBLE PRECISION.

TOLB

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

ALPHA is DOUBLE PRECISION array, dimension (N).

BETA

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

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

LDU is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.

V

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

LDV is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.

Q

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

LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.

WORK

WORK is DOUBLE PRECISION 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.

🔧 Internal Parameters 🔧

the source has a little =================== underline under this one, and then the MAXIT tea:

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.

Also the raw comment block has a tiny .fi at the end — yes, really: .fi. 💀

👑 Author 👑

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

📚 Further Details 📚

okay so bestie, here's the actual math part — DTGSJA 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 this form:

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

where U1, V1 and Q1 are orthogonal matrices, and Z**T is the transpose of Z (just so we're clear ✨). C1 and S1 are diagonal matrices satisfying:

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

and R1 is an L-by-L nonsingular upper triangular matrix. iconic behavior, honestly 💅.

Definition at line 375 of file dtgsja.f.

👑 Author 👑

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and that's the tea ☕ — this page is from LAPACK Version 3.12.0, file SRC/dtgsja.f(3). slay. 💖

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