CTGSJA(3)

📦 version 3.12.0, in opensuse-leap 16.0

CTGSJA(3) — Library Functions Manual — from the ctgsja.f(3) page, bestie 💅

okay so welcome to part one, and it's giving “short but iconic” energy ✨

🌸 NAME 🌸

the NAME section is literally just this source-file path, no cap:

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

that's it. that's the whole NAME vibe 🎀

✨ SYNOPSIS ✨

📚 Functions/Subroutines 📚

the synopsis is serving one subroutine, and its name is CTGSJA — main character energy fr fr 🌟

here's the exact signature, character-for-character, because we do not play with code 💖

subroutine CTGSJA (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)
    CTGSJA

and yes, that trailing CTGSJA line is part of the synopsis too — keeping every detail is the whole point 🫶

💫 Function/Subroutine Documentation 💫

okay so here's the actual signature, bestie — copy it exactly, no cap:

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

✨ CTGSJA ✨

🎯 Purpose 🎯

bestie, this routine computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. no cap, it's a whole thing 💅

On entry, it's assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine CGGSVP 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. periodt.

On exit,

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

where U, V and Q are unitary 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. slay!

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 in B(M-K+1:L,N+M-K-L+1:N) on exit. iconic.

The computation of the unitary 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. iykyk

📋 Parameters 📋

  • JOBU is CHARACTER*1
    • 'U': U must contain a unitary matrix U1 on entry, and the product U1*U is returned.
    • 'I': U is initialized to the unit matrix, and the unitary matrix U is returned.
    • 'N': U is not computed. no thoughts, just vibes.
  • JOBV is CHARACTER*1
    • 'V': V must contain a unitary matrix V1 on entry, and the product V1*V is returned.
    • 'I': V is initialized to the unit matrix, and the unitary matrix V is returned.
    • 'N': V is not computed.
  • JOBQ is CHARACTER*1
    • 'Q': Q must contain a unitary matrix Q1 on entry, and the product Q1*Q is returned.
    • 'I': Q is initialized to the unit matrix, and the unitary 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 CTGSJA. See Further Details. (yes, that double comma is in the original, don't come for me)

  • A is COMPLEX 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 COMPLEX 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
  • BETA(K+L+1:N) = 0. the way that C2 + S2 = I is just so iconic.
  • U is COMPLEX array, dimension (LDU,M)
    • On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by CGGSVP).
    • On exit,
      • if JOBU = 'I', U contains the unitary 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 COMPLEX array, dimension (LDV,P)
    • On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by CGGSVP).
    • On exit,
      • if JOBV = 'I', V contains the unitary 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 COMPLEX array, dimension (LDQ,N)
    • On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by CGGSVP).
    • On exit,
      • if JOBQ = 'I', Q contains the unitary 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 COMPLEX 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. yikes, but okay.

🔧 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.

👑 Author 👑

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🌈 Further Details 🌈

CTGSJA essentially uses a variant of Kogbetliantz algorithm to reduce min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and L-by-L matrix B13 to the form:

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

where U1, V1 and Q1 are unitary matrix. C1 and S1 are diagonal matrices satisfying

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

and R1 is an L-by-L nonsingular upper triangular matrix.

Definition at line 376 of file ctgsja.f.

✨ Author ✨

Ok so this part is literally the humble-brag credits section, and honestly? It gives "avatar: the last airbender ending card" energy 🥹

Generated automatically by Doxygen for LAPACK from the source code.

That's it. That's the whole author section, bestie. No mysterious lone genius with a tragic backstory and a hoodie — just Doxygen doing its little robot thing and pulling this page straight out of the source code like a girlboss 💅 The real main characters are the LAPACK devs who wrote the actual .f file, but they're letting the tool take the credit on this one 🎀

ngl, don't @ me for citation purposes, okay? The source of truth is the source code, periodt. This page is a generated artifact, so if something looks weird, go read the .f file — that's the actual icon 👑

And that's a wrap on this whole page, babes! 🦋 You just read the LAPACK Version 3.12.0 docs, generated straight from /home/abui...RC/ctgsja.f(3) (yes, the path with the chaotic ellipsis in it, we don't make the rules 🌈). Go forth, reduce those upper triangular matrix elements, and slay the day fr fr 💜🍓🧚

📦 where this page comes from