SRC/ctgsja.f(3)

💅 SRC/ctgsja.f — Library Functions Manual, section 3 🎀

okay bestie, welcome to the Library Functions Manual corner of the internet, specifically section 3 — the one where all the library calls live. today's main character is SRC/ctgsja.f, and she is a whole moment. ✨

---

🌸 NAME 🌸

SRC/ctgsja.f — that's it, that's the name, no further notes. the file itself is the vibe. 💖

not me giving this entry a personality before the DESCRIPTION section even shows up 😭 — but we're going in order, so sit tight, the good stuff is coming.

---

🔥 SYNOPSIS 🔥

💫 Functions/Subroutines 💫

so the star of the show, the one, the only:

       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

like… look at that parameter list. twenty-five arguments, no cap. she's highkey extra and we love that for her. 🫶

let's do a lil vibe check on the cast, in the exact order they walk in (no descriptions yet — we're just doing introductions, the lore drops in the next part):

• 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 the name on the door is CTGSJA, capital letters, all caps, main character energy only. 👑

🦋 quick heads up bestie: this page is long, so the DESCRIPTION and everything after it is chilling in the next part. don't touch that dial. 🎀

💖 Function/Subroutine Documentation 💖

   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 🌸

Okay bestie, here’s the tea: CTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. Iconic behavior, honestly.

On entry, it is 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.

On exit, ✨ the whole vibe transforms ✨:

       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.

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 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.

---

🎀 Parameters 🎀

• JOBU — JOBU is CHARACTER*1. Choose your fighter:
  • = '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.
• JOBV — JOBV is CHARACTER*1. Same energy, different letter:
  • = '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 — JOBQ is CHARACTER*1. Q gets a turn too:
  • = '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 — 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 CTGSJA. See Further Details.

• A — 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 — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• B — 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 — 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
  BETA(K+L+1:N)  = 0.

• U — 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 — LDU is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
• V — 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 — LDV is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
• Q — 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 — LDQ is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — WORK is COMPLEX 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 🔥

• MAXIT — 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 ✨

Okay so real talk — nobody actually wrote this page by hand, bestie. It was generated automatically by Doxygen for LAPACK straight from the source code. 💅 Like, the robots did the typing, we just added the vibes. 🎀

Generated automatically by Doxygen for LAPACK from the source code.

So if you're out here like "who do I thank for this masterpiece?" — it's the source code and a doc generator, no cap. 🫶

---

And that's the whole page, bestie — you made it to the end! 🌈 This iconic little number is from LAPACK, Version 3.12.0, and it lives its best life at SRC/ctgsja.f(3). Go forth and compute, you absolute main character. 👑💖