SRC/ztgsja.f(3)

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🌸 `SRC/ztgsja.f(3)` — Library Functions Manual 🌸

okay bestie, this is the (3) manual page for SRC/ztgsja.f, and it's giving big numerical linear algebra energy. No cap, we're keeping the whole routine, every flag, every matrix, every warning. 🫶

✨ NAME ✨

SRC/ztgsja.f

🎀 SYNOPSIS 🎀

Functions/Subroutines

subroutine ztgsja (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)
    ZTGSJA

💅 Function/Subroutine Documentation 💅

subroutine ztgsja (character jobu, character jobv, character jobq, integer
    m, integer p, integer n, integer k, integer l, complex*16, dimension(
    lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
    double precision tola, double precision tolb, double precision,
    dimension( * ) alpha, double precision, dimension( * ) beta, complex*16,
    dimension( ldu, * ) u, integer ldu, complex*16, dimension( ldv, * ) v,
    integer ldv, complex*16, dimension( ldq, * ) q, integer ldq, complex*16,
    dimension( * ) work, integer ncycle, integer info)
    ZTGSJA

Purpose

ZTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. Main character math, honestly. 💖

On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine ZGGSVP 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. iykyk, the shape lore is real. 🦋

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. periodt, that detail stays. ✨

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. obsessed with the flexibility, honestly. 💜

🌈 Parameters 🌈

JOBU — 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.

JOBV — 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 — 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 — 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 ZTGSJA. See Further Details. yes, that double comma is literally in the original — not me, bestie. 😭

A — A is COMPLEX*16 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. the way that output slice is doing the most. 🍓
LDA — LDA is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
B — B is COMPLEX*16 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 COMPLEX*16 array, dimension (LDU,M). On entry, if JOBU = 'U', U must contain a matrix U1 (usually the unitary matrix returned by ZGGSVP). 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*16 array, dimension (LDV,P). On entry, if JOBV = 'V', V must contain a matrix V1 (usually the unitary matrix returned by ZGGSVP). 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*16 array, dimension (LDQ,N). On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the unitary matrix returned by ZGGSVP). 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*16 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 — 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. bestie, that's the "it did not slay" exit. 💀

👑 Author 👑

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

🦋 Further Details 🦋

(no body in this part of the source page)

okay so bestie, here's the actual tea on what ZTGSJA is doing under the hood — it 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**H *A13*Q1 = C1*R1; V1**H *B13*Q1 = S1*R1,

where U1, V1 and Q1 are unitary matrix. (Yes, the original says "matrix" — we're keeping it exactly like that, no cap. 🫶)

C1 and S1 are diagonal matrices satisfying

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

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

Definition at line 376 of file ztgsja.f.

✨ Author ✨

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📦 where this page comes from

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