ZTGSJA(3)

📦 version 3.12.0, in opensuse-leap 16.0

💅 ZTGSJA(3) 💅

okay so welcome to ZTGSJA(3) — the LAPACK page for the routine that does the most: it computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices. yes that's a mouthful, yes it's iconic, no you don't have to say it out loud. let's get into it ✨

💖 NAME 💖

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

that's the file, that's the name, that's the whole vibe 💅

🎀 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 — the main character of this entire page, no cap 🫶

🌈 Function/Subroutine Documentation 🌈

🌟 subroutine ZTGSJA 🌟

here's the full signature bestie, every type and every dimension spec, exactly as it is:

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. like, that's literally the job, and it ate 🔥

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. got it? slay 🌸

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.

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. your call, bestie, both are valid 💅

🦋 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. (that's it, that's the whole description, she's mysterious like that 🙈)
  • 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 in there is literally in the original, we don't touch it, periodt 💀)
  • 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.
  • 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. (iconic, no notes)
  • 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). (that's the whole description, she said what she said 🎀)
  • 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. so like, that's the deadline, and she WILL report you if you miss it 😭

🍓 Author 🍓

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

🧚 Further Details 🧚

(okay bestie this section is literally empty in the original — the further details are giving nothing yet, more incoming in part 2 👀✨💖)

...and that's the whole vibe of the reduction 💖 bestie, let me break the math down for you because it's lowkey iconic:

ZTGSJA 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 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. Periodt. ✨

Ngl the way C1**2 + S1**2 = I just works out is giving main character energy — it's literally the whole point of the reduction, so don't sleep on it 🫶

Definition at line 376 of file ztgsja.f. (iykyk — that's where the magic lives, fr fr 📍)

🎀 Author 🎀

Generated automatically by Doxygen for LAPACK from the source code. Not me being a Doxygen-generated girlypop 💅 — yes, the docs wrote themselves, we just made them cute.

✨ cute little sign-off ✨

That's a wrap bestie! 🦋 This lil masterpiece is from LAPACK, Version 3.12.0, straight out of /home/abui...RC/ztgsja.f(3) — go forth and reduce those matrices, you absolute icon 💜🌸🎀

📦 where this page comes from