tgsja(3)

📦
version 3.12.1 · 9 versions
, in arch rolling

🎀 NAME 🎀

okay bestie, welcome to tgsja(3) from the Library Functions Manual — it’s literally tgsja: generalized SVD of trapezoidal matrices, step in ggsvd3. slay. 💅

💅 SYNOPSIS 💅

✨ Functions ✨

Four subroutines, all iconic, all here:

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
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
subroutine stgsja (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)
    STGSJA
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

🌸 DETAILED DESCRIPTION 🌸

This is the part where the page gets into it, bestie. The full function docs are below, and ctgsja is the main character of this chunk. 🫶

🔥 FUNCTION DOCUMENTATION 🔥

🦋 ctgsja 🦋

Here’s the signature exactly as it is, 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 ✨

CTGSJA computes the generalized singular value decomposition (GSVD) of two complex upper triangular (or trapezoidal) matrices A and B. obsessed, 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,

       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.

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. iykyk. 🦋

🌈 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 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:

this is where part 1 stops, bestie — the next part picks up right here and keeps the function-doc energy going. 💜

  • MAXIT — is an INTEGER. It says how many total loops the iterative procedure is allowed to take. If after MAXIT cycles the routine still hasn’t converged, bestie, 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.

🧮 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. Slay. ✨

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. 🫶

On exit,

       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 “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 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. 👑

🎀 Parameters

  • 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 — 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 — 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 — 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 DTGSJA. See Further Details. 🌈
  • 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 — is INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • 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 — is INTEGER. The leading dimension of the array B. LDB >= max(1,P).
  • TOLA — is DOUBLE PRECISION.
  • 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 — is DOUBLE PRECISION array, dimension (N).
  • 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 — 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 — is INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
  • 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 — is INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
  • 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 — is INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
  • WORK — is DOUBLE PRECISION array, dimension (2*N).
  • NCYCLE — is INTEGER. The number of cycles required for convergence.
  • INFO —

INFO 💜

  • INFO — INFO is INTEGER, and here’s the whole mood:
= 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, and it sets the whole loop budget:
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 ✨

Okay so DTGSJA is out here essentially using a variant of the Kogbetliantz algorithm to reduce the min(L,M-K)-by-L triangular (or trapezoidal) matrix A23 and the 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 matrix, and Z**T is the transpose of Z. 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 375 of file dtgsja.f.

subroutine stgsja 👑

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

STGSJA 👑

Purpose 💖

Bestie, listen: STGSJA computes the generalized singular value decomposition (GSVD) of two real upper triangular (or trapezoidal) matrices A and B.

On entry, it is assumed that matrices A and B have the following forms, which may be obtained by the preprocessing subroutine SGGSVP 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,

       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 ``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 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.

Parameters 🔥

  • JOBU — JOBU is CHARACTER*1, and it decides what happens with U:
= '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, same energy but for V:
= '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, and it runs the Q situation:
= '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. It’s the number of rows of the matrix A. M >= 0.
  • P — P is INTEGER. It’s the number of rows of the matrix B. P >= 0.
  • N — N is INTEGER. It’s the number of columns of the matrices A and B. N >= 0.
  • K — K is INTEGER.
  • L — L is INTEGER, and together with K it specifies the subblocks in the input matrices A and B:
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 STGSJA.
See Further Details.
  • A — A is REAL 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. It’s the leading dimension of the array A. LDA >= max(1,M).
  • B — B is REAL 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. It’s 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 and
BETA(K+L+1:N)  = 0.
  • U — U is REAL array, dimension (LDU,M). On entry, if JOBU = 'U', U must contain a matrix U1 (usually the orthogonal matrix returned by SGGSVP). 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. It’s the leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
  • V — V is REAL array, dimension (LDV,P). On entry, if JOBV = 'V', V must contain a matrix V1 (usually the orthogonal matrix returned by SGGSVP). 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. It’s the leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
  • Q — Q is REAL array, dimension (LDQ,N). On entry, if JOBQ = 'Q', Q must contain a matrix Q1 (usually the orthogonal matrix returned by SGGSVP). 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. It’s the leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
  • WORK — WORK is REAL array, dimension (2*N).
  • NCYCLE
  • NCYCLE — INTEGER. Bestie, this is the number of cycles required for convergence. ✨
  • INFO — INTEGER. Exit report card, no cap:
    • = 0: successful exit, we love to see it 💖
    • < 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. Like, it tried its best, but the vibe check said no. 🫶

👑 Author 👑

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

🌈 Further Details 🌈

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

where U1, V1 and Q1 are orthogonal matrix, and Z**T is the transpose of Z. 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 375 of file stgsja.f.

🧚 ztgsja — the complex*16 GSVD sibling 🧚

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. 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 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.

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. Choose your adventure, bestie. 💅

🎀 Parameters 🎀

  • JOBU — 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. Not invited to the function, literally.
  • JOBV — 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 — 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 — INTEGER. The number of rows of the matrix A. M >= 0.
  • P — INTEGER. The number of rows of the matrix B. P >= 0.
  • N — INTEGER. The number of columns of the matrices A and B. N >= 0.
  • K — INTEGER.
  • L — 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. 🦋
  • A — 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
  • B — 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,P).
  • TOLA — DOUBLE PRECISION.
  • TOLB — 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 — DOUBLE PRECISION array, dimension (N).
  • BETA — 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 — 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 — INTEGER. The leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
  • V — 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 — INTEGER. The leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
  • Q — 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 an INTEGER, and it's the leading dimension of the array Q. The rule is LDQ >= max(1,N) if JOBQ = 'Q'; otherwise LDQ >= 1. Slay. 💅
  • WORK — WORK is a COMPLEX*16 array with dimension (2*N). That's it, that's the whole vibe. ✨
  • NCYCLE — is an INTEGER, and it tells you the number of cycles required for convergence. Lowkey important. 🔁
  • INFO — is an INTEGER, and its whole personality is:
    • = 0: successful exit, we love a happy ending 🫶
    • < 0: if INFO = -i, the i-th argument had an illegal value — not iconic, fix your args 😭
    • = 1: the procedure does not converge after MAXIT cycles — it just did not slay 💀

🧮 Internal Parameters 🧮

  • MAXIT — is an INTEGER. It specifies the total loops that the iterative procedure may take. If after MAXIT cycles the routine fails to converge, we return INFO = 1. So yeah, it's the patience meter, bestie. ⏳

👑 Author 👑

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

These are the legends behind it, periodt. 🎀

🔍 Further Details 🔍

ZTGSJA 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. Math girlie ate that. 🔥

Definition at line 376 of file ztgsja.f.

🤖 Author 🤖

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