SRC/zggsvd3.f(3)

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💖 NAME 💖

SRC/zggsvd3.f — yes bestie, that's the whole NAME entry: minimal, clean, no notes 💅

✨ SYNOPSIS ✨

🌸 Functions/Subroutines

       subroutine zggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
           alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, rwork, iwork,
           info)

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. Slay.

🎀 Function/Subroutine Documentation 🎀

🦋 subroutine zggsvd3

   subroutine zggsvd3 (character jobu, character jobv, character jobq, integer
       m, integer n, integer p, integer k, integer l, complex*16, dimension(
       lda, * ) a, integer lda, complex*16, dimension( ldb, * ) b, integer ldb,
       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 lwork, double
       precision, dimension( * ) rwork, integer, dimension( * ) iwork, integer
       info)

ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices.

🔥 Purpose

Okay so the tea ☕: ZGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B:

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

where U, V and Q are unitary matrices. Let K+L = the effective numerical rank of the matrix (A**H,B**H)**H, then R is a (K+L)-by-(K+L) nonsingular upper triangular matrix, D1 and D2 are M-by-(K+L) and P-by-(K+L) matrices and of the following structures, respectively:

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 )
            L (  0    0   R22 )
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.

  (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 routine computes C, S, R, and optionally the unitary transformation matrices U, V and Q.

In particular, if B is an N-by-N nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of A*inv(B):

                     A*inv(B) = U*(D1*inv(D2))*V**H.

If ( A**H,B**H)**H has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:

                     A**H*A x = lambda* B**H*B x.

In some literature, the GSVD of A and B is presented in the form

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

where U and V are orthogonal and X is nonsingular, and D1 and D2 are ``diagonal''. The former GSVD form can be converted to the latter form by taking the nonsingular matrix X as

                      X = Q*(  I   0    )
                            (  0 inv(R) )

💜 Parameters

• JOBU — JOBU is CHARACTER*1; = 'U': Unitary matrix U is computed;; = 'N': U is not computed.
• JOBV — JOBV is CHARACTER*1; = 'V': Unitary matrix V is computed;; = 'N': V is not computed.
• JOBQ — JOBQ is CHARACTER*1; = 'Q': Unitary matrix Q is computed;; = 'N': Q is not computed.
• M — M is INTEGER; The number of rows of the matrix A. M >= 0.
• N — N is INTEGER; The number of columns of the matrices A and B. N >= 0.
• P — P is INTEGER; The number of rows of the matrix B. P >= 0.
• K — K is INTEGER
• L — L is INTEGER; On exit, K and L specify the dimension of the subblocks described in Purpose. K + L = effective numerical rank of (A**H,B**H)**H.
• A — A is COMPLEX*16 array, dimension (LDA,N); On entry, the M-by-N matrix A. On exit, A 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, B contains part of the triangular matrix R if M-K-L < 0. See Purpose for details.
• LDB — LDB is INTEGER; The leading dimension of the array B. LDB >= max(1,P).
• 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) = C,
• BETA(K+1:K+L) = 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
• and
• ALPHA(K+L+1:N) = 0
• BETA(K+L+1:N) = 0

• U — U is COMPLEX*16 array, dimension (LDU,M); If JOBU = 'U', U contains the M-by-M unitary matrix 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); If JOBV = 'V', V contains the P-by-P unitary matrix 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); If JOBQ = 'Q', Q contains the N-by-N unitary matrix 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 (MAX(1,LWORK)); On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — LWORK is INTEGER; The dimension of the array WORK. LWORK >= 1.

If LWORK = -1, then a workspace query is assumed; the routine only calculates the optimal size of the WORK array, returns this value as the first entry of the WORK array, and no error message related to LWORK is issued by XERBLA.

• RWORK — RWORK is DOUBLE PRECISION array, dimension (2*N)
• IWORK — IWORK is INTEGER array, dimension (N); On exit, IWORK stores the sorting information. More precisely, the following loop will sort ALPHA:

for I = K+1, min(M,K+L)
    swap ALPHA(I) and ALPHA(IWORK(I))
endfor

such that ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

• INFO — INFO is INTEGER
  • = 0: successful exit.
  • < 0: if INFO = -i, the i-th argument had an illegal value.
  • > 0: if INFO = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine ZTGSJA.

🌈 Internal Parameters

• TOLA — DOUBLE PRECISION
• TOLB — DOUBLE PRECISION

TOLA and TOLB are the thresholds to determine the effective rank of (A**H,B**H)**H. Generally, they are set to

         TOLA = MAX(M,N)*norm(A)*MACHEPS,
         TOLB = MAX(P,N)*norm(B)*MACHEPS.

The size of TOLA and TOLB may affect the size of backward errors of the decomposition.

👑 Author

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

🦋 Contributors

Ming Gu and Huan Ren, Computer Science Division, University of California at Berkeley, USA

⭐ Further Details

ZGGSVD3 replaces the deprecated subroutine ZGGSVD.

Definition at line 350 of file zggsvd3.f.

👑 Author

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