/home/abui.../cggsvd3.f(3)

💖 CGGSVD3 — Library Functions Manual 💖

Hiii bestie! This is the Library Functions Manual page for /home/abui.../cggsvd3.f(3), and CGGSVD3 is literally the main character here. Buckle up, because we're doing linear algebra with group-chat energy. 🫶

🎀 NAME 🎀

/home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/cggsvd3.f

That's the file path, periodt. No cap, that's the whole NAME section.

🌈 SYNOPSIS 🌈

🛠️ Functions/Subroutines 🛠️

subroutine CGGSVD3 (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)

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices — yes, OTHER is in all caps and we respect it. ✨

📚 Function/Subroutine Documentation 📚

Here's the full signature, with types and all the dimension(...) drama:

subroutine CGGSVD3 (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, complex, dimension( lda,
    * ) a, integer lda, complex, dimension( ldb, * ) b, integer ldb, 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 lwork, real, dimension( * ) rwork, integer, dimension( * )
    iwork, integer info)

CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices — yep, same iconic one-liner. 💅

💫 Purpose: 💫

Okay so CGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and a 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 equal 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, and D1 and D2 are M-by-(K+L) and P-by-(K+L) matrices with 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 as ( 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 — CHARACTER*1. Set it to 'U' and the unitary matrix U is computed; set it to 'N' and U is not computed.
• JOBV — CHARACTER*1. 'V' means the unitary matrix V is computed; 'N' means V is not computed.
• JOBQ — CHARACTER*1. 'Q' means the unitary matrix Q is computed; 'N' means Q is not computed.
• M — INTEGER. The number of rows of the matrix A. M >= 0.
• N — INTEGER. The number of columns of the matrices A and B. N >= 0.
• P — INTEGER. The number of rows of the matrix B. P >= 0.
• K — INTEGER.
• L — 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 — COMPLEX 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 — INTEGER. The leading dimension of the array A. LDA >= max(1,M).
• B — COMPLEX 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 — INTEGER. The leading dimension of the array B. LDB >= max(1,P).
• ALPHA — REAL array, dimension (N).
• BETA — REAL array, dimension (N). On exit, ALPHA and BETA contain the generalized singular value pairs of A and B; the exact pattern is:

            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 — COMPLEX array, dimension (LDU,M). If JOBU = 'U', U contains the M-by-M unitary matrix 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 array, dimension (LDV,P). If JOBV = 'V', V contains the P-by-P unitary matrix 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 array, dimension (LDQ,N). If JOBQ = 'Q', Q contains the N-by-N unitary matrix Q. If JOBQ = 'N', Q is not referenced.
• LDQ — INTEGER. The leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — COMPLEX array, dimension (MAX(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — INTEGER. The dimension of the array WORK. 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 — REAL array, dimension (2*N).
• IWORK — 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

That loop makes ALPHA(1) >= ALPHA(2) >= ... >= ALPHA(N).

• INFO — 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 CTGSJA.

🔬 Internal Parameters: 🔬

• TOLA — REAL
• TOLB — REAL

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: 🌟

CGGSVD3 replaces the deprecated subroutine CGGSVD. 🚨

Definition at line 351 of file cggsvd3.f.

📝 AUTHOR 📝

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