/home/abui...D/cggsvd.f(3)

🦋 CGGSVD — the deprecated-but-iconic generalized SVD 💅

okay so this page lives at /home/abui...D/cggsvd.f(3) — it's a Library Functions Manual page, and honestly the vibes are immaculate but the routine is, like, retired. Anyway bestie, buckle up, we're going on a linear algebra journey ✨

🌸 NAME 🌸

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

(yes, the DEPRECATED is literally in the path. the universe was warning us. 💀)

✨ SYNOPSIS ✨

Functions/Subroutines

subroutine CGGSVD (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb, alpha, beta, u, ldu, v, ldv, q, ldq, work, rwork, iwork, info) — CGGSVD computes the singular value decomposition (SVD) for OTHER matrices 🫶

🎀 Function/Subroutine Documentation 🎀

subroutine CGGSVD (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, real, dimension( * ) rwork, integer, dimension( * ) iwork, integer info)

CGGSVD computes the singular value decomposition (SVD) for OTHER matrices 💖

Purpose: 🧚

⚠️ This routine is deprecated and has been replaced by routine CGGSVD3.

Real talk first, bestie: this routine is deprecated, and it's been replaced by CGGSVD3. So like, if you're writing new code... maybe don't? But the math is still gorgeous, so let's go 👑

CGGSVD 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 (that's it, that's the whole entry, she's mysterious like that 🙈).
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 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 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 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) = 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 array, dimension (LDU,M). If JOBU = 'U', U contains the M-by-M unitary matrix U. If JOBU = 'N', U is not referenced. (she's not even invited to the party 🫥)
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). 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 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 array, dimension (max(3*N,M,P)+N). 🫶 the scratch space, doing the unglamorous work so the main characters can shine.
RWORK — RWORK is REAL 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 CTGSJA.

🔥 Internal Parameters: 🔥

these two don't get a public option list, they're inner-circle only ✨

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. (lowkey important, ngl 👀)

👑 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

Definition at line 335 of file cggsvd.f.

⭐ Author ⭐

Generated automatically by Doxygen for LAPACK from the source code. (yes the robot wrote this bit, we stan a self-documenting queen 🤖💅)

✨ and that's the whole page, bestie ✨

<center>LAPACK — Version 3.12.0 — /home/abui...D/cggsvd.f(3) 🦋💖🍓</center>

k bye love u, go use CGGSVD3 instead fr fr 💀💜

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