/home/abui.../zggsvd3.f(3)

Library Functions Manual page for /home/abui.../zggsvd3.f(3) — yes bestie, we’re in the manual, and it’s iconic. 💖

✨ NAME ✨

The source file for this page is literally:

/home/abuild/rpmbuild/BUILD/lapack-
static-3.12.0-build/lapack-3.12.0/SRC/zggsvd3.f

🦋 SYNOPSIS 🦋

Functions/Subroutines

Here’s the call signature, character-for-character, no cap:

       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)

And the official one-liner is: ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. That’s the vibe. 🫶

💅 Function/Subroutine Documentation 💅

Full signature time, bestie:

   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. Same iconic line, still true. ✨

🎀 Purpose 🎀

Okay so the actual lore: ZGGSVD3 computes the generalized singular value decomposition (GSVD) of an M-by-N complex matrix A and P-by-N complex matrix B. Here’s the exact math, because the matrices are the main characters and we do not skip their dialogue:

           !>
           !> 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) )
           !>

That’s the whole mathematical vibe, and honestly? It ate. 🔥

🌸 Parameters 🌸

• JOBU — CHARACTER*1; 'U': Unitary matrix U is computed; 'N': U is not computed.
• JOBV — CHARACTER*1; 'V': Unitary matrix V is computed; 'N': V is not computed.
• JOBQ — CHARACTER*1; 'Q': Unitary matrix Q is computed; 'N': 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*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 — 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, 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 — 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) = 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*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 — 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); 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*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 — INTEGER; the leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
• WORK — COMPLEX*16 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 — DOUBLE PRECISION 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

such that 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 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. Not me adding drama — that’s just what the page says. 💀

👑 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. Periodt. 💅

⭐ Definition ⭐

Definition at line 350 of file zggsvd3.f.

💜 Author 💜

Generated automatically by Doxygen for LAPACK from the source code.

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Okay bestie, that’s the whole page 💖 — signing off from LAPACK Version 3.12.0 on the /home/abui.../zggsvd3.f(3) page. Stay slay. 💅✨