/home/abui.../dggsvd3.f(3)

💅 DGGSVD3 — LAPACK Library Functions Manual 💅

okay so bestie, welcome to the LAPACK Library Functions Manual page for /home/abui.../dggsvd3.f(3). This is the full DGGSVD3 lore, and we are keeping every technical crumb. No cap. 💖

🌸 NAME 🌸

• /home/abuild/rpmbuild/BUILD/lapack-static-3.12.0-build/lapack-3.12.0/SRC/dggsvd3.f

✨ SYNOPSIS ✨

🧚 Functions/Subroutines

subroutine DGGSVD3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)
     DGGSVD3 computes the singular value decomposition (SVD) for OTHER
    matrices

The one-liner is literally: DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices.

💖 Function/Subroutine Documentation 💖

subroutine DGGSVD3 (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, double precision,
    dimension( lda, * ) a, integer lda, double precision, dimension( ldb, *
    ) b, integer ldb, 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 lwork, integer, dimension( * ) iwork,
    integer info)

DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices

🎯 Purpose:

Here’s the exact math tea, character-for-character — yes, including the !> source-comment crumbs, because we keep every single detail:

!>
!> DGGSVD3 computes the generalized singular value decomposition (GSVD)
!> of an M-by-N real matrix A and P-by-N real matrix B:
!>
!>       U**T*A*Q = D1*( 0 R ),    V**T*B*Q = D2*( 0 R )
!>
!> where U, V and Q are orthogonal matrices.
!> Let K+L = the effective numerical rank of the matrix (A**T,B**T)**T,
!> 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 orthogonal
!> 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**T.
!> If ( A**T,B**T)**T  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**T*A x = lambda* B**T*B x.
!> In some literature, the GSVD of A and B is presented in the form
!>                  U**T*A*X = ( 0 D1 ),   V**T*B*X = ( 0 D2 )
!> where U and V are orthogonal and X is nonsingular, 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': Orthogonal matrix U is computed; = 'N': U is not computed. 💅
• JOBV — JOBV is CHARACTER*1. = 'V': Orthogonal matrix V is computed; = 'N': V is not computed. ✨
• JOBQ — JOBQ is CHARACTER*1. = 'Q': Orthogonal 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 the whole note for K, bestie — no extra guesses. 🫶
• 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**T,B**T)**T. 🔥
• A — A is DOUBLE PRECISION 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 DOUBLE PRECISION array, dimension (LDB,N). On entry, the P-by-N matrix B. On exit, B contains 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. The exact exit values are:

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 DOUBLE PRECISION array, dimension (LDU,M). If JOBU = 'U', U contains the M-by-M orthogonal 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 DOUBLE PRECISION array, dimension (LDV,P). If JOBV = 'V', V contains the P-by-P orthogonal 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 DOUBLE PRECISION array, dimension (LDQ,N). If JOBQ = 'Q', Q contains the N-by-N orthogonal 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 DOUBLE PRECISION 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.

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. 🦋

• 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 DTGSJA. 🔥

🧪 Internal Parameters:

• TOLA — DOUBLE PRECISION
• TOLB — DOUBLE PRECISION

TOLA and TOLB are the thresholds to determine the effective rank of (A**T,B**T)**T. 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:

🚨 deprecated alert: DGGSVD3 replaces the deprecated subroutine DGGSVD.

Definition at line 346 of file dggsvd3.f. 📜

📜 Author

Generated automatically by Doxygen for LAPACK from the source code.

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mwah 💖 cute little sign-off: this page is from LAPACK Version 3.12.0, page /home/abui.../dggsvd3.f(3) — bye bestie, go slay those matrices! periodt 🌈✨