SRC/sggsvp3.f(3)

💖 SRC/sggsvp3.f 💖

bestie, welcome to the Library Functions Manual page for SRC/sggsvp3.f — aka the SGGSVP3 routine, and she is booked and busy doing GSVD prep. ✨

✨ NAME ✨

SRC/sggsvp3.f

🌸 SYNOPSIS 🌸

🧩 Functions/Subroutines

       subroutine sggsvp3 (jobu, jobv, jobq, m, p, n, a, lda, b, ldb, tola,
           tolb, k, l, u, ldu, v, ldv, q, ldq, iwork, tau, work, lwork, info)
           SGGSVP3

🎀 Function/Subroutine Documentation 🎀

   subroutine sggsvp3 (character jobu, character jobv, character jobq, integer
       m, integer p, integer n, real, dimension( lda, * ) a, integer lda, real,
       dimension( ldb, * ) b, integer ldb, real tola, real tolb, integer k,
       integer l, real, dimension( ldu, * ) u, integer ldu, real, dimension(
       ldv, * ) v, integer ldv, real, dimension( ldq, * ) q, integer ldq,
       integer, dimension( * ) iwork, real, dimension( * ) tau, real,
       dimension( * ) work, integer lwork, integer info)
       SGGSVP3

🎯 Purpose:

okay so the whole point: SGGSVP3 computes orthogonal matrices U, V and Q such that the block structure goes off like this:

!>
!> SGGSVP3 computes orthogonal matrices U, V and Q such that
!>
!>                    N-K-L  K    L
!>  U**T*A*Q =     K ( 0    A12  A13 )  if M-K-L >= 0;
!>                 L ( 0     0   A23 )
!>             M-K-L ( 0     0    0  )
!>
!>                  N-K-L  K    L
!>         =     K ( 0    A12  A13 )  if M-K-L < 0;
!>             M-K ( 0     0   A23 )
!>
!>                  N-K-L  K    L
!>  V**T*B*Q =   L ( 0     0   B13 )
!>             P-L ( 0     0    0  )
!>
!> where the K-by-K matrix A12 and L-by-L matrix B13 are nonsingular
!> upper triangular; A23 is L-by-L upper triangular if M-K-L >= 0,
!> otherwise A23 is (M-K)-by-L upper trapezoidal.  K+L = the effective
!> numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T.
!>
!> This decomposition is the preprocessing step for computing the
!> Generalized Singular Value Decomposition (GSVD), see subroutine
!> SGGSVD3.
!>

translation for the group chat: A12 is K-by-K and B13 is L-by-L, both nonsingular upper triangular. A23 is L-by-L upper triangular if M-K-L >= 0; otherwise A23 is (M-K)-by-L upper trapezoidal. And K+L = the effective numerical rank of the (M+P)-by-N matrix (A**T,B**T)**T. This decomposition is literally the preprocessing step for computing the Generalized Singular Value Decomposition (GSVD), so go see subroutine SGGSVD3. 💅

📋 Parameters

• JOBU — CHARACTER*1; = 'U': orthogonal matrix U is computed; = 'N': U is not computed.
• JOBV — CHARACTER*1; = 'V': orthogonal matrix V is computed; = 'N': V is not computed.
• JOBQ — CHARACTER*1; = 'Q': orthogonal matrix Q is computed; = 'N': Q is not computed.
• M — INTEGER; the number of rows of the matrix A. M >= 0.
• P — INTEGER; the number of rows of the matrix B. P >= 0.
• N — INTEGER; the number of columns of the matrices A and B. N >= 0.
• A — REAL array, dimension (LDA,N); on entry, the M-by-N matrix A. On exit, A contains the triangular (or trapezoidal) matrix described in the Purpose section.
• LDA — INTEGER; the leading dimension of the array A. LDA >= max(1,M).
• B — REAL array, dimension (LDB,N); on entry, the P-by-N matrix B. On exit, B contains the triangular matrix described in the Purpose section.
• LDB — INTEGER; the leading dimension of the array B. LDB >= max(1,P).
• TOLA — REAL.
• TOLB — REAL. TOLA and TOLB are the thresholds to determine the effective numerical rank of matrix B and a subblock of A. 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.
• K — INTEGER.
• L — INTEGER. On exit, K and L specify the dimension of the subblocks described in the Purpose section. K + L = effective numerical rank of (A**T,B**T)**T.
• U — REAL array, dimension (LDU,M); if JOBU = 'U', U contains the orthogonal 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 — REAL array, dimension (LDV,P); if JOBV = 'V', V contains the orthogonal 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 — REAL array, dimension (LDQ,N); if JOBQ = 'Q', Q contains the orthogonal 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.
• IWORK — INTEGER array, dimension (N).
• TAU — REAL array, dimension (N).
• WORK — REAL array, dimension (MAX(1,LWORK)); on exit, if INFO = 0, WORK(1) returns the optimal LWORK.
• LWORK — 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.
• INFO — INTEGER; = 0: successful exit; < 0: if INFO = -i, the i-th argument had an illegal value.

👑 Author

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

🌈 Further Details:

• The subroutine uses LAPACK subroutine SGEQP3 for the QR factorization with column pivoting to detect the effective numerical rank of the a matrix. It may be replaced by a better rank determination strategy.
• SGGSVP3 replaces the deprecated subroutine SGGSVP. 🫶

📍 Definition at line 269 of file sggsvp3.f.

🧚 Author 🧚

Generated automatically by Doxygen for LAPACK from the source code. 💖

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byeee bestie! source: SRC/sggsvp3.f(3), version: LAPACK Version 3.12.0 — go off and decompose those generalized singular values, mwah 💋🌈