ggsvd3(3)

📦

version 3.12.1 · 9 versions

3.12.1 · arch rolling (latest)
3.12.0 · opensuse-leap 16.0
3.12.1 · ubuntu stonking
3.12.1 · ubuntu questing
3.12.0 · ubuntu noble
3.12.0 · ubuntu noble
3.12.0 · centos-stream 10, fedora 43, fedora 44
3.12.1 · ubuntu resolute
3.12.1 · debian trixie

💅 NAME

okay so this is ggsvd3(3) from the Library Functions Manual, and the NAME line is literally ggsvd3 - ggsvd3: SVD, QR iteration — so yeah, bestie, it’s giving SVD and QR iteration energy. ✨

🌸 SYNOPSIS

✨ Functions

Here’s the function lineup, no cap:

cggsvd3 — CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices, fr fr. Signature:

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)

dggsvd3 — DGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. Signature:

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)

sggsvd3 — SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. Signature:

subroutine sggsvd3 (jobu, jobv, jobq, m, n, p, k, l, a, lda, b, ldb,
    alpha, beta, u, ldu, v, ldv, q, ldq, work, lwork, iwork, info)

zggsvd3 — ZGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. Signature:

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)

🦋 Detailed Description

okay so this is where the actual function documentation starts, and we’re kicking it off with the complex single-precision queen, cggsvd3. 💖

🎀 Function Documentation

💖 subroutine cggsvd3

and yes, CGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices — it’s giving main character energy. 👑

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)

Purpose:

bestie, here’s the whole purpose, and it’s a lot but we’re gonna get through it together 🫶 CGGSVD3 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.

So the routine computes C, S, R, and optionally the unitary transformation matrices U, V and Q — slay. 💅

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

okay here come the parameters, and yes, there are a lot — but every single one matters, bestie. 💖

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.
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.
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(1,LWORK)). On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
LWORK — LWORK is 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.
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:

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. periodt — bestie, use the new one, the old one is deprecated for a reason 💅

Definition at line 351 of file cggsvd3.f.

🌟 subroutine `dggsvd3`

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. okay bestie, it's giving generalized-SVD main character energy ✨

🎀 Purpose

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. like, it does the whole glow-up and lets you choose which transformation receipts you want back 💖

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 — 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 matrix A. M >= 0.
N — INTEGER; the number of columns of matrices A and B. N >= 0.
P — INTEGER; the number of rows of 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**T,B**T)**T.
A — 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 — INTEGER; the leading dimension of the array A. LDA >= max(1,M).
B — 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 — 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 — 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 — INTEGER; the leading dimension of the array U. LDU >= max(1,M) if JOBU = 'U'; LDU >= 1 otherwise.
V — 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 — INTEGER; the leading dimension of the array V. LDV >= max(1,P) if JOBV = 'V'; LDV >= 1 otherwise.
Q — 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 — INTEGER; the leading dimension of the array Q. LDQ >= max(1,N) if JOBQ = 'Q'; LDQ >= 1 otherwise.
WORK — DOUBLE PRECISION 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.
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 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. so yeah, these thresholds are not just cute little numbers, they matter 🫶

👑 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

DGGSVD3 replaces the deprecated subroutine DGGSVD. no cap, leave the deprecated one in the past where it belongs 💀

Definition at line 346 of file dggsvd3.f.

🦋 subroutine `sggsvd3`

subroutine sggsvd3 (character jobu, character jobv, character jobq, integer
    m, integer n, integer p, integer k, integer l, real, dimension( lda, * )
    a, integer lda, real, dimension( ldb, * ) b, integer ldb, real,
    dimension( * ) alpha, real, dimension( * ) beta, real, dimension( ldu, *
    ) u, integer ldu, real, dimension( ldv, * ) v, integer ldv, real,
    dimension( ldq, * ) q, integer ldq, real, dimension( * ) work, integer
    lwork, integer, dimension( * ) iwork, integer info)

SGGSVD3 computes the singular value decomposition (SVD) for OTHER matrices. same GSVD energy, but make it single precision 🌸

🌈 Purpose

okay so the math keeps going and it’s lowkey iconic, bestie — here’s the whole GSVD structure laid out exactly:

SGGSVD3 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) ).

that’s the whole mathematical situation, and it’s giving main character energy 🔥 now the parameters, because bestie you need every single one:

🌸 Parameters 🌸

JOBU — JOBU is CHARACTER*1; it’s the flag that says whether U gets computed.
- = 'U': Orthogonal matrix U is computed;
- = 'N': U is not computed.
JOBV — JOBV is CHARACTER*1; it’s the flag that says whether V gets computed.
- = 'V': Orthogonal matrix V is computed;
- = 'N': V is not computed.
JOBQ — JOBQ is CHARACTER*1; it’s the flag that says whether Q gets computed.
- = '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
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 REAL 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 REAL 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 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 REAL 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 REAL 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 REAL 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 REAL 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. 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.

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 STGSJA.

🔍 Internal Parameters 🔍

TOLA — REAL
TOLB — REAL

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: 💅

SGGSVD3 replaces the deprecated subroutine SGGSVD.

Definition at line 346 of file sggsvd3.f.

🦋 subroutine zggsvd3 🦋

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

🎯 Purpose: 🎯

✨ PARAMETERS ✨

okay bestie, this is the part where every single argument gets its moment 💅 no skipping, no summarizing, we're doing the whole roster. 🫶

JOBU — it's CHARACTER*1, and the vibe is: 'U' means the unitary matrix U is computed; 'N' means U is not computed. 🦋
JOBV — also CHARACTER*1: 'V' means the unitary matrix V is computed; 'N' means V is not computed. ✨
JOBQ — CHARACTER*1 again: '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*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 no cap, that's the whole mapping 😭
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. 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. 💅
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. Here's the status tea:
- = 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. 🫶

✨ 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. 🫡 (yes bestie, deprecated means it's the old one — use the new one, periodt.)

oh and btw, this whole thing is defined at line 350 of file zggsvd3.f. 🧚

and that's the whole page, bestie 💅✨ the footer is giving LAPACK Version 3.12.0 on the ggsvd3(3) page, and it was generated automatically by Doxygen for LAPACK from the source code. no date on this one, just version energy. periodt. 🎀🫶

📦 where this page comes from

arch rolling: lapack-doc 3.12.1-2 · custom
💖 girlified by deepseek-ai/DeepSeek-V4.1-Flash-1017 on 2026-11-04: a language model's rewrite of the original, under its terms

ggsvd3(3)

💅 NAME

🌸 SYNOPSIS

✨ Functions

🦋 Detailed Description

🎀 Function Documentation

💖 subroutine cggsvd3

🫶 Contributors

✨ Further Details

🌟 subroutine dggsvd3

🎀 Purpose

💅 Parameters

🔒 Internal Parameters

👑 Author

🫶 Contributors

✨ Further Details

🦋 subroutine sggsvd3

🌈 Purpose

🌸 Parameters 🌸

🔍 Internal Parameters 🔍

👑 Author 👑

🫶 Contributors: 🫶

💅 Further Details: 💅

🦋 subroutine zggsvd3 🦋

🎯 Purpose: 🎯

✨ PARAMETERS ✨

✨ INTERNAL PARAMETERS ✨

✨ AUTHOR ✨

✨ CONTRIBUTORS ✨

✨ FURTHER DETAILS ✨

🌟 subroutine `dggsvd3`

🦋 subroutine `sggsvd3`