---
name: "cggsvd.f"
section: "3"
platform: "linux"
package: "lapack-doc"
version: "3.12.1-2"
release: "Arch Linux 2026-11-01"
platforms: ["linux"]
url: "https://new.carta.tech/cggsvd.f"
---

# cggsvd.f(3)

## NAME

SRC/DEPRECATED/cggsvd.f

## 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 This routine is deprecated and has been replaced by routine CGGSVD3.
!>
!> 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
!>
```

*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(3*N,M,P)+N)
!>
```

*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

Definition at line **335** of file **cggsvd.f**.

## Author

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