---
name: "dggqrf.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/dggqrf.f"
---

# dggqrf.f(3)

## NAME

SRC/dggqrf.f

## SYNOPSIS

### Functions/Subroutines

subroutine **dggqrf** (n, m, p, a, lda, taua, b, ldb, taub, work, lwork, info)\
 **DGGQRF**

## Function/Subroutine Documentation

### subroutine dggqrf (integer n, integer m, integer p, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( \* ) taua, double precision, dimension( ldb, \* ) b, integer ldb, double precision, dimension( \* ) taub, double precision, dimension( \* ) work, integer lwork, integer info)

**DGGQRF**

**Purpose:**

```
!>
!> DGGQRF computes a generalized QR factorization of an N-by-M matrix A
!> and an N-by-P matrix B:
!>
!>             A = Q*R,        B = Q*T*Z,
!>
!> where Q is an N-by-N orthogonal matrix, Z is a P-by-P orthogonal
!> matrix, and R and T assume one of the forms:
!>
!> if N >= M,  R = ( R11 ) M  ,   or if N < M,  R = ( R11  R12 ) N,
!>                 (  0  ) N-M                         N   M-N
!>                    M
!>
!> where R11 is upper triangular, and
!>
!> if N <= P,  T = ( 0  T12 ) N,   or if N > P,  T = ( T11 ) N-P,
!>                  P-N  N                           ( T21 ) P
!>                                                      P
!>
!> where T12 or T21 is upper triangular.
!>
!> In particular, if B is square and nonsingular, the GQR factorization
!> of A and B implicitly gives the QR factorization of inv(B)*A:
!>
!>              inv(B)*A = Z**T*(inv(T)*R)
!>
!> where inv(B) denotes the inverse of the matrix B, and Z**T denotes the
!> transpose of the matrix Z.
!>
```

**Parameters**

*N*

```
!>          N is INTEGER
!>          The number of rows of the matrices A and B. N >= 0.
!>
```

*M*

```
!>          M is INTEGER
!>          The number of columns of the matrix A.  M >= 0.
!>
```

*P*

```
!>          P is INTEGER
!>          The number of columns of the matrix B.  P >= 0.
!>
```

*A*

```
!>          A is DOUBLE PRECISION array, dimension (LDA,M)
!>          On entry, the N-by-M matrix A.
!>          On exit, the elements on and above the diagonal of the array
!>          contain the min(N,M)-by-M upper trapezoidal matrix R (R is
!>          upper triangular if N >= M); the elements below the diagonal,
!>          with the array TAUA, represent the orthogonal matrix Q as a
!>          product of min(N,M) elementary reflectors (see Further
!>          Details).
!>
```

*LDA*

```
!>          LDA is INTEGER
!>          The leading dimension of the array A. LDA >= max(1,N).
!>
```

*TAUA*

```
!>          TAUA is DOUBLE PRECISION array, dimension (min(N,M))
!>          The scalar factors of the elementary reflectors which
!>          represent the orthogonal matrix Q (see Further Details).
!>
```

*B*

```
!>          B is DOUBLE PRECISION array, dimension (LDB,P)
!>          On entry, the N-by-P matrix B.
!>          On exit, if N <= P, the upper triangle of the subarray
!>          B(1:N,P-N+1:P) contains the N-by-N upper triangular matrix T;
!>          if N > P, the elements on and above the (N-P)-th subdiagonal
!>          contain the N-by-P upper trapezoidal matrix T; the remaining
!>          elements, with the array TAUB, represent the orthogonal
!>          matrix Z as a product of elementary reflectors (see Further
!>          Details).
!>
```

*LDB*

```
!>          LDB is INTEGER
!>          The leading dimension of the array B. LDB >= max(1,N).
!>
```

*TAUB*

```
!>          TAUB is DOUBLE PRECISION array, dimension (min(N,P))
!>          The scalar factors of the elementary reflectors which
!>          represent the orthogonal matrix Z (see Further Details).
!>
```

*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. LWORK >= max(1,N,M,P).
!>          For optimum performance LWORK >= max(N,M,P)*max(NB1,NB2,NB3),
!>          where NB1 is the optimal blocksize for the QR factorization
!>          of an N-by-M matrix, NB2 is the optimal blocksize for the
!>          RQ factorization of an N-by-P matrix, and NB3 is the optimal
!>          blocksize for a call of DORMQR.
!>
!>          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*

```
!>          INFO is 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 matrix Q is represented as a product of elementary reflectors
!>
!>     Q = H(1) H(2) . . . H(k), where k = min(n,m).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - taua * v * v**T
!>
!>  where taua is a real scalar, and v is a real vector with
!>  v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is stored on exit in A(i+1:n,i),
!>  and taua in TAUA(i).
!>  To form Q explicitly, use LAPACK subroutine DORGQR.
!>  To use Q to update another matrix, use LAPACK subroutine DORMQR.
!>
!>  The matrix Z is represented as a product of elementary reflectors
!>
!>     Z = H(1) H(2) . . . H(k), where k = min(n,p).
!>
!>  Each H(i) has the form
!>
!>     H(i) = I - taub * v * v**T
!>
!>  where taub is a real scalar, and v is a real vector with
!>  v(p-k+i+1:p) = 0 and v(p-k+i) = 1; v(1:p-k+i-1) is stored on exit in
!>  B(n-k+i,1:p-k+i-1), and taub in TAUB(i).
!>  To form Z explicitly, use LAPACK subroutine DORGRQ.
!>  To use Z to update another matrix, use LAPACK subroutine DORMRQ.
!>
```

Definition at line **213** of file **dggqrf.f**.

## Author

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