---
name: "cgelsy.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/cgelsy.f"
---

# cgelsy.f(3)

## NAME

SRC/cgelsy.f

## SYNOPSIS

### Functions/Subroutines

subroutine **cgelsy** (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, rwork, info)\
  **CGELSY solves overdetermined or underdetermined systems for GE matrices**

## Function/Subroutine Documentation

### subroutine cgelsy (integer m, integer n, integer nrhs, complex, dimension( lda, \* ) a, integer lda, complex, dimension( ldb, \* ) b, integer ldb, integer, dimension( \* ) jpvt, real rcond, integer rank, complex, dimension( \* ) work, integer lwork, real, dimension( \* ) rwork, integer info)

 **CGELSY solves overdetermined or underdetermined systems for GE matrices**

**Purpose:**

```
!>
!> CGELSY computes the minimum-norm solution to a complex linear least
!> squares problem:
!>     minimize || A * X - B ||
!> using a complete orthogonal factorization of A.  A is an M-by-N
!> matrix which may be rank-deficient.
!>
!> Several right hand side vectors b and solution vectors x can be
!> handled in a single call; they are stored as the columns of the
!> M-by-NRHS right hand side matrix B and the N-by-NRHS solution
!> matrix X.
!>
!> The routine first computes a QR factorization with column pivoting:
!>     A * P = Q * [ R11 R12 ]
!>                 [  0  R22 ]
!> with R11 defined as the largest leading submatrix whose estimated
!> condition number is less than 1/RCOND.  The order of R11, RANK,
!> is the effective rank of A.
!>
!> Then, R22 is considered to be negligible, and R12 is annihilated
!> by unitary transformations from the right, arriving at the
!> complete orthogonal factorization:
!>    A * P = Q * [ T11 0 ] * Z
!>                [  0  0 ]
!> The minimum-norm solution is then
!>    X = P * Z**H [ inv(T11)*Q1**H*B ]
!>                 [        0         ]
!> where Q1 consists of the first RANK columns of Q.
!>
!> This routine is basically identical to the original xGELSX except
!> three differences:
!>   o The permutation of matrix B (the right hand side) is faster and
!>     more simple.
!>   o The call to the subroutine xGEQPF has been substituted by the
!>     the call to the subroutine xGEQP3. This subroutine is a Blas-3
!>     version of the QR factorization with column pivoting.
!>   o Matrix B (the right hand side) is updated with Blas-3.
!>
```

**Parameters**

*M*

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

*N*

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

*NRHS*

```
!>          NRHS is INTEGER
!>          The number of right hand sides, i.e., the number of
!>          columns of matrices B and X. NRHS >= 0.
!>
```

*A*

```
!>          A is COMPLEX array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, A has been overwritten by details of its
!>          complete orthogonal factorization.
!>
```

*LDA*

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

*B*

```
!>          B is COMPLEX array, dimension (LDB,NRHS)
!>          On entry, the M-by-NRHS right hand side matrix B.
!>          On exit, the N-by-NRHS solution matrix X.
!>          If M = 0 or N = 0, B is not referenced.
!>
```

*LDB*

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

*JPVT*

```
!>          JPVT is INTEGER array, dimension (N)
!>          On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
!>          to the front of AP, otherwise column i is a free column.
!>          On exit, if JPVT(i) = k, then the i-th column of A*P
!>          was the k-th column of A.
!>
```

*RCOND*

```
!>          RCOND is REAL
!>          RCOND is used to determine the effective rank of A, which
!>          is defined as the order of the largest leading triangular
!>          submatrix R11 in the QR factorization with pivoting of A,
!>          whose estimated condition number < 1/RCOND.
!>
```

*RANK*

```
!>          RANK is INTEGER
!>          The effective rank of A, i.e., the order of the submatrix
!>          R11.  This is the same as the order of the submatrix T11
!>          in the complete orthogonal factorization of A.
!>          If NRHS = 0, RANK = 0 on output.
!>
```

*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.
!>          The unblocked strategy requires that:
!>            LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
!>          where MN = min(M,N).
!>          The block algorithm requires that:
!>            LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
!>          where NB is an upper bound on the blocksize returned
!>          by ILAENV for the routines CGEQP3, CTZRZF, CTZRQF, CUNMQR,
!>          and CUNMRZ.
!>
!>          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)
!>
```

*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.

**Contributors:**

A. Petitet, Computer Science Dept., Univ. of Tenn., Knoxville, USA \
 E. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \
 G. Quintana-Orti, Depto. de Informatica, Universidad Jaime I, Spain \

Definition at line **210** of file **cgelsy.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.