---
name: "dgelsx.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/dgelsx.f"
---

# dgelsx.f(3)

## NAME

SRC/DEPRECATED/dgelsx.f

## SYNOPSIS

### Functions/Subroutines

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

## Function/Subroutine Documentation

### subroutine dgelsx (integer m, integer n, integer nrhs, double precision, dimension( lda, \* ) a, integer lda, double precision, dimension( ldb, \* ) b, integer ldb, integer, dimension( \* ) jpvt, double precision rcond, integer rank, double precision, dimension( \* ) work, integer info)

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

**Purpose:**

```
!>
!> This routine is deprecated and has been replaced by routine DGELSY.
!>
!> DGELSX computes the minimum-norm solution to a real 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 orthogonal 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**T [ inv(T11)*Q1**T*B ]
!>                 [        0         ]
!> where Q1 consists of the first RANK columns of Q.
!>
```

**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 DOUBLE PRECISION 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 DOUBLE PRECISION 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 >= n and RANK = n, the residual sum-of-squares for
!>          the solution in the i-th column is given by the sum of
!>          squares of elements N+1:M in that column.
!>
```

*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 an
!>          initial column, otherwise it is a free column.  Before
!>          the QR factorization of A, all initial columns are
!>          permuted to the leading positions; only the remaining
!>          free columns are moved as a result of column pivoting
!>          during the factorization.
!>          On exit, if JPVT(i) = k, then the i-th column of A*P
!>          was the k-th column of A.
!>
```

*RCOND*

```
!>          RCOND is DOUBLE PRECISION
!>          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.
!>
```

*WORK*

```
!>          WORK is DOUBLE PRECISION array, dimension
!>                      (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),
!>
```

*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.

Definition at line **176** of file **dgelsx.f**.

## Author

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