---
name: "sgelsd.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/sgelsd.f"
---

# sgelsd.f(3)

## NAME

SRC/sgelsd.f

## SYNOPSIS

### Functions/Subroutines

subroutine **sgelsd** (m, n, nrhs, a, lda, b, ldb, s, rcond, rank, work, lwork, iwork, info)\
  **SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices**

## Function/Subroutine Documentation

### subroutine sgelsd (integer m, integer n, integer nrhs, real, dimension( lda, \* ) a, integer lda, real, dimension( ldb, \* ) b, integer ldb, real, dimension( \* ) s, real rcond, integer rank, real, dimension( \* ) work, integer lwork, integer, dimension( \* ) iwork, integer info)

 **SGELSD computes the minimum-norm solution to a linear least squares problem for GE matrices**

**Purpose:**

```
!>
!> SGELSD computes the minimum-norm solution to a real linear least
!> squares problem:
!>     minimize 2-norm(| b - A*x |)
!> using the singular value decomposition (SVD) 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 problem is solved in three steps:
!> (1) Reduce the coefficient matrix A to bidiagonal form with
!>     Householder transformations, reducing the original problem
!>     into a  (BLS)
!> (2) Solve the BLS using a divide and conquer approach.
!> (3) Apply back all the Householder transformations to solve
!>     the original least squares problem.
!>
!> The effective rank of A is determined by treating as zero those
!> singular values which are less than RCOND times the largest singular
!> value.
!>
!>
```

**Parameters**

*M*

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

*N*

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

*NRHS*

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

*A*

```
!>          A is REAL array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A.
!>          On exit, A has been destroyed.
!>
```

*LDA*

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

*B*

```
!>          B is REAL array, dimension (LDB,NRHS)
!>          On entry, the M-by-NRHS right hand side matrix B.
!>          On exit, B is overwritten by 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,max(M,N)).
!>
```

*S*

```
!>          S is REAL array, dimension (min(M,N))
!>          The singular values of A in decreasing order.
!>          The condition number of A in the 2-norm = S(1)/S(min(m,n)).
!>
```

*RCOND*

```
!>          RCOND is REAL
!>          RCOND is used to determine the effective rank of A.
!>          Singular values S(i) <= RCOND*S(1) are treated as zero.
!>          If RCOND < 0, machine precision is used instead.
!>
```

*RANK*

```
!>          RANK is INTEGER
!>          The effective rank of A, i.e., the number of singular values
!>          which are greater than RCOND*S(1).
!>
```

*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 must be at least 1.
!>          The exact minimum amount of workspace needed depends on M,
!>          N and NRHS. As long as LWORK is at least
!>              12*N + 2*N*SMLSIZ + 8*N*NLVL + N*NRHS + (SMLSIZ+1)**2,
!>          if M is greater than or equal to N or
!>              12*M + 2*M*SMLSIZ + 8*M*NLVL + M*NRHS + (SMLSIZ+1)**2,
!>          if M is less than N, the code will execute correctly.
!>          SMLSIZ is returned by ILAENV and is equal to the maximum
!>          size of the subproblems at the bottom of the computation
!>          tree (usually about 25), and
!>             NLVL = MAX( 0, INT( LOG_2( MIN( M,N )/(SMLSIZ+1) ) ) + 1 )
!>          For good performance, LWORK should generally be larger.
!>
!>          If LWORK = -1, then a workspace query is assumed; the routine
!>          only calculates the optimal size of the array WORK and the
!>          minimum size of the array IWORK, and returns these values as
!>          the first entries of the WORK and IWORK arrays, and no error
!>          message related to LWORK is issued by XERBLA.
!>
```

*IWORK*

```
!>          IWORK is INTEGER array, dimension (MAX(1,LIWORK))
!>          LIWORK >= max(1, 3*MINMN*NLVL + 11*MINMN),
!>          where MINMN = MIN( M,N ).
!>          On exit, if INFO = 0, IWORK(1) returns the minimum LIWORK.
!>
```

*INFO*

```
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -i, the i-th argument had an illegal value.
!>          > 0:  the algorithm for computing the SVD failed to converge;
!>                if INFO = i, i off-diagonal elements of an intermediate
!>                bidiagonal form did not converge to zero.
!>
```

**Author**

Univ. of Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Contributors:**

Ming Gu and Ren-Cang Li, Computer Science Division, University of California at Berkeley, USA \
 Osni Marques, LBNL/NERSC, USA \

Definition at line **202** of file **sgelsd.f**.

## Author

Generated automatically by Doxygen for LAPACK from the source code.