---
name: "sgels"
section: "3lapack"
platform: "linux"
package: "liblapack-doc"
version: "3.2.1-8"
release: "Debian 6.0 (Squeeze)"
platforms: ["linux"]
url: "https://new.carta.tech/sgels/3lapack"
---

# sgels(3lapack)

solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A

## NAME

SGELS - solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A

## SYNOPSIS

- SUBROUTINE SGELS(

  TRANS, M, N, NRHS, A, LDA, B, LDB, WORK, LWORK, INFO )

- CHARACTER

  TRANS

- INTEGER

  INFO, LDA, LDB, LWORK, M, N, NRHS

- REAL

  A( LDA, \* ), B( LDB, \* ), WORK( \* )

## PURPOSE

SGELS solves overdetermined or underdetermined real linear systems involving an M-by-N matrix A, or its transpose, using a QR or LQ factorization of A.  It is assumed that A has full rank. The following options are provided: \
 1\. If TRANS = 'N' and m \>= n:  find the least squares solution of\
    an overdetermined system, i.e., solve the least squares problem\
                 minimize \|\| B - A\*X \|\|.\
 2\. If TRANS = 'N' and m \< n:  find the minimum norm solution of\
    an underdetermined system A \* X = B.\
 3\. If TRANS = 'T' and m \>= n:  find the minimum norm solution of\
    an undetermined system A\*\*T \* X = B.\
 4\. If TRANS = 'T' and m \< n:  find the least squares solution of\
    an overdetermined system, i.e., solve the least squares problem\
                 minimize \|\| B - A\*\*T \* X \|\|.\
 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.\

## ARGUMENTS

- TRANS (input) CHARACTER\*1

  \= 'N': the linear system involves A;\
   \= 'T': the linear system involves A\*\*T.

- M (input) INTEGER

  The number of rows of the matrix A.  M \>= 0.

- N (input) INTEGER

  The number of columns of the matrix A.  N \>= 0.

- NRHS (input) INTEGER

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

- A (input/output) REAL array, dimension (LDA,N)

  On entry, the M-by-N matrix A. On exit, if M \>= N, A is overwritten by details of its QR factorization as returned by SGEQRF; if M \<  N, A is overwritten by details of its LQ factorization as returned by SGELQF.

- LDA (input) INTEGER

  The leading dimension of the array A.  LDA \>= max(1,M).

- B (input/output) REAL array, dimension (LDB,NRHS)

  On entry, the matrix B of right hand side vectors, stored columnwise; B is M-by-NRHS if TRANS = 'N', or N-by-NRHS if TRANS = 'T'.   On exit, if INFO = 0, B is overwritten by the solution vectors, stored columnwise: if TRANS = 'N' and m \>= n, rows 1 to n of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements N+1 to M in that column; if TRANS = 'N' and m \< n, rows 1 to N of B contain the minimum norm solution vectors; if TRANS = 'T' and m \>= n, rows 1 to M of B contain the minimum norm solution vectors; if TRANS = 'T' and m \< n, rows 1 to M of B contain the least squares solution vectors; the residual sum of squares for the solution in each column is given by the sum of squares of elements M+1 to N in that column.

- LDB (input) INTEGER

  The leading dimension of the array B. LDB \>= MAX(1,M,N).

- WORK (workspace/output) REAL array, dimension (MAX(1,LWORK))

  On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

- LWORK (input) INTEGER

  The dimension of the array WORK. LWORK \>= max( 1, MN + max( MN, NRHS ) ). For optimal performance, LWORK \>= max( 1, MN + max( MN, NRHS )\*NB ). where MN = min(M,N) and NB is the optimum block size. 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 (output) INTEGER

  \= 0:  successful exit\
   \< 0:  if INFO = -i, the i-th argument had an illegal value\
   \> 0:  if INFO =  i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank; the least squares solution could not be computed.