# dgelsy.f - Man Page

SRC/dgelsy.f

## Synopsis

### Functions/Subroutines

subroutine dgelsy (m, n, nrhs, a, lda, b, ldb, jpvt, rcond, rank, work, lwork, info)
DGELSY solves overdetermined or underdetermined systems for GE matrices

## Function/Subroutine Documentation

### subroutine dgelsy (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 lwork, integer info)

DGELSY solves overdetermined or underdetermined systems for GE matrices

Purpose:

``` DGELSY 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.

This routine is basically identical to the original xGELSX except
three differences:
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.
o The permutation of matrix B (the right hand side) is faster and
more simple.```
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 = 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 AP
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.
If NRHS = 0, RANK = 0 on output.```

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.
The unblocked strategy requires that:
LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
where MN = min( M, N ).
The block algorithm requires that:
LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
where NB is an upper bound on the blocksize returned
by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
and DORMRZ.

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

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 204 of file dgelsy.f.

## Author

Generated automatically by Doxygen for LAPACK from the source code.

## Referenced By

The man page dgelsy(3) is an alias of dgelsy.f(3).

Tue Nov 28 2023 12:08:41 Version 3.12.0 LAPACK