sgelsy.f −

**Functions/Subroutines**

subroutine **SGELSY** (M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK, WORK, LWORK, INFO)

SGELSY solves overdetermined or underdetermined systems for GE matrices

**subroutine SGELSY (integerM, integerN, integerNRHS, real, dimension( lda, * )A, integerLDA, real, dimension( ldb, * )B, integerLDB, integer, dimension( * )JPVT, realRCOND, integerRANK, real, dimension( * )WORK, integerLWORK, integerINFO)
SGELSY solves overdetermined or underdetermined systems for GE matrices**

**Purpose:**

SGELSY 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 REAL 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 REAL array, dimension (LDB,NRHS)

On entry, the M-by-NRHS right hand side matrix B.

On exit, the N-by-NRHS solution matrix X.

*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 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.

*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.

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 SGEQP3, STZRZF, STZRQF, SORMQR,

and SORMRZ.

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

Univ. of Colorado Denver

NAG Ltd.

**Date:**

November 2011

**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 sgelsy.f.

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