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# sgelsx

```
NAME
SGELSX - compute the minimum-norm solution to a real linear
least squares problem

SYNOPSIS
SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND,
RANK, WORK, INFO )

INTEGER        INFO, LDA, LDB, M, N, NRHS, RANK

REAL           RCOND

INTEGER        JPVT( * )

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

PURPOSE
SGELSX 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 annihi-
lated 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' [ inv(T11)*Q1'*B ]
[        0       ]
where Q1 consists of the first RANK columns of Q.

ARGUMENTS
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 matrices B and X. NRHS >= 0.

A       (input/output) 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     (input) INTEGER
The leading dimension of the array A.  LDA >=
max(1,M).

B       (input/output) 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.  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     (input) INTEGER
The leading dimension of the array B. LDB >=
max(1,M,N).

JPVT    (input/output) 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   (input) 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    (output) INTEGER
The effective rank of A, i.e., the order of the sub-
matrix R11.  This is the same as the order of the
submatrix T11 in the complete orthogonal factoriza-
tion of A.

WORK    (workspace) REAL array, dimension
(max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -i, the i-th argument had an illegal
value
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