## MD03BX

### QR factorization with column pivoting for a standard nonlinear least squares problem

[Specification] [Arguments] [Method] [References] [Comments] [Example]

Purpose

```  To compute the QR factorization with column pivoting of an
m-by-n matrix J (m >= n), that is, J*P = Q*R, where Q is a matrix
with orthogonal columns, P a permutation matrix, and R an upper
trapezoidal matrix with diagonal elements of nonincreasing
magnitude, and to apply the transformation Q' on the error
vector e (in-situ). The 1-norm of the scaled gradient is also
returned. The matrix J could be the Jacobian of a nonlinear least
squares problem.

```
Specification
```      SUBROUTINE MD03BX( M, N, FNORM, J, LDJ, E, JNORMS, GNORM, IPVT,
\$                   DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
INTEGER           INFO, LDJ, LDWORK, M, N
DOUBLE PRECISION  FNORM, GNORM
C     .. Array Arguments ..
INTEGER           IPVT(*)
DOUBLE PRECISION  DWORK(*), E(*), J(*), JNORMS(*)

```
Arguments

Input/Output Parameters

```  M       (input) INTEGER
The number of rows of the Jacobian matrix J.  M >= 0.

N       (input) INTEGER
The number of columns of the Jacobian matrix J.
M >= N >= 0.

FNORM   (input) DOUBLE PRECISION
The Euclidean norm of the vector e.  FNORM >= 0.

J       (input/output) DOUBLE PRECISION array, dimension (LDJ, N)
On entry, the leading M-by-N part of this array must
contain the Jacobian matrix J.
On exit, the leading N-by-N upper triangular part of this
array contains the upper triangular factor R of the
Jacobian matrix. Note that for efficiency of the later
calculations, the matrix R is delivered with the leading
dimension MAX(1,N), possibly much smaller than the value
of LDJ on entry.

LDJ     (input/output) INTEGER
The leading dimension of array J.
On entry, LDJ >= MAX(1,M).
On exit,  LDJ >= MAX(1,N).

E       (input/output) DOUBLE PRECISION array, dimension (M)
On entry, this array must contain the error vector e.
On exit, this array contains the updated vector Q'*e.

JNORMS  (output) DOUBLE PRECISION array, dimension (N)
This array contains the Euclidean norms of the columns of
the Jacobian matrix, considered in the initial order.

GNORM   (output) DOUBLE PRECISION
If FNORM > 0, the 1-norm of the scaled vector
J'*Q'*e/FNORM, with each element i further divided by
JNORMS(i) (if JNORMS(i) is nonzero).
If FNORM = 0, the returned value of GNORM is 0.

IPVT    (output) INTEGER array, dimension (N)
This array defines the permutation matrix P such that
J*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, if INFO = 0, DWORK(1) returns the optimal value
of LDWORK.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= 1,      if N = 0 or M = 1;
LDWORK >= 4*N+1,  if N > 1.
For optimum performance LDWORK should be larger.

```
Error Indicator
```  INFO    INTEGER
= 0:  successful exit;
< 0:  if INFO = -i, the i-th argument had an illegal
value.

```
Method
```  The algorithm uses QR factorization with column pivoting of the
matrix J, J*P = Q*R, and applies the orthogonal matrix Q' to the
vector e.

```
```  None
```
Example

Program Text

```  None
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Program Data
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Program Results
```  None
```