## MD03BY

### Levenberg-Marquardt parameter for a standard nonlinear least squares problem

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

Purpose

```  To determine a value for the parameter PAR such that if x solves
the system

A*x = b ,     sqrt(PAR)*D*x = 0 ,

in the least squares sense, where A is an m-by-n matrix, D is an
n-by-n nonsingular diagonal matrix, and b is an m-vector, and if
DELTA is a positive number, DXNORM is the Euclidean norm of D*x,
then either PAR is zero and

( DXNORM - DELTA ) .LE. 0.1*DELTA ,

or PAR is positive and

ABS( DXNORM - DELTA ) .LE. 0.1*DELTA .

It is assumed that a QR factorization, with column pivoting, of A
is available, that is, A*P = Q*R, where P is a permutation matrix,
Q has orthogonal columns, and R is an upper triangular matrix
with diagonal elements of nonincreasing magnitude.
The routine needs the full upper triangle of R, the permutation
matrix P, and the first n components of Q'*b (' denotes the
transpose). On output, MD03BY also provides an upper triangular
matrix S such that

P'*(A'*A + PAR*D*D)*P = S'*S .

Matrix S is used in the solution process.

```
Specification
```      SUBROUTINE MD03BY( COND, N, R, LDR, IPVT, DIAG, QTB, DELTA, PAR,
\$                   RANK, X, RX, TOL, DWORK, LDWORK, INFO )
C     .. Scalar Arguments ..
CHARACTER         COND
INTEGER           INFO, LDR, LDWORK, N, RANK
DOUBLE PRECISION  DELTA, PAR, TOL
C     .. Array Arguments ..
INTEGER           IPVT(*)
DOUBLE PRECISION  DIAG(*), DWORK(*), QTB(*), R(LDR,*), RX(*), X(*)

```
Arguments

Mode Parameters

```  COND    CHARACTER*1
Specifies whether the condition of the matrices R and S
should be estimated, as follows:
= 'E' :  use incremental condition estimation for R and S;
= 'N' :  do not use condition estimation, but check the
diagonal entries of R and S for zero values;
= 'U' :  use the rank already stored in RANK (for R).

```
Input/Output Parameters
```  N       (input) INTEGER
The order of the matrix R.  N >= 0.

R       (input/output) DOUBLE PRECISION array, dimension (LDR, N)
On entry, the leading N-by-N upper triangular part of this
array must contain the upper triangular matrix R.
On exit, the full upper triangle is unaltered, and the
strict lower triangle contains the strict upper triangle
(transposed) of the upper triangular matrix S.

LDR     INTEGER
The leading dimension of array R.  LDR >= MAX(1,N).

IPVT    (input) INTEGER array, dimension (N)
This array must define the permutation matrix P such that
A*P = Q*R. Column j of P is column IPVT(j) of the identity
matrix.

DIAG    (input) DOUBLE PRECISION array, dimension (N)
This array must contain the diagonal elements of the
matrix D.  DIAG(I) <> 0, I = 1,...,N.

QTB     (input) DOUBLE PRECISION array, dimension (N)
This array must contain the first n elements of the
vector Q'*b.

DELTA   (input) DOUBLE PRECISION
An upper bound on the Euclidean norm of D*x.  DELTA > 0.

PAR     (input/output) DOUBLE PRECISION
On entry, PAR must contain an initial estimate of the
Levenberg-Marquardt parameter.  PAR >= 0.
On exit, it contains the final estimate of this parameter.

RANK    (input or output) INTEGER
On entry, if COND = 'U', this parameter must contain the
(numerical) rank of the matrix R.
On exit, this parameter contains the numerical rank of
the matrix S.

X       (output) DOUBLE PRECISION array, dimension (N)
This array contains the least squares solution of the
system A*x = b, sqrt(PAR)*D*x = 0.

RX      (output) DOUBLE PRECISION array, dimension (N)
This array contains the matrix-vector product -R*P'*x.

```
Tolerances
```  TOL     DOUBLE PRECISION
If COND = 'E', the tolerance to be used for finding the
rank of the matrices R and S. If the user sets TOL > 0,
then the given value of TOL is used as a lower bound for
the reciprocal condition number;  a (sub)matrix whose
estimated condition number is less than 1/TOL is
considered to be of full rank.  If the user sets TOL <= 0,
then an implicitly computed, default tolerance, defined by
TOLDEF = N*EPS,  is used instead, where EPS is the machine
precision (see LAPACK Library routine DLAMCH).
This parameter is not relevant if COND = 'U' or 'N'.

```
Workspace
```  DWORK   DOUBLE PRECISION array, dimension (LDWORK)
On exit, the first N elements of this array contain the
diagonal elements of the upper triangular matrix S.

LDWORK  INTEGER
The length of the array DWORK.
LDWORK >= 4*N, if COND =  'E';
LDWORK >= 2*N, if COND <> 'E'.

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

```
Method
```  The algorithm computes the Gauss-Newton direction. A least squares
solution is found if the Jacobian is rank deficient. If the Gauss-
Newton direction is not acceptable, then an iterative algorithm
obtains improved lower and upper bounds for the parameter PAR.
Only a few iterations are generally needed for convergence of the
algorithm. If, however, the limit of ITMAX = 10 iterations is
reached, then the output PAR will contain the best value obtained
so far. If the Gauss-Newton step is acceptable, it is stored in x,
and PAR is set to zero, hence S = R.

```
References
```  [1] More, J.J., Garbow, B.S, and Hillstrom, K.E.
User's Guide for MINPACK-1.
Applied Math. Division, Argonne National Laboratory, Argonne,
Illinois, Report ANL-80-74, 1980.

```
Numerical Aspects
```                            2
The algorithm requires 0(N ) operations and is backward stable.

```
```  This routine is a LAPACK-based modification of LMPAR from the
MINPACK package [1], and with optional condition estimation.
The option COND = 'U' is useful when dealing with several
right-hand side vectors, but RANK should be reset.
If COND = 'E', but the matrix S is guaranteed to be nonsingular
and well conditioned relative to TOL, i.e., rank(R) = N, and
min(DIAG) > 0, then its condition is not estimated.

```
Example

Program Text

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Program Data
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Program Results
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