e04rmc is a part of the NAG optimization modelling suite and defines the number of residuals in a sum of squares objective function (nonlinear least squares problems) and, optionally, the sparsity structure of their first derivatives.

2 Specification

#include <nag.h>

void	e04rmc (void handle, Integer nres, Integer isparse, Integer nnzrd, const Integer irowrd[], const Integer icolrd[], NagError fail)

The function may be called by the names: e04rmc or nag_opt_handle_set_nlnls.

3 Description

After the initialization function e04rac has been called and unless the objective function has already been defined, e04rmc may be used to declare the objective function of the optimization problem as a sum of squares. It will typically be used in data fitting or calibration problems of the form

\begin{array}{l} \underset{x \in ℝ^{n}}{minimize} & f (x) = \sum_{j = 1}^{m_{r}} {r_{j} (x)}^{2} \\ subject to & l_{x} \leq x \leq u_{x}, \end{array}

where

x

is an

n

-dimensional variable vector and

r_{i} (x)

are nonlinear residuals (see Section 2.2.3 in the E04 Chapter Introduction). The values of the residuals, and possibly their derivatives, will be communicated to the solver by a user-supplied function. e04rmc also allows the structured first derivative matrix

{[\frac{\partial r_{j} (x)}{\partial x_{i}}]}_{i = 1, \dots, n, ​ j = 1, \dots, m_{r}}

to be declared as being dense or sparse. If declared as sparse, its sparsity structure must be specified here.

See Section 4.1 in the E04 Chapter Introduction for more details about the NAG optimization modelling suite.

4 References

None.

5 Arguments

1: $handle$ – void * Input

On entry: the handle to the problem. It needs to be initialized by e04rac and must not be changed between calls to the NAG optimization modelling suite.

2: $nres$ – Integer Input

On entry:

m_{r}

, the number of residuals in the objective function.

nres = 0

, no objective function will be defined and irowrd and icolrd will not be referenced and may be NULL.

Constraint:

nres \geq 0

3: $isparse$ – Integer Input

On entry: is a flag indicating if the nonzero structure of the first derivative matrix is dense or sparse.

$isparse = 0$

The first derivative matrix is considered dense and irowrd and icolrd will not be referenced and may be specified as NULL. The ordering is assumed to be column-wise, namely the function will behave as if

nnzrd = n \times m_{r}

and the vectors irowrd and icolrd filled as:

$irowrd = (1, 2, \dots, n, 1, 2, \dots, n, \dots, 1, 2, \dots, n)$ ;
$icolrd = (1, 1, \dots, 1, 2, 2, \dots, 2, \dots, m_{r}, m_{r}, \dots, m_{r})$ .

$isparse = 1$

The sparsity structure of the first derivative matrix will be supplied by nnzrd, irowrd and icolrd.

Constraint:

isparse = 0

1

4: $nnzrd$ – Integer Input

On entry: the number of nonzeros in the first derivative matrix.

Constraint: if

isparse = 1

and

nres > 0

nnzrd > 0

5: $irowrd [nnzrd]$ – const Integer Input

6: $icolrd [nnzrd]$ – const Integer Input

On entry: arrays irowrd and icolrd store the sparsity structure (pattern) of the first derivative matrix as nnzrd nonzeros in coordinate storage (CS) format (see Section 2.1.1 in the F11 Chapter Introduction). The matrix has dimensions

n \times m_{r}

. irowrd specifies one-based row indices and icolrd specifies one-based column indices. No particular order of elements is expected, but elements should not repeat and the same order should be used when the first derivative matrix is evaluated for the solver.

Constraints:

$1 \leq irowrd [l - 1] \leq n$ , for $l = 1, 2, \dots, nnzrd$ ;
$1 \leq icolrd [l - 1] \leq nres$ , for $l = 1, 2, \dots, nnzrd$ .

7: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_ALREADY_DEFINED: The objective function has already been defined.
NE_BAD_PARAM: On entry, argument $〈value〉$ had an illegal value.
NE_HANDLE: The supplied handle does not define a valid handle to the data structure for the NAG optimization modelling suite. It has not been initialized by e04rac or it has been corrupted.
NE_INT: On entry, $isparse = 〈value〉$ .
Constraint: $isparse = 0$ or $1$ .

On entry, $nnzrd = 〈value〉$ .
Constraint: $nnzrd > 0$ .

On entry, $nres = 〈value〉$ .
Constraint: $nres \geq 0$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_INVALID_CS: On entry, $i = 〈value〉$ , $icolrd [i - 1] = 〈value〉$ and $nres = 〈value〉$ .
Constraint: $1 \leq icolrd [i - 1] \leq nres$ .

On entry, $i = 〈value〉$ , $irowrd [i - 1] = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq irowrd [i - 1] \leq n$ .

On entry, more than one element of first derivative matrix has row index $〈value〉$ and column index $〈value〉$ .
Constraint: each element of first derivative matrix must have a unique row and column index.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_PHASE: The Hessians of nonlinear functions have already been defined, a nonlinear objective cannot be added.

The problem cannot be modified in this phase any more, the solver has already been called.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rmc is not threaded in any implementation.

9 Further Comments

None.

10 Example

In this example, we demonstrate how to declare a least squares problem through e04rmc and solve it with e04ffc on a very simple example. Here

n = 2

m_{r} = 3

and the residuals are computed by:

\begin{array}{l} r_{1} (x) & = & x (1) + x (2) - 0.9 \\ r_{2} (x) & = & 2 x (1) + x (2) - 1.9 \\ r_{3} (x) & = & 3 x (1) + x (2) - 3.0 \end{array}

The expected result is:

x = (0.95, 0.10)

with an objective value of

0.015

10.1 Program Text

Program Text (e04rmce.c)

10.2 Program Data

None.

10.3 Program Results

Program Results (e04rmce.r)

NAG Library Manual, Mark 27

Interfaces: FL CL AD

NAG CL Interface Introduction

E04 (Opt) Chapter Contents

E04 (Opt) Chapter Introduction

e04rm: FL CL AD

NAG CL Interfacee04rmc (handle_​set_​nlnls)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
e04rmc (handle_set_nlnls)