e04rmc is a part of the NAG optimization modelling suite and defines the residual functions for nonlinear regression problems (such as nonlinear least squares and general nonlinear data fitting) with the given number of residuals 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 handle has been initialized (e.g., e04rac has been called), e04rmc may be used to define the residual functions in the objective function of nonlinear least squares or general nonlinear data fitting problems. If the objective function has already been defined, it will be overwritten. 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}} χ (r_{j} (x)) \\ subject to & l_{x} \leq x \leq u_{x}, \end{array}

where

x

is an

n

-dimensional variable vector,

r_{j} (x)

are nonlinear residuals (see Section 2.2.3 in the E04 Chapter Introduction), and

χ

is a type of loss function. For example, the model of a least squares problem can be written as

\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}

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 by e04rmc. If e04rmc is called with

m_{r} = 0

, any existing objective function is removed, no new one is added and the problem will be solved as a feasible point problem. Note that it is possible to temporarily disable and enable individual residuals in the model by e04tcc and e04tbc, respectively.

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 (e.g., 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_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 properly initialized 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 problem cannot be modified right now, the solver is running.

7 Accuracy

Not applicable.

8 Parallelism and Performance

e04rmc is not threaded in any implementation.

9 Further Comments

9.1 Internal Changes

Internal changes have been made to this function as follows:

At Mark 27.1: Previously, it was not possible to modify the objective function once it was set or to edit the model once a solver had been called. These limitations have been removed and the associated error codes were removed.

For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.