NAG Library Routine Document

e04rmf 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

Fortran Interface

Subroutine e04rmf (

handle, nres, isparse, nnzrd, irowrd, icolrd, ifail)

Integer, Intent (In)	::	nres, isparse, nnzrd, irowrd(nnzrd), icolrd(nnzrd)
Integer, Intent (Inout)	::	ifail
Type (c_ptr), Intent (In)	::	handle

C Header Interface

#include <nagmk26.h>

void	e04rmf_ (void *handle, const Integer nres, const Integer isparse, const Integer nnzrd, const Integer irowrd[], const Integer icolrd[], Integer *ifail)

3

Description

After the initialization routine e04raf has been called and unless the objective function has already been defined, e04rmf 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. e04rmf also allows the user to declare the structured first derivative matrix

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

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

See e04raf for more details.

4

References

None.

5

Arguments

1: $handle$ – Type (c_ptr)Input

On entry: the handle to the problem. It needs to be initialized by e04raf and must not be changed.

2: $nres$ – IntegerInput

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.

Constraint:

nres \geq 0

3: $isparse$ – IntegerInput

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. The ordering is assumed to be column-wise, namely the routine 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: $nnzrd$ – IntegerInput

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

Constraint: if

nres > 0

nnzrd > 0

5: $irowrd (nnzrd)$ – Integer arrayInput

6: $icolrd (nnzrd)$ – Integer arrayInput

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:

$1 \leq irowrd (l) \leq n$ , for $l = 1, 2, \dots, nnzrd$ ;
$1 \leq icolrd (l) \leq nres$ , for $l = 1, 2, \dots, nnzrd$ .

7: $ifail$ – IntegerInput/Output

On entry: ifail must be set to

0

- 1 or 1

. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 or 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, the recommended value is

- 1

. When the value $- 1 or 1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6

Error Indicators and Warnings

If on entry

ifail = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: 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 e04raf or it has been corrupted.

$ifail = 2$: 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.

$ifail = 3$: The objective function has already been defined.

$ifail = 6$: On entry, $isparse = 〈value〉$ .
Constraint: $isparse = 0$ or $1$ .

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

On entry, $nres = 〈value〉$ .
Constraint: $nres \geq 0$ .

$ifail = 8$: On entry, $i = 〈value〉$ , $icolrd (i) = 〈value〉$ and $nres = 〈value〉$ .
Constraint: $1 \leq icolrd (i) \leq nres$ .

On entry, $i = 〈value〉$ , $irowrd (i) = 〈value〉$ and $n = 〈value〉$ .
Constraint: $1 \leq irowrd (i) \leq n$ .

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.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

Not applicable.

8

Parallelism and Performance

e04rmf 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 e04rmf and solve it with e04fff 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 (e04rmfe.f90)

10.2

Program Data

None.

10.3

Program Results

Program Results (e04rmfe.r)

NAG Library Manual, Mark 26

NAG AD Library Manual, Mark 26

NAG C Library Manual, Mark 26

E04 (opt) Chapter Contents

E04 (opt) Chapter Introduction

NAG Library Routine Document

e04rmf (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

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