# NAG Library Routine Document

## 1Purpose

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.

## 2Specification

Fortran Interface
 Subroutine e04rmf ( nres,
 Integer, Intent (In) :: nres, isparse, nnzrd, irowrd(nnzrd), icolrd(nnzrd) Integer, Intent (Inout) :: ifail Type (c_ptr), Intent (In) :: handle
#include nagmk26.h
 void e04rmf_ (void **handle, const Integer *nres, const Integer *isparse, const Integer *nnzrd, const Integer irowrd[], const Integer icolrd[], Integer *ifail)

## 3Description

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
 $minimize x∈ℝn fx= ∑ j=1 mr rj x 2 subject to lx≤x≤ux ,$
where $x$ is an $n$-dimensional variable vector and ${r}_{i}\left(x\right)$ 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
 $∂rjx ∂xi i=1,…,n , ​ j=1,…,mr$
as being dense or sparse. If declared as sparse, its sparsity structure must be specified here.
See e04raf for more details.

None.

## 5Arguments

1:     $\mathbf{handle}$ – Type (c_ptr)Input
On entry: the handle to the problem. It needs to be initialized by e04raf and must not be changed.
2:     $\mathbf{nres}$ – IntegerInput
On entry: ${m}_{r}$, the number of residuals in the objective function.
If ${\mathbf{nres}}=0$, no objective function will be defined and irowrd and icolrd will not be referenced.
Constraint: ${\mathbf{nres}}\ge 0$.
3:     $\mathbf{isparse}$ – IntegerInput
On entry: is a flag indicating if the nonzero structure of the first derivative matrix is dense or sparse.
${\mathbf{isparse}}=0$
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 ${\mathbf{nnzrd}}=n×{m}_{r}$ and the vectors irowrd and icolrd filled as:
• ${\mathbf{irowrd}}=\left(1,2,\dots ,n,1,2,\dots ,n,\dots ,1,2,\dots ,n\right)$;
• ${\mathbf{icolrd}}=\left(1,1,\dots ,1,2,2,\dots ,2,\dots ,{m}_{r},{m}_{r},\dots ,{m}_{r}\right)$.
${\mathbf{isparse}}=1$
The sparsity structure of the first derivative matrix will be supplied by nnzrd, irowrd and icolrd.
Constraint: ${\mathbf{isparse}}=0$ or $1$.
4:     $\mathbf{nnzrd}$ – IntegerInput
On entry: the number of nonzeros in the first derivative matrix.
Constraint: if ${\mathbf{nres}}>0$, ${\mathbf{nnzrd}}>0$.
5:     $\mathbf{irowrd}\left({\mathbf{nnzrd}}\right)$ – Integer arrayInput
6:     $\mathbf{icolrd}\left({\mathbf{nnzrd}}\right)$ – 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 $n×{m}_{r}$. 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:
• $1\le {\mathbf{irowrd}}\left(\mathit{l}\right)\le n$, for $\mathit{l}=1,2,\dots ,{\mathbf{nnzrd}}$;
• $1\le {\mathbf{icolrd}}\left(\mathit{l}\right)\le {\mathbf{nres}}$, for $\mathit{l}=1,2,\dots ,{\mathbf{nnzrd}}$.
7:     $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $-1\text{​ or ​}1$. 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 $-1\text{​ or ​}1$ 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 $-\mathbf{1}\text{​ or ​}1$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
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.
${\mathbf{ifail}}=2$
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.
${\mathbf{ifail}}=3$
The objective function has already been defined.
${\mathbf{ifail}}=6$
On entry, ${\mathbf{isparse}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{isparse}}=0$ or $1$.
On entry, ${\mathbf{nnzrd}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nnzrd}}>0$.
On entry, ${\mathbf{nres}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nres}}\ge 0$.
${\mathbf{ifail}}=8$
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{icolrd}}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$ and ${\mathbf{nres}}=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{icolrd}}\left(\mathit{i}\right)\le {\mathbf{nres}}$.
On entry, $i=〈\mathit{\text{value}}〉$, ${\mathbf{irowrd}}\left(\mathit{i}\right)=〈\mathit{\text{value}}〉$ and $n=〈\mathit{\text{value}}〉$.
Constraint: $1\le {\mathbf{irowrd}}\left(\mathit{i}\right)\le n$.
On entry, more than one element of first derivative matrix has row index $〈\mathit{\text{value}}〉$ and column index $〈\mathit{\text{value}}〉$.
Constraint: each element of first derivative matrix must have a unique row and column index.
${\mathbf{ifail}}=-99$
See Section 3.9 in How to Use the NAG Library and its Documentation for further information.
${\mathbf{ifail}}=-399$
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.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

Not applicable.

## 8Parallelism and Performance

e04rmf is not threaded in any implementation.

None.

## 10Example

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:
 $r1x = x1+ x2- 0.9 r2x = 2x1+ x2- 1.9 r3x = 3x1+ x2- 3.0$
The expected result is:
 $x=0.95,0.10$
with an objective value of $0.015$.

### 10.1Program Text

Program Text (e04rmfe.f90)

None.

### 10.3Program Results

Program Results (e04rmfe.r)

© The Numerical Algorithms Group Ltd, Oxford, UK. 2017