# NAG Library Routine Document

## 1Purpose

d02agf solves a two-point boundary value problem for a system of ordinary differential equations, using initial value techniques and Newton iteration; it generalizes d02haf to include the case where parameters other than boundary values are to be determined.

## 2Specification

Fortran Interface
 Subroutine d02agf ( h, e, c, n, n1, m1, aux, mat, copy,
 Integer, Intent (In) :: n, n1, m1 Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: e(n), parerr(n1), copy(1,1) Real (Kind=nag_wp), Intent (Inout) :: h, param(n1) Real (Kind=nag_wp), Intent (Out) :: c(m1,n), mat(n1,n1), wspace(n,9), wspac1(n), wspac2(n) External :: aux, bcaux, raaux, prsol
#include nagmk26.h
 void d02agf_ (double *h, const double e[], const double parerr[], double param[], double c[], const Integer *n, const Integer *n1, const Integer *m1, void (NAG_CALL *aux)(double f[], const double y[], const double *x, const double param[]),void (NAG_CALL *bcaux)(double g0[], double g1[], const double param[]),void (NAG_CALL *raaux)(double *x0, double *x1, double *r, const double param[]),void (NAG_CALL *prsol)(const double param[], const double *res, const Integer *n1, const double err[]),double mat[], const double copy[], double wspace[], double wspac1[], double wspac2[], Integer *ifail)

## 3Description

d02agf solves a two-point boundary value problem by determining the unknown parameters ${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ of the problem. These parameters may be, but need not be, boundary values (as they are in d02haf); they may include eigenvalue parameters in the coefficients of the differential equations, length of the range of integration, etc. The notation and methods used are similar to those of d02haf and you are advised to study this first. (There the parameters ${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ correspond to the unknown boundary conditions.) It is assumed that we have a system of $\mathit{n}$ first-order ordinary differential equations of the form
 $dyi dx =fix,y1,y2,…,yn, i=1,2,…,n,$
and that derivatives ${f}_{i}$ are evaluated by aux. The system, including the boundary conditions given by bcaux, and the range of integration and matching point, $r$, given by raaux, involves the ${\mathit{n}}_{1}$ unknown parameters ${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$ which are to be determined, and for which initial estimates must be supplied. The number of unknown parameters ${\mathit{n}}_{1}$ must not exceed the number of equations $\mathit{n}$. If ${\mathit{n}}_{1}<\mathit{n}$, we assume that $\left(\mathit{n}-{\mathit{n}}_{1}\right)$ equations of the system are not involved in the matching process. These are usually referred to as ‘driving equations’; they are independent of the parameters and of the solutions of the other ${\mathit{n}}_{1}$ equations. In numbering the equations for aux, the driving equations must be put last.
The estimated values of the parameters are corrected by a form of Newton iteration. The Newton correction on each iteration is calculated using a matrix whose $\left(i,j\right)$th element depends on the derivative of the $i$th component of the solution, ${y}_{i}$, with respect to the $j$th parameter, ${p}_{j}$. This matrix is calculated by a simple numerical differentiation technique which requires ${\mathit{n}}_{1}$ evaluations of the differential system.

None.

## 5Arguments

You are strongly recommended to read Sections 3 and 9 in conjunction with this section.
1:     $\mathbf{h}$ – Real (Kind=nag_wp)Input/Output
On entry: h must be set to an estimate of the step size, $h$, needed for integration.
On exit: the last step length used.
2:     $\mathbf{e}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{e}}\left(i\right)$ must be set to a small quantity to control the $i$th solution component. The element ${\mathbf{e}}\left(i\right)$ is used:
 (i) in the bound on the local error in the $i$th component of the solution ${y}_{i}$ during integration, (ii) in the convergence test on the $i$th component of the solution ${y}_{i}$ at the matching point in the Newton iteration.
The elements ${\mathbf{e}}\left(i\right)$ should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.
3:     $\mathbf{parerr}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{parerr}}\left(i\right)$ must be set to a small quantity to control the $i$th parameter component. The element ${\mathbf{parerr}}\left(i\right)$ is used:
 (i) in the convergence test on the $i$th parameter in the Newton iteration, (ii) in perturbing the $i$th parameter when approximating the derivatives of the components of the solution with respect to the $i$th parameter, for use in the Newton iteration.
The elements ${\mathbf{parerr}}\left(i\right)$ should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.
4:     $\mathbf{param}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{param}}\left(\mathit{i}\right)$ must be set to an estimate for the $\mathit{i}$th parameter, ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{n1}}$.
On exit: the corrected value for the $i$th parameter, unless an error has occurred, when it contains the last calculated value of the parameter (possibly perturbed by ${\mathbf{parerr}}\left(i\right)×\left(1+\left|{\mathbf{param}}\left(i\right)\right|\right)$ if the error occurred when calculating the approximate derivatives).
5:     $\mathbf{c}\left({\mathbf{m1}},{\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the solution when ${\mathbf{m1}}>1$ (see m1).
If ${\mathbf{m1}}=1$, the elements of c are not used.
6:     $\mathbf{n}$ – IntegerInput
On entry: $\mathit{n}$, the total number of differential equations.
7:     $\mathbf{n1}$ – IntegerInput
On entry: ${\mathit{n}}_{1}$, the number of parameters.
If ${\mathbf{n1}}<{\mathbf{n}}$, the last ${\mathbf{n}}-{\mathbf{n1}}$ differential equations (in aux) are driving equations (see Section 3).
Constraint: ${\mathbf{n1}}\le {\mathbf{n}}$.
8:     $\mathbf{m1}$ – IntegerInput
On entry: determines whether or not the final solution is computed as well as the parameter values.
${\mathbf{m1}}=1$
The final solution is not calculated;
${\mathbf{m1}}>1$
The final values of the solution at interval (length of range)/$\left({\mathbf{m1}}-1\right)$ are calculated and stored sequentially in the array c starting with the values of ${y}_{i}$ evaluated at the first end point (see raaux) stored in ${\mathbf{c}}\left(1,i\right)$.
9:     $\mathbf{aux}$ – Subroutine, supplied by the user.External Procedure
aux must evaluate the functions ${f}_{i}$ (i.e., the derivatives ${y}_{i}^{\prime }$) for given values of its arguments, $x,{y}_{1},\dots ,{y}_{\mathit{n}}$, ${p}_{1},\dots ,{p}_{{\mathit{n}}_{1}}\text{.}$
The specification of aux is:
Fortran Interface
 Subroutine aux ( f, y, x,
 Real (Kind=nag_wp), Intent (In) :: y(*), x, param(*) Real (Kind=nag_wp), Intent (Out) :: f(*)
#include nagmk26.h
 void aux (double f[], const double y[], const double *x, const double param[])
In the description of the arguments of d02agf below, $\mathit{n}$ and $\mathit{n1}$ denote the numerical values of n and n1 in the call of d02agf.
1:     $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the value of ${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
2:     $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.
3:     $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the value of the argument.
4:     $\mathbf{param}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$, the value of the parameters.
aux must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02agf is called. Arguments denoted as Input must not be changed by this procedure.
Note: aux should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02agf. If your code inadvertently does return any NaNs or infinities, d02agf is likely to produce unexpected results.
10:   $\mathbf{bcaux}$ – Subroutine, supplied by the user.External Procedure
bcaux must evaluate the values of ${y}_{i}$ at the end points of the range given the values of ${p}_{1},\dots ,{p}_{{\mathit{n}}_{1}}$.
The specification of bcaux is:
Fortran Interface
 Subroutine bcaux ( g0, g1,
 Real (Kind=nag_wp), Intent (In) :: param(*) Real (Kind=nag_wp), Intent (Out) :: g0(*), g1(*)
#include nagmk26.h
 void bcaux (double g0[], double g1[], const double param[])
In the description of the arguments of d02agf below, $\mathit{n}$ and $\mathit{n1}$ denote the numerical values of n and n1 in the call of d02agf.
1:     $\mathbf{g0}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, at the boundary point ${x}_{0}$ (see raaux).
2:     $\mathbf{g1}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, at the boundary point ${x}_{1}$ (see raaux).
3:     $\mathbf{param}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the parameters.
bcaux must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02agf is called. Arguments denoted as Input must not be changed by this procedure.
Note: bcaux should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02agf. If your code inadvertently does return any NaNs or infinities, d02agf is likely to produce unexpected results.
11:   $\mathbf{raaux}$ – Subroutine, supplied by the user.External Procedure
raaux must evaluate the end points, ${x}_{0}$ and ${x}_{1}$, of the range and the matching point, $r$, given the values ${p}_{1},{p}_{2},\dots ,{p}_{{\mathit{n}}_{1}}$.
The specification of raaux is:
Fortran Interface
 Subroutine raaux ( x0, x1, r,
 Real (Kind=nag_wp), Intent (In) :: param(*) Real (Kind=nag_wp), Intent (Out) :: x0, x1, r
#include nagmk26.h
 void raaux (double *x0, double *x1, double *r, const double param[])
In the description of the arguments of d02agf below, $\mathit{n1}$ denotes the numerical value of n1 in the call of d02agf.
1:     $\mathbf{x0}$ – Real (Kind=nag_wp)Output
On exit: must contain the left-hand end of the range, ${x}_{0}$.
2:     $\mathbf{x1}$ – Real (Kind=nag_wp)Output
On exit: must contain the right-hand end of the range ${x}_{1}$.
3:     $\mathbf{r}$ – Real (Kind=nag_wp)Output
On exit: must contain the matching point, $r$.
4:     $\mathbf{param}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$, the value of the parameters.
raaux must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02agf is called. Arguments denoted as Input must not be changed by this procedure.
Note: raaux should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02agf. If your code inadvertently does return any NaNs or infinities, d02agf is likely to produce unexpected results.
12:   $\mathbf{prsol}$ – Subroutine, supplied by the user.External Procedure
prsol is called at each iteration of the Newton method and can be used to print the current values of the parameters ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$, their errors, ${e}_{i}$, and the sum of squares of the errors at the matching point, $r$.
The specification of prsol is:
Fortran Interface
 Subroutine prsol ( res, n1, err)
 Integer, Intent (In) :: n1 Real (Kind=nag_wp), Intent (In) :: param(n1), res, err(n1)
#include nagmk26.h
 void prsol (const double param[], const double *res, const Integer *n1, const double err[])
1:     $\mathbf{param}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${p}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$, the current value of the parameters.
2:     $\mathbf{res}$ – Real (Kind=nag_wp)Input
On entry: the sum of squares of the errors in the arguments, $\sum _{i=1}^{{\mathit{n}}_{1}}{e}_{i}^{2}$.
3:     $\mathbf{n1}$ – IntegerInput
On entry: ${\mathit{n}}_{1}$, the number of parameters.
4:     $\mathbf{err}\left({\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayInput
On entry: the errors in the parameters, ${e}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,{\mathit{n}}_{1}$.
prsol must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02agf is called. Arguments denoted as Input must not be changed by this procedure.
13:   $\mathbf{mat}\left({\mathbf{n1}},{\mathbf{n1}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14:   $\mathbf{copy}\left(1,1\right)$ – Real (Kind=nag_wp) arrayInput
15:   $\mathbf{wspace}\left({\mathbf{n}},9\right)$ – Real (Kind=nag_wp) arrayWorkspace
16:   $\mathbf{wspac1}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
17:   $\mathbf{wspac2}\left({\mathbf{n}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
18:   $\mathbf{ifail}$ – 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, if you are not familiar with this argument, the recommended value is $0$. When the value $-\mathbf{1}\text{​ or ​}\mathbf{1}$ 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:
${\mathbf{ifail}}=1$
This indicates that ${\mathbf{n1}}>{\mathbf{n}}$ on entry, that is the number of parameters is greater than the number of differential equations.
${\mathbf{ifail}}=2$
As for ${\mathbf{ifail}}={\mathbf{4}}$ except that the integration failed while calculating the matrix for use in the Newton iteration.
${\mathbf{ifail}}=3$
The current matching point $r$ does not lie between the current end points ${x}_{0}$ and ${x}_{1}$. If the values ${x}_{0}$, ${x}_{1}$ and $r$ depend on the parameters ${p}_{i}$, this may occur at any time in the Newton iteration if care is not taken to avoid it when coding raaux.
${\mathbf{ifail}}=4$
The step length for integration h has halved more than $13$ times (or too many steps were needed to reach the end of the range of integration) in attempting to control the local truncation error whilst integrating to obtain the solution corresponding to the current values ${p}_{i}$. If, on failure, h has the sign of $r-{x}_{0}$ then failure has occurred whilst integrating from ${x}_{0}$ to $r$, otherwise it has occurred whilst integrating from ${x}_{1}$ to $r$.
${\mathbf{ifail}}=5$
The matrix of the equations to be solved for corrections to the variable parameters in the Newton method is singular (as determined by f07adf (dgetrf)).
${\mathbf{ifail}}=6$
A satisfactory correction to the parameters was not obtained on the last Newton iteration employed. A Newton iteration is deemed to be unsatisfactory if the sum of the squares of the residuals (which can be printed using prsol) has not been reduced after three iterations using a new Newton correction.
${\mathbf{ifail}}=7$
Convergence has not been obtained after $12$ satisfactory iterations of the Newton method.
${\mathbf{ifail}}=-99$
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.
A further discussion of these errors and the steps which might be taken to correct them is given in Section 9.

## 7Accuracy

If the process converges, the accuracy to which the unknown parameters are determined is usually close to that specified by you; and the solution, if requested, is usually determined to the accuracy specified.

## 8Parallelism and Performance

d02agf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02agf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by d02agf depends on the complexity of the system, and on the number of iterations required. In practice, integration of the differential equations is by far the most costly process involved.
There may be particular difficulty in integrating the differential equations in one direction (indicated by ${\mathbf{ifail}}={\mathbf{2}}$ or ${\mathbf{4}}$). The value of $r$ should be adjusted to avoid such difficulties.
If the matching point $r$ is at one of the end points ${x}_{0}$ or ${x}_{1}$ and some of the parameters are used only to determine the boundary values at this point, then good initial estimates for these parameters are not required, since they are completely determined by the routine (for example, see ${p}_{2}$ in EX1 of Section 10).
Wherever they occur in the procedure, the error parameters contained in the arrays e and parerr are used in ‘mixed’ form; that is ${\mathbf{e}}\left(i\right)$ always occurs in expressions of the form ${\mathbf{e}}\left(i\right)×\left(1+\left|{y}_{i}\right|\right)$, and ${\mathbf{parerr}}\left(i\right)$ always occurs in expressions of the form ${\mathbf{parerr}}\left(i\right)×\left(1+\left|{p}_{i}\right|\right)$. Though not ideal for every application, it is expected that this mixture of absolute and relative error testing will be adequate for most purposes.
Note that convergence is not guaranteed. You are strongly advised to provide an output prsol, as shown in EX1 of Section 10, in order to monitor the progress of the iteration. Failure of the Newton iteration to converge (see ${\mathbf{ifail}}={\mathbf{6}}$ or ${\mathbf{7}}$) usually results from poor starting approximations to the parameters, though occasionally such failures occur because the elements of one or both of the arrays parerr or e are too small. (It should be possible to distinguish these cases by studying the output from prsol.) Poor starting approximations can also result in the failure described under ${\mathbf{ifail}}={\mathbf{4}}$ and ${\mathbf{5}}$ in Section 6 (especially if these errors occur after some Newton iterations have been completed, that is, after two or more calls of prsol). More frequently, a singular matrix in the Newton method (monitored as ${\mathbf{ifail}}={\mathbf{5}}$) occurs because the mathematical problem has been posed incorrectly. The case ${\mathbf{ifail}}={\mathbf{4}}$ usually occurs because $h$ or $r$ has been poorly estimated, so these values should be checked first. If ${\mathbf{ifail}}={\mathbf{2}}$ is monitored, the solution ${y}_{1},{y}_{2},\dots ,{y}_{\mathit{n}}$ is sensitive to perturbations in the parameters ${p}_{i}$. Reduce the size of one or more values ${\mathbf{parerr}}\left(i\right)$ to reduce the perturbations. Since only one value ${p}_{i}$ is perturbed at any time when forming the matrix, the perturbation which is too large can be located by studying the final output from prsol and the values of the parameters returned by d02agf. If this change leads to other types of failure improve the initial values of ${p}_{i}$ by other means.
The computing time for integrating the differential equations can sometimes depend critically on the quality of the initial estimates for the parameters ${p}_{i}$. If it seems that too much computing time is required and, in particular, if the values ${\mathbf{err}}\left(i\right)$ (available on each call of prsol) are much larger than the expected values of the solution at the matching point $r$, then the coding of aux, bcaux and raaux should be checked for errors. If no errors can be found, an independent attempt should be made to improve the initial estimates for ${\mathbf{param}}\left(i\right)$.
The subroutine can be used to solve a very wide range of problems, for example:
 (a) eigenvalue problems, including problems where the eigenvalue occurs in the boundary conditions; (b) problems where the differential equations depend on some parameters which are to be determined so as to satisfy certain boundary conditions (see EX1 in Section 10); (c) problems where one of the end points of the range of integration is to be determined as the point where a variable ${y}_{i}$ takes a particular value (see EX2 in Section 10); (d) singular problems and problems on infinite ranges of integration where the values of the solution at ${x}_{0}$ or ${x}_{1}$ or both are determined by a power series or an asymptotic expansion (or a more complicated expression) and where some of the coefficients in the expression are to be determined (see EX1 in Section 10); and (e) differential equations with certain terms defined by other independent (driving) differential equations.

## 10Example

For this routine two examples are presented. There is a single example program for d02agf, with a main program and the code to solve the two example problems given in Example 1 (EX1) and Example 2 (EX2).
Example 1 (EX1)
This example finds the solution of the differential equation
 $y′′=y3-y′ 2x$
on the range $0\le x\le 16$, with boundary conditions $y\left(0\right)=0.1$ and $y\left(16\right)=1/6$.
We cannot use the differential equation at $x=0$ because it is singular, so we take the truncated series expansion
 $yx=110+p1x10+x100$
near the origin (which is correct to the number of terms given in this case). Here ${p}_{1}$ is one of the parameters to be determined. We choose the range as $\left[0.1,16\right]$ and setting ${p}_{2}={y}^{\prime }\left(16\right)$, we can determine all the boundary conditions. We take the matching point to be $16$, the end of the range, and so a good initial guess for ${p}_{2}$ is not necessary. We write $y={\mathbf{y}}\left(1\right)$, ${y}^{\prime }={\mathbf{y}}\left(2\right)$, and estimate ${p}_{1}={\mathbf{param}}\left(1\right)=0.2$, ${p}_{2}={\mathbf{param}}\left(2\right)=0.0$.
Example 2 (EX2)
This example finds the gravitational constant ${p}_{1}$ and the range ${p}_{2}$ over which a projectile must be fired to hit the target with a given velocity. The differential equations are
 $y′=tan⁡ϕ v′= -p1sin⁡ϕ+0.00002⁢v2 vcos⁡ϕ ϕ′=-p1v2k$
on the range $0 with boundary conditions
 $y=0, v=500, ϕ=0.5 at x=0 y=0, v=450, ϕ=p3 at x=p2.$
We write $y={\mathbf{y}}\left(1\right)$, $v={\mathbf{y}}\left(2\right)$, $\varphi ={\mathbf{y}}\left(3\right)$, and we take the matching point $r={p}_{2}$. We estimate ${p}_{1}={\mathbf{param}}\left(1\right)=32$, ${p}_{2}={\mathbf{param}}\left(2\right)=6000$ and ${p}_{3}={\mathbf{param}}\left(3\right)=0.54$ (though this estimate is not important).

### 10.1Program Text

Program Text (d02agfe.f90)

### 10.2Program Data

Program Data (d02agfe.d)

### 10.3Program Results

Program Results (d02agfe.r)

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