NAG Library Routine Document

D02AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

+− 9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

1 Purpose

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.

2 Specification

SUBROUTINE D02AGF (

H, E, PARERR, PARAM, C, N, N1, M1, AUX, BCAUX, RAAUX, PRSOL, MAT, COPY, WSPACE, WSPAC1, WSPAC2, IFAIL)

INTEGER	N, N1, M1, IFAIL
REAL (KIND=nag_wp)	H, E(N), PARERR(N1), PARAM(N1), C(M1,N), MAT(N1,N1), COPY(1,1), WSPACE(N,9), WSPAC1(N), WSPAC2(N)
EXTERNAL	AUX, BCAUX, RAAUX, PRSOL

3 Description

D02AGF solves a two-point boundary value problem by determining the unknown parameters

p_{1}, p_{2}, \dots, p_{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_{n_{1}}

correspond to the unknown boundary conditions.) It is assumed that we have a system of

n

first-order ordinary differential equations of the form

\frac{d y_{i}}{d x} = f_{i} (x, y_{1}, y_{2}, \dots, y_{n}), i = 1, 2, \dots, 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

n_{1}

unknown parameters

p_{1}, p_{2}, \dots, p_{n_{1}}

which are to be determined, and for which initial estimates must be supplied. The number of unknown parameters

n_{1}

must not exceed the number of equations

n

. If

n_{1} < n

, we assume that

(n - n_{1})

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

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

(i, j)

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

n_{1}

evaluations of the differential system.

4 References

None.

5 Parameters

You are strongly recommended to read Sections 3 and 8 in conjunction with this section.

1: 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: E(N) – REAL (KIND=nag_wp) arrayInput

On entry:

E (i)

must be set to a small quantity to control the

i

th solution component. The element

E (i)

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

E (i)

should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.

3: PARERR(N1) – REAL (KIND=nag_wp) arrayInput

On entry:

PARERR (i)

must be set to a small quantity to control the

i

th parameter component. The element

PARERR (i)

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

PARERR (i)

should not be chosen too small. They should usually be several orders of magnitude larger than machine precision.

4: PARAM(N1) – REAL (KIND=nag_wp) arrayInput/Output

On entry:

PARAM (i)

must be set to an estimate for the

i

th parameter,

p_{i}

, for

i = 1, 2, \dots, 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

PARERR (i) \times (1 + |PARAM (i)|)

if the error occurred when calculating the approximate derivatives).

5: C(M1,N) – REAL (KIND=nag_wp) arrayOutput

On exit: the solution when

M1 > 1

(see M1).

M1 = 1

, the elements of C are not used.

6: N – INTEGERInput

On entry:

n

, the total number of differential equations.

7: N1 – INTEGERInput

On entry:

n_{1}

, the number of parameters.

N1 < N

, the last

N - N1

differential equations (in AUX) are driving equations (see Section 3).

Constraint:

N1 \leq N

8: M1 – INTEGERInput

On entry: determines whether or not the final solution is computed as well as the parameter values.

$M1 = 1$: The final solution is not calculated;
$M1 > 1$: The final values of the solution at interval (length of range)/ $(M1 - 1)$ 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 $C (1, i)$ .

9: AUX – SUBROUTINE, supplied by the user.External Procedure

AUX must evaluate the functions

f_{i}

(i.e., the derivatives

y_{i}^{'}

) for given values of its arguments,

x, y_{1}, \dots, y_{n}

p_{1}, \dots, p_{n_{1}} .

The specification of AUX is:

SUBROUTINE AUX (

F, Y, X, PARAM)

REAL (KIND=nag_wp)

F(*), Y(*), X, PARAM(*)

In the description of the parameters of D02AGF below,

n

and

n1

denote the numerical values of N and N1 in the call of D02AGF.

1: F( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the value of $f_{i}$ , for $i = 1, 2, \dots, n$ .
2: Y( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $y_{i}$ , for $i = 1, 2, \dots, n$ , the value of the argument.
3: X – REAL (KIND=nag_wp)Input: On entry: $x$ , the value of the argument.
4: PARAM( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ , 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. Parameters denoted as Input must not be changed by this procedure.

10: 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_{n_{1}}

The specification of BCAUX is:

SUBROUTINE BCAUX (

G0, G1, PARAM)

REAL (KIND=nag_wp)

G0(*), G1(*), PARAM(*)

In the description of the parameters of D02AGF below,

n

and

n1

denote the numerical values of N and N1 in the call of D02AGF.

1: G0( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the values $y_{i}$ , for $i = 1, 2, \dots, n$ , at the boundary point $x_{0}$ (see RAAUX).
2: G1( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the values $y_{i}$ , for $i = 1, 2, \dots, n$ , at the boundary point $x_{1}$ (see RAAUX).
3: PARAM( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $p_{i}$ , for $i = 1, 2, \dots, n$ , 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. Parameters denoted as Input must not be changed by this procedure.

11: 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_{n_{1}}

The specification of RAAUX is:

SUBROUTINE RAAUX (

X0, X1, R, PARAM)

REAL (KIND=nag_wp)

X0, X1, R, PARAM(*)

In the description of the parameters of D02AGF below,

n1

denotes the numerical value of N1 in the call of D02AGF.

1: X0 – REAL (KIND=nag_wp)Output: On exit: must contain the left-hand end of the range, $x_{0}$ .
2: X1 – REAL (KIND=nag_wp)Output: On exit: must contain the right-hand end of the range $x_{1}$ .
3: R – REAL (KIND=nag_wp)Output: On exit: must contain the matching point, $r$ .
4: PARAM( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ , 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. Parameters denoted as Input must not be changed by this procedure.

12: 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_{i}

, for

i = 1, 2, \dots, n_{1}

, their errors,

e_{i}

, and the sum of squares of the errors at the matching point,

r

The specification of PRSOL is:

SUBROUTINE PRSOL (

PARAM, RES, N1, ERR)

INTEGER	N1
REAL (KIND=nag_wp)	PARAM(N1), RES, ERR(N1)

1: PARAM(N1) – REAL (KIND=nag_wp) arrayInput: On entry: $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ , the current value of the parameters.
2: RES – REAL (KIND=nag_wp)Input: On entry: the sum of squares of the errors in the parameters, $\sum_{i = 1}^{n_{1}} e_{i}^{2}$ .
3: N1 – INTEGERInput: On entry: $n_{1}$ , the number of parameters.
4: ERR(N1) – REAL (KIND=nag_wp) arrayInput: On entry: the errors in the parameters, $e_{i}$ , for $i = 1, 2, \dots, n_{1}$ .

PRSOL must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02AGF is called. Parameters denoted as Input must not be changed by this procedure.

13: MAT(N1,N1) – REAL (KIND=nag_wp) arrayWorkspace

14: COPY( $1$ , $1$ ) – REAL (KIND=nag_wp) arrayInput

15: WSPACE(N, $9$ ) – REAL (KIND=nag_wp) arrayWorkspace

16: WSPAC1(N) – REAL (KIND=nag_wp) arrayWorkspace

17: WSPAC2(N) – REAL (KIND=nag_wp) arrayWorkspace

18: IFAIL – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction 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, if you are not familiar with this parameter, the recommended value is

0

. 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:

$IFAIL = 1$: This indicates that $N1 > N$ on entry, that is the number of parameters is greater than the number of differential equations.

$IFAIL = 2$: As for $IFAIL = 4$ except that the integration failed while calculating the matrix for use in the Newton iteration.

$IFAIL = 3$: 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.

$IFAIL = 4$: 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 $r$ .

$IFAIL = 5$: 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)).

$IFAIL = 6$: 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.

$IFAIL = 7$: Convergence has not been obtained after $12$ satisfactory iterations of the Newton method.

A further discussion of these errors and the steps which might be taken to correct them is given in Section 8.

7 Accuracy

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.

8 Further Comments

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

IFAIL = 2

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}

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 9).

Wherever they occur in the procedure, the error parameters contained in the arrays E and PARERR are used in ‘mixed’ form; that is

E (i)

always occurs in expressions of the form

E (i) \times (1 + |y_{i}|)

, and

PARERR (i)

always occurs in expressions of the form

PARERR (i) \times (1 + |p_{i}|)

. 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 9, in order to monitor the progress of the iteration. Failure of the Newton iteration to converge (see

IFAIL = 6

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

IFAIL = 4

and

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

IFAIL = 5

) occurs because the mathematical problem has been posed incorrectly. The case

IFAIL = 4

usually occurs because

h

r

has been poorly estimated, so these values should be checked first. If

IFAIL = 2

is monitored, the solution

y_{1}, y_{2}, \dots, y_{n}

is sensitive to perturbations in the parameters

p_{i}

. Reduce the size of one or more values

PARERR (i)

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

ERR (i)

(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

PARAM (i)

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 9);
(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 9);
(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 9); and
(e)	differential equations with certain terms defined by other independent (driving) differential equations.

9 Example

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^{''} = \frac{y^{3} - y^{'}}{2 x}

on the range

0 \leq x \leq 16

, with boundary conditions

y (0) = 0.1

and

y (16) = 1 / 6

We cannot use the differential equation at

x = 0

because it is singular, so we take the truncated series expansion

y (x) = \frac{1}{10} + p_{1} \frac{\sqrt{x}}{10} + \frac{x}{100}

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

[0.1, 16]

and setting

p_{2} = y^{'} (16)

, 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 = Y (1)

y^{'} = Y (2)

, and estimate

p_{1} = PARAM (1) = 0.2

p_{2} = PARAM (2) = 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

\begin{array}{l} y^{'} = \tan ϕ \\ v^{'} = \frac{- (p_{1} \sin ϕ + 0.00002 v^{2})}{v \cos ϕ} \\ ϕ^{'} = \frac{- p_{1}}{v^{2}} k \end{array}

on the range

0 < x < p_{2}

with boundary conditions

\begin{array}{l} y = 0, & v = 500, & ϕ = 0.5 & at & x = 0 \\ y = 0, & v = 450, & ϕ = p_{3} & at & x = p_{2} . \end{array}

We write

y = Y (1)

v = Y (2)

ϕ = Y (3)

, and we take the matching point

r = p_{2}

. We estimate

p_{1} = PARAM (1) = 32

p_{2} = PARAM (2) = 6000

and

p_{3} = PARAM (3) = 0.54

(though this estimate is not important).

NAG Library Routine DocumentD02AGF

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Further Comments

9 Example

9.1 Program Text

9.2 Program Data

9.3 Program Results

NAG Library Routine Document

D02AGF