NAG Library Routine Document

D02HBF

+− 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

D02HBF solves a two-point boundary value problem for a system of ordinary differential equations, using initial value techniques and Newton iteration; it generalizes subroutine D02HAF to include the case where parameters other than boundary values are to be determined.

2 Specification

SUBROUTINE D02HBF (

P, N1, PE, E, N, SOLN, M1, FCN, BC, RANGE, W, SDW, IFAIL)

INTEGER	N1, N, M1, SDW, IFAIL
REAL (KIND=nag_wp)	P(N1), PE(N1), E(N), SOLN(N,M1), W(N,SDW)
EXTERNAL	FCN, BC, RANGE

3 Description

D02HBF 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; 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. (The parameters

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

correspond precisely to the unknown boundary conditions in D02HAF.) 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 the derivatives

f_{i}

are evaluated by FCN. The system, including the boundary conditions given by BC and the range of integration given by RANGE, 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 FCN, the driving equations must be put first.

The estimated values of the parameters are corrected by a form of Newton iteration. The Newton correction on each iteration is calculated using a Jacobian 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.

If the parameter IFAIL is set appropriately, the routine automatically prints messages to inform you of the flow of the calculation. These messages are discussed in detail in Section 8.

D02HBF is a simplified version of D02SAF which is described in detail in Gladwell (1979).

4 References

Gladwell I (1979) The development of the boundary value codes in the ordinary differential equations chapter of the NAG Library Codes for Boundary Value Problems in Ordinary Differential Equations. Lecture Notes in Computer Science (eds B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76 Springer–Verlag

5 Parameters

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

1: P(N1) – REAL (KIND=nag_wp) arrayInput/Output

On entry: an estimate for the

i

th parameter,

p_{i}

, for

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

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.

2: N1 – INTEGERInput

On entry:

n_{1}

, the number of parameters.

Constraint:

1 \leq N1 \leq N

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

On entry: the elements of PE must be given small positive values. The element

PE (i)

is used

(i)	in the convergence test on the $i$ th parameter in the Newton iteration, and
(ii)	in perturbing the $i$ th parameter when approximating the derivatives of the components of the solution with respect to this parameter for use in the Newton iteration.

The elements

PE (i)

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

Constraint:

PE (i) > 0.0

, for

i = 1, 2, \dots, N1

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

On entry: the elements of E must be given positive values. The element

E (i)

is used in the bound on the local error in the

i

th component of the solution

y_{i}

during integration.

The elements

E (i)

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

Constraint:

E (i) > 0.0

, for

i = 1, 2, \dots, N

5: N – INTEGERInput

On entry:

n

, the total number of differential equations.

Constraint:

N \geq 2

6: SOLN(N,M1) – REAL (KIND=nag_wp) arrayOutput

On exit: the solution when

M1 > 1

7: M1 – INTEGERInput

On entry: a value which controls exit 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 SOLN starting with the values of the solutions evaluated at the first end point (see RANGE) stored in the first column of SOLN.

Constraint:

M1 \geq 1

8: FCN – SUBROUTINE, supplied by the user.External Procedure

FCN must evaluate the functions

f_{i}

(i.e., the derivatives

y_{i}^{'}

), for

i = 1, 2, \dots, n

, at a general point

x

The specification of FCN is:

SUBROUTINE FCN (

X, Y, F, P)

REAL (KIND=nag_wp)

X, Y(*), F(*), P(*)

In the description of the parameters of D02HBF below,

n

and

n1

denote the numerical values of N and N1 in the call of D02HBF.

1: X – REAL (KIND=nag_wp)Input: On entry: $x$ , the value of the argument.
2: Y( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: $y_{i}$ , for $i = 1, 2, \dots, n$ , the value of the argument.
3: F( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the value of $f_{i}$ , for $i = 1, 2, \dots, n$ . The $f_{i}$ may depend upon the parameters $p_{j}$ , for $j = 1, 2, \dots, n_{1}$ . If there are any driving equations (see Section 3) then these must be numbered first in the ordering of the components of F in FCN.
4: P( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: the current estimate of the parameter $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ .

FCN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02HBF is called. Parameters denoted as Input must not be changed by this procedure.

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

BC must place in G1 and G2 the boundary conditions at

a

and

b

respectively (see RANGE).

The specification of BC is:

SUBROUTINE BC (

G1, G2, P)

REAL (KIND=nag_wp)

G1(*), G2(*), P(*)

In the description of the parameters of D02HBF below,

n

and

n1

denote the numerical values of N and N1 in the call of D02HBF.

1: G1( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the value of $y_{i} (a)$ , (where this may be a known value or a function of the parameters $p_{j}$ , for $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, n_{1}$ ).
2: G2( $*$ ) – REAL (KIND=nag_wp) arrayOutput: On exit: the value of $y_{i} (b)$ , for $i = 1, 2, \dots, n$ , (where these may be known values or functions of the parameters $p_{j}$ , for $j = 1, 2, \dots, n_{1}$ ). If $n > n_{1}$ , so that there are some driving equations, then the first $n - n_{1}$ values of G2 need not be set since they are never used.
3: P( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: an estimate of the parameter $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ .

BC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02HBF is called. Parameters denoted as Input must not be changed by this procedure.

10: RANGE – SUBROUTINE, supplied by the user.External Procedure

RANGE must evaluate the boundary points

a

and

b

, each of which may depend on the parameters

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

. The integrations in the shooting method are always from

a

b

The specification of RANGE is:

SUBROUTINE RANGE (

A, B, P)

REAL (KIND=nag_wp)

A, B, P(*)

In the description of the parameters of D02HBF below,

n1

denotes the actual value of N1 in the call of D02HBF.

1: A – REAL (KIND=nag_wp)Output: On exit: $a$ , one of the boundary points.
2: B – REAL (KIND=nag_wp)Output: On exit: the second boundary point, $b$ . Note that $B > A$ forces the direction of integration to be that of increasing $x$ . If A and B are interchanged the direction of integration is reversed.
3: P( $*$ ) – REAL (KIND=nag_wp) arrayInput: On entry: the current estimate of the $i$ th parameter, $p_{i}$ , for $i = 1, 2, \dots, n_{1}$ .

RANGE must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02HBF is called. Parameters denoted as Input must not be changed by this procedure.

11: W(N,SDW) – REAL (KIND=nag_wp) arrayOutput

Used mainly for workspace.

On exit: with

IFAIL = 2

3

4

5

(see Section 6),

W (i, 1)

, for

i = 1, 2, \dots, n

, contains the solution at the point

x

when the error occurred.

W (1, 2)

contains

x

12: SDW – INTEGERInput

On entry: the second dimension of the array W as declared in the (sub)program from which D02HBF is called.

Constraint:

SDW \geq 3 N + 14 + \max (11, N)

13: IFAIL – INTEGERInput/Output

For this routine, the normal use of IFAIL is extended to control the printing of error and warning messages as well as specifying hard or soft failure (see Section 3.3 in the Essential Introduction).

On entry: IFAIL must be set to a value with the decimal expansion

cba

, where each of the decimal digits

c

b

and

a

must have a value of

0

1

$a = 0$	specifies hard failure, otherwise soft failure;
$b = 0$	suppresses error messages, otherwise error messages will be printed (see Section 6);
$c = 0$	suppresses warning messages, otherwise warning messages will be printed (see Section 6).

The recommended value for inexperienced users is

110

(i.e., hard failure with all messages printed).

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:

$IFAIL = 1$: One or more of the parameters N, N1, M1, SDW, E or PE is incorrectly set.

$IFAIL = 2$: The step length for the integration became too short whilst calculating the residual (see Section 8).

$IFAIL = 3$: No initial step length could be chosen for the integration whilst calculating the residual.

Note:

IFAIL = 2

3

can occur due to choosing too small a value for E or due to choosing the wrong direction of integration. Try varying E and interchanging

a

and

b

. These error exits can also occur for very poor initial choices of the parameters in the array P and, in extreme cases, because D02HBF cannot be used to solve the problem posed.

$IFAIL = 4$: As for $IFAIL = 2$ but the error occurred when calculating the Jacobian.

$IFAIL = 5$: As for $IFAIL = 3$ but the error occurred when calculating the Jacobian.

$IFAIL = 6$: The calculated Jacobian has an insignificant column. This can occur because a parameter $p_{i}$ is incorrectly entered when posing the problem.

Note:

IFAIL = 4

5

6

usually indicate a badly scaled problem. You may vary the size of PE. Otherwise the use of the more general D02SAF which affords more control over the calculations is advised.

$IFAIL = 7$: The linear algebra routine used (F08KBF (DGESVD)) has failed. This error exit should not occur and can be avoided by changing the initial estimates $p_{i}$ .

$IFAIL = 8$: The Newton iteration has failed to converge. This can indicate a poor initial choice of parameters $p_{i}$ or a very difficult problem. Consider varying the elements $PE (i)$ if the residuals are small in the monitoring output. If the residuals are large, try varying the initial parameters $p_{i}$ .

$IFAIL = 9$
$IFAIL = 10$
$IFAIL = 11$
$IFAIL = 12$
$IFAIL = 13$: Indicates that a serious error has occurred in an internal call. Check all array subscripts and subroutine parameter lists in the call to D02HBF. Seek expert help.

7 Accuracy

If the process converges, the accuracy to which the unknown parameters are determined is usually close to that specified by you; the solution, if requested, may be determined to a required accuracy by varying E.

8 Further Comments

The time taken by D02HBF 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.

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

E (i)

always occurs in expressions of the form

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

and

PE (i)

always occurs in expressions of the form

PE (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.

You may determine a suitable direction of integration

a

b

and suitable values for

E (i)

by integrations with D02PEF. The best direction of integration is usually the direction of decreasing solutions. You are strongly recommended to set IFAIL to obtain self-explanatory error messages, and also monitoring information about the course of the computation. You may select the channel numbers on which this output is to appear by calls of X04AAF (for error messages) or X04ABF (for monitoring information) – see Section 9 for an example. Otherwise the default channel numbers will be used, as specified in the Users' Note. The monitoring information produced at each iteration includes the current parameter values, the residuals and

2

-norms: a basic norm and a current norm. At each iteration the aim is to find parameter values which make the current norm less than the basic norm. Both these norms should tend to zero as should the residuals. (They would all be zero if the exact parameters were used as input.) For more details, in particular about the other monitoring information printed, you are advised to consult the specification of D02SAF, and especially the description of the parameter MONIT there.

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 of the residuals printed by the monitoring routine are much larger than the expected values of the solution at

b

, then the coding of FCN, BC and RANGE should be checked for errors. If no errors can be found, an independent attempt should be made to improve the initial estimates for

p_{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 Example 2 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 Example 2 in Section 9);
(d)	singular problems and problems on infinite ranges of integration where the values of the solution at $a$ or $b$ 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 Example 1 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 D02HBF, 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^{''} = (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 a truncated power series expansion

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

near the origin where

p_{1}

is one of the parameters to be determined. We choose the interval as

[0.1, 16]

and setting

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

, we can determine all the boundary conditions. We take

X1 = 16

. We write

y = Y (1)

y^{'} = Y (2)

, and estimate

PARAM (1) = 0.2

PARAM (2) = 0.0

. Note the call to X04ABF before the call to D02HBF.

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

on the range

0 < x < p_{2}

, with boundary conditions

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

We write

y = Y (1)

v = Y (2)

ϕ = Y (3)

. We estimate

p_{1} = PARAM (1) = 32

p_{2} = PARAM (2) = 6000

and

p_{3} = PARAM (3) = 0.54

(though this last estimate is not important).

NAG Library Routine DocumentD02HBF

+− 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

D02HBF