d02gaf solves a two-point boundary value problem with assigned boundary values for a system of ordinary differential equations, using a deferred correction technique and a Newton iteration.

2 Specification

Fortran Interface

Subroutine d02gaf (

u, v, n, a, b, tol, fcn, mnp, x, y, np, w, lw, iw, liw, ifail)

Integer, Intent (In)	::	n, mnp, lw, liw
Integer, Intent (Inout)	::	np, ifail
Integer, Intent (Out)	::	iw(liw)
Real (Kind=nag_wp), Intent (In)	::	u(n,2), v(n,2), a, b, tol
Real (Kind=nag_wp), Intent (Inout)	::	x(mnp)
Real (Kind=nag_wp), Intent (Out)	::	y(n,mnp), w(lw)
External	::	fcn

C Header Interface

#include <nag.h>

void

d02gaf_ (const double u[], const double v[], const Integer *n, const double *a, const double *b, const double *tol,
void (NAG_CALL *fcn)(const double *x, const double y[], double f[]),
const Integer *mnp, double x[], double y[], Integer *np, double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

The routine may be called by the names d02gaf or nagf_ode_bvp_fd_nonlin_fixedbc.

3 Description

d02gaf solves a two-point boundary value problem for a system of

n

differential equations in the interval [

a, b

]. The system is written in the form:

y_{i}^{'} = f_{i} (x, y_{1}, y_{2}, \dots, y_{n}), i = 1, 2, \dots, n

(1)

and the derivatives

f_{i}

are evaluated by fcn. Initially,

n

boundary values of the variables

y_{i}

must be specified, some at

a

and some at

b

. You must supply estimates of the remaining

n

boundary values and all the boundary values are used in constructing an initial approximation to the solution. This approximate solution is corrected by a finite difference technique with deferred correction allied with a Newton iteration to solve the finite difference equations. The technique used is described fully in Pereyra (1979). The Newton iteration requires a Jacobian matrix

\frac{\partial f_{i}}{\partial y_{j}}

and this is calculated by numerical differentiation using an algorithm described in Curtis et al. (1974).

You supply an absolute error tolerance and may also supply an initial mesh for the construction of the finite difference equations (alternatively a default mesh is used). The algorithm constructs a solution on a mesh defined by adding points to the initial mesh. This solution is chosen so that the error is everywhere less than your tolerance and so that the error is approximately equidistributed on the final mesh. The solution is returned on this final mesh.

If the solution is required at a few specific points then these should be included in the initial mesh. If on the other hand the solution is required at several specific points then you should use the interpolation routines provided in Chapter E01 if these points do not themselves form a convenient mesh.

4 References

Curtis A R, Powell M J D and Reid J K (1974) On the estimation of sparse Jacobian matrices J. Inst. Maths. Applics. 13 117–119

Pereyra V (1979) PASVA3: An adaptive finite-difference Fortran program for first order nonlinear, ordinary boundary problems 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 Arguments

1: $u (n, 2)$ – Real (Kind=nag_wp) array Input

On entry:

u (i, 1)

must be set to the known or estimated value of

y_{i}

a

and

u (i, 2)

must be set to the known or estimated value of

y_{i}

b

, for

i = 1, 2, \dots, n

2: $v (n, 2)$ – Real (Kind=nag_wp) array Input

On entry:

v (i, j)

must be set to

0.0

u (i, j)

is a known value and to

1.0

u (i, j)

is an estimated value, for

i = 1, 2, \dots, n

and

j = 1, 2

Constraint: precisely

n

of the

v (i, j)

must be set to

0.0

, i.e., precisely

n

of the

u (i, j)

must be known values, and these must not be all at

a

or all at

b

3: $n$ – Integer Input

On entry:

n

, the number of equations.

Constraint:

n \geq 2

4: $a$ – Real (Kind=nag_wp) Input

On entry:

a

, the left-hand boundary point.

5: $b$ – Real (Kind=nag_wp) Input

On entry:

b

, the right-hand boundary point.

Constraint:

b > a

6: $tol$ – Real (Kind=nag_wp) Input

On entry: a positive absolute error tolerance. If

a = x_{1} < x_{2} < \dots < x_{np} = b

is the final mesh,

z_{j} (x_{i})

is the

j

th component of the approximate solution at

x_{i}

, and

y_{j} (x)

is the

j

th component of the true solution of equation (1) (see Section 3) and the boundary conditions, then, except in extreme cases, it is expected that

| z_{j} (x_{i}) - y_{j} (x_{i}) | \leq tol, i = 1, 2, \dots, np ​ and ​ j = 1, 2, \dots, n .

(2)

Constraint:

tol > 0.0

7: $fcn$ – 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:

Fortran Interface

Subroutine fcn (

x, y, f)

Real (Kind=nag_wp), Intent (In)	::	x, y(*)
Real (Kind=nag_wp), Intent (Out)	::	f(*)

C Header Interface

void	fcn (const double *x, const double y[], double f[])

In the description of the arguments of d02gaf below,

n

denotes the actual value of n in the call of d02gaf.

1: $x$ – Real (Kind=nag_wp) Input: On entry: $x$ , the value of the argument.
2: $y (*)$ – Real (Kind=nag_wp) array Input: On entry: $y_{i}$ , for $i = 1, 2, \dots, n$ , the value of the argument.
3: $f (*)$ – Real (Kind=nag_wp) array Output: On exit: the values of $f_{i}$ , for $i = 1, 2, \dots, n$ .

fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02gaf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02gaf. If your code inadvertently does return any NaNs or infinities, d02gaf is likely to produce unexpected results.

8: $mnp$ – Integer Input

On entry: the maximum permitted number of mesh points.

Constraint:

mnp \geq 32

9: $x (mnp)$ – Real (Kind=nag_wp) array Input/Output

On entry: if

np \geq 4

(see np), the first np elements must define an initial mesh. Otherwise the elements of x need not be set.

Constraint:

a = x (1) < x (2) < \dots < x (np) = b, np \geq 4 .

(3)

On exit:

x (1), x (2), \dots, x (np)

define the final mesh (with the returned value of np) satisfying the relation (3).

10: $y (n, mnp)$ – Real (Kind=nag_wp) array Output

On exit: the approximate solution

z_{j} (x_{i})

satisfying (2), on the final mesh, that is

y (j, i) = z_{j} (x_{i}), i = 1, 2, \dots, np ​ and ​ j = 1, 2, \dots, n,

where np is the number of points in the final mesh.

The remaining columns of y are not used.

11: $np$ – Integer Input/Output

On entry: determines whether a default or user-supplied mesh is used.

$np = 0$: A default value of $4$ for np and a corresponding equispaced mesh $x (1), x (2), \dots, x (np)$ are used.
$np \geq 4$: You must define an initial mesh using the array x as described.

Constraint:

np = 0

4 \leq np \leq mnp

On exit: the number of points in the final (returned) mesh.

12: $w (lw)$ – Real (Kind=nag_wp) array Workspace

13: $lw$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which d02gaf is called.

Constraint:

lw \geq mnp \times (3 n^{2} + 6 n + 2) + 4 n^{2} + 4 n

14: $iw (liw)$ – Integer array Workspace

15: $liw$ – Integer Input

On entry: the dimension of the array iw as declared in the (sub)program from which d02gaf is called.

Constraint:

liw \geq mnp \times (2 n + 1) + n^{2} + 4 n + 2

16: $ifail$ – Integer Input/Output

This routine uses an ifail input value codification that differs from the normal case to distinguish between errors and warnings (see Section 4 in the Introduction to the NAG Library FL Interface).

On entry: ifail must be set to one of the values below to set behaviour on detection of an error; these values have no effect when no error is detected. The behaviour relate to whether or not program execution is halted and whether or not messages are printed when an error or warning is detected.

ifail	Execution	Error Printing	Warning Printed
$0$	halted	No	No
$1$	continue	No	No
$10$	halted	Yes	No
$11$	continue	Yes	No
$100$	halted	No	Yes
$101$	continue	No	Yes
$110$	halted	Yes	Yes
$111$	continue	Yes	Yes

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

1

11

101

111

is recommended. If the printing of messages is undesirable, then the value

1

is recommended. Otherwise, the recommended value is

110

. When the value $1$ , $11$ , $101$ or $111$ 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:

$ifail = 1$: On entry, $a = ⟨ value ⟩$ and $b = ⟨ value ⟩$ .
Constraint: $b > a$ .

On entry, $liw = ⟨ value ⟩$ .
Constraint: $liw \geq mnp \times (2 \times n + 1) + n^{2} + 4 \times n + 2$ ; that is, $⟨ value ⟩$ .

On entry, $lw = ⟨ value ⟩$ .
Constraint: $lw \geq mnp \times (3 \times n^{2} + 6 \times n + 2) + 4 \times n^{2} + 4 \times n$ ; that is, $⟨ value ⟩$ .

On entry, $mnp = ⟨ value ⟩$ .
Constraint: $mnp \geq 32$ .

On entry, $n = ⟨ value ⟩$ .
Constraint: $n \geq 2$ .

On entry, $np = ⟨ value ⟩$ .
Constraint: $np = 0$ or $np \geq 4$ .

On entry, $np = ⟨ value ⟩$ and $mnp = ⟨ value ⟩$ .
Constraint: $np \leq mnp$ .

On entry, $tol = ⟨ value ⟩$ .
Constraint: $tol > 0.0$ .

On entry: $a = ⟨ value ⟩$ and $x (1) = ⟨ value ⟩$ .
Constraint: $a = x (1) < x (2) < \dots < x (np) = b, np \geq 4$ .

On entry: $b = ⟨ value ⟩$ and $x (np) = ⟨ value ⟩$ .
Constraint: $a = x (1) < x (2) < \dots < x (np) = b, np \geq 4$ .

The number of known left boundary values must be less than the number of equations: the number of known left boundary values $= ⟨ value ⟩$ , the number of equations $= ⟨ value ⟩$ .

The number of known right boundary values must be less than the number of equations: the number of known right boundary values $= ⟨ value ⟩$ , the number of equations $= ⟨ value ⟩$ .

The sequence x is not strictly increasing. For $i = ⟨ value ⟩$ , $x (i) = ⟨ value ⟩$ and $x (i + 1) = ⟨ value ⟩$ .

The sum of known left and right boundary values must equal the number of equations: the number of known left boundary values $= ⟨ value ⟩$ , the number of known right boundary values $= ⟨ value ⟩$ , the number of equations $= ⟨ value ⟩$ .

$ifail = 2$: The Newton iteration has failed to converge.
This could be due to there being too few points in the initial mesh or to the initial approximate solution being too inaccurate. If this latter reason is suspected or you cannot make changes to prevent this error, you should use the routine with a continuation facility instead.

$ifail = 3$: Newton iteration has reached round-off level.
If desired accuracy has not been reached, tol is too small for this problem and this machine precision.

$ifail = 4$: A finer mesh is required for the accuracy requested; that is, $mnp = ⟨ value ⟩$ is not large enough.

$ifail = 5$: A serious error occurred in a call to the internal integrator.
The error code internally was $⟨ value ⟩$ . Please contact NAG.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

The solution returned by the routine will be accurate to your tolerance as defined by the relation (2) except in extreme circumstances. If too many points are specified in the initial mesh, the solution may be more accurate than requested and the error may not be approximately equidistributed.

8 Parallelism and Performance

d02gaf is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.

d02gaf 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.

9 Further Comments

The time taken by d02gaf depends on the difficulty of the problem, the number of mesh points (and meshes) used, the number of Newton iterations and the number of deferred corrections.

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 unit numbers on which this output is to appear by calls of x04aaf (for error messages) or x04abf (for monitoring information) – see Section 10 for an example. Otherwise the default unit numbers will be used, as specified in the Users' Note.

A common cause of convergence problems in the Newton iteration is that you have specified too few points in the initial mesh. Although the routine adds points to the mesh to improve accuracy it is unable to do so until the solution on the initial mesh has been calculated in the Newton iteration.

If you specify zero known and estimated boundary values, the routine constructs a zero initial approximation and in many cases the Jacobian is singular when evaluated for this approximation, leading to the breakdown of the Newton iteration.

You may be unable to provide a sufficiently good choice of initial mesh and estimated boundary values, and hence the Newton iteration may never converge. In this case the continuation facility provided in d02raf is recommended.

In the case where you wish to solve a sequence of similar problems, the final mesh from solving one case is strongly recommended as the initial mesh for the next.

10 Example

This example solves the differential equation

y^{'''} = - y y^{''} - β (1 - {y^{'}}^{2})

with boundary conditions

y (0) = y^{'} (0) = 0, y^{'} (10) = 1

for

β = 0.0

and

β = 0.2

to an accuracy specified by

tol = 1.0E−3

. We solve first the simpler problem with

β = 0.0

using an equispaced mesh of

26

points and then we solve the problem with

β = 0.2

using the final mesh from the first problem.

Note the call to x04abf prior to the call to d02gaf.

d02ga: FL CL CPP AD

NAG FL Interfaced02gaf (bvp_​fd_​nonlin_​fixedbc)

▸▿ 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

NAG FL Interface
d02gaf (bvp_fd_nonlin_fixedbc)