d02tgf : NAG Library, Mark 26

d02tgf solves a system of linear ordinary differential equations by least squares fitting of a series of Chebyshev polynomials using collocation.

Fortran Interface

Subroutine d02tgf (

n, m, l, x0, x1, k1, kp, c, ldc, coeff, bdyc, w, lw, iw, liw, ifail)

Integer, Intent (In)	::	n, m(n), l(n), k1, kp, ldc, lw, liw
Integer, Intent (Inout)	::	ifail
Integer, Intent (Out)	::	iw(liw)
Real (Kind=nag_wp), Intent (In)	::	x0, x1
Real (Kind=nag_wp), Intent (Inout)	::	c(ldc,n)
Real (Kind=nag_wp), Intent (Out)	::	w(lw)
External	::	coeff, bdyc

C Header Interface

#include nagmk26.h

void

d02tgf_ ( const Integer *n, const Integer m[], const Integer l[], const double *x0, const double *x1, const Integer *k1, const Integer *kp, double c[], const Integer *ldc,
void (NAG_CALL *coeff)( const double *x, const Integer *i, double a[], const Integer *ia, const Integer *ia1, double *rhs),
void (NAG_CALL *bdyc)( double *x, const Integer *i, const Integer *j, double a[], const Integer *ia, const Integer *ia1, double *rhs),
double w[], const Integer *lw, Integer iw[], const Integer *liw, Integer *ifail)

d02tgf calculates an approximate solution of a linear or linearized system of ordinary differential equations as a Chebyshev series. Suppose there are

n

differential equations for

n

variables

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

, over the range

(x_{0}, x_{1})

. Let the

i

th equation be

\sum_{j = 1}^{m_{i} + 1} \sum_{k = 1}^{n} f_{k j}^{i} (x) y_{k}^{(j - 1)} (x) = r^{i} (x)

where

y_{k}^{(j)} (x) = \frac{d^{j} y_{k} (x)}{d x^{j}}

. coeff evaluates the coefficients

f_{k j}^{i} (x)

and the right-hand side

r^{i} (x)

for each

i

,

1 \leq i \leq n

, at any point

x

. The boundary conditions may be applied either at the end points or at intermediate points; they are written in the same form as the differential equations, and specified by bdyc. For example the

j

th boundary condition out of those associated with the

i

th differential equation takes the form

\sum_{j = 1}^{l_{i} + 1} \sum_{k = 1}^{n} f_{k j}^{i j} (x^{i j}) y_{k}^{(j - 1)} (x^{i j}) = r^{i j} (x^{i j}),

where

x^{i j}

lies between

x_{0}

and

x_{1}

. It is assumed in this routine that certain of the boundary conditions are associated with each differential equation. This is for your convenience; the grouping does not affect the results.

The degree of the polynomial solution must be the same for all variables. You specify the degree required,

k_{1} - 1

, and the number of collocation points,

k_{p}

, in the range. The routine sets up a system of linear equations for the Chebyshev coefficients, with

n

equations for each collocation point and one for each boundary condition. The collocation points are chosen at the extrema of a shifted Chebyshev polynomial of degree

k_{p} - 1

. The boundary conditions are satisfied exactly, and the remaining equations are solved by a least squares method. The result produced is a set of Chebyshev coefficients for the

n

functions

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

, with the range normalized to

[- 1, 1]

.

e02akf can be used to evaluate the components of the solution at any point on the range

[x_{0}, x_{1}]

(see Section 10 for an example). e02ahf and e02ajf may be used to obtain Chebyshev series representations of derivatives and integrals (respectively) of the components of the solution.

Picken S M (1970) Algorithms for the solution of differential equations in Chebyshev-series by the selected points method Report Math. 94 National Physical Laboratory

If on entry

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:

Estimates of the accuracy of the solution may be obtained by using the checks described in Section 9. The Chebyshev coefficients are calculated by a stable numerical method.

d02tgf 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 d02tgf depends on the complexity of the system of differential equations, the degree of the polynomial solution and the number of matching points.

If the number of matching points

k_{p}

is equal to the number of coefficients

k_{1}

minus the average number of boundary conditions

\frac{1}{n} \sum_{i = 1}^{n} l_{i}

, then the least squares solution reduces to simple solution of linear equations and true collocation results. The accuracy of the solution may be checked by repeating the calculation with different values of

k_{1}

. If the Chebyshev coefficients decrease rapidly, the size of the last two or three gives an indication of the error. If they do not decrease rapidly, it may be desirable to use a different method. Note that the Chebyshev coefficients are calculated for the range normalized to

[- 1, 1]

.

Generally the number of boundary conditions required is equal to the sum of the orders of the

n

differential equations. However, in some cases fewer boundary conditions are needed, because the assumption of a polynomial solution is equivalent to one or more boundary conditions (since it excludes singular solutions).

A system of nonlinear differential equations must be linearized before using the routine. The calculation is repeated iteratively. On each iteration the linearized equation is used. In the example in Section 10, the

y

variables are to be determined at the current iteration whilst the

z

variables correspond to the solution determined at the previous iteration, (or the initial approximation on the first iteration). For a starting approximation, we may take, say, a linear function, and set up the appropriate Chebyshev coefficients before starting the iteration. For example, if

y_{1} = a x + b

in the range

(x_{0}, x_{1})

, we set

B

, the array of coefficients,

$B (1, 1) = a \times (x_{0} + x_{1}) + 2 \times b$ ,
$B (1, 2) = a \times (x_{1} - x_{0}) / 2$ ,
and the remainder of the entries to zero.

In some cases a better initial approximation may be needed and can be obtained by using e02adf or e02aff to obtain a Chebyshev series for an approximate solution. The coefficients of the current iterate must be communicated to coeff and bdyc, for example using global variables. (See Section 10.) The convergence of the (Newton) iteration cannot be guaranteed in general, though it is usually satisfactory from a good starting approximation.

This example solves the nonlinear system

\begin{matrix} 2 y_{1}^{'} + (y_{2}^{2} - 1) y_{1} + y_{2} = 0, \\ 2 y_{2}^{''} - y_{1}^{'} = 0, \end{matrix}

in the range

(- 1, 1)

, with

y_{1} = 0

,

y_{2} = 3

,

y_{2}^{'} = 0

at

x = - 1

.

Suppose an approximate solution is

z_{1}

,

z_{2}

such that

y_{1} \sim z_{1}

,

y_{2} \sim z_{2}

: then the first equation gives, on linearizing,

2 y_{1}^{'} + (z_{2}^{2} - 1) y_{1} + (2 z_{1} z_{2} + 1) y_{2} = 2 z_{1} z_{2}^{2} .

The starting approximation is taken to be

z_{1} = 0

,

z_{2} = 3

. In the program below, the array

B

is used to hold the coefficients of the previous iterate (or of the starting approximation). We iterate until the Chebyshev coefficients converge to five figures. e02akf is used to calculate the solution from its Chebyshev coefficients.

On entry,	$n < 1$ ,
or	$m (i) < 1$ for some $i$ ,
or	$l (i) < 0$ for some $i$ ,
or	$x0 \geq x1$ ,
or	$k1 < 1 + m (i)$ for some $i$ ,
or	$n \times kp < n \times k1 + \sum_{i = 1}^{n} l (i)$ ,
or	$ldc < k1$ .

On entry,	lw is too small (see Section 5),
or	$liw < n \times k1$ .

NAG Library Routine Document

d02tgf (bvp_coll_nth_comp)

▸▿ 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 Library Routine Document

d02tgf (bvp_coll_nth_comp)

▸▿ 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