NAG Library Routine Document

D02JAF

, and the number of collocation points, KP. The routine sets up a system of linear equations for the Chebyshev coefficients, one equation for each collocation point and one for each boundary condition. The boundary conditions are solved exactly, and the remaining equations are then solved by a least squares method. The result produced is a set of coefficients for a Chebyshev series solution of the differential equation on an interval normalized to

(- 1, 1)

E02AKF can be used to evaluate the solution at any point on the interval

(x_{0}, x_{1})

– see Section 9 for an example. E02AHF followed by E02AKF can be used to evaluate its derivatives.

4 References

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

5 Parameters

1: N – INTEGERInput

On entry:

n

, the order of the differential equation.

Constraint:

N \geq 1

2: CF – REAL (KIND=nag_wp) FUNCTION, supplied by the user.External Procedure

CF defines the differential equation (see Section 3). It must return the value of a function

f_{j} (x)

at a given point

x

, where, for

1 \leq j \leq n + 1

f_{j} (x)

is the coefficient of

y^{(j - 1)} (x)

in the equation, and

f_{0} (x)

is the right-hand side.

The specification of CF is:

FUNCTION CF (

J, X)

REAL (KIND=nag_wp) CF

INTEGER	J
REAL (KIND=nag_wp)	X

1: J – INTEGERInput: On entry: the index of the function $f_{j}$ to be evaluated.
2: X – REAL (KIND=nag_wp)Input: On entry: the point at which $f_{j}$ is to be evaluated.

CF must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02JAF is called. Parameters denoted as Input must not be changed by this procedure.

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

BC defines the boundary conditions, each of which has the form

y^{(k - 1)} (x_{1}) = s_{k}

y^{(k - 1)} (x_{0}) = s_{k}

. The boundary conditions may be specified in any order.

The specification of BC is:

SUBROUTINE BC (

I, J, RHS)

INTEGER	I, J
REAL (KIND=nag_wp)	RHS

1: I – INTEGERInput: On entry: the index of the boundary condition to be defined.
2: J – INTEGEROutput: On exit: must be set to $- k$ if the boundary condition is $y^{(k - 1)} (x_{0}) = s_{k}$ , and to $+ k$ if it is $y^{(k - 1)} (x_{1}) = s_{k}$ .
J must not be set to the same value $k$ for two different values of I.
3: RHS – REAL (KIND=nag_wp)Output: On exit: must be set to the value $s_{k}$ .

BC must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D02JAF is called. Parameters denoted as Input must not be changed by this procedure.

4: X0 – REAL (KIND=nag_wp)Input

5: X1 – REAL (KIND=nag_wp)Input

On entry: the left- and right-hand boundaries,

x_{0}

and

x_{1}

, respectively.

Constraint:

X1 > X0

6: K1 – INTEGERInput

On entry: the number of coefficients to be returned in the Chebyshev series representation of the solution (hence the degree of the polynomial approximation is

K1 - 1

Constraint:

K1 \geq N + 1

7: KP – INTEGERInput

On entry: the number of collocation points to be used.

Constraint:

KP \geq K1 - N

8: C(K1) – REAL (KIND=nag_wp) arrayOutput

On exit: the computed Chebyshev coefficients; that is, the computed solution is:

{\sum^{'}}_{i = 1}^{K1} C (i) T_{i - 1} (x)

where

T_{i} (x)

is the

i

th Chebyshev polynomial of the first kind, and

\sum^{'}

denotes that the first coefficient,

C (1)

, is halved.

9: W(LW) – REAL (KIND=nag_wp) arrayWorkspace

10: LW – INTEGERInput

On entry: the dimension of the array W as declared in the (sub)program from which D02JAF is called.

Constraint:

LW \geq 2 \times (KP + N) \times (K1 + 1) + 7 \times K1

11: IW(K1) – INTEGER arrayWorkspace

12: 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:

$IFAIL = 1$

On entry,	$N < 1$ ,
or	$X0 \geq X1$ ,
or	$K1 < N + 1$ ,
or	$KP < K1 - N$ .

$IFAIL = 2$

On entry,

LW < 2 \times (KP + N) \times (K1 + 1) + 7 \times K1

(insufficient workspace).

$IFAIL = 3$: Either the boundary conditions are not linearly independent (that is, in BC the variable J is set to the same value $k$ for two different values of I), or the rank of the matrix of equations for the coefficients is less than the number of unknowns. Increasing KP may overcome this latter problem.

$IFAIL = 4$: The least squares routine F04AMF has failed to correct the first approximate solution (see F04AMF).

7 Accuracy

The Chebyshev coefficients are determined by a stable numerical method. The accuracy of the approximate solution may be checked by varying the degree of the polynomial and the number of collocation points (see Section 8).

8 Further Comments

The time taken by D02JAF depends on the complexity of the differential equation, the degree of the polynomial solution, and the number of matching points.

The collocation points in the interval

(x_{0}, x_{1})

are chosen to be the extrema of the appropriate shifted Chebyshev polynomial. If

KP = K1 - N

, then the least squares solution reduces to the solution of a system of linear equations, and true collocation results.

The accuracy of the solution may be checked by repeating the calculation with different values of K1 and with KP fixed but

KP ≫ K1 - N

. If the Chebyshev coefficients decrease rapidly (and consistently for various K1 and KP), the size of the last two or three gives an indication of the error. If the Chebyshev coefficients do not decay rapidly, it is likely that the solution cannot be well-represented by Chebyshev series. Note that the Chebyshev coefficients are calculated for the interval

(- 1, 1)

Systems of regular linear differential equations can be solved using D02JBF. It is necessary before using D02JBF to write the differential equations as a first-order system. Linear systems of high-order equations in their original form, singular problems, and, indirectly, nonlinear problems can be solved using D02TGF.

9 Example

This example solves the equation

y^{''} + y = 1

with boundary conditions

y (- 1) = y (1) = 0 .

We use

K1 = 4

6

and

8

, and

KP = 10

and

15

, so that the different Chebyshev series may be compared. The solution for

K1 = 8

and

KP = 15

is evaluated by E02AKF at nine equally spaced points over the interval

(- 1, 1)

NAG Library Routine DocumentD02JAF

+− 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

D02JAF