NAG Library Routine Document
D02JAF
1 Purpose
D02JAF solves a regular linear two-point boundary value problem for a single nth-order ordinary differential equation by Chebyshev series using collocation and least-squares.
2 Specification
| SUBROUTINE D02JAF ( | N, CF, BC, X0, X1, K1, KP, C, W, LW, IW, IFAIL) |
| INTEGER | N, K1, KP, LW, IW(K1), IFAIL |
| double precision | CF, X0, X1, C(K1), W(LW) |
| EXTERNAL | CF, BC |
3 Description
D02JAF calculates the solution of a regular two-point boundary value problem for a single
nth-order linear ordinary differential equation as a Chebyshev series in the interval
x0,x1. The differential equation
is defined by
CF, and the boundary conditions at the points
x0 and
x1 are defined by
BC.
You specify the degree of Chebyshev series required,
K1-1, 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
x0,x1 – 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≥1.
- 2: CF – double precision FUNCTION, supplied by the user.External Procedure
-
CF defines the differential equation (see
Section 3). It must return the value of a function
fjx at a given point
x, where, for
1≤j≤n+1,
fjx is the coefficient of
y j-1 x in the equation, and
f0x is the right-hand side.
The specification of
CF is:
| double precision FUNCTION CF ( | J, X) |
| INTEGER | J |
| double precision | X |
- 1: J – INTEGERInput
-
On entry: the index of the function fj to be evaluated.
- 2: X – double precisionInput
-
On entry: the point at which fj is to be evaluated.
CF must be 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-1x1=sk or
yk-1x0=sk. The boundary conditions may be specified in any order.
The specification of
BC is:
| SUBROUTINE BC ( | I, J, RHS) |
| INTEGER | I, J |
| double precision | 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 x0=sk, and to
+k if it is
y k-1 x1=sk.
J must not be set to the same value
k for two different values of
I.
- 3: RHS – double precisionOutput
-
On exit: must be set to the value sk.
BC must be 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 – double precisionInput
- 5: X1 – double precisionInput
-
On entry: the left- and right-hand boundaries, x0 and x1, 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≥N+1.
- 7: KP – INTEGERInput
-
On entry: the number of collocation points to be used.
Constraint:
KP≥K1-N.
- 8: C(K1) – double precision arrayOutput
-
On exit: the computed Chebyshev coefficients; that is, the computed solution is:
where
Tix is the
ith Chebyshev polynomial of the first kind, and
∑' denotes that the first coefficient,
C1, is halved.
- 9: W(LW) – double precision 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≥2×KP+N×K1+1+7×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.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
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.
6 Error Indicators and Warnings
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:
- IFAIL=1
-
| On entry, | N<1, |
| or | X0≥X1, |
| or | K1<N+1, |
| or | KP<K1-N. |
- IFAIL=2
-
| On entry, | LW<2×KP+N×K1+1+7×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 x0,x1 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
with boundary conditions
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.
9.1 Program Text
Program Text (d02jafe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d02jafe.r)
