NAG Library Routine Document
D02JBF
1 Purpose
D02JBF solves a regular linear two-point boundary value problem for a system of ordinary differential equations by Chebyshev series using collocation and least-squares.
2 Specification
| SUBROUTINE D02JBF ( | N, CF, BC, X0, X1, K1, KP, C, LDC, W, LW, IW, LIW, IFAIL) |
| INTEGER | N, K1, KP, LDC, LW, IW(LIW), LIW, IFAIL |
| double precision | CF, X0, X1, C(LDC,N), W(LW) |
| EXTERNAL | CF, BC |
3 Description
D02JBF calculates the solution of a regular two-point boundary value problem for a regular linear
nth-order system of first-order ordinary differential equations 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,
n equations 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 for each component of the solution of the system of differential equations on an interval normalized to
-1,1.
E02AKF can be used to evaluate the components of 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 their 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 system of differential equations.
Constraint:
N≥1.
- 2: CF – double precision FUNCTION, supplied by the user.External Procedure
-
CF defines the system of differential equations (see
Section 3). It must return the value of a coefficient function
ai,jx, of
A, at a given point
x, or of a right-hand side function
rix if
J=0.
The specification of
CF is:
| double precision FUNCTION CF ( | I, J, X) |
| INTEGER | I, J |
| double precision | X |
- 1: I – INTEGERInput
- 2: J – INTEGERInput
-
On entry: indicate the function to be evaluated, namely ai,jx if 1≤J≤n, or rix if J=0.
1≤I≤n, 0≤J≤n.
- 3: X – double precisionInput
-
On entry: the point at which the function is to be evaluated.
CF must be declared as EXTERNAL in the (sub)program from which D02JBF 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
n boundary conditions, which have the form
ykx0=s or
ykx1=s. 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
ith boundary condition is
ykx0=s, or to
+k if it is
ykx1=s.
J must not be set to the same value
k for two different values of
I.
- 3: RHS – double precisionOutput
-
On exit: the value s.
BC must be declared as EXTERNAL in the (sub)program from which D02JBF 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 components of the solution (hence the degree of the polynomial approximation is K1-1).
Constraint:
K1≥2.
- 7: KP – INTEGERInput
-
On entry: the number of collocation points to be used.
Constraint:
KP≥K1-1.
- 8: C(LDC,N) – double precision arrayOutput
-
On exit: the computed Chebyshev coefficients of the
kth component of the solution,
yk; that is, the computed solution is:
where
Tix is the
ith Chebyshev polynomial of the first kind, and
∑' denotes that the first coefficient,
C1k, is halved.
- 9: LDC – INTEGERInput
-
On entry: the first dimension of the array
C as declared in the (sub)program from which D02JBF is called.
Constraint:
LDC≥K1.
- 10: W(LW) – double precision arrayWorkspace
- 11: LW – INTEGERInput
-
On entry: the dimension of the array
W as declared in the (sub)program from which D02JBF is called.
Constraint:
LW≥2×N×KP+1×N×K1+1+7×N×K1.
- 12: IW(LIW) – INTEGER arrayWorkspace
- 13: LIW – INTEGERInput
-
On entry: the dimension of the array
IW as declared in the (sub)program from which D02JBF is called.
Constraint:
LIW≥N×K1+2.
- 14: 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<2, |
| or | KP<K1-1, |
| or | LDC<K1. |
- IFAIL=2
-
| On entry, | LW<2×N×KP+1×N×K1+1+7×N×K1, |
| or | LIW<N×K1+2 (i.e., 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 polynomials and the number of collocation points (see
Section 8).
8 Further Comments
The time taken by D02JBF depends on the size and complexity of the differential system, 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=K1-1, 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≫K1-1. If the Chebyshev coefficients decrease rapidly for each component (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.
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
The equation is written as the first-order system
for solution by D02JBF and the boundary conditions are written
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 (d02jbfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d02jbfe.r)
