NAG Library Routine Document
D02TGF
1 Purpose
D02TGF solves a system of linear ordinary differential equations by least-squares fitting of a series of Chebyshev polynomials using collocation.
2 Specification
| SUBROUTINE D02TGF ( | N, M, L, X0, X1, K1, KP, C, LDC, COEFF, BDYC, W, LW, IW, LIW, IFAIL) |
| INTEGER | N, M(N), L(N), K1, KP, LDC, LW, IW(LIW), LIW, IFAIL |
| double precision | X0, X1, C(LDC,N), W(LW) |
| EXTERNAL | COEFF, BDYC |
3 Description
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
y1,y2,…,yn, over the range
x0,x1. Let the
ith equation be
where
yk
j
x=djykx
dxj
.
COEFF evaluates the coefficients
fkjix and the right-hand side
rix for each
i,
1≤i≤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
jth boundary condition out of those associated with the
ith differential equation takes the form
where
xij lies between
x0 and
x1. 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, k1-1, and the number of collocation points, kp, 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 kp-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 y1,y2,…,yn, with the range normalized to -1,1.
E02AKF can be used to evaluate the components of the solution at any point on the range
x0,x1 (see
Section 9 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.
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 number of differential equations in the system.
Constraint:
N≥1.
- 2: M(N) – INTEGER arrayInput
-
On entry: Mi must be set to the highest order derivative occurring in the ith equation, for i=1,2,…,n.
Constraint:
Mi≥1, for i=1,2,…,n.
- 3: L(N) – INTEGER arrayInput
-
On entry: Li must be set to the number of boundary conditions associated with the ith equation, for i=1,2,…,n.
Constraint:
Li≥0, for i=1,2,…,n.
- 4: X0 – double precisionInput
-
On entry: the left-hand boundary, x0.
- 5: X1 – double precisionInput
-
On entry: the right-hand boundary, x1.
Constraint:
X1>X0.
- 6: K1 – INTEGERInput
-
On entry:
the number of coefficients, k1, to be returned in the Chebyshev series representation of the solution (hence, the degree of the polynomial approximation is K1-1).
Constraint:
K1≥1+
max
1≤i≤N
Mi.
- 7: KP – INTEGERInput
-
On entry: the number of collocation points to be used, kp.
Constraint:
N×KP≥N×K1+ ∑ i=1 N Li .
- 8: C(LDC,N) – double precision arrayOutput
-
On exit: the
kth column of
C contains the computed Chebyshev coefficients of the
kth component of the solution,
yk; that is, the computed solution is:
where
Tix is the 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 D02TGF is called.
Constraint:
LDC≥K1.
- 10: COEFF – SUBROUTINE, supplied by the user.External Procedure
-
COEFF defines the system of differential equations (see
Section 3). It must evaluate the coefficient functions
fkjix and the right-hand side function
rix of the
ith equation at a given point. Only nonzero entries of the array
A and
RHS need be specifically assigned, since all elements are set to zero by D02TGF before calling
COEFF.
The specification of
COEFF is:
| SUBROUTINE COEFF ( | X, I, A, IA, IA1, RHS) |
| INTEGER | I, IA, IA1 |
| double precision | X, A(IA,IA1), RHS |
Important: the dimension declaration for
A must contain the variable
IA, not an integer constant.
- 1: X – double precisionInput
-
On entry: x, the point at which the functions must be evaluated.
- 2: I – INTEGERInput
-
On entry: the equation for which the coefficients and right-hand side are to be evaluated.
- 3: A(IA,IA1) – double precision arrayInput/Output
-
On entry: all elements of
A are set to zero.
On exit: Akj must contain the value fkjix, for 1≤k≤n, 1≤j≤mi+1.
- 4: IA – INTEGERInput
- 5: IA1 – INTEGERInput
-
On entry: the first and second dimensions of
A, respectively.
- 6: RHS – double precisionInput/Output
-
On entry: is set to zero.
On exit: it must contain the value rix.
COEFF must be declared as EXTERNAL in the (sub)program from which D02TGF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 11: BDYC – SUBROUTINE, supplied by the user.External Procedure
-
BDYC defines the boundary conditions (see
Section 3). It must evaluate the coefficient functions
fkj
ij and right-hand side function
rij in the
jth boundary condition associated with the
ith equation, at the point
xij at which the boundary condition is applied. Only nonzero entries of the array
A and
RHS need be specifically assigned, since all elements are set to zero by D02TGF before calling
BDYC.
The specification of
BDYC is:
| SUBROUTINE BDYC ( | X, I, J, A, IA, IA1, RHS) |
| INTEGER | I, J, IA, IA1 |
| double precision | X, A(IA,IA1), RHS |
Important: the dimension declaration for
A must contain the variable
IA, not an integer constant.
- 1: X – double precisionOutput
-
On exit: xij, the value at which the boundary condition is applied.
- 2: I – INTEGERInput
-
On entry: the differential equation with which the condition is associated.
- 3: J – INTEGERInput
-
On entry: the boundary condition for which the coefficients and right-hand side are to be evaluated.
- 4: A(IA,IA1) – double precision arrayInput/Output
-
On entry: all elements of
A are set to zero.
On exit: the value
f
kj
ij
xij
, for 1≤k≤n, 1≤j≤mi+1.
- 5: IA – INTEGERInput
- 6: IA1 – INTEGERInput
-
On entry: the first and second dimensions of
A, respectively.
- 7: RHS – double precisionInput/Output
-
On entry: is set to zero.
On exit: the value rij xij .
BDYC must be declared as EXTERNAL in the (sub)program from which D02TGF is called. Parameters denoted as
Input must
not be changed by this procedure.
- 12: W(LW) – double precision arrayWorkspace
- 13: LW – INTEGERInput
-
On entry: the dimension of the array
W as declared in the (sub)program from which D02TGF is called.
Constraint:
LW≥2× N×KP+NL × N×K1+1 + 7×N×K1 , where NL= ∑ i=1 n Li .
- 14: IW(LIW) – INTEGER arrayWorkspace
- 15: LIW – INTEGERInput
-
On entry: the dimension of the array
IW as declared in the (sub)program from which D02TGF is called.
Constraint:
LIW≥N×K1+1.
- 16: 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 | Mi<1 for some i, |
| or | Li<0 for some i, |
| or | X0≥X1, |
| or | K1<1+Mi for some i, |
| or |
N×KP<N×K1+
∑
i=1
n
Li
, |
| or | LDC<K1. |
- IFAIL=2
-
- IFAIL=3
-
Either the boundary conditions are not linearly independent, 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). Increasing
KP may remove this difficulty.
7 Accuracy
Estimates of the accuracy of the solution may be obtained by using the checks described in
Section 8. The Chebyshev coefficients are calculated by a stable numerical method.
8 Further Comments
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 kp is equal to the number of coefficients k1 minus the average number of boundary conditions
1n
∑
i=1
n
li
, 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 k1. 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 9, 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
y1=ax+b in the range
x0,x1, we set
B, the array of coefficients,
- B1,1=a×x0+x1+2×b,
- B1,2=a×x1-x0/2,
- 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, e.g., in COMMON. (See
Section 9.) The convergence of the (Newton) iteration cannot be guaranteed in general, though it is usually satisfactory from a good starting approximation.
9 Example
This example solves the nonlinear system
in the range
-1,1, with
y1=0,
y2=3,
y2′=0 at
x=-1.
Suppose an approximate solution is
z1,
z2 such that
y1∼z1,
y2∼z2: then the first equation gives, on linearizing,
The starting approximation is taken to be
z1=0,
z2=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.
9.1 Program Text
Program Text (d02tgfe.f)
9.2 Program Data
None.
9.3 Program Results
Program Results (d02tgfe.r)
