NAG Library Routine Document
D02JBF
1 Purpose
D02JBF solves a regular linear twopoint 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 
REAL (KIND=nag_wp) 
CF, X0, X1, C(LDC,N), W(LW) 
EXTERNAL 
CF, BC 

3 Description
D02JBF calculates the solution of a regular twopoint boundary value problem for a regular linear
$n$thorder system of firstorder ordinary differential equations as a Chebyshev series in the interval
$\left({x}_{0},{x}_{1}\right)$. The differential equation
is defined by
CF, and the boundary conditions at the points
${x}_{0}$ and
${x}_{1}$ are defined by
BC.
You specify the degree of Chebyshev series required,
${\mathbf{K1}}1$, 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
$\left(1,1\right)$.
E02AKF can be used to evaluate the components of the solution at any point on the interval
$\left({x}_{0},{x}_{1}\right)$ – 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 Chebyshevseries 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:
${\mathbf{N}}\ge 1$.
 2: CF – REAL (KIND=nag_wp) 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
${a}_{i,j}\left(x\right)$, of
$A$, at a given point
$x$, or of a righthand side function
${r}_{i}\left(x\right)$ if
${\mathbf{J}}=0$.
The specification of
CF is:
INTEGER 
I, J 
REAL (KIND=nag_wp) 
X 

 1: I – INTEGERInput
 2: J – INTEGERInput
On entry: indicate the function to be evaluated, namely ${a}_{i,j}\left(x\right)$ if $1\le {\mathbf{J}}\le n$, or ${r}_{i}\left(x\right)$ if ${\mathbf{J}}=0$.
$1\le {\mathbf{I}}\le n$, $0\le {\mathbf{J}}\le n$.
 3: X – REAL (KIND=nag_wp)Input
On entry: the point at which the function is to be evaluated.
CF must either be a module subprogram USEd by, or 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
${y}_{k}\left({x}_{0}\right)=s$ or
${y}_{k}\left({x}_{1}\right)=s$. 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
$i$th boundary condition is
${y}_{k}\left({x}_{0}\right)=s$, or to
$+k$ if it is
${y}_{k}\left({x}_{1}\right)=s$.
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: the value $s$.
BC must either be a module subprogram USEd by, or 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 – REAL (KIND=nag_wp)Input
 5: X1 – REAL (KIND=nag_wp)Input
On entry: the left and righthand boundaries, ${x}_{0}$ and ${x}_{1}$, respectively.
Constraint:
${\mathbf{X1}}>{\mathbf{X0}}$.
 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 ${\mathbf{K1}}1$).
Constraint:
${\mathbf{K1}}\ge 2$.
 7: KP – INTEGERInput
On entry: the number of collocation points to be used.
Constraint:
${\mathbf{KP}}\ge {\mathbf{K1}}1$.
 8: C(LDC,N) – REAL (KIND=nag_wp) arrayOutput
On exit: the computed Chebyshev coefficients of the
$k$th component of the solution,
${y}_{k}$; that is, the computed solution is:
where
${T}_{i}\left(x\right)$ is the
$i$th Chebyshev polynomial of the first kind, and
${\sum}^{\prime}$ denotes that the first coefficient,
${\mathbf{C}}\left(1,k\right)$, 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:
${\mathbf{LDC}}\ge {\mathbf{K1}}$.
 10: W(LW) – REAL (KIND=nag_wp) arrayWorkspace
 11: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which D02JBF is called.
Constraint:
${\mathbf{LW}}\ge 2\times {\mathbf{N}}\times \left({\mathbf{KP}}+1\right)\times \left({\mathbf{N}}\times {\mathbf{K1}}+1\right)+7\times {\mathbf{N}}\times {\mathbf{K1}}$.
 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:
${\mathbf{LIW}}\ge {\mathbf{N}}\times \left({\mathbf{K1}}+2\right)$.
 14: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ 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 $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{N}}<1$, 
or  ${\mathbf{X0}}\ge {\mathbf{X1}}$, 
or  ${\mathbf{K1}}<2$, 
or  ${\mathbf{KP}}<{\mathbf{K1}}1$, 
or  ${\mathbf{LDC}}<{\mathbf{K1}}$. 
 ${\mathbf{IFAIL}}=2$
On entry,  ${\mathbf{LW}}<2\times {\mathbf{N}}\times \left({\mathbf{KP}}+1\right)\times \left({\mathbf{N}}\times {\mathbf{K1}}+1\right)+7\times {\mathbf{N}}\times {\mathbf{K1}}$, 
or  ${\mathbf{LIW}}<{\mathbf{N}}\times \left({\mathbf{K1}}+2\right)$ (i.e., insufficient workspace). 
 ${\mathbf{IFAIL}}=3$
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.
 ${\mathbf{IFAIL}}=4$

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).
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 $\left({x}_{0},{x}_{1}\right)$ are chosen to be the extrema of the appropriate shifted Chebyshev polynomial. If ${\mathbf{KP}}={\mathbf{K1}}1$, 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
${\mathbf{KP}}\gg {\mathbf{K1}}1$. 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 wellrepresented by Chebyshev series. Note that the Chebyshev coefficients are calculated for the interval
$\left(1,1\right)$.
Linear systems of highorder 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 firstorder system
for solution by D02JBF and the boundary conditions are written
We use
${\mathbf{K1}}=4$,
$6$ and
$8$, and
${\mathbf{KP}}=10$ and
$15$, so that the different Chebyshev series may be compared. The solution for
${\mathbf{K1}}=8$ and
${\mathbf{KP}}=15$ is evaluated by
E02AKF at nine equally spaced points over the interval
$\left(1,1\right)$.
9.1 Program Text
Program Text (d02jbfe.f90)
9.2 Program Data
Program Data (d02jbfe.d)
9.3 Program Results
Program Results (d02jbfe.r)