NAG Library Routine Document
D02JAF
1 Purpose
D02JAF solves a regular linear twopoint boundary value problem for a single $n$thorder 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 
REAL (KIND=nag_wp) 
CF, X0, X1, C(K1), W(LW) 
EXTERNAL 
CF, BC 

3 Description
D02JAF calculates the solution of a regular twopoint boundary value problem for a single
$n$thorder linear ordinary differential equation 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, 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
$\left(1,1\right)$.
E02AKF can be used to evaluate the solution at any point on the interval
$\left({x}_{0},{x}_{1}\right)$ – see
Section 10 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 Chebyshevseries by the selected points method Report Math. 94 National Physical Laboratory
5 Parameters
 1: $\mathrm{N}$ – INTEGERInput

On entry: $n$, the order of the differential equation.
Constraint:
${\mathbf{N}}\ge 1$.
 2: $\mathrm{CF}$ – 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
${f}_{j}\left(x\right)$ at a given point
$x$, where, for
$1\le j\le n+1$,
${f}_{j}\left(x\right)$ is the coefficient of
${y}^{\left(j1\right)}\left(x\right)$ in the equation, and
${f}_{0}\left(x\right)$ is the righthand side.
The specification of
CF is:
INTEGER 
J 
REAL (KIND=nag_wp) 
X 

 1: $\mathrm{J}$ – INTEGERInput

On entry: the index of the function ${f}_{j}$ to be evaluated.
 2: $\mathrm{X}$ – 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: $\mathrm{BC}$ – SUBROUTINE, supplied by the user.External Procedure

BC defines the boundary conditions, each of which has the form
${y}^{\left(k1\right)}\left({x}_{1}\right)={s}_{k}$ or
${y}^{\left(k1\right)}\left({x}_{0}\right)={s}_{k}$. 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: $\mathrm{I}$ – INTEGERInput

On entry: the index of the boundary condition to be defined.
 2: $\mathrm{J}$ – INTEGEROutput

On exit: must be set to
$k$ if the boundary condition is
${y}^{\left(k1\right)}\left({x}_{0}\right)={s}_{k}$, and to
$+k$ if it is
${y}^{\left(k1\right)}\left({x}_{1}\right)={s}_{k}$.
J must not be set to the same value
$k$ for two different values of
I.
 3: $\mathrm{RHS}$ – 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: $\mathrm{X0}$ – REAL (KIND=nag_wp)Input
 5: $\mathrm{X1}$ – REAL (KIND=nag_wp)Input

On entry: the left and righthand boundaries, ${x}_{0}$ and ${x}_{1}$, respectively.
Constraint:
${\mathbf{X1}}>{\mathbf{X0}}$.
 6: $\mathrm{K1}$ – 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 ${\mathbf{K1}}1$).
Constraint:
${\mathbf{K1}}\ge {\mathbf{N}}+1$.
 7: $\mathrm{KP}$ – INTEGERInput

On entry: the number of collocation points to be used.
Constraint:
${\mathbf{KP}}\ge {\mathbf{K1}}{\mathbf{N}}$.
 8: $\mathrm{C}\left({\mathbf{K1}}\right)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the computed Chebyshev coefficients; 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,
${\mathbf{C}}\left(1\right)$, is halved.
 9: $\mathrm{W}\left({\mathbf{LW}}\right)$ – REAL (KIND=nag_wp) arrayWorkspace
 10: $\mathrm{LW}$ – INTEGERInput

On entry: the dimension of the array
W as declared in the (sub)program from which D02JAF is called.
Constraint:
${\mathbf{LW}}\ge 2\times \left({\mathbf{KP}}+{\mathbf{N}}\right)\times \left({\mathbf{K1}}+1\right)+7\times {\mathbf{K1}}$.
 11: $\mathrm{IW}\left({\mathbf{K1}}\right)$ – INTEGER arrayWorkspace

 12: $\mathrm{IFAIL}$ – 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  ${\mathbf{K1}}<{\mathbf{N}}+1$, 
or  ${\mathbf{KP}}<{\mathbf{K1}}{\mathbf{N}}$. 
 ${\mathbf{IFAIL}}=2$

On entry,  ${\mathbf{LW}}<2\times \left({\mathbf{KP}}+{\mathbf{N}}\right)\times \left({\mathbf{K1}}+1\right)+7\times {\mathbf{K1}}$ (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).
 ${\mathbf{IFAIL}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.8 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.7 in the Essential Introduction for further information.
 ${\mathbf{IFAIL}}=999$
Dynamic memory allocation failed.
See
Section 3.6 in the Essential Introduction for further information.
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 9).
8 Parallelism and Performance
D02JAF is not threaded by NAG in any implementation.
D02JAF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the
X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the
Users' Note for your implementation for any additional implementationspecific information.
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 $\left({x}_{0},{x}_{1}\right)$ are chosen to be the extrema of the appropriate shifted Chebyshev polynomial. If ${\mathbf{KP}}={\mathbf{K1}}{\mathbf{N}}$, 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
${\mathbf{KP}}\gg {\mathbf{K1}}{\mathbf{N}}$. 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 wellrepresented by Chebyshev series. Note that the Chebyshev coefficients are calculated for the interval
$\left(1,1\right)$.
Systems of regular linear differential equations can be solved using
D02JBF. It is necessary before using
D02JBF to write the differential equations as a firstorder system. Linear systems of highorder equations in their original form, singular problems, and, indirectly, nonlinear problems can be solved using
D02TGF.
10 Example
This example solves the equation
with boundary conditions
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
$\left(1,1\right)$.
10.1 Program Text
Program Text (d02jafe.f90)
10.2 Program Data
Program Data (d02jafe.d)
10.3 Program Results
Program Results (d02jafe.r)