NAG Library Routine Document
d02uef (bvp_ps_lin_solve)
1
Purpose
d02uef finds the solution of a linear constant coefficient boundary value problem by using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid.
2
Specification
Fortran Interface
Subroutine d02uef ( 
n, a, b, m, c, bmat, y, bvec, f, uc, resid, ifail) 
Integer, Intent (In)  ::  n, m  Integer, Intent (Inout)  ::  ifail  Real (Kind=nag_wp), Intent (In)  ::  a, b, c(n+1), y(m), bvec(m)  Real (Kind=nag_wp), Intent (Inout)  ::  bmat(m,m+1), f(m+1)  Real (Kind=nag_wp), Intent (Out)  ::  uc(n+1,m+1), resid 

C Header Interface
#include <nagmk26.h>
void 
d02uef_ (const Integer *n, const double *a, const double *b, const Integer *m, const double c[], double bmat[], const double y[], const double bvec[], double f[], double uc[], double *resid, Integer *ifail) 

3
Description
d02uef solves the constant linear coefficient ordinary differential problem
subject to a set of
$m$ linear constraints at points
${y}_{\mathit{i}}\in \left[a,b\right]$, for
$\mathit{i}=1,2,\dots ,m$:
where
$1\le m\le 4$,
$B$ is an
$m\times \left(m+1\right)$ matrix of constant coefficients and
${\beta}_{i}$ are constants. The points
${y}_{i}$ are usually either
$a$ or
$b$.
The function
$f\left(x\right)$ is supplied as an array of Chebyshev coefficients
${c}_{j}$,
$j=0,1,\dots ,n$ for the function discretized on
$n+1$ Chebyshev Gauss–Lobatto points (as returned by
d02ucf); the coefficients are normally obtained by a previous call to
d02uaf. The solution and its derivatives (up to order
$m$) are returned, in the form of their Chebyshev series representation, as arrays of Chebyshev coefficients; subsequent calls to
d02ubf will return the corresponding function and derivative values at the Chebyshev Gauss–Lobatto discretization points on
$\left[a,b\right]$. Function and derivative values can be obtained on any uniform grid over the same range
$\left[a,b\right]$ by calling the interpolation routine
d02uwf.
4
References
Clenshaw C W (1957) The numerical solution of linear differential equations in Chebyshev series Proc. Camb. Phil. Soc. 53 134–149
Coutsias E A, Hagstrom T and Torres D (1996) An efficient spectral method for ordinary differential equations with rational function coefficients Mathematics of Computation 65(214) 611–635
Greengard L (1991) Spectral integration and twopoint boundary value problems SIAM J. Numer. Anal. 28(4) 1071–80
Lundbladh A, Hennigson D S and Johannson A V (1992) An efficient spectral integration method for the solution of the Navier–Stokes equations Technical report FFA–TN 1992–28 Aeronautical Research Institute of Sweden
Muite B K (2010) A numerical comparison of Chebyshev methods for solving fourthorder semilinear initial boundary value problems Journal of Computational and Applied Mathematics 234(2) 317–342
5
Arguments
 1: $\mathbf{n}$ – IntegerInput

On entry: $n$, where the number of grid points is $n+1$.
Constraint:
${\mathbf{n}}\ge 8$ and
n is even.
 2: $\mathbf{a}$ – Real (Kind=nag_wp)Input

On entry: $a$, the lower bound of domain $\left[a,b\right]$.
Constraint:
${\mathbf{a}}<{\mathbf{b}}$.
 3: $\mathbf{b}$ – Real (Kind=nag_wp)Input

On entry: $b$, the upper bound of domain $\left[a,b\right]$.
Constraint:
${\mathbf{b}}>{\mathbf{a}}$.
 4: $\mathbf{m}$ – IntegerInput

On entry: the order, $m$, of the boundary value problem to be solved.
Constraint:
$1\le {\mathbf{m}}\le 4$.
 5: $\mathbf{c}\left({\mathbf{n}}+1\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the Chebyshev coefficients
${c}_{j}$,
$j=0,1,\dots ,n$, for the righthand side of the boundary value problem. Usually these are obtained by a previous call of
d02uaf.
 6: $\mathbf{bmat}\left({\mathbf{m}},{\mathbf{m}}+1\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry:
${\mathbf{bmat}}\left(\mathit{i},\mathit{j}+1\right)$ must contain the coefficients
${B}_{\mathit{i},\mathit{j}+1}$, for
$\mathit{i}=1,2,\dots ,m$ and
$\mathit{j}=0,1,\dots ,m$, in the problem formulation of
Section 3.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain $\left[1,1\right]$.
 7: $\mathbf{y}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the points,
${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$, where the boundary conditions are discretized.
 8: $\mathbf{bvec}\left({\mathbf{m}}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the values,
${\beta}_{\mathit{i}}$, for
$\mathit{i}=1,2,\dots ,m$, in the formulation of the boundary conditions given in
Section 3.
 9: $\mathbf{f}\left({\mathbf{m}}+1\right)$ – Real (Kind=nag_wp) arrayInput/Output

On entry: the coefficients,
${f}_{\mathit{j}}$, for
$\mathit{j}=1,2,\dots ,m+1$, in the formulation of the linear boundary value problem given in
Section 3. The highest order term,
${\mathbf{f}}\left({\mathbf{m}}+1\right)$, needs to be nonzero to have a well posed problem.
On exit: the coefficients have been scaled to form an equivalent problem defined on the domain $\left[1,1\right]$.
 10: $\mathbf{uc}\left({\mathbf{n}}+1,{\mathbf{m}}+1\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: the Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The $n+1$ elements ${\mathbf{uc}}\left(1:{\mathbf{n}}+1,1\right)$ contain the coefficients representing the solution
$U\left({x}_{\mathit{i}}\right)$, for $\mathit{i}=0,1,\dots ,n$.
${\mathbf{uc}}\left(1:{\mathbf{n}}+1,\mathit{j}+1\right)$ contains the coefficients representing the $\mathit{j}$th derivative of $U$, for $\mathit{j}=1,2,\dots ,m$.
 11: $\mathbf{resid}$ – Real (Kind=nag_wp)Output

On exit: the maximum residual resulting from substituting the solution vectors returned in
uc into both linear equations of
Section 3 representing the linear boundary value problem and associated boundary conditions. That is
 12: $\mathbf{ifail}$ – IntegerInput/Output

On entry:
ifail must be set to
$0$,
$1\text{or}1$. If you are unfamiliar with this argument you should refer to
Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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
${\mathbf{ifail}}=0$ or
$1$, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
Errors or warnings detected by the routine:
 ${\mathbf{ifail}}=1$

On entry,
${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint:
n is even.
On entry, ${\mathbf{n}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{n}}\ge 8$.
 ${\mathbf{ifail}}=2$

On entry, ${\mathbf{a}}=\u2329\mathit{\text{value}}\u232a$ and ${\mathbf{b}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: ${\mathbf{a}}<{\mathbf{b}}$.
 ${\mathbf{ifail}}=3$

On entry, ${\mathbf{f}}\left({\mathbf{m}}+1\right)=0.0$.
 ${\mathbf{ifail}}=6$

On entry, ${\mathbf{m}}=\u2329\mathit{\text{value}}\u232a$.
Constraint: $1\le {\mathbf{m}}\le 4$.
 ${\mathbf{ifail}}=7$

Internal error while unpacking matrix during iterative refinement.
Please contact
NAG.
 ${\mathbf{ifail}}=8$

Singular matrix encountered during iterative refinement.
Please check that your system is well posed.
 ${\mathbf{ifail}}=9$

During iterative refinement, the maximum number of iterations was reached.
$\text{Number of iterations}=\u2329\mathit{\text{value}}\u232a$ and $\text{residual achieved}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=10$

During iterative refinement, convergence was achieved, but the residual is more than $100\times \mathit{machineprecision}$. $\text{Residual achieved on convergence}=\u2329\mathit{\text{value}}\u232a$.
 ${\mathbf{ifail}}=99$
An unexpected error has been triggered by this routine. Please
contact
NAG.
See
Section 3.9 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=399$
Your licence key may have expired or may not have been installed correctly.
See
Section 3.8 in How to Use the NAG Library and its Documentation for further information.
 ${\mathbf{ifail}}=999$
Dynamic memory allocation failed.
See
Section 3.7 in How to Use the NAG Library and its Documentation for further information.
7
Accuracy
The accuracy should be close to machine precision for well conditioned boundary value problems.
8
Parallelism and Performance
d02uef is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
d02uef 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 number of operations is of the order $n\mathrm{log}\left(n\right)$ and the memory requirements are $\mathit{O}\left(n\right)$; thus the computation remains efficient and practical for very fine discretizations (very large values of $n$). Collocation methods will be faster for small problems, but the method of d02uef should be faster for larger discretizations.
10
Example
This example solves the thirdorder problem $4{U}_{xxx}+3{U}_{xx}+2{U}_{x}+U=2\mathrm{sin}x2\mathrm{cos}x$ on $\left[\pi /2,\pi /2\right]$ subject to the boundary conditions $U\left[\pi /2\right]=0$, $3{U}_{xx}\left[\pi /2\right]+2{U}_{x}\left[\pi /2\right]+U\left[\pi /2\right]=2$, and $3{U}_{xx}\left[\pi /2\right]+2{U}_{x}\left[\pi /2\right]+U\left[\pi /2\right]=2$ using the Chebyshev integration formulation on a Chebyshev Gauss–Lobatto grid of order $16$.
10.1
Program Text
Program Text (d02uefe.f90)
10.2
Program Data
Program Data (d02uefe.d)
10.3
Program Results
Program Results (d02uefe.r)