NAG Library Routine Document

D02UEF

1  Purpose

2  Specification

3  Description

4  References

5  Parameters

1:     $\mathrm{N}$ – INTEGERInput

On entry: $n$, where the number of grid points is $n+1$.

Constraint: $\mathbf{N}\ge 8$ and N is even.

2:     $\mathrm{A}$ – REAL (KIND=nag_wp)Input

On entry: $a$, the lower bound of domain $\left[a,b\right]$.

Constraint: $\mathbf{A}<\mathbf{B}$.

3:     $\mathrm{B}$ – REAL (KIND=nag_wp)Input

On entry: $b$, the upper bound of domain $\left[a,b\right]$.

Constraint: $\mathbf{B}>\mathbf{A}$.

4:     $\mathrm{M}$ – INTEGERInput

On entry: the order, $m$, of the boundary value problem to be solved.

Constraint: $1\le \mathbf{M}\le 4$.

5:     $\mathrm{C}\left(\mathbf{N}+1\right)$ – REAL (KIND=nag_wp) arrayInput

On entry: the Chebyshev coefficients $c_j$, $j=0,1,\dots ,n$, for the right hand side of the boundary value problem. Usually these are obtained by a previous call of D02UAF.

6:     $\mathrm{BMAT}\left(\mathbf{M},\mathbf{M}+1\right)$ – 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:     $\mathrm{Y}\left(\mathbf{M}\right)$ – 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:     $\mathrm{BVEC}\left(\mathbf{M}\right)$ – 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:     $\mathrm{F}\left(\mathbf{M}+1\right)$ – 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:   $\mathrm{UC}\left(\mathbf{N}+1,\mathbf{M}+1\right)$ – 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:   $\mathrm{RESID}$ – 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

$$\max \left\{\underset{i=1,m}{\max }\left(\left|\sum _{\mathit{j}=0}^m{B}_{i,j+1}{\left(\frac{d^{j}u}{d{x}^j}\right)}_{\left(x=y_i\right)}-{\beta }_i\right|\right),\underset{i=1,n+1}{\max }\left(\left|\sum _{\mathit{j}=0}^m{f}_{j+1}{\left(\frac{d^{j}u}{{dx}^j}\right)}_{\left(x=x_i\right)}-f\left(x\right)\right|\right)\right\}\text{.}$$

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

$\mathbf{IFAIL}=1$

On entry, $\mathbf{N}=〈\mathit{\text{value}}〉$.
Constraint: N is even.

On entry, $\mathbf{N}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{N}\ge 8$.

$\mathbf{IFAIL}=2$

On entry, $\mathbf{A}=〈\mathit{\text{value}}〉$ and $\mathbf{B}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{A}<\mathbf{B}$.

$\mathbf{IFAIL}=3$

On entry, $\mathbf{F}\left(\mathbf{M}+1\right)=0.0$.

$\mathbf{IFAIL}=6$

On entry, $\mathbf{M}=〈\mathit{\text{value}}〉$.
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.
$\text{Number of iterations}=〈\mathit{\text{value}}〉$ and $\text{residual achieved}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=10$

During iterative refinement, convergence was achieved, but the residual is more than $100\times \mathit{machine precision}$. $\text{Residual achieved on convergence}=〈\mathit{\text{value}}〉$.

$\mathbf{IFAIL}=-99$

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

8  Parallelism and Performance

9  Further Comments

10  Example

10.1  Program Text

Program Text (d02uefe.f90)

10.2  Program Data

Program Data (d02uefe.d)

10.3  Program Results

Program Results (d02uefe.r)