NAG Library Function Document

nag_ode_bvp_ps_lin_solve (d02uec)

1

Purpose

2

Specification

3

Description

4

References

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:    $\mathbf{a}$ – doubleInput

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

Constraint: $\mathbf{a}<\mathbf{b}$.

3:    $\mathbf{b}$ – doubleInput

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:    $\mathbf{c}\left[\mathbf{n}+1\right]$ – const doubleInput

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 nag_ode_bvp_ps_lin_coeffs (d02uac).

6:    $\mathbf{bmat}\left[\mathbf{m}\times \left(\mathbf{m}+1\right)\right]$ – doubleInput/Output

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{bmat}\left[\left(j-1\right)\times \mathbf{m}+i-1\right]$.

On entry: $\mathbf{bmat}\left[\mathit{j}\times \mathbf{m}+\mathit{i}-1\right]$ 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:    $\mathbf{y}\left[\mathbf{m}\right]$ – const doubleInput

On entry: the points, $y_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,m$, where the boundary conditions are discretized.

8:    $\mathbf{bvec}\left[\mathbf{m}\right]$ – const doubleInput

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:    $\mathbf{f}\left[\mathbf{m}+1\right]$ – doubleInput/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, $\mathbf{f}\left[\mathbf{m}\right]$, 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:  $\mathbf{uc}\left[\left(\mathbf{n}+1\right)\times \left(\mathbf{m}+1\right)\right]$ – doubleOutput

Note: the $\left(i,j\right)$th element of the matrix is stored in $\mathbf{uc}\left[\left(j-1\right)\times \left(\mathbf{n}+1\right)+i-1\right]$.

On exit: the Chebyshev coefficients in the Chebyshev series representations of the solution and derivatives of the solution to the boundary value problem. The coefficients of $U$ are stored as the first $n+1$ elements of uc, the first derivative coefficients are stored as the next $n+1$ elements of uc, and so on.

11:  $\mathbf{resid}$ – double *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:  $\mathbf{fail}$ – NagError *Input/Output

The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

6

Error Indicators and Warnings

NE_ALLOC_FAIL

Dynamic memory allocation failed.
See Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.

NE_BAD_PARAM

On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.

NE_CONVERGENCE

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

NE_INT

On entry, $\mathbf{m}=〈\mathit{\text{value}}〉$.
Constraint: $1\le \mathbf{m}\le 4$.

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

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

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.

Internal error while unpacking matrix during iterative refinement.
Please contact NAG.

NE_NO_LICENCE

Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.

NE_REAL_2

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

NE_REAL_ARRAY

On entry, $\mathbf{f}\left[\mathbf{m}\right]=0.0$.

NE_SINGULAR_MATRIX

Singular matrix encountered during iterative refinement.
Please check that your system is well posed.

NE_TOO_MANY_ITER

During iterative refinement, the maximum number of iterations was reached.
$\text{Number of iterations}=〈\mathit{\text{value}}〉$ and $\text{residual achieved}=〈\mathit{\text{value}}〉$.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d02uece.c)

10.2

Program Data

Program Data (d02uece.d)

10.3

Program Results

Program Results (d02uece.r)