NAG FL Interface
d02tgf (bvp_​coll_​nth_​comp)

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

1: $\mathbf{n}$ – Integer Input

On entry: $n$, the number of differential equations in the system.

Constraint: $\mathbf{n}\ge 1$.

2: $\mathbf{m}\left(\mathbf{n}\right)$ – Integer array Input

On entry: $\mathbf{m}\left(\mathit{i}\right)$ must be set to the highest order derivative occurring in the $\mathit{i}$th equation, for $\mathit{i}=1,2,\dots ,n$.

Constraint: $\mathbf{m}\left(\mathit{i}\right)\ge 1$, for $\mathit{i}=1,2,\dots ,n$.

3: $\mathbf{l}\left(\mathbf{n}\right)$ – Integer array Input

On entry: $\mathbf{l}\left(\mathit{i}\right)$ must be set to the number of boundary conditions associated with the $\mathit{i}$th equation, for $\mathit{i}=1,2,\dots ,n$.

Constraint: $\mathbf{l}\left(\mathit{i}\right)\ge 0$, for $\mathit{i}=1,2,\dots ,n$.

4: $\mathbf{x0}$ – Real (Kind=nag_wp) Input

On entry: the left-hand boundary, $x_0$.

5: $\mathbf{x1}$ – Real (Kind=nag_wp) Input

On entry: the right-hand boundary, $x_1$.

Constraint: $\mathbf{x1}>\mathbf{x0}$.

6: $\mathbf{k1}$ – Integer Input

On entry: the number of coefficients, $k_1$, to be returned in the Chebyshev series representation of the solution (hence, the degree of the polynomial approximation is $\mathbf{k1}-1$).

Constraint: $\mathbf{k1}\ge 1+{\displaystyle \underset{1\le i\le \mathbf{n}}{\max }}\mathbf{m}\left(i\right)$.

7: $\mathbf{kp}$ – Integer Input

On entry: the number of collocation points to be used, $k_p$.

Constraint: $\mathbf{n}\times \mathbf{kp}\ge \mathbf{n}\times \mathbf{k1}+{\displaystyle \sum _{i=1}^{\mathbf{n}}}\mathbf{l}\left(i\right)$.

8: $\mathbf{c}(\mathbf{ldc},\mathbf{n})$ – Real (Kind=nag_wp) array Output

On exit: the $k$th column of c contains the computed Chebyshev coefficients of the $k$th component of the solution, $y_k$; that is, the computed solution is:

$$y_k=\underset{i=1}{\overset{k_1}{{\sum }^{\prime }}}\mathbf{c}(i,k)T_{i-1}\left(x\right)\text{, \hspace{1em}}1\le k\le n\text{,}$$

where $T_i\left(x\right)$ is the Chebyshev polynomial of the first kind and ${\sum }^{\prime }$ denotes that the first coefficient, $\mathbf{c}(1,k)$, is halved.

9: $\mathbf{ldc}$ – Integer Input

On entry: the first dimension of the array c as declared in the (sub)program from which d02tgf is called.

Constraint: $\mathbf{ldc}\ge \mathbf{k1}$.

10: $\mathbf{coeff}$ – Subroutine, supplied by the user. External Procedure

coeff defines the system of differential equations (see Section 3). It must evaluate the coefficient functions $f_{kj}^i\left(x\right)$ and the right-hand side function $r^i\left(x\right)$ of the $i$th equation at a given point. Only nonzero entries of the array a and rhs need be specifically assigned, since all elements are set to zero by d02tgf before calling coeff.

The specification of coeff is:

Fortran Interface copy

Subroutine coeff ( x, i, a, ia, ia1, rhs) Integer, Intent (In) :: i, ia, ia1 Real (Kind=nag_wp), Intent (In) :: x Real (Kind=nag_wp), Intent (Inout) :: a(ia,ia1), rhs

C Header Interface copy

void  coeff (const double *x, const Integer *i, double a[], const Integer *ia, const Integer *ia1, double *rhs)

C++ Header Interface copy

#include <nag.h> extern "C" { void  coeff (const double &x, const Integer &i, double a[], const Integer &ia, const Integer &ia1, double &rhs) }

Important: the dimension declaration for a must contain the variable ia, not an integer constant.

1: $\mathbf{x}$ – Real (Kind=nag_wp) Input

On entry: $x$, the point at which the functions must be evaluated.

2: $\mathbf{i}$ – Integer Input

On entry: the equation for which the coefficients and right-hand side are to be evaluated.

3: $\mathbf{a}(\mathbf{ia},\mathbf{ia1})$ – Real (Kind=nag_wp) array Input/Output

On entry: all elements of a are set to zero.

On exit: $\mathbf{a}(k,j)$ must contain the value $f_{kj}^i\left(x\right)$, for $1\le k\le n$, $1\le j\le m_i+1$.

4: $\mathbf{ia}$ – Integer Input

5: $\mathbf{ia1}$ – Integer Input

On entry: the first dimension of the array a and the second dimension of the array a as declared in the (sub)program from which d02tgf is called.

6: $\mathbf{rhs}$ – Real (Kind=nag_wp) Input/Output

On entry: is set to zero.

On exit: it must contain the value $r^i\left(x\right)$.

coeff must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02tgf is called. Arguments denoted as Input must not be changed by this procedure.

Note: coeff should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tgf. If your code inadvertently does return any NaNs or infinities, d02tgf is likely to produce unexpected results.

11: $\mathbf{bdyc}$ – Subroutine, supplied by the user. External Procedure

bdyc defines the boundary conditions (see Section 3). It must evaluate the coefficient functions $f_{kj}^{ij}$ and right-hand side function $r^{ij}$ in the $j$th boundary condition associated with the $i$th equation, at the point $x^{ij}$ at which the boundary condition is applied. Only nonzero entries of the array a and rhs need be specifically assigned, since all elements are set to zero by d02tgf before calling bdyc.

The specification of bdyc is:

Fortran Interface copy

Subroutine bdyc ( x, i, j, a, ia, ia1, rhs) Integer, Intent (In) :: i, j, ia, ia1 Real (Kind=nag_wp), Intent (Inout) :: a(ia,ia1), rhs Real (Kind=nag_wp), Intent (Out) :: x

C Header Interface copy

void  bdyc (double *x, const Integer *i, const Integer *j, double a[], const Integer *ia, const Integer *ia1, double *rhs)

C++ Header Interface copy

#include <nag.h> extern "C" { void  bdyc (double &x, const Integer &i, const Integer &j, double a[], const Integer &ia, const Integer &ia1, double &rhs) }

Important: the dimension declaration for a must contain the variable ia, not an integer constant.

1: $\mathbf{x}$ – Real (Kind=nag_wp) Output

On exit: $x^{ij}$, the value at which the boundary condition is applied.

2: $\mathbf{i}$ – Integer Input

On entry: the differential equation with which the condition is associated.

3: $\mathbf{j}$ – Integer Input

On entry: the boundary condition for which the coefficients and right-hand side are to be evaluated.

4: $\mathbf{a}(\mathbf{ia},\mathbf{ia1})$ – Real (Kind=nag_wp) array Input/Output

On entry: all elements of a are set to zero.

On exit: the value $f_{kj}^{ij}\left(x^{ij}\right)$, for $1\le k\le n$, $1\le j\le m_i+1$.

5: $\mathbf{ia}$ – Integer Input

6: $\mathbf{ia1}$ – Integer Input

On entry: the first dimension of the array a and the second dimension of the array a as declared in the (sub)program from which d02tgf is called.

7: $\mathbf{rhs}$ – Real (Kind=nag_wp) Input/Output

On entry: is set to zero.

On exit: the value $r^{ij}\left(x^{ij}\right)$.

bdyc must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02tgf is called. Arguments denoted as Input must not be changed by this procedure.

Note: bdyc should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02tgf. If your code inadvertently does return any NaNs or infinities, d02tgf is likely to produce unexpected results.

12: $\mathbf{w}\left(\mathbf{lw}\right)$ – Real (Kind=nag_wp) array Workspace

13: $\mathbf{lw}$ – Integer Input

On entry: the dimension of the array w as declared in the (sub)program from which d02tgf is called.

Constraint: $\mathbf{lw}\ge 2\times (\mathbf{n}\times \mathbf{kp}+\mathit{NL})\times (\mathbf{n}\times \mathbf{k1}+1)+7\times \mathbf{n}\times \mathbf{k1}$, where $\mathit{NL}={\displaystyle \sum _{i=1}^n}\mathbf{l}\left(i\right)$.

14: $\mathbf{iw}\left(\mathbf{liw}\right)$ – Integer array Workspace

15: $\mathbf{liw}$ – Integer Input

On entry: the dimension of the array iw as declared in the (sub)program from which d02tgf is called.

Constraint: $\mathbf{liw}\ge \mathbf{n}\times \mathbf{k1}+1$.

16: $\mathbf{ifail}$ – Integer Input/Output

On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.

If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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, a negative number of boundary conditions was set for one of the system equations.

On entry, $\mathbf{k1}=⟨\mathit{\text{value}}⟩$ and $\max \mathbf{m}\left(i\right)=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{k1}\ge 1+\max \mathbf{m}\left(i\right)$.

On entry, $\mathbf{kp}=⟨\mathit{\text{value}}⟩$ and $\mathbf{k1}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{kp}+1\ge \mathbf{k1}$.

On entry, $\mathbf{ldc}=⟨\mathit{\text{value}}⟩$ and $\mathbf{k1}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{ldc}\ge \mathbf{k1}$.

On entry, $\mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{n}\ge 1$.

On entry, one of the equations is of order less than $1$.

On entry, $\mathbf{x1}=⟨\mathit{\text{value}}⟩$ and $\mathbf{x0}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{x1}>\mathbf{x0}$.

$\mathbf{ifail}=2$

On entry, $\mathbf{liw}=⟨\mathit{\text{value}}⟩$ and $\mathbf{n}\times \mathbf{k1}+2\times \mathbf{n}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{liw}\ge \mathbf{n}\times \mathbf{k1}+2\times \mathbf{n}$.

On entry, $\mathbf{lw}=⟨\mathit{\text{value}}⟩$ and $2\times \mathbf{n}\times (\mathbf{kp}+1)\times (\mathbf{n}\times \mathbf{k1}+1)+7\times \mathbf{n}\times \mathbf{k1}=⟨\mathit{\text{value}}⟩$.
Constraint: $\mathbf{lw}\ge 2\times \mathbf{n}\times (\mathbf{kp}+1)\times (\mathbf{n}\times \mathbf{k1}+1)+7\times \mathbf{n}\times \mathbf{k1}$.

$\mathbf{ifail}=3$

Either the boundary conditions are not linearly independent, or the coefficient matrix is rank deficient. Increasing the number of collocation points may overcome this latter problem.

$\mathbf{ifail}=4$

Iterative refinement in the least squares solution has failed to converge. The coefficient matrix is too ill-conditioned.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results