The routine may be called by the names d02tyf or nagf_ode_bvp_coll_nlin_interp.
3Description
d02tyf and its associated routines (d02tlf,d02tvf,d02txfandd02tzf) solve the two-point boundary value problem for a nonlinear mixed order system of ordinary differential equations
over an interval $[a,b]$ subject to $p$ ($\text{}>0$) nonlinear boundary conditions at $a$ and $q$ ($\text{}>0$) nonlinear boundary conditions at $b$, where $p+q={\displaystyle \sum _{i=1}^{n}}{m}_{i}$. Note that ${y}_{i}^{\left(m\right)}\left(x\right)$ is the $m$th derivative of the $i$th solution component. Hence ${y}_{i}^{\left(0\right)}\left(x\right)={y}_{i}\left(x\right)$. The left boundary conditions at $a$ are defined as
First, d02tvf must be called to specify the initial mesh, error requirements and other details. Then, d02tlf can be used to solve the boundary value problem. After successful computation, d02tzf can be used to ascertain details about the final mesh and other details of the solution procedure, and d02tyf can be used to compute the approximate solution anywhere on the interval $[a,b]$ using interpolation.
Ascher U M and Bader G (1987) A new basis implementation for a mixed order boundary value ODE solver SIAM J. Sci. Stat. Comput.8 483–500
Ascher U M, Christiansen J and Russell R D (1979) A collocation solver for mixed order systems of boundary value problems Math. Comput.33 659–679
Ascher U M, Mattheij R M M and Russell R D (1988) Numerical Solution of Boundary Value Problems for Ordinary Differential Equations Prentice–Hall
Grossman C (1992) Enclosures of the solution of the Thomas–Fermi equation by monotone discretization J. Comput. Phys.98 26–32
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York
5Arguments
1: $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the independent variable.
Constraint:
$a\le {\mathbf{x}}\le b$, i.e., not outside the range of the original mesh specified in the initialization call to d02tvf.
2: $\mathbf{y}({\mathbf{neq}},{\mathbf{mmax}})$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{y}}(\mathit{i},\mathit{j})$ contains an approximation to ${y}_{\mathit{i}}^{\left(\mathit{j}\right)}\left(x\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$ and $\mathit{j}=0,1,\dots ,{m}_{\mathit{i}}-1$. The remaining elements of y (where ${m}_{i}<{\mathbf{mmax}}$) are initialized to $0.0$.
3: $\mathbf{neq}$ – IntegerInput
On entry: the number of differential equations.
Constraint:
${\mathbf{neq}}$ must be the same value as supplied to d02tvf.
4: $\mathbf{mmax}$ – IntegerInput
On entry: the maximal order of the differential equations,
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left({m}_{\mathit{i}}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{neq}}$.
Constraint:
${\mathbf{mmax}}$ must contain the maximum value of the components of the argument m as supplied to d02tvf.
5: $\mathbf{rcomm}(*)$ – Real (Kind=nag_wp) arrayCommunication Array
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument rcomm in the previous call to d02tlf.
On entry: this must be the same array as supplied to d02tlf and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated routines.
Note: the dimension of this array is dictated by the requirements of associated functions that must have been previously called. This array must be the same array passed as argument icomm in the previous call to d02tlf.
On entry: this must be the same array as supplied to d02tlf and must remain unchanged between calls.
On exit: contains information about the solution for use on subsequent calls to associated routines.
7: $\mathbf{ifail}$ – IntegerInput/Output
On entry: ifail must be set to $0$, $\mathrm{-1}$ 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 $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $\mathrm{-1}$ is recommended since useful values can be provided in some output arguments even when ${\mathbf{ifail}}\ne {\mathbf{0}}$ on exit. 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
Note: in some cases d02tyf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{mmax}}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{max}\phantom{\rule{0.25em}{0ex}}{\mathbf{m}}\left(i\right)=\u27e8\mathit{\text{value}}\u27e9$ in d02tvf.
Constraint: ${\mathbf{mmax}}=\mathrm{max}\phantom{\rule{0.25em}{0ex}}{\mathbf{m}}\left(i\right)$ in d02tvf.
On entry, ${\mathbf{neq}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{neq}}=\u27e8\mathit{\text{value}}\u27e9$ in d02tvf.
Constraint: ${\mathbf{neq}}={\mathbf{neq}}$ in d02tvf.
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\le \u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{x}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{x}}\ge \u27e8\mathit{\text{value}}\u27e9$.
The solver routine did not produce any results suitable for interpolation.
The solver routine does not appear to have been called.
${\mathbf{ifail}}=2$
The solver routine did not converge to a suitable solution.
A converged intermediate solution has been used.
Interpolated values should be treated with caution.
The solver routine did not satisfy the error requirements.
Interpolated values should be treated with caution.
${\mathbf{ifail}}=-99$
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.
7Accuracy
If d02tyf returns the value ${\mathbf{ifail}}={\mathbf{0}}$, the computed values of the solution components ${y}_{i}$ should be of similar accuracy to that specified by the argument tols of d02tvf. Note that during the solution process the error in the derivatives ${y}_{i}^{\left(\mathit{j}\right)}$, for $\mathit{j}=1,2,\dots ,{m}_{i}-1$, has not been controlled and that the derivative values returned by d02tyf are computed via differentiation of the piecewise polynomial approximation to ${y}_{i}$. See also Section 9.
8Parallelism and Performance
d02tyf 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 implementation-specific information.
9Further Comments
If d02tyf returns the value
${\mathbf{ifail}}={\mathbf{2}}$ in this routine and ${\mathbf{ifail}}={\mathbf{5}}$ in d02tlf,
then the accuracy of the interpolated values may be proportional to the quantity ermx as returned by d02tzf.
If d02tlf returned a value for
ifail
other than ${\mathbf{ifail}}={\mathbf{0}}$, then nothing can be said regarding either the quality or accuracy of the values computed by d02tyf.
10Example
The following example is used to illustrate that a system with singular coefficients can be treated without modification of the system definition. See also d02tlf,d02tvf,d02txfandd02tzf, for the illustration of other facilities.
Consider the Thomas–Fermi equation used in the investigation of potentials and charge densities of ionized atoms. See Grossman (1992), for example, and the references therein. The equation is
The coefficient ${x}^{-1/2}$ implies a singularity at the left-hand boundary $x=0$.
We use the initial approximation $y\left(x\right)=1-x/a$, which satisfies the boundary conditions, on a uniform mesh of six points. For illustration we choose $a=1$, as in Grossman (1992). Note that in ffun and fjac (see d02tlf) we have taken the precaution of setting the function value and Jacobian value to $0.0$ in case a value of $y$ becomes negative, although starting from our initial solution profile this proves unnecessary during the solution phase. Of course the true solution $y\left(x\right)$ is positive for all $x<a$.