The routine may be called by the names d02tzf or nagf_ode_bvp_coll_nlin_diag.
3Description
d02tzf and its associated routines (d02tlf,d02tvf,d02txfandd02tyf) 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. 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
Cole J D (1968) Perturbation Methods in Applied Mathematics Blaisdell, Waltham, Mass.
Keller H B (1992) Numerical Methods for Two-point Boundary-value Problems Dover, New York
5Arguments
1: $\mathbf{mxmesh}$ – IntegerInput
On entry: the maximum number of points allowed in the mesh.
Constraint:
this must be identical to the value supplied for the argument mxmesh in the prior call to d02tvf.
2: $\mathbf{nmesh}$ – IntegerOutput
On exit: the number of points in the mesh last used by d02tlf.
3: $\mathbf{mesh}\left({\mathbf{mxmesh}}\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: ${\mathbf{mesh}}\left(\mathit{i}\right)$ contains the $\mathit{i}$th point of the mesh last used by d02tlf, for $\mathit{i}=1,2,\dots ,{\mathbf{nmesh}}$. ${\mathbf{mesh}}\left(1\right)$ will contain $a$ and ${\mathbf{mesh}}\left({\mathbf{nmesh}}\right)$ will contain $b$. The remaining elements of mesh are not initialized.
On exit: ${\mathbf{ipmesh}}\left(\mathit{i}\right)$ specifies the nature of the point ${\mathbf{mesh}}\left(\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,{\mathbf{nmesh}}$, in the final mesh computed by d02tlf.
${\mathbf{ipmesh}}\left(i\right)=1$
Indicates that the $i$th point is a fixed point and was used by the solver before an extrapolation-like error test.
${\mathbf{ipmesh}}\left(i\right)=2$
Indicates that the $i$th point was used by the solver before an extrapolation-like error test.
${\mathbf{ipmesh}}\left(i\right)=3$
Indicates that the $i$th point was used by the solver only as part of an extrapolation-like error test.
The remaining elements of ipmesh are initialized to $-1$.
See Section 9 for advice on how these values may be used in conjunction with a continuation process.
5: $\mathbf{ermx}$ – Real (Kind=nag_wp)Output
On exit: an estimate of the maximum error in the solution computed by d02tlf, that is
where ${v}_{i}$ is the approximate solution for the $i$th solution component. If d02tlf returned successfully with ${\mathbf{ifail}}={\mathbf{0}}$, ermx will be less than ${\mathbf{tols}}\left({\mathbf{ijermx}}\right)$ in d02tvf where tols contains the error requirements as specified in Sections 3 and 5 in d02tvf.
If d02tlf returned with ${\mathbf{ifail}}={\mathbf{5}}$, ermx will be greater than ${\mathbf{tols}}\left({\mathbf{ijermx}}\right)$ in d02tvf.
If d02tlf returned any other value for ifail then an error estimate is not available and ermx is initialized to $0.0$.
6: $\mathbf{iermx}$ – IntegerOutput
On exit: indicates the mesh sub-interval where the value of ermx has been computed, that is $[{\mathbf{mesh}}\left({\mathbf{iermx}}\right),{\mathbf{mesh}}\left({\mathbf{iermx}}+1\right)]$.
If an estimate of the error is not available then iermx is initialized to $0$.
7: $\mathbf{ijermx}$ – IntegerOutput
On exit: indicates the component $i$ ($\text{}={\mathbf{ijermx}}$) of the solution for which ermx has been computed, that is the approximation of ${y}_{i}$ on $[{\mathbf{mesh}}\left({\mathbf{iermx}}\right),{\mathbf{mesh}}\left({\mathbf{iermx}}+1\right)]$ is estimated to have the largest error of all components ${y}_{i}$ over mesh sub-intervals defined by mesh.
If an estimate of the error is not available then ijermx is initialized to $0$.
8: $\mathbf{rcomm}(*)$ – 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.
10: $\mathbf{ifail}$ – IntegerInput/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 $-1$ 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 $-1$, 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 d02tzf may return useful information.
${\mathbf{ifail}}=1$
On entry, ${\mathbf{mxmesh}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mxmesh}}=\u27e8\mathit{\text{value}}\u27e9$ in d02tvf.
Constraint: ${\mathbf{mxmesh}}={\mathbf{mxmesh}}$ in d02tvf.
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.
Error estimate information is not available.
The solver routine did not satisfy the error requirements.
Information has been supplied on the last mesh used.
${\mathbf{ifail}}=-99$
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
Not applicable.
8Parallelism and Performance
d02tzf is not threaded in any implementation.
9Further Comments
Note that:
if d02tlf returned ${\mathbf{ifail}}={\mathbf{0}}$, ${\mathbf{4}}$ or ${\mathbf{5}}$ then it will always be the case that ${\mathbf{ipmesh}}\left(1\right)={\mathbf{ipmesh}}\left({\mathbf{nmesh}}\right)=1$;
if d02tlf returned ${\mathbf{ifail}}={\mathbf{0}}$ or ${\mathbf{5}}$ then it will always be the case that
${\mathbf{ipmesh}}\left(\mathit{i}\right)=3$, for $\mathit{i}=2,4,\dots ,{\mathbf{nmesh}}-1$ (even $i$) and
${\mathbf{ipmesh}}\left(\mathit{i}\right)=1$ or $2$, for $\mathit{i}=3,5,\dots ,{\mathbf{nmesh}}-2$ (odd $i$);
if d02tlf returned ${\mathbf{ifail}}={\mathbf{4}}$ then it will always be the case that
${\mathbf{ipmesh}}\left(\mathit{i}\right)=1$ or $2$, for $\mathit{i}=2,3,\dots ,{\mathbf{nmesh}}-1$.
If d02tzf returns ${\mathbf{ifail}}={\mathbf{0}}$, then examination of the mesh may provide assistance in determining a suitable starting mesh for d02tvf in any subsequent attempts to solve similar problems.
If the problem being treated by d02tlf is one of a series of related problems (for example, as part of a continuation process), then the values of ipmesh and mesh may be suitable as input arguments to d02txf. Using the mesh points not involved in the extrapolation error test is usually appropriate. ipmesh and mesh should be passed unchanged to d02txf but nmesh should be replaced by $({\mathbf{nmesh}}+1)/2$.
If d02tzf returns ${\mathbf{ifail}}={\mathbf{2}}$, nothing can be said regarding the quality of the mesh returned. However, it may be a useful starting mesh for d02tvf in any subsequent attempts to solve the same problem.
If d02tlf returns ${\mathbf{ifail}}={\mathbf{5}}$, this corresponds to the solver requiring more than mxmesh mesh points to satisfy the error requirements. If mxmesh can be increased and the preceding call to d02tlf was not part, or was the first part, of a continuation process then the values in mesh may provide a suitable mesh with which to initialize a subsequent attempt to solve the same problem. If it is not possible to provide more mesh points then relaxing the error requirements by setting ${\mathbf{tols}}\left({\mathbf{ijermx}}\right)$ to ermx might lead to a successful solution. It may be necessary to reset the other components of tols. Note that resetting the tolerances can lead to a different sequence of meshes being computed and hence to a different solution being computed.
10Example
The following example is used to illustrate the use of fixed mesh points, simple continuation and numerical approximation of a Jacobian. See also d02tlf,d02tvf,d02txfandd02tyf, for the illustration of other facilities.
where $\epsilon $ is small and positive. The nature of the solution depends markedly on the values of $\alpha ,\beta $. See Cole (1968).
We choose $\alpha =-\frac{1}{3},\beta =\frac{1}{3}$ for which the solution is known to have corner layers at $x=\frac{1}{3},\frac{2}{3}$. We choose an initial mesh of seven points $[0.0,0.15,0.3,0.5,0.7,0.85,1.0]$ and ensure that the points $x=0.3,0.7$ near the corner layers are fixed, that is the corresponding elements of the array
ipmesh are set to $1$. First we compute the solution for $\epsilon =\text{1.0E\u22124}$ using in guess the initial approximation $y\left(x\right)=\alpha +(\beta -\alpha )x$ which satisfies the boundary conditions. Then we use simple continuation to compute the solution for $\epsilon =\text{1.0E\u22125}$. We use the suggested values for nmesh, ipmesh and mesh in the call to d02txf prior to the continuation call, that is only every second point of the preceding mesh is used and the fixed mesh points are retained.
Although the analytic Jacobian for this system is easy to evaluate, for illustration the procedure fjac uses central differences and calls to ffun to compute a numerical approximation to the Jacobian.