d02gaf solves a two-point boundary value problem with assigned boundary values for a system of ordinary differential equations, using a deferred correction technique and a Newton iteration.
The routine may be called by the names d02gaf or nagf_ode_bvp_fd_nonlin_fixedbc.
3Description
d02gaf solves a two-point boundary value problem for a system of $\mathit{n}$ differential equations in the interval [$a,b$]. The system is written in the form:
and the derivatives ${f}_{i}$ are evaluated by fcn. Initially, $\mathit{n}$ boundary values of the variables ${y}_{i}$ must be specified, some at $a$ and some at $b$. You must supply estimates of the remaining $\mathit{n}$ boundary values and all the boundary values are used in constructing an initial approximation to the solution. This approximate solution is corrected by a finite difference technique with deferred correction allied with a Newton iteration to solve the finite difference equations. The technique used is described fully in Pereyra (1979). The Newton iteration requires a Jacobian matrix $\frac{\partial {f}_{i}}{\partial {y}_{j}}$ and this is calculated by numerical differentiation using an algorithm described in Curtis et al. (1974).
You supply an absolute error tolerance and may also supply an initial mesh for the construction of the finite difference equations (alternatively a default mesh is used). The algorithm constructs a solution on a mesh defined by adding points to the initial mesh. This solution is chosen so that the error is everywhere less than your tolerance and so that the error is approximately equidistributed on the final mesh. The solution is returned on this final mesh.
If the solution is required at a few specific points then these should be included in the initial mesh. If on the other hand the solution is required at several specific points then you should use the interpolation routines provided in Chapter E01 if these points do not themselves form a convenient mesh.
4References
Curtis A R, Powell M J D and Reid J K (1974) On the estimation of sparse Jacobian matrices J. Inst. Maths. Applics.13 117–119
Pereyra V (1979) PASVA3: An adaptive finite-difference Fortran program for first order nonlinear, ordinary boundary problems Codes for Boundary Value Problems in Ordinary Differential Equations. Lecture Notes in Computer Science (eds B Childs, M Scott, J W Daniel, E Denman and P Nelson) 76 Springer–Verlag
5Arguments
1: $\mathbf{u}({\mathbf{n}},2)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{u}}(\mathit{i},1)$ must be set to the known or estimated value of ${y}_{\mathit{i}}$ at $a$ and ${\mathbf{u}}(\mathit{i},2)$ must be set to the known or estimated value of ${y}_{\mathit{i}}$ at $b$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
2: $\mathbf{v}({\mathbf{n}},2)$ – Real (Kind=nag_wp) arrayInput
On entry: ${\mathbf{v}}(\mathit{i},\mathit{j})$ must be set to $0.0$ if ${\mathbf{u}}(\mathit{i},\mathit{j})$ is a known value and to $1.0$ if ${\mathbf{u}}(\mathit{i},\mathit{j})$ is an estimated value, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=1,2$.
Constraint:
precisely $\mathit{n}$ of the ${\mathbf{v}}(i,j)$ must be set to $0.0$, i.e., precisely $\mathit{n}$ of the ${\mathbf{u}}(i,j)$ must be known values, and these must not be all at $a$ or all at $b$.
is the final mesh, ${z}_{j}\left({x}_{i}\right)$ is the $j$th component of the approximate solution at ${x}_{i}$, and ${y}_{j}\left(x\right)$ is the $j$th component of the true solution of equation (1) (see Section 3) and the boundary conditions, then, except in extreme cases, it is expected that
7: $\mathbf{fcn}$ – Subroutine, supplied by the user.External Procedure
fcn must evaluate the functions
${f}_{\mathit{i}}$ (i.e., the derivatives ${y}_{\mathit{i}}^{\prime}$), for $\mathit{i}=1,2,\dots ,\mathit{n}$, at a general point $x$.
In the description of the arguments of d02gaf below, $\mathit{n}$ denotes the actual value of n in the call of d02gaf.
1: $\mathbf{x}$ – Real (Kind=nag_wp)Input
On entry: $x$, the value of the argument.
2: $\mathbf{y}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.
3: $\mathbf{f}\left(*\right)$ – Real (Kind=nag_wp) arrayOutput
On exit: the values of
${f}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
fcn must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d02gaf is called. Arguments denoted as Input must not be changed by this procedure.
Note:fcn should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d02gaf. If your code inadvertently does return any NaNs or infinities, d02gaf is likely to produce unexpected results.
8: $\mathbf{mnp}$ – IntegerInput
On entry: the maximum permitted number of mesh points.
Constraint:
${\mathbf{mnp}}\ge 32$.
9: $\mathbf{x}\left({\mathbf{mnp}}\right)$ – Real (Kind=nag_wp) arrayInput/Output
On entry: if ${\mathbf{np}}\ge 4$ (see np), the first np elements must define an initial mesh. Otherwise the elements of x need not be set.
On exit: ${\mathbf{x}}\left(1\right),{\mathbf{x}}\left(2\right),\dots ,{\mathbf{x}}\left({\mathbf{np}}\right)$ define the final mesh (with the returned value of np) satisfying the relation (3).
10: $\mathbf{y}({\mathbf{n}},{\mathbf{mnp}})$ – Real (Kind=nag_wp) arrayOutput
On exit: the approximate solution ${z}_{j}\left({x}_{i}\right)$ satisfying (2), on the final mesh, that is
On entry: determines whether a default or user-supplied mesh is used.
${\mathbf{np}}=0$
A default value of $4$ for np and a corresponding equispaced mesh ${\mathbf{x}}\left(1\right),{\mathbf{x}}\left(2\right),\dots ,{\mathbf{x}}\left({\mathbf{np}}\right)$ are used.
${\mathbf{np}}\ge 4$
You must define an initial mesh using the array x as described.
Constraint:
${\mathbf{np}}=0$ or $4\le {\mathbf{np}}\le {\mathbf{mnp}}$.
On exit: the number of points in the final (returned) mesh.
12: $\mathbf{w}\left({\mathbf{lw}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
13: $\mathbf{lw}$ – IntegerInput
On entry: the dimension of the array w as declared in the (sub)program from which d02gaf is called.
This routine uses an ifail input value codification that differs from the normal case to distinguish between errors and warnings (see Section 4 in the Introduction to the NAG Library FL Interface).
On entry: ifail must be set to one of the values below to set behaviour on detection of an error; these values have no effect when no error is detected. The behaviour relate to whether or not program execution is halted and whether or not messages are printed when an error or warning is detected.
For environments where it might be inappropriate to halt program execution when an error is detected, the value $1$, $11$, $101$ or $111$ is recommended. If the printing of messages is undesirable, then the value $1$ is recommended. Otherwise, the recommended value is $110$. When the value $\mathbf{1}$, $\mathbf{11}$, $\mathbf{101}$ or $\mathbf{111}$ 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:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{a}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{b}}>{\mathbf{a}}$.
On entry, ${\mathbf{liw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{liw}}\ge {\mathbf{mnp}}\times (2\times {\mathbf{n}}+1)+{{\mathbf{n}}}^{2}+4\times {\mathbf{n}}+2$; that is, $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{lw}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lw}}\ge {\mathbf{mnp}}\times (3\times {{\mathbf{n}}}^{2}+6\times {\mathbf{n}}+2)+4\times {{\mathbf{n}}}^{2}+4\times {\mathbf{n}}$; that is, $\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{mnp}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{mnp}}\ge 32$.
On entry, ${\mathbf{n}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{n}}\ge 2$.
On entry, ${\mathbf{np}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{np}}=0$ or ${\mathbf{np}}\ge 4$.
On entry, ${\mathbf{np}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{mnp}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{np}}\le {\mathbf{mnp}}$.
On entry, ${\mathbf{tol}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{tol}}>0.0$.
On entry: ${\mathbf{a}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left(1\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{a}}={\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{np}}\right)={\mathbf{b}}\text{, \hspace{1em}}{\mathbf{np}}\ge 4$.
On entry: ${\mathbf{b}}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left({\mathbf{np}}\right)=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{a}}={\mathbf{x}}\left(1\right)<{\mathbf{x}}\left(2\right)<\cdots <{\mathbf{x}}\left({\mathbf{np}}\right)={\mathbf{b}}\text{, \hspace{1em}}{\mathbf{np}}\ge 4$.
The number of known left boundary values must be less than the number of equations: the number of known left boundary values $\text{}=\u27e8\mathit{\text{value}}\u27e9$, the number of equations $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
The number of known right boundary values must be less than the number of equations: the number of known right boundary values $\text{}=\u27e8\mathit{\text{value}}\u27e9$, the number of equations $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
The sequence x is not strictly increasing. For $i=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{x}}\left(i\right)=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{x}}\left(i+1\right)=\u27e8\mathit{\text{value}}\u27e9$.
The sum of known left and right boundary values must equal the number of equations: the number of known left boundary values $\text{}=\u27e8\mathit{\text{value}}\u27e9$, the number of known right boundary values $\text{}=\u27e8\mathit{\text{value}}\u27e9$, the number of equations $\text{}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=2$
The Newton iteration has failed to converge.
This could be due to there being too few points in the initial mesh or to the initial approximate solution being too inaccurate. If this latter reason is suspected or you cannot make changes to prevent this error, you should use the routine with a continuation facility instead.
${\mathbf{ifail}}=3$
Newton iteration has reached round-off level.
If desired accuracy has not been reached, tol is too small for this problem and this machine precision.
${\mathbf{ifail}}=4$
A finer mesh is required for the accuracy requested; that is, ${\mathbf{mnp}}=\u27e8\mathit{\text{value}}\u27e9$ is not large enough.
${\mathbf{ifail}}=5$
A serious error occurred in a call to the internal integrator.
The error code internally was $\u27e8\mathit{\text{value}}\u27e9$. Please contact NAG.
${\mathbf{ifail}}=-99$
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
The solution returned by the routine will be accurate to your tolerance as defined by the relation (2) except in extreme circumstances. If too many points are specified in the initial mesh, the solution may be more accurate than requested and the error may not be approximately equidistributed.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d02gaf is not thread safe and should not be called from a multithreaded user program. Please see Section 1 in FL Interface Multithreading for more information on thread safety.
d02gaf 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
The time taken by d02gaf depends on the difficulty of the problem, the number of mesh points (and meshes) used, the number of Newton iterations and the number of deferred corrections.
You are strongly recommended to set ifail to obtain self-explanatory error messages, and also monitoring information about the course of the computation. You may select the unit numbers on which this output is to appear by calls of x04aaf (for error messages) or x04abf (for monitoring information) – see Section 10 for an example. Otherwise the default unit numbers will be used, as specified in the Users' Note.
A common cause of convergence problems in the Newton iteration is that you have specified too few points in the initial mesh. Although the routine adds points to the mesh to improve accuracy it is unable to do so until the solution on the initial mesh has been calculated in the Newton iteration.
If you specify zero known and estimated boundary values, the routine constructs a zero initial approximation and in many cases the Jacobian is singular when evaluated for this approximation, leading to the breakdown of the Newton iteration.
You may be unable to provide a sufficiently good choice of initial mesh and estimated boundary values, and hence the Newton iteration may never converge. In this case the continuation facility provided in d02raf is recommended.
In the case where you wish to solve a sequence of similar problems, the final mesh from solving one case is strongly recommended as the initial mesh for the next.
for $\beta =0.0$ and $\beta =0.2$ to an accuracy specified by ${\mathbf{tol}}=\text{1.0E\u22123}$. We solve first the simpler problem with $\beta =0.0$ using an equispaced mesh of $26$ points and then we solve the problem with $\beta =0.2$ using the final mesh from the first problem.
Note the call to x04abf prior to the call to d02gaf.