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\title{User's Guide for {TWPBVP}:\\
A Code for Solving Two-Point Boundary Value Problems}

\author{J.\ R.\ Cash\\
{\small Department of Mathematics}\\
{\small Imperial College}\\
{\small London, England}\\
{\small e-mail: j.cash{\rm\small $@$}ic.ac.uk}
\and
Margaret H. Wright\\ 
{\small AT\&amp;T Bell Laboratories}\\
{\small Murray Hill, New Jersey}\\
{\small USA}\\
{\small e-mail: mhw{\rm\small $@$}research.att.com}
}

\date{}              

\begin{document}

\def\norm#1{\Vert#1\Vert}
\def\real{\cal R}
\def\half  {{\textstyle{1\over 2}}}
\def\TWPBVP{\small TWPBVP}

\maketitle \thispagestyle{empty}

\section{Introduction}

The code {\TWPBVP} (written in Fortran 77) is designed for the 
numerical solution of the two-point boundary value problem
\begin{eqnarray}
{du\over dx}  = f(x,u),&amp; &amp; a \le x \le b, \label{eqn-prob1}\\
g_i(u(a)) = 0, \quad i = 1,\dots p; &amp;  &amp;g_i(u(b)) = 0, \quad i = p+1,\dots, n,
\label{eqn-prob2}
\end{eqnarray}
where $x\in\real$, 
$u\in{\real}^n$, $g_i\in{\real}$ for $1\le i \le n$, and 
$0 \le p \le n$.
The functions $f$ and $\{g_i\}$ are assumed to be differentiable.
The method used in {\TWPBVP} is a deferred correction method
based on mono-implicit Runge-Kutta formulas and adaptive mesh
refinement.  For details about the method, see
\cite{Cash86,Cash88,CW1,CW2}.

Two aspects of the formulation (\ref{eqn-prob1})--(\ref{eqn-prob2})
should be noted:
\begin{enumerate}
\item The problem must be posed as a first-order system.
This requirement is not unduly restrictive, however, since
standard techniques can be used to convert an $n$th-order
equation to a system of $n$ first-order equations.
For example, the second-order problem
$$
  y'' = f(x,y,y'),\quad 0\le x \le 1,\quad
  y(0) = \alpha,\quad y(1) = \beta,
$$
can be converted to the following first-order system:
\begin{eqnarray*}
u' = z &amp; &amp; u(0) = \alpha\\
z' = f(x,u,z) &amp; &amp; u(1) = \beta.
\end{eqnarray*}
\item The boundary conditions must be separated.
{\TWPBVP} allows all the boundary conditions to be
imposed at one end of the interval $[a,b]$, so that 
initial value problems are handled as a special case.  It is hoped to
extend {\TWPBVP} to handle non-separated
boundary conditions; see \cite{AMR} for a discussion of
techniques that allow some non-separated
boundary conditions to be converted to the separated form needed here.
\end{enumerate}

\section{Formal parameters}\label{sec-params}

The calling sequence for {\TWPBVP} is

\begin{verbatim}
      subroutine twpbvp(ncomp, nlbc, nucol, aleft, aright,
     *       nfxpnt, fixpnt, ntol, ltol, tol,
     *       linear, givmsh, giveu, nmsh,
     *       xx, nudim, u, nmax,
     *       lwrkfl, wrk, lwrkin, iwrk, 
     *       fsub, dfsub, gsub, dgsub, iflbvp)
\end{verbatim}

We now describe the formal parameters.

\begin{description}
\item{\tt ncomp -}  Integer, input.  {\tt ncomp} is the number of 
first-order differential 
equations ($n$ in (\ref{eqn-prob1})),
and is the number of components of $u$ at each mesh point.
{\tt ncomp} must  be positive.

\item{\tt nlbc -}  Integer, input.  {\tt nlbc} is the number of 
boundary conditions at the left endpoint ($p$ in (\ref{eqn-prob2})).
{\tt nlbc} must be nonnegative and must
not exceed {\tt ncomp}.  If ${\tt nlbc} = {\tt ncomp}$, all the 
boundary conditions occur at the left endpoint,
so the problem is an initial value problem.
If {\tt nlbc} is zero, all the boundary conditions occur at 
the right endpoint.

\item{\tt nucol - }  Integer, input.  {\tt nucol}
is the declared column dimension 
of the two-dimensional array {\tt u} used to store the computed solution
(see the discussion below of the parameter {\tt u}).  {\tt nucol} must be
positive, and is an upper bound on {\tt nmax}, the maximum allowed 
number of mesh points.  In the specification of {\tt nmax}, below, the
actual formula for computing {\tt nmax} is given. In order to make full
use of the floating-point workspace, the user should select {\tt nucol} 
equal to {\tt nmax}.

\item{\tt aleft - } Floating-point, input.  The left endpoint of 
the interval of interest (the quantity $a$ in (\ref{eqn-prob1})).

\item{\tt  aright - } Floating-point, input.  The right endpoint of the
interval of interest (the value $b$ in (\ref{eqn-prob1})).
{\tt aright} must strictly exceed {\tt aleft}.


\item{\tt nfxpnt - }  Integer, input.  The number of ``fixed points'', i.e.,
points that must be included in every mesh in addition to the endpoints
$a$ and $b$. {\tt nfxpnt} must be
nonnegative, and may be zero.  If ${\tt nfxpnt} = 0$, only the
two endpoints are required to appear in every mesh.


\item{\tt fixpnt - }  Floating-point array of size at least {\tt nfxpnt}, 
input.  If ${\tt nfxpnt} = 0$, the {\tt fixpnt}
array must still be declared, say as {\tt dimension fixpnt(1)}, 
but is never accessed.  If ${\tt nfxpnt} &gt; 0$,
the {\tt fixpnt} array  contains
{\tt nfxpnt} fixed points that the user wishes to be included in
every mesh.  ${\tt fixpnt}(1)$ must strictly exceed {\tt aleft};
for $1\le i \le {\tt nfxpnt}-1$,
${\tt fixpnt}(i+1)$ must strictly exceed ${\tt fixpnt}(i)$;
and ${\tt fixpnt}({\tt nfxpnt})$ must be strictly less than {\tt aright}. 

\item{\tt  ntol - }  Integer, input.  The number of tolerances
used to determine termination of {\tt twpbvp}, i.e., the
number of components whose estimated accuracy is to be tested.
{\tt ntol} must be positive.

\item{\tt ltol - }  Integer array of size {\tt ntol}, input.  
For each $i$, ${\tt ltol}(i)$ gives the index of the component of the
computed solution $u$ controlled by the $i$th tolerance.
For example, if ${\tt ntol} = 2$, ${\tt ltol}(1) = 2$
and ${\tt ltol}(2)=3$, then component $2$ of $u$ is controlled by
${\tt tol}(1)$ and component $3$ is controlled by ${\tt tol}(2)$;
see below for a description of {\tt tol}.
Each element of {\tt ltol} must be an integer between $1$ and {\tt ncomp}.


\item{\tt tol - } Floating-point array of size {\tt ntol}, input.  
For each $i$, ${\tt tol}(i)$ gives the $i$th error tolerance used 
in performing  termination tests.  For each $i = 1,\dots,{\tt ntol}$, 
the code attempts at each mesh point $x_k$ to approximate the true
(unknown) solution $u(x_k)$ by a quantity $u_k$ such that
\begin{equation}
  {{|u_k^j - u^j(x_k)|}\over{\max(1, |u_k^j|)}}
 &lt; {\tt tol}\,(i)
\label{eqn-errortest}
\end{equation}
for each $i = 1,\dots, {\tt ntol}$ and $j = {\tt ltol}(i)$,
where $u_k^j$ denotes component $j$ of $u_k$.  Relation (\ref{eqn-errortest})
is a mixed relative/absolute error criterion of the type normally
used for solving differential equations.


\item{\tt linear - }  Logical, input.  If {\tt linear} is {\tt .true.}, 
the problem is treated as linear.  If {\tt linear} is {\tt .false}, 
the problem is solved as nonlinear.  Note that different algorithmic
strategies are used for linear and nonlinear problems; hence, if one
solves the same linear problem twice, with {\tt linear} set to 
{\tt .true.} and then to {\tt .false.}, the computed
solutions and meshes may well be different, although
both solutions should satisfy the same overall error criterion.

\item{\tt givmsh - }  Logical, input.  If {\tt givmsh}
is {\tt .true.}, the user must
define two parameters: {\tt nmsh} and the {\tt xx} array;
see below.  If {\tt givmsh} is {\tt true.},
{\tt nmsh} must contain a positive number of initial 
mesh points, and {\tt xx} must contain these
mesh points.  If {\tt givmsh} is {\tt .false.}, the user need not specify
{\tt nmsh} or {\tt xx}, which are chosen by default; see the explanations
below of {\tt nmsh} and {\tt xx}.

\item{\tt giveu - }  Logical, input.  If {\tt giveu}
is {\tt .false}, the code sets the
initial trial solution at all mesh points to the constant
value {\tt uval0}, which is contained in the labeled
common area {\tt algprs}; the default value of {\tt uval0}, set 
in a block data routine, is zero.  To change {\tt uval0}, the user 
should include the labeled common area {\tt algprs} in the 
calling routine and set {\tt uval0} to the desired value;
see Section~\ref{sec-otherpars}.

If {\tt giveu} is {\tt .true.}, {\tt givmsh} must also be set to {\tt .true.}
When {\tt giveu} is {\tt .true.}, the user must set {\tt nmsh} 
to the initial number of mesh points, {\tt xx} to the initial 
array of mesh points, and ${\tt u}(i,j)$, $i=1, \dots, {\tt ncomp}$ and 
$j = 1, \dots, {\tt nmsh}$, to the initial trial
solution at these mesh points.  However, if the first Newton procedure 
fails with these
values for {\tt xx} and {\tt u}, the code reverts 
to setting all components of the initial trial solution to {\tt uval0}
for the next mesh.

\item{\tt  nmsh - } Integer, optional input and output.  
If the parameter {\tt givmsh}
is {\tt .true.}, the user must set {\tt nmsh} to the positive initial
number of mesh points (including the two endpoints).
If {\tt givmsh} is {\tt .false.}, the user need not set {\tt nmsh}.
If {\tt givmsh} is {\tt .false.}, the initial value of {\tt nmsh} is set 
to the greater of {\tt nfxpnt}+2  and {\tt nminit}, an integer in
the labeled common area {\tt algprs}.  (The default value of 
{\tt nminit} is set in a block data routine to 7;
see Section~\ref{sec-otherpars}.)
To specify another initial value for {\tt nmsh} without specifying the
initial mesh points, the user can include the labeled common 
{\tt algprs} in the calling routine and set {\tt nminit} to the desired 
value; see Section~\ref{sec-otherpars}.

On output, {\tt nmsh} contains the final number of mesh points.

\item{\tt  xx - }  Floating-point array, of size {\tt nmax} (see below; 
nmax cannot exceed {\tt nucol}), optional input and output.  
When {\tt givmsh} is {\tt .false.}, the initial mesh {\tt xx}
is defined as follows.
If ${\tt nfxpnt} = 0$, the initial mesh contains {\tt nmsh} points equally
spaced in $[{\tt aleft}, {\tt aright}]$.  If ${\tt nfxpnt} &gt; 0$, the 
initial mesh contains (if possible) a total of {\tt nmsh} points.  These points
are equally spaced in each subinterval $[{\tt aleft}, {\tt fixpnt}(1)]$, 
$[{\tt fixpnt}(1), {\tt fixpnt}(2)]$, \dots, 
$[{\tt fixpnt}({\tt nfxpnt}), {\tt aright}]$.

If {\tt givmsh} is {\tt .true.}, the user must define the {\tt xx} array
on input as an initial mesh of strictly increasing values,
with ${\tt xx}(1) = {\tt aleft}$ and ${\tt xx}({\tt nmsh}) = {\tt aright}$.  
If ${\tt nfxpnt} &gt; 0$
and the user sets the initial mesh, the desired fixed points
{\em must} be included in the {\tt xx} array, since the code performs
no tests to check for their inclusion!
If {\tt givmsh} is {\tt .false.}, {\tt nmsh} is set to the maximum 
of {\tt nfxpnt}+2
and {\tt nminit} as defined in the labeled common area {\tt algprs}. 
 
On output, {\tt xx} contains the final array of {\tt nmsh} mesh points.

\item{\tt  nudim - }  Integer, input.  {\tt nudim} is the declared row 
dimension of the array {\tt u}.  {\tt nudim} must be greater than 
or equal to {\tt ncomp}.

\item{\tt  u - }  Floating-point array, of declared size 
{\tt (nudim, nucol)}, and
of conceptual size {\tt (ncomp, nmsh)}, where {\tt ncomp} is the number
of components and {\tt nmsh} is the number of mesh points.
The {\tt u} array is an optional input parameter, and an output parameter.
If {\tt giveu} (see above) is {\tt .false.}, the {\tt u}
array need not be set by the
user.  

If {\tt giveu} is {.true.}, the user must provide an initial
array ${\tt u}(i,j)$, $i=1,\dots {\tt ncomp}$ and
$j = 1, \dots {\tt nmsh}$, corresponding to the
desired initial trial solution, where {\tt nmsh} is the user-specified
number of initial mesh points.  In this case, ${\tt u}(*,1)$ corresponds 
to the estimated solution at {\tt aleft}, and ${\tt u}(*, {\tt nmsh})$ 
corresponds to the estimated solution at {\tt aright}.

\item{\tt  nmax - }  Integer, output.  {\tt nmax} is the maximum number of mesh 
points possible with the given values of {\tt ncomp}, {\tt ntol}, 
{\tt lwrkfl} (size of floating-point workspace),
{\tt lwrkin} (size of integer workspace) and {\tt nucol}.   
(See below for the descriptions 
of {\tt lwrkfl} and {\tt lwrkin}.)  {\tt nmax} can be no larger than 
the input parameter {\tt nucol}.
When the value of {\tt lwrkfl} is much greater than ${\tt ncomp*ncomp}$, 
the maximum 
number of mesh points allowed by the floating-point workspace limit
is 
$$
  {  {{\tt lwrkfl} - 5({\tt ncomp*ncomp}) -2{\tt ntol} - 9{\tt ncomp}} \over
   {4({\tt ncomp*ncomp}) + 12{\tt ncomp}  + 3}  }
$$
When {\tt lwrkin} is much greater than {\tt ncomp}, the maximum number of
mesh points allowed by the integer workspace limit is
$({\tt lwrkin} - {\tt ncomp})/({\tt ncomp} + 2)$.

The value of {\tt nmax} is calculated in {\tt twpbvp}, and is not allowed 
to exceed {\tt nucol}.  The user can determine the exact maximum 
number of mesh points corresponding to specified values of {\tt ncomp}, 
{\tt lwrkfl} and {\tt lwrkin} by calling {\tt twpbvp} with 
${\tt nucol} = 1$ and the 
parameter {\tt iprint} set to $0$ (its default value) or $1$; see below 
for a discussion of {\tt iprint}. 

\item{\tt lwrkfl - }  Integer, input.  The size of the user-provided 
floating-point workspace array {\tt wrk}.

\item{\tt  wrk - }  Floating-point array of size {\tt lwrkfl}, input.  
User-provided floating-point workspace, used to store intermediate 
values.

\item{\tt lwrkin - }  Integer, input.  The size of the user-provided integer
workspace array {\tt iwrk}.

\item{\tt iwrk - }  Integer array of size {\tt lwrkin}, input.  User-provided 
integer workspace, used to store intermediate values.

\item{\tt  fsub - }  Name of user-provided subroutine.  {\tt fsub}
{\em must} be declared
{\tt external} in the calling routine.  {\tt fsub} defines the function
$f$ in the formulation (\ref{eqn-prob1}) of the system of 
first-order differential  equations.
{\tt fsub} must have the following declaration:
\begin{description}
\item{$\null$} {\tt subroutine fsub(ncomp, x, u, f)}
\item{$\null$} {\tt ncomp} (input to {\tt fsub})
is the number of components of {\tt u};
\item{$\null$} {\tt x} (input to {\tt fsub}) is a floating-point scalar;
\item{$\null$} {\tt u} (input to {\tt fsub}) is a floating-point array 
of size {\tt ncomp};
\item{$\null$} {\tt f} (output from {\tt fsub}) is a floating-point array 
of dimension {\tt ncomp}. On exit from {\tt fsub}, the array 
{\tt f} must contain the {\tt ncomp}-dimensional vector {\tt f}
whose $i$th component
is the value of the $i$th component of the vector $f$ of (\ref{eqn-prob1})
evaluated at {\tt x} and {\tt u}.
\end{description}


\item{\tt  dfsub - }  Name of user-provided subroutine.  {\tt dfsub}
{\em must} be 
declared {\tt external} in the calling routine.  {\tt dfsub} calculates
the Jacobian matrix of {\tt f} as defined by the subroutine {\tt fsub}.  
{\tt dfsub} must have the following declaration:
\begin{description}
\item{$\null$} {\tt subroutine dfsub(ncomp, x, u, df)}
\item{$\null$} {\tt ncomp} (input to {\tt dfsub}) is the number of 
components of {\tt u}; 
\item{$\null$} {\tt x} (input to {\tt dfsub}) is a floating-point scalar;
\item{$\null$} {\tt u} (input to {\tt dfsub}) is a floating-point 
array of dimension {\tt ncomp};
\item{$\null$} {\tt df} (output from {\tt dfsub}) is an 
{\tt ncomp} by {\tt ncomp} floating-point
array whose declared row dimension in the routine {\tt dfsub}
must be {\tt ncomp}.  On exit from {\tt dfsub}, the 
array {\tt df} must contain the {\tt ncomp} by {\tt ncomp} Jacobian matrix
of $f$ from (\ref{eqn-prob1}) ({\tt f} of {\tt fsub})
with respect to $u$, namely ${\tt df}(i,j)$ is the partial derivative
of the $i$th function $f_i$ with respect to the $j$th component 
of $u$.
\end{description}

\item{\tt  gsub - }  Name of user-provided subroutine.  {\tt gsub}
{\em must} be declared {\tt external} in the calling routine.  
{\tt gsub} is called {\tt ncomp} times
each time the boundary conditions are calculated;
the $i$th call to {\tt gsub} defines the $i$th function
$g_i(\cdot)$ in the boundary conditions of (\ref{eqn-prob2}).
{\tt gsub} must have the following declaration:
\begin{description}
\item{$\null$} {\tt subroutine gsub(i, ncomp, u, g)}
\item{$\null$} {\tt i} (input to {\tt gsub}) is an integer 
ranging from 1 to {\tt ncomp};
\item{$\null$} {\tt ncomp} (input to {\tt gsub}) is the 
number of components of {\tt u};
\item{$\null$} {\tt u} (input to {\tt gsub}) is a floating-point 
array of dimension {\tt ncomp};
\item{$\null$} {\tt g} (output from {\tt gsub}) is a floating-point scalar.  
On exit from the $i$th call to
{\tt gsub}, the value of {\tt g} must
contain the value of the function $g_i$ from (\ref{eqn-prob2})
that defines the $i$th boundary condition evaluated at $u$.
\end{description}

\item{\tt dgsub - }  Name of user-provided subroutine.  {\tt dgsub} 
{\em must} be 
declared {\tt external} in the calling routine.  {\tt dgsub} calculates
the Jacobian matrices corresponding to the functions $g_i$ 
of (\ref{eqn-prob2}) and {\tt gsub} that define the boundary conditions. 
{\tt dgsub} must have the following declaration:
\begin{description}
\item{} {\tt subroutine dgsub(i, ncomp, u, dg)}
\item {\tt i} (input to {\tt dgsub}) is an integer ranging from 
1 to {\tt ncomp};
\item{$\null$} {\tt ncomp} (input to {\tt dgsub}) is the number 
of components of {\tt u};
\item{$\null$} {\tt u} (input to {\tt dgsub}) is a floating-point 
array of dimension {\tt ncomp};
\item{$\null$} {\tt dg} (output from {\tt dgsub}) is a floating-point 
array of dimension {\tt ncomp}.  
On exit from the $i$th call of {\tt dgsub}, the 
vector {\tt dg }must contain the {\tt ncomp} partial derivatives of the
$i$th function $g_i$ of (\ref{eqn-prob2}) with respect to $u$.
\end{description}

\item{\tt iflbvp - }  Integer, output.  {\tt iflbvp} indicates the result of
{\tt twpbvp}.  If the routine has terminated with apparent success,
${\tt iflbvp} = 0$.  If one of the input parameters is invalid,
${\tt iflbvp} = -1$.  If the number of mesh points needed for the
next iteration would exceed {\tt nmax}, then ${\tt iflbvp} = 1$.

\end{description}

\section{Related parameters}\label{sec-otherpars}

Some other parameters that may be of interest to the user, but do
not occur in the
formal declaration of {\tt twpbvp}, are stored in the labeled common
area {\tt algprs}, whose declaration is
\begin{verbatim}
        logical pdebug
        common/algprs/ nminit, pdebug, iprint, idum, uval0
\end{verbatim}

Any of the parameters in {\tt algprs} may be changed by the user
by including this labeled common in the calling routine
and setting the value as desired.

The value of the integer {\tt nminit} is discussed above under the
formal parameter {\tt nmsh}.  {\tt nminit}
specifies the default initial number of mesh points, and, if not
altered, is set to $7$.

{\tt pdebug} is a logical variable whose default value is {\tt .false.};
if {\tt pdebug} is set to {\tt .true.}, an extensive debug printout will
occur during execution of {\tt twpbvp}.

{\tt iprint} is an integer variable controlling the amount of
non-debug printout.  Its default value is zero, which means that
some intermediate printing occurs (each Newton iteration,
each change of mesh, and so on; see the example output in
Section~\ref{sec-sampout}).  If {\tt iprint} is set to $1$, 
more printing occurs.  If {\tt iprint} is set to $-1$, no printing at 
all occurs in {\tt twpbvp}.  Values of {\tt iprint}
other than $0$, $1$ and $-1$ are invalid.

{\tt idum} is a dummy integer value needed only for common alignment.
It serves no purpose in the algorithm.

{\tt uval0} is a floating-point value that defines the initial
value of the trial solution when {\tt giveu} is {\tt .false.}, or in some
instances when an intermediate Newton process fails.  Unless changed
by the user, the
default value of {\tt uval0} is zero.  The user may wish to change
{\tt uval0} if $u=0$ is inappropriate for some
reason---for example, the function $f$ in (\ref{eqn-prob1})
is undefined for $u=0$.

The user should also look at the routine d1mach (which is provided
automatically via netlib). The variables specified in this routine 
describe machine precision and may need to be changed depending on
which machine is being used.

\section{Suggestions for difficult problems}

For difficult, nonlinear singular perturbation problems, the code
{\TWPBVP} may experience severe difficulties in obtaining convergence for the
Newton iteration scheme.  In such circumstances two approaches
may be helpful.  The first is to experiment with
different initial meshes.  This can involve using a different
number of equally spaced points (by changing the default value of
the parameter {\tt nminit}; see Section \ref{sec-otherpars}),
or, alternatively, trying a graded mesh in which 
extra mesh points are placed in regions of known or suspected difficulty.

A second possibility is to use {\sl continuation}, which is  particularly
appropriate if a small parameter is associated with the given problem. 
For example, a large class of important singular perturbation
problems appear in the form
$$
\epsilon y'' = f(x,y,y'),
$$
where $\epsilon$ is a very small parameter.  
An often-successful strategy for solving such problems is 
to begin with a relatively large value of $\epsilon$,
and then to solve a sequence of problems with $\epsilon$ steadily decreasing
to the required value.  The power of this approach comes from the fact
that information regarding meshes and solution values can be passed
from one problem to the next.  Continuation is fully described in
\cite{AMR}.  J.\ R.\ Cash, one of the authors of {\TWPBVP}, and
R.\ W.\ Wright have developed an automatic continuation code that can be 
made available (with no guarantees) to interested users. For a copy of
this continuation code e-mail either j.cash or r.wright@ic.ac.uk.


\section{A sample problem}\label{sec-sample}


\subsection{Mathematical formulation}

Consider a second-order boundary value problem
parameterized in terms of $\epsilon$:
\begin{eqnarray}
  \epsilon u'' &amp;  = &amp; -e^u u' + \half \pi \sin(\half \pi x)
 e^{2u}, \quad 0 \le x \le 1,\label{eqn-samprob}\\
  u(0) &amp; = &amp; 0, \quad u(1) = 0.\nonumber
\end{eqnarray}
(As $\epsilon$ decreases toward zero, (\ref{eqn-samprob})
becomes more difficult to solve numerically.)
In order to apply {\TWPBVP} to (\ref{eqn-samprob}), it 
must be converted to a system of two first-order equations:
\begin{equation}
 u'  =  z,\qquad
 z'  =  {1\over\epsilon}\,
          {\Bigl( -e^u z + \half \pi\, \sin(\half \pi x) \,e^{2u}\Bigr)},
\label{eqn-newprob}
\end{equation}
with $u(0) = 0$ and $u(1) = 0$.

Suppose now that we wish to solve (\ref{eqn-newprob}) with 
$\epsilon = 10^{-2}$, and to achieve
an accuracy of approximately $10^{-6}$ in the value of $u$ only.
We shall provide neither an initial mesh nor an initial guess for
the solution.  The problem is to be solved on a machine with
IEEE arithmetic, and hence double precision is used (approximately
16 decimal digits).

\subsection{Sample Fortran routines}

The following Fortran 77 main program calls
{\tt twpbvp} with the user-specified parameters given above.
The subroutines {\tt fsub},
{\tt dfsub}, {\tt gsub} and {\tt dgsub} define the function $f$
and the boundary conditions $g_1$ and $g_2$ for problem (\ref{eqn-newprob}).

{\small

\begin{verbatim}


      implicit double precision (a-h,o-z)

      dimension  fixpnt(2), ltol(2), tol(2)
      dimension  u(2, 1000), xx(1000)
      dimension wrk(10000), iwrk(6000)
      character*30 pname

      logical linear, giveu, givmsh
      external fsub, dfsub, gsub, dgsub

*  The value of eps represents the parameter epsilon
*  appearing in the test problem.

      common/probs/eps, pi

      logical pdebug
      common/algprs/ nminit, pdebug, iprint, idum, uval0

*  blas: dload
*  double precision datan

      parameter (one = 1.0d+0, zero = 0.0d+0, ten = 10.0d+0)

      pi = 4.0d+0*datan(one)

      pname = `Test problem with eps = .01'

      eps = 1.0d-2

      nudim = 2
      lwrkfl = 10000
      lwrkin = 6000
      nucol = 1000

      ncomp = 2
      nlbc = 1
      ntol = 1
      ltol(1) = 1
      tol(1) = 1.0d-2
      nfxpnt = 0

      aleft = zero
      aright = one

      linear = .false.
      giveu = .false.
      givmsh = .false.


      call twpbvp(ncomp, nlbc, nucol, aleft, aright,
     *       nfxpnt, fixpnt, ntol, ltol, tol,
     *       linear, givmsh, giveu, nmsh,
     *       xx, nudim, u, nmax,
     *       lwrkfl, wrk, lwrkin, iwrk, 
     *       fsub, dfsub, gsub, dgsub, iflbvp)


      write(6,771) pname
      write(6,772) ncomp, nlbc, ntol
      write(6,773) aleft, aright
      if (linear) write (6,774)
      if (.not. linear) write(6,775)
      if (nfxpnt .gt. 0) write(6,776) nfxpnt
      write(6,777) (ltol(i), i=1,ntol)
      write(6,778) (tol(i), i=1,ntol)
      if (iflbvp .eq. 0) then
         write(6,901) nmsh
         if (nmsh .lt. 25) then
            iminc = 1
         elseif (nmsh .lt. 75) then
            iminc = 5
         elseif (nmsh .lt. 200) then
            iminc = 10
         elseif (nmsh .lt. 1000) then
            iminc = 20
         else
            iminc = 50
         endif
         do 100 i = 1, nmsh-1, iminc
            write(6,900) i, xx(i), (u(j,i), j=1,ncomp)
  100    continue
         write(6,900) nmsh, xx(nmsh), (u(j,nmsh), j=1,ncomp)
      endif

      stop
  771 format(1h ,a30)
  772 format(1h ,'ncomp=',i5,5x,'nlbc=',i5,5x,'ntol=',i5)
  773 format(1h ,'aleft =',1pe11.3,5x,'aright=',1pe11.3)
  774 format(1h ,'solved as a linear problem')
  775 format(1h ,'solved as a nonlinear problem')
  776 format(1h ,'nfxpnt =',i5)
  777 format(1h ,'ltol',5x,5i5)
  778 format(1h ,'tolerances',5(1pe11.3))
  900 format(1h ,i4,5(1pe14.6))
  901 format(1h ,'final solution, nmsh =',i8)
      end

      subroutine fsub(ncomp, x, u, f)
      implicit double precision (a-h,o-z)
      dimension u(*), f(*)
      common/probs/eps, pi

      parameter (half = 0.5d+0, two = 2.0d+0)

*  double precision dexp, dsin

*  nonlinear problem 1

      f(1) = u(2)
      f(2) = (-dexp(u(1))*u(2) + 
     *           half*pi*dsin(half*pi*x)*dexp(two*u(1)))/eps
 
      return
      end

      subroutine dfsub(ncomp, x, u, df)
      implicit double precision (a-h,o-z)
      dimension u(*), df(ncomp,*)
      common/probs/eps, pi
      parameter (half = 0.5d+0, one = 1.0d+0, zero = 0.0d+0)
      parameter (two = 2.0d+0)

*  double precision dexp, dsin

      df(1,1) = zero
      df(1,2) = one
      df(2,1) = -(dexp(u(1))*u(2) 
     *            - pi*dsin(pi*half*x)*dexp(two*u(1)))/eps
      df(2,2) = -dexp(u(1))/eps
      return
      end

      subroutine gsub(i, ncomp, u, g)
      implicit double precision (a-h,o-z)
      dimension u(*)
      g = u(1)
      return
      end


      subroutine dgsub(i, ncomp, u, dg)
      implicit double precision (a-h,o-z)
      dimension u(*), dg(*)
      parameter (one = 1.0d+0, zero = 0.0d+0)

      dg(1) = one
      dg(2) = zero
      return
      end

\end{verbatim}
}

\subsection{Sample output}\label{sec-sampout}

The complete output produced by running this example on an
SGI 4D/240S with 25 MHz MIPS R3000 cpu chip, under
IRIX System V release 3.3.1, with binary IEEE arithmetic, is
given next.

{\small
\begin{verbatim}

 Test problem with eps = .01   
 ncomp =    2     nlbc =    1     ntol =    1
 aleft =  0.000E+00     aright =  1.000E+00
 solved as a nonlinear problem
 ltol         1
 tolerances  1.000E-02
 nmax from workspace =     231
 unimsh.  nmsh =    7
 initu, uval0    0.00000D+00
  iter       alfa      merit       rnsq
     0  2.315E-01  1.043E+03  2.122E+03
     1  1.558E-02  1.722E+04  2.060E+03
     2  4.499E-03  1.044E+05  2.043E+03
  dblmsh.  the doubled mesh has       13 points.
  iter       alfa      merit       rnsq
     0  2.404E-01  5.077E+01  4.582E+03
     1  4.809E-01  1.320E+02  1.488E+03
     2  9.617E-01  7.193E+02  1.743E+02
     3  1.000E-02  8.279E+04  1.722E+02
     4  3.271E-03  4.623E+05  1.719E+02
 smpmsh, new nmsh =     40
  iter       alfa      merit       rnsq
     0  1.000E+00  1.506E+02  6.596E+01
     1  1.000E+00  5.433E+00  4.495E+00
     2  1.000E+00  2.004E-03  8.292E-04
     3  1.000E+00  1.627E-10  6.306E-11
 Convergence, iter =    4    rnsq =   6.306E-11
 fixed Jacobian convergence    1  1.187E-10

 final solution, nmsh =      40

    i  mesh point        u(1)          u(2)
    1  0.000000E+00  0.000000E+00 -4.904288E+01
    6  4.166667E-02 -6.130136E-01 -3.138167E+00
   11  8.333333E-02 -6.638966E-01 -2.759269E-01
   16  1.250000E-01 -6.655629E-01  9.117284E-02
   21  1.666667E-01 -6.596432E-01  1.807697E-01
   26  2.083333E-01 -6.508858E-01  2.382272E-01
   31  2.500000E-01 -6.398296E-01  2.923964E-01
   36  6.666667E-01 -3.986648E-01  8.886575E-01
   40  1.000000E+00 -2.509872E-30  1.546497E+00

\end{verbatim}

}

\begin{thebibliography}{9999999999}

\bibitem[AMR88]{AMR}
 U.\ Ascher, R.\ M.\ M.\ Mattheij, and R.\ D.\ Russell,
{\it Numerical Solution of Boundary Value Problems for Ordinary
Differential Equations},
Prentice-Hall, Englewood Cliffs, NJ, 1988.

\bibitem[Cash86]{Cash86}
J.\ R.\ Cash (1986), On the numerical integration of nonlinear 
two-point boundary value problems using iterated deferred 
corrections, Part 1: A survey and comparison of some 
one-step formulae, {\sl Comput.\ Math.\ Appl.}\ 12a, 1029--1048.

\bibitem[Cash88]{Cash88}
J.\ R.\ Cash (1988),
On the numerical integration of nonlinear two-point boundary
value problems using iterated deferred corrections, Part 2:
The development and analysis of highly stable deferred correction
formulae, {\sl SIAM J.\ Numer.\ Anal.}\ 25, 862--882.

\bibitem[CW90]{CW1}
J.\ R.\ Cash and M.\ H.\ Wright (1990),
Implementation issues in solving nonlinear equations for
two-point boundary value problems,
{\sl Computing} 45, 17--37.

\bibitem[CW91]{CW2}
J.\ R.\ Cash and M.\ H.\ Wright (1991),
A deferred correction method for nonlinear two-point boundary
value problems: Implementation and numerical evaluation,
{\sl SIAM J.\ Sci.\ Stat.\ Comput.}\ 12, 971--989.

\end{thebibliography}

\end{document}

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