NAG Library Routine Document
D02GAF
1 Purpose
D02GAF solves a twopoint boundary value problem with assigned boundary values for a system of ordinary differential equations, using a deferred correction technique and a Newton iteration.
2 Specification
SUBROUTINE D02GAF ( 
U, V, N, A, B, TOL, FCN, MNP, X, Y, NP, W, LW, IW, LIW, IFAIL) 
INTEGER 
N, MNP, NP, LW, IW(LIW), LIW, IFAIL 
REAL (KIND=nag_wp) 
U(N,2), V(N,2), A, B, TOL, X(MNP), Y(N,MNP), W(LW) 
EXTERNAL 
FCN 

3 Description
D02GAF solves a twopoint 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.
4 References
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 finitedifference 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
5 Parameters
 1: U(N,$2$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{U}}\left(\mathit{i},1\right)$ must be set to the known or estimated value of ${y}_{\mathit{i}}$ at $a$ and ${\mathbf{U}}\left(\mathit{i},2\right)$ must be set to the known or estimated value of ${y}_{\mathit{i}}$ at $b$, for $\mathit{i}=1,2,\dots ,\mathit{n}$.
 2: V(N,$2$) – REAL (KIND=nag_wp) arrayInput
On entry: ${\mathbf{V}}\left(\mathit{i},\mathit{j}\right)$ must be set to $0.0$ if ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ is a known value and to $1.0$ if ${\mathbf{U}}\left(\mathit{i},\mathit{j}\right)$ is an estimated value, for $\mathit{i}=1,2,\dots ,\mathit{n}$ and $\mathit{j}=1,2$.
Constraint:
precisely $\mathit{n}$ of the ${\mathbf{V}}\left(i,j\right)$ must be set to $0.0$, i.e., precisely $\mathit{n}$ of the ${\mathbf{U}}\left(i,j\right)$ must be known values, and these must not be all at $a$ or all at $b$.
 3: N – INTEGERInput
On entry: $\mathit{n}$, the number of equations.
Constraint:
${\mathbf{N}}\ge 2$.
 4: A – REAL (KIND=nag_wp)Input
On entry: $a$, the lefthand boundary point.
 5: B – REAL (KIND=nag_wp)Input
On entry: $b$, the righthand boundary point.
Constraint:
${\mathbf{B}}>{\mathbf{A}}$.
 6: TOL – REAL (KIND=nag_wp)Input
On entry: a positive absolute error tolerance. If
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
Constraint:
${\mathbf{TOL}}>0.0$.
 7: 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
$x$.
The specification of
FCN is:
SUBROUTINE FCN ( 
X, Y, F) 
REAL (KIND=nag_wp) 
X, Y(*), F(*) 

In the description of the parameters of D02GAF below,
$\mathit{n}$ denotes the actual value of
N in the call of D02GAF.
 1: X – REAL (KIND=nag_wp)Input
On entry: $x$, the value of the argument.
 2: Y($*$) – REAL (KIND=nag_wp) arrayInput
On entry: ${y}_{\mathit{i}}$, for $\mathit{i}=1,2,\dots ,\mathit{n}$, the value of the argument.
 3: F($*$) – 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. Parameters denoted as
Input must
not be changed by this procedure.
 8: MNP – INTEGERInput
On entry: the maximum permitted number of mesh points.
Constraint:
${\mathbf{MNP}}\ge 32$.
 9: X(MNP) – 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: Y(N,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
where
NP is the number of points in the final mesh.
The remaining columns of
Y are not used.
 11: NP – INTEGERInput/Output
On entry: determines whether a default or usersupplied 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: W(LW) – REAL (KIND=nag_wp) arrayWorkspace
 13: LW – INTEGERInput
On entry: the dimension of the array
W as declared in the (sub)program from which D02GAF is called.
Constraint:
${\mathbf{LW}}\ge {\mathbf{MNP}}\times \left(3{{\mathbf{N}}}^{2}+6{\mathbf{N}}+2\right)+4{{\mathbf{N}}}^{2}+4{\mathbf{N}}$.
 14: IW(LIW) – INTEGER arrayWorkspace
 15: LIW – INTEGERInput
On entry: the dimension of the array
IW as declared in the (sub)program from which D02GAF is called.
Constraint:
${\mathbf{LIW}}\ge {\mathbf{MNP}}\times \left(2{\mathbf{N}}+1\right)+{{\mathbf{N}}}^{2}+4{\mathbf{N}}+2$.
 16: IFAIL – INTEGERInput/Output

For this routine, the normal use of
IFAIL is extended to control the printing of error and warning messages as well as specifying hard or soft failure (see
Section 3.3 in the Essential Introduction).
On entry:
IFAIL must be set to a value with the decimal expansion
$\mathit{cba}$, where each of the decimal digits
$c$,
$b$ and
$a$ must have a value of
$0$ or
$1$.
$a=0$ 
specifies hard failure, otherwise soft failure; 
$b=0$ 
suppresses error messages, otherwise error messages will be printed (see Section 6); 
$c=0$ 
suppresses warning messages, otherwise warning messages will be printed (see Section 6). 
The recommended value for inexperienced users is $110$ (i.e., hard failure with all messages printed).
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
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
One or more of the parameters
N,
TOL,
NP,
MNP,
LW or
LIW has been incorrectly set, or
${\mathbf{B}}\le {\mathbf{A}}$, or the condition
(3) on
X is not satisfied, or the number of known boundary values (specified by
V) is not
N.
 ${\mathbf{IFAIL}}=2$
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 you should use
D02RAF instead. If the warning ‘Jacobian matrix is singular’ is printed this could be due to specifying zero estimated boundary values and these should be varied. This warning could also be printed in the unlikely event of the Jacobian matrix being calculated inaccurately. If you cannot make changes to prevent the warning then
D02RAF should be used.
 ${\mathbf{IFAIL}}=3$
The Newton iteration has reached roundoff level. It could be, however, that the answer returned is satisfactory. This error might occur if too much accuracy is requested.
 ${\mathbf{IFAIL}}=4$
A finer mesh is required for the accuracy requested; that is
MNP is not large enough.
 ${\mathbf{IFAIL}}=5$
A serious error has occurred in a call to D02GAF. Check all array subscripts and subroutine parameter lists in calls to D02GAF. Seek expert help.
7 Accuracy
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.
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 selfexplanatory error messages, and also monitoring information about the course of the computation. You may select the channel numbers on which this output is to appear by calls of
X04AAF (for error messages) or
X04ABF (for monitoring information) – see
Section 9 for an example. Otherwise the default channel 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.
9 Example
This example solves the differential equation
with boundary conditions
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.
9.1 Program Text
Program Text (d02gafe.f90)
9.2 Program Data
Program Data (d02gafe.d)
9.3 Program Results
Program Results (d02gafe.r)