D03MAF places a triangular mesh over a given twodimensional region. The region may have any shape, including one with holes.
D03MAF begins with a uniform triangular grid as shown in
Figure 1 and assumes that the region to be triangulated lies within the rectangle given by the inequalities
This rectangle is drawn in bold in
Figure 1. The region is specified by the
ISIN which must determine whether any given point
$\left(x,y\right)$ lies in the region. The uniform grid is processed columnwise, with
$\left({x}_{1},{y}_{1}\right)$ preceding
$\left({x}_{2},{y}_{2}\right)$ if
${x}_{1}<{x}_{2}$ or
${x}_{1}={x}_{2}$,
${y}_{1}<{y}_{2}$. Points near the boundary are moved onto it and points well outside the boundary are omitted. The direction of movement is chosen to avoid pathologically thin triangles. The points accepted are numbered in exactly the same order as the corresponding points of the uniform grid were scanned. The output consists of the
$x,y$ coordinates of all grid points and integers indicating whether they are internal and to which other points they are joined by triangle sides.
Further details of the algorithm are given in the references.
Reid J K (1970) Fortran subroutines for the solutions of Laplace's equation over a general routine in two dimensions Harwell Report TP422
Reid J K (1972) On the construction and convergence of a finiteelement solution of Laplace's equation J. Instr. Math. Appl. 9 1–13
 1: H – REAL (KIND=nag_wp)Input
On entry: $h$, the required length for the sides of the triangles of the uniform mesh.
 2: M – INTEGERInput
 3: N – INTEGERInput
On entry: values
$m$ and
$n$ such that all points
$\left(x,y\right)$ inside the region satisfy the inequalities
Constraint:
${\mathbf{M}}={\mathbf{N}}>2$.
 4: NB – INTEGERInput
On entry: the number of times a triangle side is bisected to find a point on the boundary. A value of
$10$ is adequate for most purposes (see
Section 7).
Constraint:
${\mathbf{NB}}\ge 1$.
 5: NPTS – INTEGEROutput
On exit: the number of points in the triangulation.
 6: PLACES($2$,SDINDX) – REAL (KIND=nag_wp) arrayOutput
On exit: the $x$ and $y$ coordinates respectively of the $i$th point of the triangulation.
 7: INDX($4$,SDINDX) – INTEGER arrayOutput
On exit: ${\mathbf{INDX}}\left(1,i\right)$ contains $i$ if point $i$ is inside the region and $i$ if it is on the boundary. For each triangle side between points $i$ and $j$ with $j>i$, ${\mathbf{INDX}}\left(k,i\right)$, $k>1$, contains $j$ or $j$ according to whether point $j$ is internal or on the boundary. There can never be more than three such points. If there are less, then some values ${\mathbf{INDX}}\left(k,i\right)$, $k>1$, are zero.
 8: SDINDX – INTEGERInput
On entry: the second dimension of the arrays
PLACES and
INDX as declared in the (sub)program from which D03MAF is called.
Constraint:
${\mathbf{SDINDX}}\ge {\mathbf{NPTS}}$.
 9: ISIN – INTEGER FUNCTION, supplied by the user.External Procedure
ISIN must return the value
$1$ if the given point (
X,
Y) lies inside the region, and
$0$ if it lies outside.
The specification of
ISIN is:
 1: X – REAL (KIND=nag_wp)Input
 2: Y – REAL (KIND=nag_wp)Input
On entry: the coordinates of the given point.
ISIN must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which D03MAF is called. Parameters denoted as
Input must
not be changed by this procedure.
 10: DIST($4$,SDDIST) – REAL (KIND=nag_wp) arrayWorkspace
 11: SDDIST – INTEGERInput
On entry: the second dimension of the array
DIST as declared in the (sub)program from which D03MAF is called.
Constraint:
${\mathbf{SDDIST}}\ge 4{\mathbf{N}}$.
 12: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ 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).
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).
Points are moved onto the boundary by bisecting a triangle side
NB times. The accuracy is therefore
$h\times {2}^{{\mathbf{NB}}}$.