D105 Integration over a Triangle

Function subprograms RTRINT and DTRINT compute an approximate value of the integral

$I\; =\; \int \int f(x,y)\hspace{0.17em}dx\; dy,$

evaluated over the interior of an arbitrary triangle $\Delta$ in the xy-plane. An attempted accuracy may, optionally, be specified.

On computers other than CDC or Cray, only the double-precision version DTRINT is available. On CDC and Cray computers, only the single-precision version RTRINT is available.

Structure:

FUNCTION subprograms
User Entry Names: RTRINT, DTRINT
Obsolete User Entry Names: TRIINT $\equiv$ RTRINT
Files Referenced: Unit 6
External References: MTLMTR (N002), ABEND (Z035)

Usage:

For $t=R$ (type REAL), $t=D$ (type DOUBLE PRECISION),

    tTRINT(F,NSD,NPT,EPS,X1,Y1,X2,Y2,X3,Y3)

F

(type according to t) Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program. This subprogram must set $F(X,Y)=\; f(X,Y)$ .

NSD

( INTEGER)
$=\; 0:$ No subdivision of the given triangle.
$=\; 1:$ Subdivision of the given triangle (see Method).

NPT

( INTEGER)
$=\; 7:$ A 7-point integration formula is used.
$=\; 25:$ A 25-point integration formula is used.
$=\; 64:$ A 64-point integration formula is used.

EPS

(type according to t) Accuracy parameter (see Accuracy).

X1,Y1

(type according to t) The coordinates of the vertices of $\Delta$ .

X2,Y2

X3,Y3

Method:

$NSD\; =\; 0:$
An approximation $I$₀ to I is found by computing the NPT-point formula for the triangle $\Delta$ . The value of EPS has no influence on the result.
$NSD\; =\; 1:$
After computing $I$₀ , the triangle $\Delta$ is subdivided into two subtriangles $\Delta \text{'}$ and $\Delta \text{'}\text{'}$ , the corresponding approximations $I\text{'}$ and $I\text{'}\text{'}$ are computed, and a test is made to see whether

$\{|I$₀-(I'+I'')|1+|I'+I''|}< EPS

If this test is satisfied, the routine terminates by setting the function value to $I$₀ . If it fails, the process of subdivision and testing continues according to a tree structure. The routine terminates either because the test is passed successfully by all the subtriangles at some level, or because a maximum number of subdivisions is reached (see Error Handling).

Accuracy:

Unless there is severe cancellation of positive and negative values of $f(x,y)$ over $\Delta$ ,the argument EPS may, if $NCD\; =\; 1$ , be considered as specifying a bound on the relative error of I in the case |I|>1, and a bound on the absolute error in the case |I|<1.

Restrictions:

"Mild" singularities are permitted if they coincide with the vertices of $\Delta$ . Any other singularity lying inside $\Delta$ or on its boundaries will most likely lead to too many subdivisions (see Error Handling), or cause a wrong result.

Error handling:

Error D105.1: $NPT\; \ne 7,\hspace{0.17em}25,\hspace{0.17em}64$ .
Error D105.2: The number of subdivisions has reached 35 without success.
In both cases, the function value is set equal to zero, and a message is written on Unit 6, unless subroutine MTLSET (N002) has been called.

References:

1. K.S. Kölbig, A Fortran program and some numerical test results for the integration over a triangle, CERN 64--32 (1964).

D106