D105: Integration over a Triangle

Author(s): K.S. Kölbig Library: MATHLIB
Submitter: Submitted: 02.05.1966
Language: Fortran Revised: 01.12.1994

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

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evaluated over the interior of an arbitrary triangle tex2html_wrap_inline101 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 tex2html_wrap_inline105 RTRINT
Files Referenced: Unit 6
External References: MTLMTR, ABEND

Usage:

For tex2html_wrap_inline107 (type REAL), tex2html_wrap_inline109 (type DOUBLE PRECISION), 

    tTRINT(F,NSD,NPT,EPS,X1,Y1,X2,Y2,X3,Y3)
has, in any arithmetic expression, the approximate value of the integral I.
F
(type according to t) Name of a user-supplied FUNCTION subprogram, declared EXTERNAL in the calling program. This subprogram must set tex2html_wrap_inline113 .
NSD
(INTEGER)
tex2html_wrap_inline115 No subdivision of the given triangle.
tex2html_wrap_inline117 Subdivision of the given triangle (see Method).
NPT
(INTEGER)
tex2html_wrap_inline119 A 7-point integration formula is used.
tex2html_wrap_inline121 A 25-point integration formula is used.
tex2html_wrap_inline123 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 tex2html_wrap_inline125 .
X2,Y2
X3,Y3

Method:

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An approximation tex2html_wrap_inline129 to I is found by computing the NPT-point formula for the triangle tex2html_wrap_inline133 . The value of EPS has no influence on the result.
tex2html_wrap_inline135
After computing tex2html_wrap_inline137 , the triangle tex2html_wrap_inline139 is subdivided into two subtriangles tex2html_wrap_inline141 and tex2html_wrap_inline143 , the corresponding approximations I' and I'' are computed, and a test is made to see whether

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If this test is satisfied, the routine terminates by setting the function value to tex2html_wrap_inline151 . 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 tex2html_wrap_inline155 ,the argument EPS may, if tex2html_wrap_inline157 , 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 tex2html_wrap_inline165 . Any other singularity lying inside tex2html_wrap_inline167 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: tex2html_wrap_inline169 .
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).
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Michel Goossens Tue Jun 4 23:36:20 METDST 1996