D108: Trapezoidal Rule Integration with an Estimated Error

Author(s): K.S. Kölbig	Library: MATHLIB
Submitter:	Submitted: 01.03.1968
Language: Fortran	Revised:

Let a function f(x) be given by its values at certain discrete points . Let the function values be accompanied by an estimated standard deviation (square root of the variance). Subroutine subprogram TRAPER then approximates the integral

by a linear combination of the using the trapezoidal rule. It calculates the standard deviation of I by

The function values f(A) and f(B) are calculated by linear interpolation.

Structure:

SUBROUTINE subprogram
User Entry Names: TRAPER

Usage:

    CALL TRAPER(X,Y,E,N,A,B,RE,SD)

X,Y,E

(REAL) Arrays of length containing , respectively.

N

(INTEGER) Number of function values

A,B

(REAL) Limits of integration.

RE

(REAL) On exit, RE contains an approximate value of the integral I.

SD

(REAL) On exit, SD contains an approximate value of the standard deviation .

If no are given, the array E should be filled with zeros.

Restrictions:

Although there are no restrictions on A and B (B may be less than A), care must be taken if one or both of them is either smaller than X(1) or bigger than X(N). In these cases or are extrapolated linearly from Y(1) and Y(2) or Y(N-1) and Y(N) respectively, which may lead to unreasonable results. If or , RE and SD will be set to zero. It is assumed that all the are distinct. No test is made for this.

Notes:

This program should only be used for the problem described above. For general-purpose numerical integration to a preassigned accuracy use GAUSS (D103).