◆ Avint()

Sub Avint	(	N As	Long,
		X() As	Double,
		Y() As	Double,
		A As	Double,
		B As	Double,
		Result As	Double,
		Info As	Long
	)

Finite interval quadrature for a function with tabulated data (approximation with overlapping parabolas)

Purpose

This routine integrates a function tabulated at arbitrarily spaced abscissas. The limits of integration need not coincide with the tabulated abscissas.
A method of overlapping parabolas fitted to the data is used provided that there are at least 3 abscissas between the limits of integration.
Avint also handles two special cases.

If the limits of integration are equal, Avint returns a result of zero regardless of the number of tabulated values.
If there are only two function values, Avint uses the trapezoid rule.

Parameters

[in]	N	Number of data. (N >= 2)
[in]	X()	Array X(LX - 1) (LX >= N) Abscissas (strictly in increasing order).
[in]	Y()	Array Y(LY - 1) (LY >= N) Ordinates (function values).
[in]	A	Lower limit of integration. (A <= B)
[in]	B	Upper limit of integration. (A <= B)
[out]	Result	Computed approximate value of integral.
[out]	Info	= 0: Successful exit. = -1: The argument N had an illegal value. (N < 2) = -2: The argument X() is invalid. = -3: The argument Y() is invalid. = -4: The argument A (or B) had an illegal value. (A > B) = 1: Less than 3 function values between the limits of integration. = 2: Abscissas not strictly in increasing order.

Reference: SLATEC

Example Program: Using the following table, compute S = integral of 1/(1 + x^2) dx [0, 4] (= atan(4)).
x 1/(1 + x^2)

----- -------------

-1 0.5

0 1

1 0.5

2 0.2

3 0.1

4 0.05882

5 0.03846

----- -------------

Sub Ex_Avint()

Const N = 7, A = 0, B = 4

Dim X(N - 1) As Double, Y(N - 1) As Double, S As Double

Dim Info As Long, I As Long

'-- Data

X(0) = -1: Y(0) = 0.5

X(1) = 0: Y(1) = 1

X(2) = 1: Y(2) = 0.5

X(3) = 2: Y(3) = 0.2

X(4) = 3: Y(4) = 0.1

X(5) = 4: Y(5) = 0.05882

X(6) = 5: Y(6) = 0.03846

'-- Compute integral 1/(1 + x^2) dx [0, 4] (= atan(4))

Call Avint(N, X(), Y(), A, B, S, Info)

Debug.Print "S =", S, "S(true) =", Atn(4)

Debug.Print "Info =", Info

End Sub

Avint
Sub Avint(N As Long, X() As Double, Y() As Double, A As Double, B As Double, Result As Double, Info As Long)
Finite interval quadrature for a function with tabulated data (approximation with overlapping parabol...

Example Results: S = 1.33197416666667 S(true) = 1.32581766366803

Info = 0