BLI Integration.

Approximate the fifth derivative by

And assume that the coefficient of 1/720 remains the same.

Then the integral to x_j can be written as

This leads to

Figure 1 Intermediate values found by BLI

Derivatives at J-1 aredetermined by the curve through J-3to J+1. Derivatives at J are determined by the curve through J-2 to J+2.

The four values of f(x_j) most relevant for the integral from x_j-1 to x_j are shown in figure 1. A Lagrange polynomial through the appropriate five points gives third derivatives at x_j-1 and x_j. ..\Derivatives\Lagrange.docx

The goal is to spread N points from x₀ to x_N such that the error term in the integral is minimal.

This is accomplished by finding values of for each interval,then adding new points in the middle of the intervals with the largest values of

..\interpolation\Bli.doc àIbli.for

Richardson’s Extrapolation

Form three integral estimates using every point, every second point, and every third point out of the xi and fi vectors returned by BLI. Ignore the possibility of using the 1/N⁴

Eqn 18

Multiply the top of equation 18 by 4 and subtract the second

Eqn 19

Multiply the top by 9 and subtract the third

Eqn 20

Multiply the middle one by 9 and the bottom by 4 and subtract

Eqn 21

DO I=1,NP

G(I)=(4*G1(I)-G2(I))/3

GC=(9*G1(I)-G3(I))/8

GCC=(9*G1(I)-4*G3(I))/5

ERRG(I)=SQRT((G(I)-GC))**2+(G(I)-GCC)**2+(GCC-GC)**2)/3)

ENDDO

Assignment

Use the BLI in to evaluate a few test integrals between 0 and 1. Compare with analytic evaluations when possible. ..\References.htm. Plot G(x),ERRG(x), G(x)+ERRG(x), and G(x)-ERRG(x)

ftest = x⁴
Integral should be 0.886226925455
Integral 0 to 1 should be 1.64493406685
error integral.
Fresnel integral. Let fun=sin((p/2)x*x) . Integrate from 0 to [Abramowitz and Stegun give 0.567822] This is a maximum of the integral. This is a Fresnel integral which arises in diffraction problems. It is also a chirp which arises when two stars collide.

Integral to 1 should be 1.0471975512