Infinite Integrals by Gaussian Quadrature

[Albert Chan]

(11-23-2022 01:55 PM)Eddie W. Shore Wrote:  The program INFGAUS calculates the integral:

∫ e^-x * f(x) dx from x = a to x = ∞

...

e^-a * Σ(( w_i * f(z_i + a)) for i=1 to 3)

where

w_1 = 0.71109390099
z_1 = 0.4157745568

w_2 = 0.2785177336
z_2 = 2.29428036

w_3 = 0.0103892565
z_3 = 6.289945083

Source

HP-45 Applications Handbook Hewlett Packard Company. 1974.

The handbook only listed the shifts and weights ... How does the numbers derived ?

I was expecting Gaussian Quadrature of z integral. But with 3 points, results are terrible.
Note: in XCas, Int(...) == quote(int(...))

XCas> Int(e^-x * f(x), x, a, inf) (x=-ln(y))

∫(f(-ln(y)), y, 0, exp(-a))

XCas> Int(f(-ln(y)), y, 0, exp(-a)) (y = (z+1)*exp(-a)/2)

exp(-a) * ∫(f(a - ln((1+z)/2))/2, z, -1, 1)