Simplex quadrature, Stroud & Secrest, error?

From: Chengiz Khan (chengiz_at_my-deja.com)
Date: 10/25/04 

Date: 25 Oct 2004 10:52:24 -0700

Hi,

In Chapter 3 of Stroud and Secrest's "Gaussian Quadrature Formulas",
it is mentioned (page that has eqn 3.6 and ff - sorry - i have copies
w/o page nos) that the 2-pt formula derived from cubes and applied to
a tetrahedron is exact for a polynomial of degree <= 3, defined as a
polynomial f(x,y,z) with x^a y^b z^c terms such that a+b+c <= 3.

However the int_-1^1 (1+x^2) f(x) dx ~ A1 f(x1) + A2 f(x2) equation
just before it *cannot be exact* if f(x) is of degree >1.

So it seems to me that the 2 point formula should be exact only if
a,b,c <= 1 and not just a+b+c <= 3. Eg. it won't do f(x,y,z) = x^2
correctly. (*)

My implementation also confirms this. Any thoughts?

(*) This seems to me to be mathematically valid as well, coz the
*real* integral for a tet is "int_0^1 int_0^1-x int_0^1-x-y
f(x,y,z) dz dy dx" which will introduce higher order terms by the time
one gets to the integral over x. This is unlike a cube where all the
integrals go 0..1 so no higher order terms are introduced.

Khan

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