From: hanson@math.uic.edu
Date: 20 Nov 2002 20:24:26 -0600
To: nthakr1@uic.edu
Subject: Re: computer problem 4

Hello Nishant,

You asked:
>I need some help with computer problem 4. I did Part A, B and C for 3 given 
>functions. I don't know how part D works. I tried to look in the book but I 
>still do not understand it. Can i come see you tommorow for this assignment if 
>you time. Please email me the time. I free anytime before 12:30 or after 2 pm. 
>Please let me know. I need some help.

Sorry, I only come in MWF and I have office hours right after class at
2pm, but I usually stay any answer questions before I go back to my
office.
>
>About GR3. I quite don't see how is it going to work with the given functions. 
>Does it have to be between -1 and 1? And if yes, which formula do we use?

The problem is that I have not finished talking about Gaussian
Quadrature with composite rules in class.   The 2XGr_3 means a double 
application of GR_3 to the interval [a,b], so you have to split that
up into two parts, [a,(a+b)/2] and [(a+b)/2,b], split at the midpoint,
applying one GR_3 to each part.  If we call these part intervals,
[a(1),b(1)] and [a(2),b(2)], respectively, that transform x to the
universal variable t by the linear transformation given in class,

x(t,j) = 0.5*(b(j)-a(j))*t + 0.5*(b(j)+a(j)) for j=1:2 and t in [-1,+1],

Then the universal function becomes,

F(t,j) = 0.5*(b(j)-a(j))*f(x(t,j)) since dx(t,j)=0.5*(b(j)-a(j)),

while the integral of f(x) on [a,b] (there are 3 such functions)
becomes,

w1*(F(x(t1,1))+F(x(t1,2)))+w2*(F(x(t2,1))+F(x(t2,2)))+w3*(F(x(t3,1))+F(x(t3,2)))

where the w's are the Gaussian weights and tj's are the Gaussian nodes 
given in part D, for j=1:2, including the double application of GR_3.

GoodLuck,
FBH

BCC: Class

============================
Another Version of the Explanation:
============================

From: hanson@math.uic.edu
Date: 26 Nov 2002 01:00:37 -0600
To: ssamra1@uic.edu
Subject: Re: Problem 4  471
Cc: hanson@math.uic.edu

Hello Samir,

you said:
and it is very confusing?  I do not know where you got this from?
>       In prevouis e-mails, you stated stated the big F is 
>F(t,j)+0.5*(b(j)-a(j))*f(x(t,j)), this tells me that there are to be
two 

OK, now I see that "+" should be an equals:

[a(1),b(1)]=[a,0.5*(a+b)] and [a(2),b(2)]=[0.5*(a+b),b]

x(t,j) = 0.5*(b(j)-a(j))*t + 0.5*(b(j)+a(j)) for j=1:2 subintevals.

F(t,j) = 0.5*(b(j)-a(j))*f(x(t,j)) for j=1:2 subintevals.

>parameters. But later on, when you define the integral for f(x) on
[a,b] which 
>is basically three functions, example w1*(F(x(t1,1)) + F(x(t1,2))),
F(x) takes 

Integral(f,[a,b]) = sum(j=1:2)[Integral(f,[a(j),b(j)])

with approximation:

2XGR3 = sum(j=1:2)[sum(i=1:3)[w_i*F(t_i,j)]],

where [w_i]_{3X1} = [w1;w2;w3] and [x_i]_{3X1} = [x1;x2;x3] are
the GR3 weights and nodes.  Here "i" counts GR3 terms and "j" counts
the two subintervals.

>only one parameter which is the result of x(t,j). I am confused on what
F(x) 
>takes as a parameter and it's defination. Thank you for your time.

GoodLuck,
FBH

BCC: Class