HW7

Matrices from "real" problems.  We are going to build 
a collection of real matrices for real problems. These
are the examples discussed in class and are mostly
also discussed in the book p190-207.
 

1) Write a function that generates the appropriate 
linear system to fit a polynomial 

y = an x^n + ... + a2 x^2 + a1 x + a0

of degree n through the n+1 points 

x0      y0
x1      y1
x2      y2
...
xn      yn

The input should be the matrix of values and the
output should be the vector

{a0, a1, a2, a3, a4, ..., an}


2) Write a function that generates the cubic spline

Y1 = a1 x^3 + b1 x^2 + c1 x + d1
Y2 = a2 x^3 + b2 x^2 + c2 x + d2
...
Yn = an x^3 + bn x^2 + cn x + dn

through the n+1 points

x0	y0
x1	y1
x2 	y2
...
xn	yn 

where 
Y1 gives the y values for x0<x<x1
Y2 gives the y values for x1<x<x2
...
etc.

The input should be the matrix of values
and the output should be the matrix of values

a1	b1 	c1	d1
a1	b1 	c1	d1
....
an	bn 	cn	dn