. Given the following linear system:
A
.
− .
.
,
¯b
− .
3.02
1.05
2.53
1.61
=
4 33
0 56
−1 78
and=
7 23
−0.83 −0.54
1.47
−3.38
Compute the condition number of the matrix
Find the solution to the system using Gaussian Elimination with pivoting and using six-digit arithmetic.
Use Gaussian Elimination with pivoting and using three digit arith-metic.
Comment about the difference between both solutions
As we did in class, some curve fitting problems can be converted to linear least square approximation problems. One example is the fitting function y = axB.
For this exercise suppose that you are given a set of points (xI , yI) 1 ≤ i ≤ n and you want to find the values of a and b for the following fitting function
1
y = (ax + b)3
Use the least squares method to find the formulas to compute a and b.
Given the following data, do a quadratic least squares polynomial fit using P2(x) = a0 + a1x + a2x2.
xI
yI
xI
yI
xI
yI
-1.0
7.904
-0.3
0.335
0.4
-0.711
-0.9
7.452
-0.2
-0.271
0.5
0.224
-0.8
5.827
-0.1
-0.963
0.6
0.689
-0.7
4.400
0
-0.847
0.7
0.861
-0.6
2.908
0.1
-1.278
0.8
1.358
-0.5
2.144
0.2
-1.335
0.9
2.613
-0.4
0.581
0.3
-0.656
1.0
4.599
1
For n, m ≥ 0 and n = m, show
TN(x)TM (x)
dx = 0
−1 1 − x2
This is called the orthogonality relation for Chebyshev polynomials.
Hint: Use TN(x) = cos(n cos−1(x)) and x = cos θ.
2
