Answer each question:
Section 7.3 The Jacobi and Gauss-Siedel Iterative Techniques
1)
Find the first two iterations of the Jacobi method for the following linear systems, using x
(0)= 0.:
d. 4x
1–x
2- x
4= 0,
-x
1
+4x
2–x
3 _x5
=5,-x
2– 4x
3– x
6= 0,
-x
1+4x
4–x
5= 6,
-x
2– x
4+4x
5–x
6
= -2,
-x
3– x
5+4x
6=6
9)
The linear system 2x
1- x
2+ x
3= -1,
2x
1+2x
2+ 2x
3= 4,
-x
1–x
2
+ 2x
3= -5
Has the solution (1, 2, -1)
t.
- Show that p(Tj) = /2 > 1.
- Show that Jacobi method with x(0)
= 0 fails to give a good approximation after 25 iterations.
- Show that p(Tg) = 1/2
- Use the Gauss-Seidel method with x(0)
= 0 to approximate the solution to the linear system to within 10-5
in the ?norm.
10)
The linear system x
1+2x
2
– 2x
3= 7,
x
1+ x
2+x
3=2,
2x
1+2x
2+x
3= 5
Has the solution (1, 2, -1)
t.
- Show that p(Tj) =0
- Use the Jacobi method with x(0)
= 0 to approximate the solution to the linear system to within 10-5
in the ?norm.
- Show that p(Tg) = 2
- Show that the Gauss-Seidel method applied as in part (b) fails to give a good approximation in 25 iterations.
Section 8.1 Discrete Least Square Approximation
3)
Find the least squares polynomials of degree 1, 2, and 3 for the data in the following table. Compute the error E in each case. Graph the data and the polynomials.
x
i0 0.15 0.31 0.5 0.6 0.75
y
i
1.0 1.004 1.031 1.117 1.223 1.422
5)
Given the data:
x
i
1.0 1.1 1.3 1.5 1.9 2.1
y
i1.84 1.96 2.21 2.45 2.94 3.18
- Construct the least squares polynomial of degree 1, and compute the error.
- Construct the least squares polynomial of degree 2, and compute the error.
- Construct the least squares polynomial of degree 3, and compute the error.
- Construct the least squares approximation of the form beax
, and compute the error.
- Construct the least squares approximation of the form bxa
, and compute the error.
Section 8.2 Chebyshev Polynomials and Economization of Power Series
11)
Use the Gram-Schmidt procedure to calculate L
1, L
2and L
3where { L
0(x), L
1(x), L
2(x), L
3(x) } is an orthogonal set of polynomials on (0, ) with respect to the weight functions (x) = e
-xand L
0(x) 1. The polynomials obtained from this procedure are called the Laguerre polynomials.