CS 323 Practice Problems for Exam 2
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CS 323
Practice Problems for Exam 2
1. Given the System of equations A
= 
:
5x1 + 2x2 
x3 = b1
x1 
7x2 + 2x3 = b2
2x1 + 3x2 
4x3 = b3
(a) Use Gaussian Elimination with no pivoting to compute A−1
(b) Use Gaussian Elimination with no pivoting to compute L, U such that A = LU (L is a lower triangular matrix, and U is an upper triangular matrix).
2. Write the equations needed to compute 
(1) given 
0 for the following system of equations:
5x1 + 2x2 
x3 = b1
x1 
7x2 + 2x3 = b2
2x1 + 3x2 
4x3 = b3
(a) In the case of Jacobi
(b) In the case of Gauss-Seidel
(c) Will an iterative solution converge? Why?
3. Given the matrices:
1 0 0 
1 1 1 
If A = LU, 
nd the solution to A
= 
using forward/backward substitution for each of the following values of 
.
(a)
= 
(b)
1
1
1
2
1
2
A−1 = 
nd the solution to A
= 
for each one of the following values of 
(a)
(b)
5. Given the points
x y 1 
0 2 
1
3 0
1
1
1
2
1
2
Use Lagrange’s method to 
nd a 2nd degree polynomial that goes through each one of the given points.
6. Given x1 = 2, x2 = 2.5 x3 = 4 x + 3 compute Neville’s Table:
|
0.5 |
|
|
|
|
0.2 |
|
|
|
|
0.25 |
7. Suppose that you are given xk = 2 + 0.1k and yk = √x
k for k = 0,...,4.
(a) Compute the divided di
erences table.
(b) Provide the approximation polynomial.
(c) Compute P4(2.15)
8. Given xi = 0.025,0.5,0.75, 1.0 and yi = cos
xi.
(a) Construct a free cubic spline that approximates the given (xi,yi) values. (free cubic spline
means that we use c0 = 0 and cn = 0 as we did in class). (b) Integrate the spline over [0, 1] and compare the result to R01 cos
x = 0.
(c) Use the derivatives of the spline to approximate f′ (0.5) and f′′ (0.5). Compare the approx- imation to the actual values.
2022-03-28