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MATH 417: Numerical Analysis
Instructor: Prof. Wolfgang Bangerth [email protected], Teaching Assistants: Dukjin Nam [email protected]
Homework assignment 4 – due 2/27/
Problem 1 (Convergence order for functions). Determine the exponent p in the following statements and explain your choice:
a) 12x^4 − 2 x^3 ln x = O(xp) as x → ∞.
b) (^) (x−^2 1) 2 − ln ln x + (^) ln ln^1 x = O((x − 1)p) as x → 1.
c) Is there an exponent p for which the statement
x ln x = O(xp) as x → ∞
is true? If not, is there an exponent for which the following holds:
x ln x = o(xp) as x → ∞.
(4 points)
Problem 2 (Gaussian elimination). Solve (on paper, showing the individ- ual steps) the following system of linear equations using Gaussian elimination:
1 2
1 3
1 4
1 5 1 3
1 4
1 5
1 6 1 4
1 5
1 6
1 7
x 1 x 2 x 3 x 4
Verify that your result is correct. (The matrix is the example is the so-called Hilbert matrix, with entries Hij = (^) i+^1 j− 1. It has a number of nasty properties that make it a good testcase for matrix algorithms.) (5 points)
(please turn over)
Problem 3 (Gaussian elimination). Using Gaussian elimination, it is sim- ple to solve the following problem
x 1 x 2 x 3
One would eliminate the occurrence of x 1 in the second equation by subtracting the first from the second equation, arriving at a diagonal matrix. Describe what happens if the system instead looked like this:
x 1 x 2 x 3
Does the algorithm still work? If not, propose a remedy. (2 points)
Problem 4 (Positive definite matrices). Positive definite matrices are those matrices for which xT^ Ax > 0 for all vectors x 6 = 0. These matrices play an important role in many applications of engineering and physics. Let us consider one of their properties. Any matrix A can be written as A = As^ + Aa, where the symmetric part As and the skew-symmetric part Aa^ of a matrix are defined as
As^ =
A + AT
, Aa^ =
A − AT
Prove that if A is positive definite then As^ is positive definite, and vice versa. (3 points)
Problem 5 (LU decomposition). Solve the linear system Ax = b with the Hilbert matrix system we already saw in Problem 2:
A =
1 2
1 3
1 4
1 5 1 3
1 4
1 5
1 6 1 4
1 5
1 6
1 7
, b =
by applying the following steps with paper and pencil:
- Compute the LU decomposition of A and write down the elimination steps.
- Use forward and backward substitution to obtain the solution x.
(5 points)
