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Ceng 375 Numerical Computing
Midterm
Nov 17, 2006 12.40-14.30
Good Luck!
- (30 Pts) Consider the function:
- i
- Use two iterations of Newton s method to estimate the root of this function between and
- ii
- Estimate the error in your answer to part i.
- iii
- Approximately how many iterations of the bisection method would have been required to achieve the same error?
- (30 Pts) Consider the function , on , defined by
- iv
- Describe the general working of a bracketing method. What are the assumptions for this family of methods? Are these assumptions satisfied for ?
- v
- Describe how the secant method determine a smaller sub-interval containing a root.
- vi
- What is the advantages and disadvantages of the secant method?
- vii
- Apply the secant method to , to reduce the size of the original interval , twice.
- (25 Pts) Find the factorization of
by Gaussian elimination (without pivoting). Clearly show how you get the entries of and .
- (25 Pts) Consider solving the following linear system by the Jacobi method.
- viii
- Write down the Jacobi iteration formula for this problem given initial guess .
- ix
- Assume that the error (vector) at iteration is denoted by and that
. How many iterations do we need before
?
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Cem Ozdogan
2006-12-28