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CSE/MATH 451 Final, Fall 1997
Name: I.D. Number:
1 2 3 4 5 total
- (25 points) Let N be a given positive number.
(1) Use Netwon’s method on the equation x^2 - N =0to derive the following iterative method for the square root of N :
where x 0 is a given initial guess.
(2) Derive a similar iterative method for the third root of N. (3) Let N = AB , i.e., write N as the product of two numbers A and B. Then a popular formula for an
approximation of is
Show that you can obtain the above formula by iterating the method in (1) twice with x 0 = A or B. What about x 0 =( A + B )/2?
- (25 points) Solve the following set of inconsistent linear equations in the least squares sense using the normal equation method
What is the 2-norm of the minimum residual?
- (20 points) Consider the IVP
Apply Euler’s method to the above IVP with constant step size h , write down the iterative scheme, and show that
yn = (1+ h ) n.
What is y (1)? Show that y (1)- yn = O ( h ).
- (15 points) Find polynomial of least degree that interpolates the set of data points:
Find another different polynomial which also interpolates the above data.
- (15 points) Write a MATLAB code fragment that computes the following matrix
where p is a positive integer, A is square n -by- n , v is a vector n -by-1, and D is diagonal chosen that the columns of A has unit 2-norm. Use the following function header
function K = Krylov(A,v,p)
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