代做Numerical Analysis代写留学生Matlab程序

1. (10 points)

(a). (5 points) Define machine epsilon.

(b). (5 points) List the following in order of increasing computational cost asmeasured by the number of operations (e.g., from least to most): (i) Gaussianelimination with full pivoting; (ii) calculation of determinant; (iii) Gaussianelimination with partial pivoting; (iv) Gaussian elimination with scaled partialpivoting.

2. (25 points)

(a). (5 points) Define quadratic convergence of a sequence pn.

(b). (20 points) Suppose that p is a zero of f(x) with multiplicity 3.  Show that thefollowing modified version of Newton's method converges at least quadratically:

Hint: You may use the results proved in class about conditions under which fixed point iteration methods converge quadratically.

3. (20 points)

(a). (10 points) Let A = Find the PLU decomposition of A, using partial pivoting. That is, use partial pivoting to determine whether rows need to be switched.

(b). (10 points) Use the PLU decomposition you found in part (a) to solve the system of equations:

4. (20 points)

(a). (5 points) Let A be a n x n matrix. What does it mean for A to be strictly diagonally dominant?

(b). (10 points) Find all a>Oand b>0 such that the matrix A =  is strictly diagonally dominant.

4 (c). (5 points) For the choices of a and b that you found in part (b), what is the permutation matrix P in the PLU decomposition of A? Justify your answer.

5. (25 points)

(a). (5 points) Let A be a n x n matrix. Define the determinant of A.

(b). (10 points) Is det  = det? Why or why not? Justify your answer.

5 (c). (10 points) Calculate det