Iterative Method Homework



Iterative Method Homework

1. Solve Au = b with [pic], [pic][pic], [pic] using conjugate gradients. Note that the true solutions is [pic].

2. Consider the boundary value problem [pic] defined over the square 0 ( x ( 6 and 0 ( y ( 6. Divide the square into three subdivisions in each direction (so that the x and y increment is 2) and suppose that the boundary data is 0 except for x=6 it is u = 10. Use the 5 point star stencil to write a system of 4 equations in 4 unknowns u1, u2, u3, u4.

3. For n=2 use calculus to prove that a potential minimum point of F(u) = (1/ 2 ) uTA u - bTu is the solution to A u = b if A is symmetric. Hint: let [pic], [pic] and [pic] so that [pic] Set the partial derivatives of F with respect to x and y to zero.

4. Watkins exercise 8.1.12 (3rd edition) or 7.1.12 (2nd edition).

5. For n = 100, b = ones(n,1); and for each of the three matrices below apply 20 steps of my cg_step code. One can do this using

k = 0; for i=1:20, cg_step, end

in Matlab. The cg_step code prints out ||r|| / ||b|| = || b – Au || / ||b|| which is a measure of how well the current guess solves the system A u = b . Compare the final values of ||r||/||b|| for the three runs. Also for each matrix calculate cond(A) and gamma = (sqrt(cond(A)) – 1 ) / ( sqrt(cond(A)) + 1 ). Discuss.

a) A = diag( 1.01 .^ (0 : n – 1 ) );

b) A = diag( 1.03 .^ (0 : n – 1 ) );

c) A = diag( 1.1 .^ (0 : n – 1 ) );

6. For each of the following values of k = cond(A) predict how many steps are required to reduce the error in conjugate gradients by 10-6 : k=4, 81, 9801, 99801. Use the formulas [pic] and [pic] . According to the second formula the error will be reduced by a factor of [pic] when [pic].

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