Problem 5: Sensitivity of solution to perturbations See Section 2.5 for the SVD and 3.3 on error analysis. We consider Ax = b where A is non-singular. Assume that we perturb matrix A and vector b by a small amount: A + 6A, 11 + 6b. The new solution x is itself perturbed as a result (A+6A) (x+6x) = 1) +617. (3) We denote the singular value decomposition of A as A = UZVT; ui (column iof U) and a,- (column 1' of V) are the left and right singular vectors of A. We denote a,- the singular values. For the questions below we chose c1 6 R\\{0} and 5x = c111,, such that x + 6x at 0. 1. 8 points. Using Eq. (3), show that ox = A-'(8b - SA(x + 8x) ). 2. 10 points. Assume that Sb = c2un and SA = 0. Find c2 such that Eq. and 118x//2 = 1/All2 |186| 2 are satisfied. 3. 10 points. Assume that Sb = C2un. Find c2 # 0 and SA # 0 such that Eq. (3) and are satisfied. For ill-conditioned matrices, IA-|/2 may be very large. This shows that a small perturbation in the input (A or b) may result in large changes in x