6. (25pt) Let A be an n-by-n lower triangular matrix. An inexpensive estimator for || A||2 can be constructed by fixing x1 = 1 and y1 = 21,1 and choosing Xi = 1, i = 2, ..., n so as to maximize the norm of y = Ar. This estimator is realized in the following algorithm: x1 = 1; y1 = 1,1 For i = 2 ton S=0 For j = 1 to i - 1 s=s+aijj end j If |aj,i + s| > |aj,i 8 then X;=1 else Xi = -1 end if Yi = ajili+s end i r = || 4||2/vn Please compute the number of floating point operations required for evaluating r (5pt). Also, construct an example showing that r is only an estimate from below for the true || A||2. You should be able to find an example using small values of n, i.e., n = 2 or n = 3. (15pt) 2 of 2 6. (25pt) Let A be an n-by-n lower triangular matrix. An inexpensive estimator for || A||2 can be constructed by fixing x1 = 1 and y1 = 21,1 and choosing Xi = 1, i = 2, ..., n so as to maximize the norm of y = Ar. This estimator is realized in the following algorithm: x1 = 1; y1 = 1,1 For i = 2 ton S=0 For j = 1 to i - 1 s=s+aijj end j If |aj,i + s| > |aj,i 8 then X;=1 else Xi = -1 end if Yi = ajili+s end i r = || 4||2/vn Please compute the number of floating point operations required for evaluating r (5pt). Also, construct an example showing that r is only an estimate from below for the true || A||2. You should be able to find an example using small values of n, i.e., n = 2 or n = 3. (15pt) 2 of 2