If a vector y Rn is used to construct an n n Vandermonde matrix V, then V will be

Chapter 7, Problem 2

(choose chapter or problem)

If a vector y Rn is used to construct an n n Vandermonde matrix V, then V will be nonsingular, provided that y1, y2, . . . , yn are all distinct. (a) Construct a Vandermonde system by setting y = rand(6, 1) and V = vander(y) Generate vectors b and s in R6 by setting b = sum(V_ ) _ and s = ones(6, 1) If V and b had been computed in exact arithmetic, then the exact solution of Vx = b would be s. Why? Explain. Solve Vx = b, using the \ operation. Compare the computed solution x with the exact solution s using the MATLAB format long. How many significant digits were lost? Determine the condition number of V. (b) The Vandermonde matrices become increasingly ill conditioned as the dimension n increases. Even for small values of n, we can make the matrix ill conditioned by taking two of the points close together. Set x(2) = x(1) + 1.0e12 and use the new value of x(2) to recompute V. For the new matrix V, set b = sum(V_ ) _ and solve the system Vz = b. How many digits of accuracy were lost? Compute the condition number of V.