Question

4. Let f(x) = 14xex+1 + 30ex+1 −7x3 −43x2 −95x−75. (a) Apply Newton’s method to find both roots of the function in the interval [−5 2, 1 2], to as much precision as possible. For each root, print out the sequence of iterates, the errors ei, and the error ratios ei+1/e_i and ei+1/e2 i. (b) In each case, determine if the error converges linearly or quadratically. Explain briefly why and what you conclude from it.

Answer 1

From the plot it is understood that the error converges quadratically.

In Newton's method of root finding, error after n+1 iterations is proportional to square of error after n iterations. Hence the error converges quadratically.