Newtons Method for Cube Root: The problem: You are required by a computer manufacturer to write a library function for the IEEE floating-point system, double-precision, to find the cube root,y13, of any given positive number y. This library function must be very efficient and it should always work. Use the fact that any floating-point number can be represented asy=a2b, where a is a normalized fraction (i.e., 0.5a <1) and is an integer exponent. For efficiency purposes it makes sense to store some useful constants ahead of computation time, e.g., the constants 21/3,23, etc., should these prove useful.

(a) (6 points) What is the iteration function of Newton method for findinga13. Simplify the expression to reduce computation. What is the flop (floating-point operations)count per iteration? (b) (6 points) Show howy13can be obtained oncea13has been calculated for the corresponding fraction in at most five additional flops.

(c) (4 points) What is a good an initial approximation toa13? Roughly how many iterations would be needed (the floating-point system has 53 bits of precision) using this approximation?

Please answer in detail for each question. Thank you.