Problem

On a certain modern computer, floating-point numbers have a 48-bit mantissa. Moreover, f...

On a certain modern computer, floating-point numbers have a 48-bit mantissa. Moreover, floating-point hardware can perform addition, subtraction, multiplication, and reciprocation, but not division. Unfortunately, the reciprocation hardware produces a result accurate to less than full precision, whereas the other operations produce results accurate to full floating-point precision.

a. Show that Newton’s method can be used to find a zero of the function f (x) = 1 − 1/(ax). This will provide an approximation to 1/a that is accurate to full floating-point precision. How many iterations are required?

b. Show how to obtain an approximation to b/a that is accurate to full floating-point precision.

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