Question

1. Questions (a) Truncations chooses the closest machine number in the direction towards zero. This implies that for positive numbers the result is less or equal than the original mumber while for negative numbers the result of truncation is larger or equal than the original number. Round to nearest chooses the closest machine number. This minimizes the absolute error imtroduced by the rounding (b) 32-bit significant: es 21-32 2-31=0.466 x 10-9. (c) In divisions the maximum relative error is the sum of the relative errors of the two operands. In subtraction relative errors can grow much more strongly if the result is close to zero. Subtraction is therefore the more problematic operation. JAzl Ayl E-(x/y) + lyl Ayl Arl E,(x-y) + -y - y (d) The condition f(a)- f(b) < 0 implies that f(a) has a different sign than f(b). This guarantees the existence of a root in the interval la, b. (e) The advantages of the bisection method are that its convergence is guaranteed and that it gives strict bounds on the root. Its disadvantages are slow convergences and the necessity to provide an interval in which the function changes sign. The secant method also requires two starting, values but there is no requirement of a sign change. The secant method converges superlinearly (with 1.62). However, it does not guarantee convergence and gives no bounds for the root. p (f) A fixed point is a value r that satisfies the equation g(r) =r.

Answer 1

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