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Unformatted text preview: COT4501 Fall 2010 Homework 1 solutions 1.3: Since the relative error = approximate value true value true value , we get r = a t t from which we obtain that a = t (1 + r ) . 1.4: We have an error in the function value due to a perturbation h in the argument x . a) The absolute error in evaluating sin( x ) is sin( x + h )- sin( x ) h cos( x ) when h is small, b) the relative error is sin( x + h ) sin( x ) sin( x ) h cot( x ) , c) the condition number is approximately equal to | xf ( x ) f ( x ) | = x cot( x ) and d) the problem is highly sensitive to values of x = 2 + k, k N since the relative error and condition number go to infinity. 1.6: (a) x y = sin( x ) y = x forward error backward error y- y x- x = arcsin( x )- x 0.1 .0998 0.1 0.002 0.0002 0.5 .4794 0.5 0.0206 0.0236 1.0 .8415 1.0 0.1585 0.5708 Table 1: Forward and backward error of the sine function when using the first term of the Taylor series (b) x y = sin( x ) y = x- x 3 / 6 forward error...
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- Spring '08