Explanation Algorithm used: To ﬁnd a solution to f ( x ) = 0 given the continuous function f on the interval [ a , b ] , where f ( a ) and f ( b ) have opposite signs: INPUT endpoints a, b ; tolerance TOL ; maximum number of iterations N 0 . OUTPUT approximate solution p or message of failure. Step 1: Set i = 1 ; F A = f ( a ) . Step 2: While i ≤ N 0 do Steps 3–6. Step 3: Set p = a + ( b − a ) 2 ; ( Compute p i .) F P = f ( p ) Step 4: If F P = 0 or ( b − a ) 2 < T O L then OUTPUT ( p ); ( Procedure completed successfully .) STOP. Step 5: Set i = i + 1 . Step 6 If F A ⋅ F P > 0 then set a = p ;( Compute a i , b i .) F A = F P else set b = p .( FA is unchanged .) Step 7 OUTPUT (‘Method failed after N 0 iterations, N 0 =’, N 0 ); ( The procedure was unsuccessful .) STOP. Theorem used: Suppose that f ∈ C [ a , b ] and f ( a ) ⋅ f ( b ) < 0 . The Bisection method generates a sequence { p n } n = 1 ∞ approximating a zero p of f with | p n − p | ≤ b − a 2 n , n ≥ 1 . Calculation: Given that the function is f ( x ) = x 4 + x − 4 on the interval [ 1 , 4 ] . Find the functional value of the end points as follows. f ( 1 ) = 1 4 + 1 − 4 = − 2 f ( 4 ) = 4 4 + 4 − 4 = 64 Since the values having opposite sign, the above theorem and algorithm can be applied. By the above theorem, | p n − p | ≤ b − a 2 n ≤ 10 − 3 . 4 − 1 2 n ≤ 10 − 3 1 , 000 ( 3 ) ≤ 2 n 3 , 000 ≤ 2 n n ≥ 12 Thus, the solution of the given function accurate to 10 − 4 is obtained in 14 iterations

10th Edition

ISBN: 9781305253667

Author: Richard L. Burden, J. Douglas Faires, Annette M. Burden

Publisher: Cengage Learning

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Chapter 2.1, Problem 18ES

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