By Abdelwahab Kharab
Praise for past Editions
Kharab and Guenther provide an enticing, transparent, error-free, and well-written creation to numerical equipment ... hugely recommended.
-J.H. Ellison, CHOICE
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Extra info for An Introduction to Numerical Methods : A MATLAB Approach, Third Edition
Write a computer program in a programming language of your choice that ﬁnds the zeros of a function f (x) on an interval [a, b] using the bisection method. Assume that there is only one simple root in this interval. Input data to the program should be a function f (x), a, b, and the error tolerance . 5, 0]. What does your program do if the interval is reduced to [−1, 0]? 2. The roots of the polynomial p(x) = (x − 1)(x − 2) . . (x − 20) − 10−8 x19 are highly sensitive to small alterations in their coeﬃcients.
Verify that the function f (x) = x2 sin x + 2x − 3 has exactly one root in (0, 2). Find this root by using the bisection method with an error of no more than 10−5 . 1. 2. Find a root of f (x) = x3 + 2x − 3 in the range 0 ≤ x ≤ 7/5 using the bisection method. 3. 31 has one root in the interval [3, 4]. How many steps of the bisection method are needed to locate the root with an error of at most 10−5 . 4. 6 has exactly one root in (1, 2). Demonstrate it graphically and determine the root with an error of at most 10−5 ?
687, a in the form a(b+c) write the quotient b−c b2 −c2 and compute it using the numbers as given, and then using three digits and rounding. 6. 14159265358979 . .. 14, what is the absolute error? Express your answer using chopping to a decimal normalized ﬂoating-point representation with 5 signiﬁcant digits. 7. 67. Find the absolute and relative error of your results. (a) Proceeding from left to right. (b) Proceeding from right to left. ✐ ✐ ✐ ✐ ✐ “k” — 2011/11/22 — 10:14 — page 31 — ✐ ROUND-OFF ERRORS ✐ 31 8.