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Unformatted text preview: 155:307 Analysis II–HW#1 September 13, 2010 1 Problem 1 We want to ﬁnd the solution to the equation sin(x) − 2 exp(−x2 ) = 0. Consider the following three approaches:
1. use successive substitution (iteration) by rearranging the solution as
xn+1 = f (xn ).
2. use Newton’s method with an analytical expression for the derivative,
3. use fzero, and
4. use fsolve
Provide printouts of all your programs and their execution showing the
sequence of iterates and the convergence from an initial guess x = 1.
Plot the values of the function sin(xn ) − 2 exp(−x2 ) as a function of the
iteration number, n. Put all the approaches listed above on a single plot
with log-log axes. Comment on the orders of convergence. 2 Problem 2 Find all the roots of the polynomial
x5 − 8x4 + 35x3 − 106x2 + 170x − 200 = 0
and verify Descartes’ rule is observed. Provide a printout of your program.
Graph the polynomial over a range showing all the real roots. 1 3 Problem 3 In problem 1.7 of the text (p. 58), the following equation is presented:
τI τp τm τv s4 + (τI τp τm + τI τp τv + τI τm τv )s3 + (1.8Kc τI τD + τI τp +
τI τv + τI τm )s2 + (τI + 1.8Kc τI )s + 1.8Kc = 0
where τI = 10, τD = 1, τp = 10, τm = 5, and τv = 5. Find the critical
value of Kc using the Matlab function fzero. Provide a printout of your
program. 2 ...
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