For problem 1, the number of iterations and how you determined that number. For
ID: 3585889 • Letter: F
Question
For problem 1, the number of iterations and how you determined that number.
For problems 2 and 3, the source code, and the output when running it on the two polynomials.
Also for problem 3, answer the discussion questions.
Fo problem 4, show your work for each iteration until you get the result correct to 4 significant digits.
1. Using the bisection starting from the initial bracket [0,2], estimate the number of iterations that would
be required to find a root of ex 4x = 0 to four decimal place accuracy?
2. Write a program that uses Newton’s method and double-precision floating point arithmetic to obtain
all the zeros of
x7 28x6 + 322x5 1,960x4 + 6,769x3 13,132x2 + 13,068x 5,040
The exact zeros are 1, 2, 3, 4, 5, 6, and 7. Starting with initial guesses of 0.9, 1.9, 2.9, 3.9, 4,9, 5.9, and
6.9, and using the termination condition |xi xi1| < 107|xi|, have your main program print a table
of values for your solutions and the actual function value to 8 places after the decimal point as well as
the number of iterations necessary to obtain that solution. Evaluate the polynomial and its derivative
using Horner’s method.
3. In problem 2, change the coefficient of the x2 term to 13,133 and repeat your process (that is, after
a change of one unit in the fifth place of one coefficient is made.) What zeros do you now find? What
is the difference in the solutions due the this small perturbation of one coefficient? Since coefficients
of polynomials of high degree are often found experimentally, what does that tell you about using this
method when there is some doubt about the accuracy of the coefficients?
4. A modification of the regula falsi method, called the secant method, retains the use of secants throughout,
but may give up the bracketing of the root.
Secant method: Given a function f(x) and two points x1, x0
for n = 0, 1, 2, . . . , until satisfied do:
calculate xn+1 = f(xn)xn1f(xn1)xn
f(xn)f(xn1)
The function f(x) = 4sinx ex has a zero on the interval [0, 0.5] Find this zero correct to four
significant digits using the secant method.
Explanation / Answer
We need 17 iterations
Due to huge rush amount in questions, we are only able to answer 1 question or 4 subquestions at a time. we request your co-operation.
low high mid f(mid) 1 0 2 1 -1.28172 2 0 1 0.5 -0.35128 3 0 0.5 0.25 0.284025 4 0.25 0.5 0.375 -0.04501 5 0.25 0.375 0.3125 0.116838 6 0.3125 0.375 0.34375 0.035226 7 0.34375 0.375 0.359375 -0.00507 8 0.34375 0.359375 0.351563 0.015037 9 0.351563 0.359375 0.355469 0.004974 10 0.355469 0.359375 0.357422 -4.9E-05 11 0.355469 0.357422 0.356445 0.002462 12 0.356445 0.357422 0.356934 0.001206 13 0.356934 0.357422 0.357178 0.000579 14 0.357178 0.357422 0.3573 0.000265 15 0.3573 0.357422 0.357361 0.000108 16 0.357361 0.357422 0.357391 2.98E-05 17 0.357391 0.357422 0.357407 -9.4E-06Related Questions
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