1. The iteration
,
converges to . For a = 2, determine
(a) Number of iterations n such that |xn+1 − xn| ≤√10−5.
(b) Determine the order of convergence assuming 2 = 1.4143.
2. Let f(x) = tan(π−x)−x and consider the equation f(x) = 0. Now, we wish to determine the approximate root for the equation in [1.6,3] using the following algorithm. Step 1: Divide the interval into n equal parts by the points
x0 = 1.6, x1 = x0 + h,...,xn = xn−1 + h = 3.
Step 2: Then determine the values of f(xk), k = 0,1,...,n and set that value of xk to be the root for which |f(xk) − 0| is minimum.
3. Consider the equation
.
Use bisection method to find an approximate root in the interval [π/2, π]. Then modify the approximation using Newton’s method which is correct up to seven decimal places.
4. Consider the equation
x
sinx = 0.
2 −
Use bisection method to find an approximate root in the interval [π/2, π]. Then modify the approximation using fix point iteration and calculate the order of convergence.
5. Consider f(x) = 0, f(x) = e−x(x2 + 5x + 2) + 1. Find an approximate root using secant method with x0 = −1 and the stopping criterion |xn+1 − xn| ≤ 10−5|xn+1|.
6. Consider f(x) = 0, f(x) = e−x(x2 + 5x + 2) + 1. Use Bisection method to find an approximation of actual root. Then modify the root using following iterative scheme
.
Determine the order of convergence.
7. Consider the equation
x
sinx = 0.
2 −
Use bisection method to find an approximate root in the interval [π/2, π]. Then modify the root using following iterative scheme
.
Determine the order of convergence.
1
8. Consider f(x) = 0, f(x) = e−x(x2 + 5x + 2) + 1. Use Bisection method to find an approximation of actual root. Then modify the root using following iterative scheme
.
Determine the order of convergence.
