[SOLVED] MA 322 - Lab1

1. The iteration

,

converges to . For a = 2, determine

• Number of iterations n such that |xn₊₁ − xn^{| ≤}√10⁻⁵.
• Determine the order of convergence assuming 2 = 1.

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.

2