Description
- Apply bisection method to find the root of the function
√ f(x) = x − 1.1
Starting from the interval [0,2], with altol = 10−8 ( absolute error tolerance).
- How many iterations are required? Does the iteration count match the expectations,based on our convergence analysis?
- Convert this into a fixed point iteration and find the approximated value of theroot of f with altol = 10−8.
- The function f(x) = tan(πx)−6 has a zero at (1/π)arctan6 ≈ 0.447431543. Let x0 = 0 and x1 = 0.48, and use ten iterations for each of the following methods to approximate this root. Which method is most successful and why?
- Bisection method.
- Secant method.
- Draw the graph of a function having the following properties:
- The function has exactly two fixed points.
- Give two choices of the initial guess x0 and y0 such that the corresponding sequences {xn} and {yn} have the properties that {xn} converges to one of the fixed point and the sequence {yn} goes away and diverges. Point out the first three terms of both the sequences on the graph.
- Use the Euler’s methods and Runge-Kutta methods of order 2 and 4 to solve the IVP
with initial condition y(0) = 1. Compare the solutions for along with the exact solution y(x) = 3exp(−x/2) + x − 2
