Question 1: Nonlinear Equations for univariate case.
a) Bisection Method is used for tfinding the roots of a continuous function, , given endpoints; . The interval contains a root, , because and have opposite signs.
Bisection method bisects the given interval at the midpoint, and then chooses a new interval based such that end of point of interval have opposite signs, i.e., if , then the new interval is set to then the interval is set to .
The above procedure is repeated until the following two conditions are simultaneously met,
The function value at the interval is sufficiently small, i.e.,
The new interval is sufficiently small, i.e.,
Implement the Bisection Method in the cell below to find the root of in the interval . Show the number of iterations the bisection method took to converge.
Use the cell below to implement your code.
Note: There is no need to write the function for Bisection Method. However, if you wish to implement the function, use the appendix.
Implement the Newton-Raphson Method in the cell below to find the root of convergence is not reached in 100 iterations, quit the algorithm by displaying an error message indicating that Newton-Raphson failed to converge.
Show the number of iterations the Newton-Raphson method took to converge.
If the convergence is not reached for Newto-Raphson in 100 iterations, quit the function by displaying an error message indicating that Newton-Raphson failed to
Find a solution of the following system of three nonlinear equations using Newton-Raphson method. , where
a) Write a MATLAB function named evaluateEquations(x), that evaluates the above equation at a given input vector.
Use the framework of the function provided in the Appendix.
b) Write a MATLAB function named evaluateJacobian(), that evaluates the Jacobian of the above equations.
Use the framework of the function provided in the Appendix.
