Numerical solutions of equations

Question 1

Explain the interval bisection method

Answer 1

You are given that a root of a function f(x) lies in the interval [a,b]. One of f(a) and f(b) will be positive, and the other negative. Until the lower and upper boundaries agree to the required accuracy, update the interval depending on whether f((a+b)/2) is positive or negative; if it is positive then (a+b)/2 replaces whichever out of a and b returns positive when put into the function, and vice versa if it is negative

Question 2

Explain the linear interpolation method

Answer 2

You are given that a root of a function f(x) lies in the interval [a,b]. One of f(a) and f(b) will be positive, and the other negative. Until two successive x values agree to the required accuracy, find where the straight line between (a,f(a)) and (b,f(b)) crosses the x axis using the equation (b-x)/(x-a)=|f(b)/f(a)|. If f(x) is positive, update the interval by replacing with x whichever out of a and b returns a positive number when put into the function, and vice versa if x is negative

Question 3

Explain the Newton—Raphson method

Answer 3

Let x₀ equal a suitable start value. Then, until two successive x values agree to the required accuracy, let xᵢ₊₁=xᵢ-(f(xᵢ)/f’(xᵢ))