Numerical Methods

Iterative Methods for Solving Equations

Understand the concept of iteration and how it can be used to solve equations. Iterative methods provide successive approximations to the solution of an equation which can be refined to any desired level of accuracy.
Master the fixed point iteration, including rearranging an equation into the form x = g(x) and using the iterative formula xₙ₊₁ = g(xₙ).
Be aware of the conditions for the convergence of the fixed point iteration and understand that the rate of convergence depends on the gradient of g(x) being less than 1 in magnitude at the root of the equation.

Newton-Raphson Method

Get to grips with the Newton-Raphson method, another iterative technique for finding approximate solutions to equations of the form f(x) = 0. This method uses the derivative of the function and can often converge more quickly than the fixed point iteration.
Know the iterative formula for the Newton-Raphson method: xₙ₊₁ = xₙ - f(xₙ)/f’(xₙ).
Understanding the geometry involved in the Newton-Raphson method can be very helpful. Each iteration essentially involves drawing a tangent to the curve at a certain point and finding where that tangent crosses the x-axis.

The Bisection Method

Learn the bisection method for locating roots of a function. This involves repeatedly dividing an interval into two equal parts (“bisections”) and selecting the subinterval in which a root is known to exist.
Understand the concept of error bounds and interval bisection, and know that the bisection method provides a way to systematically reduce these error bounds.

Numerical Integration

Learn different numerical methods for finding approximate values of definite integrals, including the Trapezium Rule and Simpson’s Rule.
Know how to perform numerical integration using simple and composite Trapezium Rule and Simpson’s Rule.
Be aware of the sources of error in numerical integration and understand the conditions under which certain methods are more accurate than others.

The above are critical concepts in numerical methods. Practice is key to mastering these techniques, so work through plenty of examples until the procedures feel comfortable and familiar. But remember that understanding the concepts is just as important as being able to perform the calculations.