← Further Maths syllabus

SingaporeFurther Maths

Numerical Methods

4 dot points across 4 inquiry questions. Click any dot point for a focused answer with worked past exam questions where available.

How does rearranging an equation into the form x = g(x) give an iterative method, and what controls whether it converges?

Solve an equation by fixed-point iteration of the form x = g(x), and use the derivative condition to decide convergence
A focused answer to the H2 Further Mathematics outcome on fixed-point iteration. Rearranging f(x) = 0 into x = g(x), the iteration x_{n+1} = g(x_n), the staircase and cobweb diagrams, and the convergence condition that the magnitude of g prime is less than one.
11 min answer →

How does the Newton-Raphson method find a root of an equation, and when does it succeed or fail?

Apply the Newton-Raphson method to find a root of an equation numerically and discuss its convergence and failure cases
A focused answer to the H2 Further Mathematics outcome on the Newton-Raphson method. The iterative formula, its geometric meaning as a tangent intercept, choosing a starting value, the quadratic convergence, and the situations in which the method fails.
11 min answer →

How do the trapezium rule and Simpson's rule approximate a definite integral, and how accurate are they?

Approximate a definite integral using the trapezium rule and Simpson's rule and comment on the accuracy of the estimate
A focused answer to the H2 Further Mathematics outcome on numerical integration. The trapezium rule and Simpson's rule, the strip width and ordinates, applying each rule, the over- or under-estimate behaviour, and which rule is more accurate.
11 min answer →

How does Euler's method step a differential equation forward numerically, and what limits its accuracy?

Use Euler's method and the improved Euler (midpoint) method to obtain a numerical solution of a first-order differential equation
A focused answer to the H2 Further Mathematics outcome on numerical solution of ODEs. Euler's method as a tangent step, the step size, accumulation of error, and the improved Euler (midpoint) method for greater accuracy.
11 min answer →