The equation f(x)=0 \, may be solved by the Newton-Raphson method: x_{n+1} = x_n - \displaystyle \frac{f(x_n)}{f'(x_n)} .
This method is based on the simple geometric idea that, starting with an initial guess at the solution, a tangent drawn from the curve at this point will hit the x axis at a point closer to the true solution.
The applet demonstrates the method.
To run the demo:
- Choose a function.
- Drag the mouse on the graph along the x axis to animate the algorithm.
Summary/Background
In numerical analysis, Newton's method (also known as the Newton–Raphson method) is an efficient algorithm for finding approximations to the zeros (or roots) of a real-valued function. As such, it is an example of a root-finding algorithm. It can also be used to find a minimum or maximum of such a function, by finding a zero in the function's first derivative, see Newton's method as an optimization algorithm.
Newton's method was described by Isaac Newton in De analysi per aequationes numero terminorum infinitas, written in 1669.
Joseph Raphson was an English mathematician known best for the Newton-Raphson method. Little is known about Raphson's life - even his exact birth and death years are unknown, though the mathematical historian Florian Cajori supplied the approximate dates 1648-1715. Raphson attended Jesus College in Cambridge and graduated with an M.A. in 1692. Raphson was made a Fellow of the Royal Society in 30 November 1689 after being proposed for membership by Edmund Halley.
Note:
- the Newton Raphson method is not always successful!
- it might not lead to the particular solution you were looking for
Software/Applets used on this page
This applet forms part of "Java Number Cruncher: The Java Programmer's Guide to Numerical Computation", Prentice-Hall, by Ronald Mak, and is provided for MathsNetAlevel-plus by that author - see
Apropos-logic
Apropos-logic
Glossary
algorithm
A set of precise instructions which, if followed, will solve a problem.
axis
One of two straight lines on a graph from which measurements are taken. One axis (the y axis) is vertical; the other (the x axis) is horizontal.
derivative
rate of change, dy/dx, f'(x), 
, Dx.
, Dx.equation
A statement that two mathematical expressions are equal.
function
A rule that connects one value in one set with one and only one value in another set.
geometric
A sequence where each term is obtained by multiplying the previous one by a constant.
graph
A diagram showing a relationship between two variables.
The diagram shows a vertical y axis and a horizontal x axis.
The diagram shows a vertical y axis and a horizontal x axis.
newton
the unit of force
newton raphson method
A method for find an approximate solution to an equation by using differentiation
solution
the answer to a problem.
tangent
1. The trigonometrical function defined as opposite/adjacent in a right-angled triangle.
2. A straight line that touches a curve at one point.
2. A straight line that touches a curve at one point.
This question appears in the following syllabi:
| Syllabus | Module | Section | Topic | Exam Year |
|---|---|---|---|---|
| AQA A-Level (UK - Pre-2017) | FP1 | Numerical Methods | Newton Raphson | - |
| AQA A2 Further Maths 2017 | Pure Maths | Numerical Methods | Newton Raphson - Extra | - |
| AQA A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| AQA AS/A2 Further Maths 2017 | Pure Maths | Numerical Methods | Newton Raphson - Extra | - |
| AQA AS/A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| CCEA A-Level (NI) | C3 | Numerical Methods | Newton Raphson | - |
| Edexcel A-Level (UK - Pre-2017) | FP1 | Numerical Methods | Newton Raphson | - |
| Edexcel A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| Edexcel AS Further Maths 2017 | Further Pure 1 | Numerical Methods | Newton Raphson | - |
| Edexcel AS/A2 Further Maths 2017 | Further Pure 1 | Numerical Methods | Newton Raphson | - |
| Edexcel AS/A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| OCR A-Level (UK - Pre-2017) | FP2 | Numerical Methods | Newton Raphson | - |
| OCR A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| OCR AS Further Maths 2017 | Pure Core | Numerical Methods - Extra | Newton Raphson | - |
| OCR MEI A2 Maths 2017 | Pure Maths | Numerical Methods | Newton-Raphson Method | - |
| OCR MEI AS Further Maths 2017 | Numerical Methods | Solution of Equations | Newton Raphson | - |
| OCR-MEI A-Level (UK - Pre-2017) | NM | Numerical Methods | Newton Raphson | - |
| Pre-U A-Level (UK) | 8 | Numerical Methods | Newton Raphson | - |
| Universal (all site questions) | N | Numerical Methods | Newton Raphson | - |
| WJEC A-Level (Wales) | FP3 | Numerical Methods | Newton Raphson | - |