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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:
  1. Choose a function.
  2. Drag the mouse on the graph along the x axis to animate the algorithm.

Summary/Background

MathsNet imageIn 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

Mak
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

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.

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.

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.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)FP1Numerical MethodsNewton Raphson-
AQA A2 Further Maths 2017Pure MathsNumerical MethodsNewton Raphson - Extra-
AQA A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
AQA AS/A2 Further Maths 2017Pure MathsNumerical MethodsNewton Raphson - Extra-
AQA AS/A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
CCEA A-Level (NI)C3Numerical MethodsNewton Raphson-
Edexcel A-Level (UK - Pre-2017)FP1Numerical MethodsNewton Raphson-
Edexcel A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
Edexcel AS Further Maths 2017Further Pure 1Numerical MethodsNewton Raphson-
Edexcel AS/A2 Further Maths 2017Further Pure 1Numerical MethodsNewton Raphson-
Edexcel AS/A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
OCR A-Level (UK - Pre-2017)FP2Numerical MethodsNewton Raphson-
OCR A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
OCR AS Further Maths 2017Pure CoreNumerical Methods - ExtraNewton Raphson-
OCR MEI A2 Maths 2017Pure MathsNumerical MethodsNewton-Raphson Method-
OCR MEI AS Further Maths 2017Numerical MethodsSolution of EquationsNewton Raphson-
OCR-MEI A-Level (UK - Pre-2017)NMNumerical MethodsNewton Raphson-
Pre-U A-Level (UK)8Numerical MethodsNewton Raphson-
Universal (all site questions)NNumerical MethodsNewton Raphson-
WJEC A-Level (Wales)FP3Numerical MethodsNewton Raphson-