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linear interpolationThe method of linear interpolation is also known as the method of false position. To find an approximate solution to the equation f(x)=0, start by choosing values for x_1 and x_2. For the method to work, these values need to be either side of the required root (marked X in the diagram); in other words y_1 and y_2 need to be of opposite sign. The possible solution is then calculated by assuming that the curve is straight between (x_1, y_1) and (x_2,y_2) (hence the term linear interpolation) and using the similar triangles ABC and XAD to find the position of X.
Because ABC and XAD are similar,
\displaystyle \frac{BC}{AC} = \frac{AD}{DX}, so
\displaystyle \frac{(y_2-y_1)}{(x_2-x_1)} = \frac{-y_1}{DX},
Our estimated solution is the x-coordinate of point X, ie., x_1 + DX, therefore
x = x_1 - \displaystyle \frac{y_1(x_2-x_1)}{(y_2-y_1)}

Summary/Background

The values you choose for x_1and x_2 need to be in the neighbourhood of the required solution. This means they need to be close enough so that they are not near other roots of the same equation.
Note the similarity between the formula above for linear interpolation, based on similar triangles:
x = x_1 - \displaystyle \frac{y_1(x_2-x_1)}{(y_2-y_1)}
and the formula for the Newton-Raphson method, based on gradient:
x_{n+1} = x_n - \displaystyle \frac{f(x_n)}{f'(x_n)}

The false position method uses the same formula as the secant method. However, the secant method does not require that the root remain bracketed like the bisection method does, and hence it does not always converge. However, it does not apply the formula on x_{n−1} and x_n, like the secant method, but on x_n and on the last iterate x_k such that f(x_k) and f(x_n) have a different sign. This means that the false position method always converges.

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

equation

A statement that two mathematical expressions are equal.

gradient

The slope of a line; the angle of its inclination to the horizontal.

interpolation

the process of estimating a value of y' for a given value of x' based on observed data where x' lies within the range of observed x.

linear

Straight, not curved. A linear equation is of the first degree, for example y = 2x+1.

secant

Trigonometry: the reciprocal of the cosine function, 1/cos x.
Coordinate geometry: a straight line that intersects two points on a curve.

solution

the answer to a problem.

straight

The type of line produced by a linear equation.

work

Equal to F x s, where F is the force in Newtons and s is the distance travelled and is measured in Joules.

Full Glossary List

This question appears in the following syllabi:

SyllabusModuleSectionTopicExam Year
AQA A-Level (UK - Pre-2017)FP1Numerical MethodsLinear interpolation-
AQA A2 Further Maths 2017Pure MathsNumerical MethodsLinear Interpolation - Extra-
AQA AS/A2 Further Maths 2017Pure MathsNumerical MethodsLinear Interpolation - Extra-
CCEA A-Level (NI)C3Numerical MethodsLinear interpolation-
Edexcel A-Level (UK - Pre-2017)FP1Numerical MethodsLinear interpolation-
Edexcel AS Further Maths 2017Further Pure 1Numerical MethodsLinear Interpolation-
Edexcel AS/A2 Further Maths 2017Further Pure 1Numerical MethodsLinear Interpolation-
Methods (UK)M2Numerical MethodsLinear interpolation-
OCR A-Level (UK - Pre-2017)FP2Numerical MethodsLinear interpolation-
OCR AS Further Maths 2017Pure CoreNumerical Methods - ExtraLinear Interpolation-
OCR MEI AS Further Maths 2017Numerical MethodsSolution of EquationsLinear Interpolation-
OCR-MEI A-Level (UK - Pre-2017)NMNumerical MethodsLinear interpolation-
Universal (all site questions)NNumerical MethodsLinear interpolation-
WJEC A-Level (Wales)FP3Numerical MethodsLinear interpolation-