19 newton-raphson iteration

19: Newton-Raphson 19: Newton-Raphson IterationIteration

© Christine Crisp

““Teach A Level Maths”Teach A Level Maths”

Vol. 2: A2 Core Vol. 2: A2 Core ModulesModules

Newton-Raphson Iteration

Module C3

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)(1 nn xgx

It isn’t always possible to find iterative formulae of the type

that will find the solution of every equation.

Another iterative method that is useful is called the Newton-Raphson method.


)(xfy

Suppose we want to find an approximate solution to the equation0)( xf

The root lies between 1 and 2.

To see how the method works, we’ll sketchusing .

)(xfy 1)( 3 xxxf

We’ll zoom in near


)(xfy

Suppose our first estimate is given by . 20 xWe draw the tangent to the curve at0x

0x

Each point , , . . . is closer to . 1x 2x

1x

Repeating . . .

2x

The point where the tangent meets the x-axis we call .1x


)(xfy

0x1x

),( 00 yxx

To carry out the iteration we need to find the points where the tangents meet the x-axis.

The grad. of the tangentx

y

in change the

in change the

10

00 xx

yx

dx

dy

at

0y

10 xx


)( 00 xfy and

10

00 xx

yx

dx

dy

at

We have and we need to find . 1x

Then,

10

00

/ )()(

xx

xfxf

Rearranging: )(

)(

0/

010

xf

xfxx

)(

)(

0/

001

xf

xfxx

Using and in the formula isn’t very

convenient, so, since we have)(xfy 0x

dx

dy at0y

)( 0/

10

00 xf

xx

yx

dx

dy

at


)(

)(

0/

001

xf

xfxx So,

We just need to alter the subscripts to find : 2x

)(

)(

1/

112

xf

xfxx

Generalising gives

)(

)(/1

n

nnn

xf

xfxx

We don’t need a diagram to use this formula but we must know how to differentiate .

)(xfConvergence is often very fast.

Newton-Raphson Iteratione.g. Use the Newton-Raphson method with

to find the root of the equation 20 x

013 xxcorrect to 4 d.p.

Solution: Let 1)( 3 xxxf

Differentiate: 13)( 2/ xxf

)(

)(/1

n

nnn

xf

xfxx

)13(

)1(2

3

1

n

nnnn

x

xxxx

Using a calculator we need:

ENTER,2

)d.p. ( 461801x

ANS

ANS(ANSANS

)13(

)12

3

Then,

Newton-Raphson IterationSUMMAR

YTo use the Newton-Raphson method to estimate a root of an equation:

rearrange the equation into the form 0)( xf

choose a suitable starting value for 0x

substitute and into the formula )(xf )(/ xf

differentiate to find )(xf )(/ xf

Tip: It saves a lot of errors if, before you type the formula into your calculator, you write the formula with ANS replacing every x.

use a calculator to iterate

)(

)(/1

n

nnn

xf

xfxx


Exercise

1. (a) Use the Newton-Raphson method to estimate the root of the following equation to 6 d.p. using the starting value given:

;022 23 xx 10 x

(b) What happens if you use ?

22 23 xxy

00 x(c) Use your calculator or a graph plotter to

sketch the graph of .(d) What is special about the graph at

and why does it explain the answer to (b) ?

00 x

2. Use the Newton-Raphson method to estimate one root of to 4 d.p. using

xx 1cos3 20 x



;022 23 xx 10 x(a)

xxxf 43)( 2/

)(

)(/1

n

nnn

xf

xfxx

)43(

)22(2

23

1

nn

nnnn

xx

xxxx

ANS)ANS

ANS(ANSANS

43(

)222

23

...,8395450,8571430,1 210 xxx

) d.p. 6( 8392870x


The iteration fails immediately.

(b) What happens if you use ?00 x

)43(

)22(2

23

1

nn

nnnn

xx

xxxx

(c)22 23 xxy

At x = 0, there is a stationary point.

We also notice that the tangent never meets the x-axis.

At a stationary point so in the formula we are dividing by 0.

0)(/ xf


Solution:

,20 x

2. Use the Newton-Raphson method to estimate one root of to 4 d.p. using

xx 1cos3 20 x

Let xxxf 1cos3)(

1sin3)(/ xxf

)(

)(/1

n

nnn

xf

xfxx

)1sin3(

)1cos3(1

n

nnnn x

xxxx

Radians!

86241,85621 21 xx) d.p. ( 4 86241x

)1sin3(

)1cos3(1

ANS

ANSANSANSxn


The Newton-Raphson method will fail if

i.e. at a stationary point 0)(/ xf

It will also sometimes fail to give the expected root if the initial value is close to a stationary point. Can you draw a graph to show what could happen in this case?

This is one example.


152 23 xxxy

With the iteration gives the root

instead of the closer root .

910 x

5761x 1870x

0x


The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied.For most purposes the slides can be printed as “Handouts” with up to 6 slides per sheet.


)(1 nn xgx

It isn’t always possible to find iterative formulae of the type

that will find the solution of every equation.

Another iterative method that is useful is called the Newton-Raphson method.

Newton-Raphson IterationSUMMAR

YTo use the Newton-Raphson method to estimate a root of an equation:

rearrange the equation into the form 0)( xf

choose a suitable starting value for 0x

substitute and into the formula )(xf )(/ xf

differentiate to find )(xf )(/ xf

Tip: It saves a lot of errors if, before you type the formula into your calculator, you write the formula with ANS replacing every x.

use a calculator to iterate

Newton-Raphson Iteratione.g. Use the Newton-Raphson method with

to find the root of the equation 20 x

013 xxcorrect to 4 d.p.


Differentiate: 13)( 2/ xxf

)(

)(/1

n

nnn

xf

xfxx

13

12

3

1

n

nnnn

x

xxxx

Using a calculator we need:

ENTER,2

)d.p. ( 461801x

ANS

ANSANSANS

13

12

3

Then,


The Newton-Raphson method will fail if

i.e. at a stationary point 0)(/ xf

It will also sometimes fail if the initial value is close to a stationary point.

19 newton-raphson iteration

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