secant method

04/22/23http://

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Secant Method

Civil Engineering Majors

Authors: Autar Kaw, Jai Paul

http://numericalmethods.eng.usf.eduTransforming Numerical Methods Education for STEM

Undergraduates

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Secant Method

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Secant Method – Derivation

)(xf)f(x - = xx

i

iii 1

f ( x )

f ( x i )

f ( x i - 1 )

x i + 2 x i + 1 x i X

ii xfx ,

1

1 )()()(

ii

iii xx

xfxfxf

)()())((

1

11

ii

iiiii xfxf

xxxfxx

Newton’s Method

Approximate the derivative

Substituting Equation (2) into Equation (1) gives the Secant method

(1)

(2)

Figure 1 Geometrical illustration of the Newton-Raphson method.


Secant Method – Derivation

)()())((

1

11

ii

iiiii xfxf

xxxfxx

The Geometric Similar Triangles

f(x)

f(xi)

f(xi-1)

xi+1 xi-1 xi X

B

C

E D A

11

1

1

)()(

ii

i

ii

i

xxxf

xxxf

DEDC

AEAB

Figure 2 Geometrical representation of the Secant method.

The secant method can also be derived from geometry:

can be written as

On rearranging, the secant method is given as


Algorithm for Secant Method


Step 1

0101

1 x

- xx = i

iia

Calculate the next estimate of the root from two initial guesses

Find the absolute relative approximate error)()())((

1

11

ii

iiiii xfxf

xxxfxx


Step 2Find if the absolute relative approximate error is greater than the prespecified relative error tolerance.

If so, go back to step 1, else stop the algorithm.

Also check if the number of iterations has exceeded the maximum number of iterations.


Example 1 You are making a bookshelf to carry books that range from 8 ½ ” to 11” in height and would take 29”of space along length. The material is wood having Young’s Modulus 3.667 Msi, thickness 3/8 ” and width 12”. You want to find the maximum vertical deflection of the bookshelf. The vertical deflection of the shelf is given by

xxxxxv 018507.010 x 66722.010 x 13533.010 x 42493.0)( 4-65-83-4

where x is the position where the deflection is maximum. Hence to findthe maximum deflection we need to find where and conduct the second derivative test.

0)( dxdvxf


Example 1 Cont.The equation that gives the position x where the deflection is maximum is given by

Use the secant method of finding roots of equations to find the position where the deflection is maximum. Conduct three iterations to estimate the root of the above equation. Find the absolute relative approximate error at the end of each iteration and the number of significant digits at least correct at the end of each iteration.

0018507.010x 12748.010 x 26689.010x 67665.0)f( 2-33-54-8 x xx x

Books

Bookshelf

Figure 2 A loaded bookshelf.


Example 1 Cont.

0018507.010x 12748.010 x 26689.010x 67665.0)f( 2-33-54-8 x xx x

Figure 3 Graph of the function f(x).

0 5 10 15 20 25 300.02

0.01

0

0.01

0.02

f(x)

0.01883

0.01851

0f x( )

290 x


Example 1 Cont.Let us take the initial guesses of the root of as and .Iteration 1The estimate of the root is

0xf 101 x 150 x

4

233548

20

330

540

80

102591.8

018507.0151012748.015106689.2151067665.0

018507.01012748.0106689.21067665.0

xxxxf

10

10001

xfxfxxxfxx

3

233548

21

331

541

81

104956.8

018507.0101012748.010106689.2101067665.0

018507.01012748.0106689.21067665.0

xxxxf

Solution


Example 1 Cont.

Figure 4 Graph of the estimated root after Iteration 1.

557.14104956.8102591.8

1015102591.815 34

4

1

x

0 5 10 15 20 25 300.03

0.02

0.01

0

0.01

0.02

0.03

f(x)x'1, (first guess)x0, (previous guess)Secant linex1, (new guess)

0.027

0.027

0

f x( )

f x( )

f x( )

secant x( )

f x( )

290 x x 0 x 1' x x 1


Example 1 Cont.The absolute relative approximate error at the end of Iteration 1 is

a

% 0433.3

100557.14

15557.14

1001

01

x

xxa

The number of significant digits at least correct is 1, because the absolute relative approximate error is less than 5%.


Example 1 Cont.Iteration 2The estimate of the root is

5

233548

21

331

541

81

109870.2

018507.0557.141012748.0557.14106689.2557.141067665.0

018507.01012748.0106689.21067665.0

xxxxf

01

01112 xfxf

xxxfxx

572.14102591.8109870.215557.14109870.215 45

5

2

x


Example 1 Cont.

Figure 5 Graph of the estimate root after Iteration 2.

0 5 10 15 20 25 300.03

0.02

0.01

0

0.01

0.02

0.03

f(x)x1 (guess)x0 (previous guess)Secant linex2 (new guess)

0.028

0.028

0

f x( )

f x( )

f x( )

secant x( )

f x( )

290 x x 1 x 0 x x 2



a

% 10611.0

100572.14

557.14572.14

1002

12

x

xxa

The number of significant digits at least correct is 2, because the absolute relative approximate error is less than 0.5%.


Example 1 Cont.Iteration 3The estimate of the root is

9

233548

22

332

542

82

100676.6

018507.0572.141012748.0572.14106689.2572.141067665.0

018507.01012748.0106689.21067665.0

xxxxf

12

12223 xfxf

xxxfxx

572.14109870.2100676.6557.14572.14100676.6572.14 59

9

2

x


Example 1 Cont.

Figure 6 Graph of the estimate root after Iteration 3.

Entered function along given interval with current and next root and the tangent line of the curve at the current root

0 5.8 11.6 17.4 23.2 290.03

0.02

0.01

0

0.01

0.02

0.03

f(x)x2 (guess)x1 (previous guess)Secant linex3 (new guess)

0.028

0.028

0

f x( )

f x( )

f x( )

secant x( )

f x( )

290 x x 2 x 1 x x 3



a

%102.1559

100572.14

572.14572.14

100

5

2

12

x

xxa

The number of significant digits at least correct is 6, because the absolute relative approximate error is less than 0.00005%.


Advantages

Converges fast, if it converges Requires two guesses that do not need

to bracket the root


Drawbacks

Division by zero

10 5 0 5 102

1

0

1

2

f(x)prev. guessnew guess

2

2

0f x( )

f x( )

f x( )

1010 x x guess1 x guess2

0 xSinxf


Drawbacks (continued)

Root Jumping

10 5 0 5 102

1

0

1

2

f(x)x'1, (first guess)x0, (previous guess)Secant linex1, (new guess)

2

2

0

f x( )

f x( )

f x( )

secant x( )

f x( )

1010 x x 0 x 1' x x 1

0Sinxxf

Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

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secant method

Documents

x http

unknown http

books bookshelf http

secant method12figure

maximum vertical deflection

findthe maximum deflection

end of iteration

maximum number of iterations