2/2/2016 1 Secant Method Electrical Engineering Majors Authors: Autar Kaw, Jai Paul

05/03/23http://

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Secant Method

Electrical Engineering Majors

Authors: Autar Kaw, Jai Paul

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

Undergraduates

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.

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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

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Algorithm for Secant Method

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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

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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.

3843 ln10775468.8ln10341077.210129241.11 RRT

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Example 1Thermistors are temperature-measuring devices based on the principle that the thermistor material exhibits a change in electrical resistance with a change in temperature. By measuring the resistance of the thermistor material, one can then determine the temperature.

Figure 3 A typical thermistor.

For a 10K3A Betatherm thermistor, the relationship between the resistance, R, of the thermistor and the temperature is given by

where T is in Kelvin and R is in ohms.

Thermally conductive epoxy coating

Tin plated copper alloy lead wires

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Example 1 Cont.For the thermistor, error of no more than ±0.01oC is acceptable. To find the range of the resistance that is within this acceptable limit at 19oC, we need to solve

and

Use the Newton-Raphson method of finding roots of equations to find the resistance R at 18.99oC.

a) Conduct three iterations to estimate the root of the above equation.

b) 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.

3843 ln10775468.8ln10341077.210129241.115.27301.19

1 RR

3843 ln10775468.8ln10341077.210129241.115.27399.18

1 RR

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Example 1 Cont.

3384 10293775.2ln10775468.8ln10341077.2)( RRRf

11000 12000 13000 14000 15000

0.00006

0.00004

0.00002

0.00002

Entered function on given interval

Figure 4 Graph of the function f(R).

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13014 107563.1105383.3

1400015000105383.315000

55

5

1

10

10001

R

RfRfRRRfRR

12000 14000 16000 18000 20000

0.00005

0.00005

0.0001

Entered function on given interval with two initial guesses and estimated root

Example 1 Cont.Initial guesses: 15000,14000 01 RR

Iteration 1The estimate of the root is

Figure 5 Graph of the estimate of the root after Iteration 1.

%257.15

10013014

1500013014a

The absolute relative approximate error is

The number of significant digits at least correct is 0.

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13083 105383.3102658.1

1500013014102658.113014

R

56

6

2

01

01112

R

RffRRRfRR

12000 14000 16000 18000 20000

0.00005

0.00005

0.0001

Entered function on given interval with two initial guesses and estimated root

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

Figure 6 Graph of the estimate of the root after Iteration 2.

The absolute relative approximate error is

The number of significant digits at least correct is 1.

%52422.0

10013083

1301413083a

12000 14000 16000 18000 20000

0.00005

0.00005

0.0001

Entered function on given interval with two initial guesses and estimated root

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13078 102658.1108911.8

1301413083108911.813083 R

R

68

8

3

12

12223

RffRRRfRR

%034415.0

10013078

1308313078a

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

Figure 7 Graph of the estimate of the root after Iteration 3.

The absolute relative approximate error is

The number of significant digits at least correct is 3.

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Advantages

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

to bracket the root

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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

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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

http://numericalmethods.eng.usf.edu/topics/secant_method.html

THE END

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