Major: All Engineering Majors. Authors: Autar Kaw, Jai Paul

Transcription

1 Secant Method Major: All Engneerng Majors Authors: Autar Kaw, Ja Paul Transormng Numercal Methods Educaton or STEM Undergraduates /0/00

2 Secant Method

3 Secant Method Dervaton () ( ) ( - ) θ + + [ ( )], Fgure Geometrcal llustraton o the Newton-Raphson method. X Newton s Method ( ) + - ( ) ( + ) ( ) ( ( ( )( ) ) ( () Appromate the dervatve () Substtutng Equaton () nto Equaton () gves the Secant method ) )

4 Secant Method Dervaton The secant method can also be derved rom geometry: () ( ) ( - ) C E D A + - B X The Geometrc Smlar Trangles AB DC AE DE can be wrtten as ( ) ( ) + + On rearrangng, the secant method s gven as Fgure Geometrcal representaton o the Secant method. + ( ( )( ) ( ) ) 4

5 Algorthm or Secant Method 5

6 6 Step 00 - a + + Calculate the net estmate o the root rom two ntal guesses Fnd the absolute relatve appromate error ) ( ) ( ) )( ( +

7 Step Fnd the absolute relatve appromate error s greater than the prespeced relatve error tolerance. I so, go back to step, else stop the algorthm. Also check the number o teratons has eceeded the mamum number o teratons. 7

8 Eample You are workng or DOWN THE TOILET COMPANY that makes loats or ABC commodes. The loatng ball has a specc gravty o 0.6 and has a radus o 5.5 cm. You are asked to nd the depth to whch the ball s submerged when loatng n water. 8 Fgure Floatng Ball Problem.

9 Eample Cont. 9 The equaton that gves the depth to whch the ball s submerged under water s gven by ( ) 4 Use the Secant method o ndng roots o equatons to nd the depth to whch the ball s submerged under water. Conduct three teratons to estmate the root o the above equaton. Fnd the absolute relatve appromate error and the number o sgncant dgts at least correct at the end o each teraton.

10 To ad n the understandng o how ths method works to nd the root o an equaton, the graph o () s shown to the rght, where Soluton ( ) Eample Cont. Fgure 4 Graph o the uncton (). 0

11 Eample Cont. Let us assume the ntal guesses o the root o as 0.0 and ( ) 0 0 Iteraton The estmate o the root s ( 0 )( 0 ) ( 0 ) ( ) ( ( 0.05) )( ) ( 0.05) ( ) ( ( ).99 0 )

12 Eample Cont. The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s 0, as you need an absolute relatve appromate error o 5% or less or one sgncant dgts to be correct n your result.

13 Eample Cont. Fgure 5 Graph o results o Iteraton.

14 Eample Cont. Iteraton The estmate o the root s ( )( 0 ) ( ) ( ) 0 4 ( ( ) )( ) ( ) ( ) ( ( ).99 0 )

15 Eample Cont. The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s, as you need an absolute relatve appromate error o 5% or less. 5

16 Eample Cont. Fgure 6 Graph o results o Iteraton. 6

17 Eample Cont. Iteraton The estmate o the root s ( )( ) ( ) ( ) 4 ( ( 0.064) )( ) ( 0.064) ( ) ( ( ).99 0 )

18 Eample Cont. The absolute relatve appromate error Iteraton s a % 00 a at the end o The number o sgncant dgts at least correct s 5, as you need an absolute relatve appromate error o 0.5% or less. 8

19 Iteraton # Fgure 7 Graph o results o Iteraton. 9

20 Advantages Converges ast, t converges Requres two guesses that do not need to bracket the root 0

21 Drawbacks ( ) ( ) ( ) () prev. guess new guess, guess, guess Dvson by zero ( ) Sn( ) 0 0

22 Drawbacks (contnued) ( ) ( ) ( ) secant( ) ( ) , 0, ',, () ', (rst guess) 0, (prevous guess) Secant lne, (new guess) Root Jumpng ( ) Sn 0 0

23 Addtonal Resources For all resources on ths topc such as dgtal audovsual lectures, prmers, tetbook chapters, multple-choce tests, worksheets n MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physcal problems, please vst thod.html

24 THE END