secant method n.
Download
Skip this Video
Loading SlideShow in 5 Seconds..
Secant Method PowerPoint Presentation
Download Presentation
Secant Method

Loading in 2 Seconds...

play fullscreen
1 / 24

Secant Method - PowerPoint PPT Presentation


  • 355 Views
  • Uploaded on

Secant Method. Civil Engineering Majors Authors: Autar Kaw, Jai Paul http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates. Secant Method http://numericalmethods.eng.usf.edu. Secant Method – Derivation. Newton’s Method. (1).

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about 'Secant Method' - Mia_John


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
secant method

Secant Method

Civil Engineering Majors

Authors: Autar Kaw, Jai Paul

http://numericalmethods.eng.usf.edu

Transforming Numerical Methods Education for STEM Undergraduates

http://numericalmethods.eng.usf.edu

secant method derivation
Secant Method – Derivation

Newton’s Method

(1)

Approximate the derivative

(2)

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

Figure 1 Geometrical illustration of the Newton-Raphson method.

http://numericalmethods.eng.usf.edu

secant method derivation1
Secant Method – Derivation

The secant method can also be derived from geometry:

The Geometric Similar Triangles

can be written as

On rearranging, the secant method is given as

Figure 2 Geometrical representation of the Secant method.

http://numericalmethods.eng.usf.edu

algorithm for secant method
Algorithm for Secant Method

http://numericalmethods.eng.usf.edu

step 1
Step 1

Calculate the next estimate of the root from two initial guesses

Find the absolute relative approximate error

http://numericalmethods.eng.usf.edu

step 2
Step 2

Find 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.

http://numericalmethods.eng.usf.edu

example 1
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

where x is the position where the deflection is maximum. Hence to find

the maximum deflection we need to find where and conduct the second derivative test.

http://numericalmethods.eng.usf.edu

example 1 cont
Example 1 Cont.

The equation that gives the position x where the deflection is maximum is given by

Figure 2 A loaded bookshelf.

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.

http://numericalmethods.eng.usf.edu

example 1 cont1
Example 1 Cont.

Figure 3 Graph of the function f(x).

http://numericalmethods.eng.usf.edu

example 1 cont2
Example 1 Cont.

Solution

Let us take the initial guesses of the root of as and .

Iteration 1

The estimate of the root is

http://numericalmethods.eng.usf.edu

example 1 cont3
Example 1 Cont.

Figure 4 Graph of the estimated root after Iteration 1.

http://numericalmethods.eng.usf.edu

example 1 cont4
Example 1 Cont.

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

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

http://numericalmethods.eng.usf.edu

example 1 cont5
Example 1 Cont.

Iteration 2

The estimate of the root is

http://numericalmethods.eng.usf.edu

example 1 cont6
Example 1 Cont.

Figure 5 Graph of the estimate root after Iteration 2.

http://numericalmethods.eng.usf.edu

example 1 cont7
Example 1 Cont.

The absolute relative approximate error at the end of Iteration 2 is

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

http://numericalmethods.eng.usf.edu

example 1 cont8
Example 1 Cont.

Iteration 3

The estimate of the root is

http://numericalmethods.eng.usf.edu

example 1 cont9
Example 1 Cont.

Figure 6 Graph of the estimate root after Iteration 3.

http://numericalmethods.eng.usf.edu

example 1 cont10
Example 1 Cont.

The absolute relative approximate error at the end of Iteration 3 is

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

http://numericalmethods.eng.usf.edu

advantages
Advantages
  • Converges fast, if it converges
  • Requires two guesses that do not need to bracket the root

http://numericalmethods.eng.usf.edu

drawbacks
Drawbacks

Division by zero

http://numericalmethods.eng.usf.edu

drawbacks continued
Drawbacks (continued)

Root Jumping

http://numericalmethods.eng.usf.edu

additional resources
Additional Resources

For 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

slide24
THE END

http://numericalmethods.eng.usf.edu