Secant Method

1 / 24

# Secant Method - PowerPoint PPT Presentation

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).

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

## PowerPoint Slideshow about 'Secant Method' - Mia_John

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

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

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 – 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

http://numericalmethods.eng.usf.edu

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

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

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.

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.

http://numericalmethods.eng.usf.edu

Example 1 Cont.

Figure 3 Graph of the function f(x).

http://numericalmethods.eng.usf.edu

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 Cont.

Figure 4 Graph of the estimated root after Iteration 1.

http://numericalmethods.eng.usf.edu

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 Cont.

Iteration 2

The estimate of the root is

http://numericalmethods.eng.usf.edu

Example 1 Cont.

Figure 5 Graph of the estimate root after Iteration 2.

http://numericalmethods.eng.usf.edu

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 Cont.

Iteration 3

The estimate of the root is

http://numericalmethods.eng.usf.edu

Example 1 Cont.

Figure 6 Graph of the estimate root after Iteration 3.

http://numericalmethods.eng.usf.edu

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

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

http://numericalmethods.eng.usf.edu

Drawbacks

Division by zero

http://numericalmethods.eng.usf.edu

Drawbacks (continued)

Root Jumping

http://numericalmethods.eng.usf.edu