Secant%20Method

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Secant Method

• Civil Engineering Majors
• Authors Autar Kaw, Jai Paul
• http//numericalmethods.eng.usf.edu
• Transforming Numerical Methods Education for STEM
  Undergraduates

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Secant Method http//numericalmethods.eng.u
sf.edu
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Secant Method Derivation
Newtons 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.
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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.
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Algorithm for Secant Method
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Step 1
Calculate the next estimate of the root from two
initial guesses
Find the absolute relative approximate error
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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.

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Example 1

• You are making a bookshelf to carry books that
  range from 8 ½ to 11 in height and would take
  29of space along length. The material is wood
  having Youngs 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.
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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.
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Example 1 Cont.
Figure 3 Graph of the function f(x).
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Example 1 Cont.
Solution

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

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Example 1 Cont.
Figure 4 Graph of the estimated root after
Iteration 1.
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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.
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Example 1 Cont.

• Iteration 2
• The estimate of the root is

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Example 1 Cont.
Figure 5 Graph of the estimate root after
Iteration 2.
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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.
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Example 1 Cont.

• Iteration 3
• The estimate of the root is

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Example 1 Cont.
Figure 6 Graph of the estimate root after
Iteration 3.
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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.
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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
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Drawbacks (continued)
Root Jumping
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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

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• THE END
• http//numericalmethods.eng.usf.edu