Secant Method
- 1. Numerical Computing Secant Method Lecture # 9 By Nasima Akhtar
- 2. Secant Method Working Rule • The secant method begins by finding two points on the curve of f(x), hopefully near to the root we seek. • A graph or a few applications of bisection might be used to determine the approximate location of the root. • we draw the line through these two points and find where it intersects the x- axis. • The line through two points on the curve is called the secant line.
- 3. Graphical Representation of Secant Method 1
- 5. Derivation Approach 1 • From the similar Triangle • ABC = ADE • 𝐷𝐸 𝐵𝐶 = 𝐴𝐷 𝐴𝐵 • 𝑓(𝑥0) 𝑓(𝑥1) = 𝑥0−𝑥2 𝑥1−𝑥2 By re-arrangement we get (I have done to whole derivation on page See the next slide. 𝑥2 = 𝑥1- f(𝑥1)* 𝑥0−𝑥1 𝑓 𝑥0 −𝑓(𝑥1) 𝑥𝑛+1 = 𝑥𝑛- f(𝑥𝑛)* 𝑥𝑛−1−𝑥𝑛 𝑓 𝑥𝑛−1 −𝑓(𝑥𝑛) A C B D E
- 7. Derivation Approach 2 From Newton Raphson Method we have, 𝑥𝑖+1 = 𝑥𝑖- 𝑓(𝑥𝑖) 𝑓′(𝑥𝑖) -----------------------1 From the equation of line we have slope, 𝑓′ (𝑥𝑖)= 𝑓 𝑥𝑖−1 −𝑓 𝑥𝑖 𝑥𝑖−1−𝑥 𝑖 ------------------------2 Putting equation 2 in 1 𝑥𝑖+1 = 𝑥𝑖- 𝑓(𝑥𝑖) 𝑓 𝑥𝑖−1 −𝑓 𝑥𝑖 𝑥𝑖−1−𝑥 𝑖 𝑥𝑖+1 = 𝑥𝑖- f(𝑥𝑖)* 𝑥𝑖−1−𝑥𝑖 𝑓 𝑥𝑖−1 −𝑓(𝑥𝑖)
- 8. Algorithm for Secant Method • To determine a root of f(x) = 0, given two values, 𝑥0 and 𝑥1 , that are near the root, If If (𝑥0 ) l < I f (𝑥1)l then Swap 𝑥0 with 𝑥1 Repeat Set 𝑥2 = 𝑥1-f(𝑥1)* 𝑥0−𝑥1 𝑓 𝑥0 −𝑓(𝑥1) Set 𝑥0 = 𝑥1 . Set 𝑥1 = 𝑥2 . Until If(𝑥2)l < tolerance value.
- 9. Graphical Representation of Secant Method 2
- 10. Matlab Code f=@(x)(x^3-5*x+1); x1=2; x2 = 2.5; i = 0; while i <= 5 val = f(x2); val1 = f(x1); temp = x2 - x1; temp1 = val - val1; nVal = temp/temp1; nVal = nVal * val; nVal = x2 - nVal; x1 = x2; x2 = nVal; i = i+1; end fprintf('Point is %fn',x2)
- 11. Example Numerical • F(x)=𝒙𝟑 − 𝟓𝒙 + 𝟏 • So we consider 𝒙𝟎 =2, 𝒙𝟏 = 𝟐. 𝟓 • F(2)=-1 • F(2.5)=4.125 • 𝒙𝒏+𝟏 = 𝒙𝒏-f(𝒙𝒏)* 𝒙𝒏−𝒙𝒏−𝟏 𝒇 𝒙𝒏 −𝒇 𝒙𝒏−𝟏 • 𝒙𝟐 = 𝒙𝟏-f(𝒙𝟏)* 𝒙𝟏−𝒙𝟎 𝒇 𝒙𝟏 −𝒇(𝒙𝟎) • 𝒙𝟐 = 𝟐. 𝟓 – 4.125* 𝟐.𝟓−𝟐 (𝟒.𝟏𝟐𝟓)−(−𝟏) =2.0975 • 𝒙𝟑 = 𝒙𝟐 - f(𝒙𝟐)* 𝒙𝟐−𝒙𝟏 𝒇 𝒙𝟐 −𝒇(𝒙𝟏) =2.1213 • 𝒙𝟒 =2.1285 • 𝒙𝟓 =2.1284 X F(x) 1 -3 2 -1 2.5 4.125 3 13
- 12. Secant Method Pros and Cons • Advantages 1. No computations of derivatives 2. Only f(x) computation each step 3. It converges faster than linear rate. • Disadvantages 1. It may not converge 2. There is no guaranteed error bound for the computed iterates
- 14. Thank You
Editor's Notes
- #5: Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around). All corresponding angles are equal All corresponding sides have the same ratio
- #6: Two triangles are Similar if the only difference is size (and possibly the need to turn or flip one around). All corresponding angles are equal All corresponding sides have the same ratio
- #10: Source: https://www.youtube.com/watch?v=HpptfoWqhds