Secant Method
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The secant method is a root-finding algorithm that uses successive secant lines to converge on a root of an equation. It begins with two initial points and finds where the secant line between those points intersects the x-axis. It then uses the intersection point as the next estimate and draws a new secant line. This process repeats until the estimate converges within a specified tolerance of the root. The secant method requires only function evaluations, unlike other methods that also require derivative evaluations. However, it may not always converge and provides no error bounds for the estimates.
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This document discusses linear multistep methods for solving initial value differential problems. It makes three key points:
1) Linear multistep methods can be viewed as fixed point iterative methods with a differential operator instead of an integral operator. They converge to a fixed point if consistent and stable.
2) The document proves that any linear multistep method defines a contraction mapping, ensuring convergence to a unique fixed point when the problem is Lipschitz continuous.
3) Examples of explicit and implicit linear multistep methods are given, along with their fixed point iterative formulations. These include Euler's method, trapezoidal method, and Adams-Moulton method.
Similar a Secant Method (20)
Applications of Differential Calculus in real life
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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
Notas del editor
- 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
- 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
- Source: https://www.youtube.com/watch?v=HpptfoWqhds