False Point Method / Regula falsi method
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This lecture contains False Point Method working rule, Graphical representation, Example, Pros and cons of this method and a Matlab Code.
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This document describes the False Position Method for finding the roots of equations. The method uses linear interpolation to estimate the root between two initial guesses that bracket it. It improves on the bisection method by choosing a "false position" where the line between the guesses crosses the x-axis, rather than the midpoint. The false position formula is derived using similar triangles. An example applying the method to find a root of x^3 - 2x - 3 = 0 is shown. The merits of the false position method are faster convergence compared to bisection, while the demirits are possible non-monotonic convergence and lack of precision guarantee.
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The false position method is a root-finding algorithm that uses linear interpolation to estimate the root of a function. It improves upon the bisection method by using the function values at the endpoints of the interval rather than just their signs. The method chooses the intercept of the secant line through the two endpoints as the next approximation of the root, and continues iteratively narrowing the interval until the root is found.
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The False-Position Method is an iterative root-finding algorithm that improves upon the bisection method. It uses the slope of a line between two points to estimate a new root, rather than always bisecting the interval. Given an initial interval where the function changes sign, it calculates a new x-value at the intersection of the x-axis and a line through two existing points. It then chooses a new interval based on where the function changes sign again. The method is similar to bisection but uses a different formula to calculate the new estimate. An example finds a root of 3x + sin(x) - exp(x) = 0 between 0 and 0.5, converging to a solution of approximately 0.
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This document describes the False Position Method for finding the roots of equations. The method uses linear interpolation to estimate the root between two initial guesses that bracket it. It improves on the bisection method by choosing a "false position" where the line between the guesses crosses the x-axis, rather than the midpoint. The false position formula is derived using similar triangles. An example applying the method to find a root of x^3 - 2x - 3 = 0 is shown. The merits of the false position method are faster convergence compared to bisection, while the demirits are possible non-monotonic convergence and lack of precision guarantee.
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Integration is used in physics to determine rates of change and distances given velocities. Numerical integration is required when the antiderivative is unknown. It involves approximating the definite integral of a function as the area under its curve between bounds. The Trapezoidal Rule approximates this area using straight lines between points, while Simpson's Rule uses quadratic or cubic functions, achieving greater accuracy with fewer points. Both methods involve dividing the area into strips and summing their widths multiplied by the function values at strip points.
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The false position method is a root-finding algorithm that uses linear interpolation to estimate the root of a function. It improves upon the bisection method by using the function values at the endpoints of the interval rather than just their signs. The method chooses the intercept of the secant line through the two endpoints as the next approximation of the root, and continues iteratively narrowing the interval until the root is found.
The False-Position Method
The False-Position Method is an iterative root-finding algorithm that improves upon the bisection method. It uses the slope of a line between two points to estimate a new root, rather than always bisecting the interval. Given an initial interval where the function changes sign, it calculates a new x-value at the intersection of the x-axis and a line through two existing points. It then chooses a new interval based on where the function changes sign again. The method is similar to bisection but uses a different formula to calculate the new estimate. An example finds a root of 3x + sin(x) - exp(x) = 0 between 0 and 0.5, converging to a solution of approximately 0.
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- 1. Numerical Computing Lecture # 11 Regula Falsi Method By Nasima Akhtar
- 2. Working Rule • The Regula–Falsi Method is a numerical method for estimating the roots of a polynomial f(x). A value x replaces the midpoint in the Bisection Method and serves as the new approximation of a root of f(x). The objective is to make convergence faster. Assume that f(x) is continuous. • This method also known as CHORD METHOD ,, LINEAR INTERPOLATION and method is one of the bracketing methods and based on intermediate value theorem
- 4. Derivation • Equation Of line gives us: • 𝑦 − 𝑦1 𝑥 − 𝑥1 = 𝑦2− 𝑦1 𝑥2− 𝑥1 • 𝑦 − 𝑦1 = 𝑦2− 𝑦1 𝑥2− 𝑥1 (𝑥 − 𝑥1) • (𝑥1, 𝑦1)=(a,f(a)), (𝑥2, 𝑦2)=(b,f(b)), (𝑥 , 𝑦 )=(𝑥0, 𝑦0) • y- f(a) = f(b)−f(a) b− 𝑎 (𝑥 −a) • 0- f(a) = f(b)−f(a) b− 𝑎 (𝑥 −a) at x-axis y=0 • 0- f(a) = f(b)−f(a) b− 𝑎 (𝑥 −a) • (b− 𝑎)(− f(a) ) f(b)−f(a) =x-a • a + (b− 𝑎)(− f(a) ) f(b)−f(a) =x • 𝑎(f(b)−f(a))+(b− 𝑎)(− f(a) ) f(b)−f(a) =x ` • 𝑎(f(b))−a(f(a))−b(f(a) )+a(f(a) ) f(b)−f(a) =x ` • 𝑎(f(b)) −b(f(a) ) f(b)−f(a) =x ` • X= 𝑎(f(b)) −b(f(a) ) f(b)−f(a) `
- 5. Algorithm 1.Find points a and b such that a < b and f(a) * f(b) < 0. 2.Take the interval [a, b] and determine the next value of x1. 3.If f(x1) = 0 then x1 is an exact root, else if f(x1) * f(b) < 0 then let a = x1, else if f(a) * f(x1) < 0 then let b = x1. 4.Repeat steps 2 & 3 until f(xi) = 0 or |f(xi)| tolerance
- 6. Example Numerical • Find Approximate root using Regula Falsi method of the equation 𝑥3-4x+1 Putting values in x= 𝑎(f(b)) −b(f(a) ) f(b)−f(a) ` X F(x) a=0 1 b=1 -2 𝑥0=0.3333 F(𝑥0)= -0.2963 𝑥1=0.25714 F(𝑥1)= -0.0115 𝑥2=0.2542 F(𝑥2)= -0.0003 𝑥3=0.2541 F(𝑥3)= -0.00001 𝑥4=0.2541
- 7. Pros and Cons Advantages • 1. It always converges. • 2. It does not require the derivative. • 3. It is a quick method. Disadvantages • 1. One of the interval definitions can get stuck. • 2. It may slowdown in unfavourable situations.
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