Bisection
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This lecture contains Bisection Method working rule, Graphical representation, Example, Pros and cons of this method and a Matlab Code.
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The document describes the bisection method, a root-finding algorithm that uses binary search to find roots (values that make a function equal to zero) of a continuous function. It works by repeatedly bisecting an interval known to contain a root and narrowing in on the root. The key steps are: (1) Start with an interval [a,b] where the function changes sign, ensuring a root exists in the interval. (2) Calculate the midpoint m of the interval. (3) Determine which subinterval [a,m] or [m,b] contains the root based on the sign change and update the interval accordingly. (4) Repeat until the interval size is small enough to approximate the root.
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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 bisection method is used to find the root of equations by repeatedly bisecting an interval and determining if the function value at the midpoint is positive or negative. The document provides examples of using the bisection method to find roots of equations like X^3-X-1, 4sinx-e^x, and X^2-4X-10. It shows calculating the function values at the endpoints of intervals, determining if the sign changes, bisecting the interval, and repeating until converging on the root.
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The document describes the false position method for finding roots of equations. It involves using linear interpolation between two initial guesses that bracket the root to find a new, improved estimate of the root. The method iteratively calculates new estimates using a false position formula until converging to within a specified tolerance of the true root. While faster than bisection, false position may converge less precisely in some cases if the graph is convex down between the initial guesses.
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The document discusses the bisection method, an iterative algorithm for finding approximations to the solutions of equations. It works by repeatedly bisecting an interval containing the solution and narrowing in on the answer. The method is demonstrated by using it to find the solution of x^2 - 2 = 0 between 1 and 2. It converges to a solution between 1.41420 and 1.41422, which is approximately √2 to 4 decimal places. The reader is then tasked with using the bisection method to solve the equation x^3 + x - 1 = 0 to 4 decimal places of accuracy by choosing an appropriate interval containing the single real solution.
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The document discusses the bisection method, a numerical method for finding the roots of a polynomial function. The bisection method works by repeatedly bisecting an interval where the function changes sign and narrowing in on a root. It involves finding an interval [a,b] where f(a) and f(b) have opposite signs, taking the midpoint x1, and discarding half of the interval based on the sign of f(x1). This process is repeated, halving the interval width each time, until a root is isolated within the desired accuracy. An example applying the bisection method to find the roots of f(x) = x^3 + 3x - 5 on the interval [1,2
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The document describes the bisection method, a root-finding algorithm that uses binary search to find roots (values that make a function equal to zero) of a continuous function. It works by repeatedly bisecting an interval known to contain a root and narrowing in on the root. The key steps are: (1) Start with an interval [a,b] where the function changes sign, ensuring a root exists in the interval. (2) Calculate the midpoint m of the interval. (3) Determine which subinterval [a,m] or [m,b] contains the root based on the sign change and update the interval accordingly. (4) Repeat until the interval size is small enough to approximate the root.
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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 bisection method is used to find the root of equations by repeatedly bisecting an interval and determining if the function value at the midpoint is positive or negative. The document provides examples of using the bisection method to find roots of equations like X^3-X-1, 4sinx-e^x, and X^2-4X-10. It shows calculating the function values at the endpoints of intervals, determining if the sign changes, bisecting the interval, and repeating until converging on the root.
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The document describes the false position method for finding roots of equations. It involves using linear interpolation between two initial guesses that bracket the root to find a new, improved estimate of the root. The method iteratively calculates new estimates using a false position formula until converging to within a specified tolerance of the true root. While faster than bisection, false position may converge less precisely in some cases if the graph is convex down between the initial guesses.
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The document discusses the bisection method, an iterative algorithm for finding approximations to the solutions of equations. It works by repeatedly bisecting an interval containing the solution and narrowing in on the answer. The method is demonstrated by using it to find the solution of x^2 - 2 = 0 between 1 and 2. It converges to a solution between 1.41420 and 1.41422, which is approximately √2 to 4 decimal places. The reader is then tasked with using the bisection method to solve the equation x^3 + x - 1 = 0 to 4 decimal places of accuracy by choosing an appropriate interval containing the single real solution.
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The document discusses numerical methods for solving algebraic and transcendental equations. It describes direct and iterative methods. Bisection, regula falsi, and Newton Raphson are iterative root-finding algorithms explained in detail with examples. The order of convergence of iterative methods is defined as the rate at which error decreases between successive approximations. The document serves as seminar material on engineering mathematics covering numerical solutions of equations.
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The document discusses the bisection method, a numerical method for finding the roots of a polynomial function. The bisection method works by repeatedly bisecting an interval where the function changes sign and narrowing in on a root. It involves finding an interval [a,b] where f(a) and f(b) have opposite signs, taking the midpoint x1, and discarding half of the interval based on the sign of f(x1). This process is repeated, halving the interval width each time, until a root is isolated within the desired accuracy. An example applying the bisection method to find the roots of f(x) = x^3 + 3x - 5 on the interval [1,2
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3) Secant method, which is faster than bisection but slower than Newton's, and also requires starting points close to the root. Newton's and secant methods can be extended to systems of nonlinear equations.
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1) The document discusses exponent rules including the zero exponent rule, expanded power rule, and negative exponent rules.
2) It provides examples of applying these rules such as 53=1, (3c2/2d3)4={81c8/16d12}, and (3/r3)-2=r6/9.
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This document discusses numerical methods for finding the roots of nonlinear equations. It covers the bisection method, Newton-Raphson method, and their applications. The bisection method uses binary search to bracket the root within intervals that are repeatedly bisected until a solution is found. The Newton-Raphson method approximates the function as a linear equation to rapidly converge on roots. Examples and real-world applications are provided for both methods.
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The document summarizes numerical methods for finding the roots of equations, including the Newton-Raphson method, secant method, false position method, and methods for handling repeated roots.
The Newton-Raphson method uses the tangent line to iteratively find better approximations to the root. It has quadratic convergence but may diverge for repeated roots. The secant method approximates the derivative using two points to overcome issues with the Newton-Raphson method. The false position method uses linear interpolation in each interval to home in on the root. For repeated roots, the modified Newton-Raphson method solves for the roots of a related function to ensure convergence.
Secent method
The secant method is a root-finding algorithm that uses successive secants of a function to linearly approximate the root. It requires two initial guesses, x0 and x1, to construct a secant line through the points (x0, f(x0)) and (x1, f(x1)). The x-intercept of this line provides the next approximation x2. Repeating this process iteratively refines the approximation until the root is found to within a desired precision. The secant method converges faster than bisection near roots and does not require evaluating derivatives, but it may fail to converge for some functions.
newton raphson method
1) Newton Raphson method is a numerical technique used to find roots of algebraic and transcendental equations. It uses successive approximations, starting from an initial guess, to find better approximations for the roots of the equations.
2) The method involves calculating the derivative of the function f(x) and determining the next approximation using the formula xn+1 = xn - f(xn)/f'(xn).
3) An example of finding the root of x3 - 2x - 5 = 0 is shown, starting from an initial guess of 2.5 and iteratively applying the Newton Raphson formula to obtain the root as 2.094551482.
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4) Using a secant method with x0 =
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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.
Introduction to Functions
- A function is a rule that maps each input to a unique output. Not every rule defines a valid function.
- For a rule to be a valid function, it must map each input to only one output. The domain is the set of valid inputs, and the range is the set of corresponding outputs.
- Functions can be represented graphically by plotting the input-output pairs. The graph of a valid function should only intersect the vertical line above each input once.
Sol78
The document summarizes the solution to finding the infinite square root of 1 - 17/16. It shows that this expression equals either 1/2 or approximately 0.073. Graphing the function f(x) = 1 - 17/16 - x reveals that starting at 1, iterations of f(x) will converge to 1/2, not 0.073, since 1/2 is a stable fixed point while 0.073 is unstable. Therefore, the infinite square root equals 1/2.
Sol78
The document summarizes the solution to finding the infinite square root of 1 - 17/16. It shows that this expression equals either 1/2 or approximately 0.073. Graphing the function f(x) = 1 - 17/16 - x reveals that starting at 1, iterations of f(x) will converge to 1/2, not 0.073, since 1/2 is a stable fixed point while 0.073 is unstable. Therefore, the infinite square root equals 1/2.
Chapter 3
The document discusses numerical methods for approximating integrals and solving non-linear equations. It introduces the trapezium rule for approximating integrals and provides examples of using the rule. It then discusses iterative methods like the iteration method and Newton-Raphson method for finding approximate roots of non-linear equations, providing examples of applying each method. The objectives are to enable students to use the trapezium rule and understand solving non-linear equations using iterative methods.
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this is document about Gauss quadrature. For more information please visit http://swiftmemberreview.com
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Fortran is a general-purpose programming language, mainly intended for mathematical computations in
science applications
this chapter is the third chapter
Basic Cal - Quarter 1 Week 1-2.pptx
The document provides an overview of key concepts in calculus limits including:
1) Limits describe the behavior of a function as its variable approaches a constant value.
2) Tables of values and graphs can be used to evaluate limits by showing how the function values change as the variable nears the constant.
3) Common limit laws are described such as addition, multiplication, and substitution which allow evaluating limits of combined functions.
Newton-Raphson Iteration marths 4 ntsm
The Newton-Raphson method estimates roots of equations by:
1) Rearranging the equation into the form f(x) = 0 and choosing an initial x-value
2) Substituting into the formula xn+1 = xn - f(xn)/f'(xn)
3) Differentiating to find f'(x) and iterating the formula using a calculator until convergence
The method may fail if the starting value is near a stationary point where f'(x) = 0, causing division by zero in the formula.
Similar to Bisection (20)
Numerical Analysis (Solution of Non-Linear Equations)
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Bisection
- 1. Numerical Computing Bisection Method Lecture # 9 By Nasima Akhtar
- 2. Bisection Method Working Rule •It begins with two values for x that bracket a root. It determines that they do in fact bracket a root because the function f(x) changes signs at these two x-values and, if f(x) is continuous. •The bisection method then successively divides the initial interval in half, finds in which half the root(s) must lie, and repeats with the endpoints of the smaller interval •Suppose, we wish to locate the root of an equation f (x) = 0 in an interval, say (x0, x1). Let f (x0) and f (x1) are of opposite signs, such that f (x0) f (x1) < 0.
- 3. Graphical Representation of Bisection Method
- 4. Formula Derivation 𝑥2= 𝑥0 + 𝑥1 2 If f (x2) = 0, then x2 is the desired root of f (x) = 0. However, if f (x2) ≠ 0 then the root may be between x0 and x2 or x2 and x1. • As we know f(𝑥0)> 0 then, we can find which value to replace for next iteration by the following condition such as • If f(𝑥0)*f(𝑥2) < 0 then, 𝑥1= 𝑥2 else 𝑥0= 𝑥2 General form of this method is given by: 𝑥𝑛 = 𝑥𝑛−2 + 𝑥𝑛−1 2
- 5. Algorithm for Bisection Method
- 6. MERITS OF BISECTION METHOD 1. The iteration using bisection method always produces a root, since the method brackets the root between two values. 2. As iterations are conducted, the length of the interval gets halved. So one can guarantee the convergence in case of the solution of the equation. 3. Bisection method is simple to program in a computer.
- 7. DEMERITS OF BISECTION METHOD 1. The convergence of bisection method is slow as it is simply based on halving the interval. 2. Cannot be applied over an interval where there is discontinuity. 3. Cannot be applied over an interval where the function takes always value of the same sign. 4. Method fails to determine complex roots (give only real roots) 5. If one of the initial guesses “𝑎0” or “𝑏0” is closer to the exact solution, it will take larger number of iterations to reach the root.
- 8. Example Numerical F(x)=𝑥3 + 3𝑥 − 5 So we consider 𝑥0 = 1 , 𝑥1 = 2 F(1)= - 1 F(2)=9 𝑥𝑛 = 𝑥𝑛−2+𝑥𝑛−1 2 ------ 1 For n=2 𝑥𝑛−2 = 1 𝑥𝑛−1=2 Putting in 1 𝑥2 = 𝑥0+𝑥1 2 = 1+2 2 = 1.5 As f( 1.5)= 2.875 i.e F(1.5)>0 X F(x) Update Value 1 -1 2 9 1.5 2.875 𝑥2 1.25 0.703125 𝑥2 1.125 -0.2011 𝑥1 1.187500 0.237061 𝑥2 1.156250 0.014557 𝑥2 1.140625 -0.094143 𝑥1 1.156250 0.040003 𝑥2
- 9. Example Numerical So replace 𝑥1with 𝑥2, 𝐼𝑛𝑡𝑒𝑟𝑣𝑎𝑙 𝑤𝑒 𝑐𝑜𝑛𝑠𝑖𝑑𝑒𝑟 𝑖𝑠 𝑥0 = 1 , 𝑥1 = 1.5 i.e [1,1.5] 𝑥2 = 𝑥0+𝑥1 2 = 1+1.5 2 = 1.25 As f( 1.25)= 2.875 i.e F(1.25)>0 𝑥3 = 𝑥1+𝑥2 2 = 1+1.25 2 = 1.125 As f( 1.125)= -0.2011 i.e F(1.125)<0 𝑥4 = 𝑥2+𝑥3 2 = 1.125+1.25 2 = 1.187500 As f(1.187500)= 0.237061 i.e. F(1.187500)>0 𝑥5 = 𝑥3+𝑥4 2 = 1.125+1.187500 2 = 1.156250 As f(1.156250)= 0.014557 i.e. F(1.156250)>0 X F(x) Up dat e Val ue 1 -1 2 9 1.5 2.875 𝑥2 1.25 0.703125 𝑥2 1.125 -0.2011 𝑥1 1.187500 0.237061 𝑥2 1.156250 0.014557 𝑥2 1.140625 -0.094143 𝑥1 1.156250 0.040003 𝑥2
- 10. Matlab Code function_x=@(x) x.^3+3*x-5; x1=1; x2=2; figure(1) fplot(function_x,[x1 x2],'b-') grid on hold on x_mid = (x1 + x2)/2; iterate=1; %fprintf('%f',abs(- 4)) %while abs(myfunction(x_mid))> 0.01 while abs(x1 - x2) > 0.01 %or you can change it to number of iterations, it can be any condition if (function_x(x2)*function_x(x _mid))<0 x1=x_mid; else x2=x_mid; end x_mid = (x1+x2)/2; %fprintf('The root of data is %gn' , x_mid); iterate=iterate+1; fprintf('%d Approximation Bracket is [%f,%f] gives function value %f,%f respectively and, mid value is %fn',iterate, x1,x2,function_x(x1),function _x(x2),x_mid); end plot(x_mid,function_x(x_mid) ,'r') fprintf('The root of data is %fn and iteration is %dn' , x_mid,iterate);
- 11. Thank You