Bisection
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The bisection method is a root-finding algorithm that uses binary search to find roots or zeroes of a function. It works by repeatedly bisecting an interval and determining whether the root lies in the upper or lower interval based on the sign of the function. The algorithm converges to a root by halving the size of the bracketing interval at each iteration. An example applies the bisection method to find the depth at which a floating ball is submerged. After 10 iterations, the estimated root is found to two significant digits.
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Bisection & Regual falsi methods
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bisection method
This document describes the bisection method for finding roots of equations numerically. It begins by classifying equations as linear, polynomial, or generally non-linear. For non-linear equations, numerical methods are required. The bisection method iteratively halves the interval that contains a root until a solution is found to within a specified tolerance. An example illustrates the step-by-step process of applying the bisection method to find the root of a sample function. MATLAB code is also presented to implement the bisection method.
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The document discusses various numerical methods for finding the roots or zeros of equations, where the root is the value of x that makes the equation f(x) equal to zero. It begins by defining roots and describing graphical and incremental search methods. It then covers closed methods like bisection and false position that evaluate the function within an interval known to contain the root. Open methods like Newton-Raphson, secant method, and fixed-point iteration are also discussed. The document provides examples of applying these root-finding methods and notes situations where roots could be missed. It concludes by listing references for further information.
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This document provides an overview of the bisection method for finding the roots of nonlinear equations. It begins with definitions of the bisection method and why it is used. The algorithm involves choosing initial values that bracket a root, then iteratively calculating the midpoint and narrowing the interval until the desired accuracy is reached. An example problem and real-life application are provided. Advantages are that the method is simple, robust, and guaranteed to converge for continuous functions. Disadvantages include slow convergence and inability to find roots if the function just touches the x-axis. In conclusion, while simple, the bisection method always converges to find roots.
Matlab lecture 5 bisection method@taj
Matlab lecture 5 bisection method@taj
.
This lecture is from my class lectures in IIT-JU.
.
Find me:
Website: https://www.tajimiitju.blogspot.com
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Bisection method
The document describes the bisection method for finding roots of equations. It provides an introduction to the bisection method and its graphical representation. It also presents the algorithm, a C program implementing the method, and examples finding roots of polynomial and trigonometric equations using bisection.
Regula Falsi (False position) Method
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This document describes the bisection method for finding roots of equations numerically. It begins by classifying equations as linear, polynomial, or generally non-linear. For non-linear equations, numerical methods are required. The bisection method iteratively halves the interval that contains a root until a solution is found to within a specified tolerance. An example illustrates the step-by-step process of applying the bisection method to find the root of a sample function. MATLAB code is also presented to implement the bisection method.
Applications of numerical methods
The document discusses numerical methods for finding roots of equations and integrating functions. It covers root-finding algorithms like the bisection method, Regula Falsi method, modified Regula Falsi, and secant method. These algorithms iteratively find roots by narrowing the interval that contains the root. The document also discusses numerical integration techniques like the trapezoidal rule to approximate the area under a curve without having a closed-form solution. It notes the tradeoffs between different root-finding algorithms in terms of speed, accuracy, and ability to guarantee convergence.
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The document discusses various numerical methods for finding the roots or zeros of equations, where the root is the value of x that makes the equation f(x) equal to zero. It begins by defining roots and describing graphical and incremental search methods. It then covers closed methods like bisection and false position that evaluate the function within an interval known to contain the root. Open methods like Newton-Raphson, secant method, and fixed-point iteration are also discussed. The document provides examples of applying these root-finding methods and notes situations where roots could be missed. It concludes by listing references for further information.
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This document provides an overview of the bisection method for finding the roots of nonlinear equations. It begins with definitions of the bisection method and why it is used. The algorithm involves choosing initial values that bracket a root, then iteratively calculating the midpoint and narrowing the interval until the desired accuracy is reached. An example problem and real-life application are provided. Advantages are that the method is simple, robust, and guaranteed to converge for continuous functions. Disadvantages include slow convergence and inability to find roots if the function just touches the x-axis. In conclusion, while simple, the bisection method always converges to find roots.
Matlab lecture 5 bisection method@taj
Matlab lecture 5 bisection method@taj
.
This lecture is from my class lectures in IIT-JU.
.
Find me:
Website: https://www.tajimiitju.blogspot.com
Linked in: https://www.linkedin.com/in/tajimiitju
Researchgate: https://www.researchgate.net/profile/Tajim_Md_Niamat_Ullah_Akhund
Youtube: https://www.youtube.com/tajimiitju?sub_confirmation=1
Slideshare: https://www.slideshare.net/TajimMdNiamatUllahAk
Facebook: https://www.facebook.com/tajim.mohammad
Gitlab: https://gitlab.com/users/tajimiitju
Google+: plus.google.com/+tajimiitju
Email: tajim.mohammad.3@gmail.com
Twitter: https://twitter.com/Tajim53
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The document describes the bisection method for finding roots of equations. It provides an introduction to the bisection method and its graphical representation. It also presents the algorithm, a C program implementing the method, and examples finding roots of polynomial and trigonometric equations using bisection.
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Solving an equation f(x) = g(x) is the same as finding the roots of the function f – g. Thus root-finding algorithms allows solving any equation.
Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points.
The behaviour of root-finding algorithms is studied in numerical analysis. Algorithms perform best when they take advantage of known characteristics of the given function, so different algorithms are used to solve different types of equations. Desirable characteristics include a rapid rate of convergence, ability to separate close roots, robustness against failures of differentiability, and low propagation rate of rounding errors.
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Visit here for getting code implementation- https://github.com/MinhasKamal/AlgorithmImplementations/blob/master/numericalMethods/equationSolving/NonLinearEquationSolvingProcess.c
Created in 2nd year of Bachelor of Science in Software Engineering (BSSE) course at Institute of Information Technology, University of Dhaka (IIT, DU).
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This document discusses methods for finding the roots of equations. It begins by explaining the importance of determining roots and how it relates to other mathematical problems. It then outlines different types of methods including graphical methods, which provide initial estimates but lack precision, and closed methods, which limit the search domain. Specific closed methods discussed include bisection, false position, fixed point, Newton-Raphson, and secant methods. Graphics are provided to illustrate each method.
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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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- Techniques for polynomials like Müller's and Bairstow's methods.
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The document discusses several numerical methods for finding the roots or zeros of equations:
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2) Bisection and incremental search methods iteratively evaluate the function across intervals to find where the sign changes, narrowing in on a root.
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The document discusses the bisection method for finding roots of equations. It begins by defining the bisection method as a root finding technique that repeatedly bisects an interval and selects a subinterval containing the root. It notes that while simple and robust, the bisection method converges slowly. The document then provides the step-by-step algorithm for implementing the bisection method and works through an example of finding the root of f(x) = x^2 - 2 between 1 and 2. It concludes by presenting the bisection method code in C++.
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This document discusses methods for finding the roots of equations. It begins by explaining the importance of determining roots and how it relates to other mathematical problems. It then outlines different types of methods including graphical methods, which provide initial estimates but lack precision, and closed methods, which limit the search domain. Specific closed methods discussed include bisection, false position, fixed point, Newton-Raphson, and secant methods. Graphics are provided to illustrate each method.
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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 discusses the bisection method for finding real roots of equations. It provides the step-by-step algorithm for applying the bisection method. The key steps are: (1) find two values a and b where the function has opposite signs, (2) compute the midpoint x0 between a and b and evaluate the function there, (3) replace either a or b with x0 depending on whether f(x0) is positive or negative, and (4) repeat until the desired accuracy is reached. The document includes an example of applying the bisection method to find the root of the equation f(x) = x^3 - 2x - 5 between 2 and 3.
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In mathematics and computing, a root-finding algorithm is an algorithm, for finding values x such that f(x) = 0, for a given continuous function f from the real numbers to real numbers or from the complex numbers to the complex numbers. Such an x is called a root or zero of the function f. As, generally, the roots may not be described exactly, they are approximated as floating point numbers, or isolated in small intervals (or disks for complex roots), an interval or disk output being equivalent to an approximate output together with an error bound.
Solving an equation f(x) = g(x) is the same as finding the roots of the function f – g. Thus root-finding algorithms allows solving any equation.
Numerical root-finding methods use iteration, producing a sequence of numbers that hopefully converge towards the root as a limit. They require one or more initial guesses of the root as starting values, then each iteration of the algorithm produces a successively more accurate approximation to the root. Since the iteration must be stopped at some point these methods produce an approximation to the root, not an exact solution. Many methods compute subsequent values by evaluating an auxiliary function on the preceding values. The limit is thus a fixed point of the auxiliary function, which is chosen for having the roots of the original equation as fixed points.
The behaviour of root-finding algorithms is studied in numerical analysis. Algorithms perform best when they take advantage of known characteristics of the given function, so different algorithms are used to solve different types of equations. Desirable characteristics include a rapid rate of convergence, ability to separate close roots, robustness against failures of differentiability, and low propagation rate of rounding errors.
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The document describes the bisection method for finding the root of an equation. It provides the theoretical basis and algorithm for the bisection method. An example problem is worked through over 3 iterations to demonstrate how the method converges on a root by narrowing the range between the lower and upper bounds. The example tracks the estimate of the root and absolute relative error at each step. Advantages and drawbacks of the bisection method are also summarized.
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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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Created in 2nd year of Bachelor of Science in Software Engineering (BSSE) course at Institute of Information Technology, University of Dhaka (IIT, DU).
Roots of equations
This document discusses methods for finding the roots of equations. It begins by explaining the importance of determining roots and how it relates to other mathematical problems. It then outlines different types of methods including graphical methods, which provide initial estimates but lack precision, and closed methods, which limit the search domain. Specific closed methods discussed include bisection, false position, fixed point, Newton-Raphson, and secant methods. Graphics are provided to illustrate each method.
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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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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 document discusses various numerical methods for finding roots of functions, including:
- Bracketing methods like bisection and false position that search between initial lower and upper bounds.
- Open methods like Newton-Raphson and secant that do not require bracketing but may not converge.
- Techniques for polynomials like Müller's and Bairstow's methods.
Examples demonstrate applying bisection, false position, and Newton-Raphson to find the mass in a falling object problem. The convergence properties and relative performance of the different methods are analyzed.
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The document discusses several numerical methods for finding the roots or zeros of equations:
1) The graphical method involves plotting the function and determining where it crosses the x-axis, approximating the root.
2) Bisection and incremental search methods iteratively evaluate the function across intervals to find where the sign changes, narrowing in on a root.
3) Methods like Newton-Raphson and secant methods use tangents or secants extended from points to intersect the x-axis and give better estimates with each iteration.
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This document discusses the application of numerical methods in real-life problems. It provides examples of using the bisection method to find the root of equations related to estimating ocean currents, modeling combustion flow, airflow patterns, and other applications. Specifically, it shows the steps to use the bisection method to estimate the depth at which a floating ball with given properties would be submerged. Over three iterations, it computes the estimated root, error, and number of significant digits estimated.
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This chapter discusses various types of errors that can occur in numerical analysis calculations, including:
- Round-off errors due to limitations in significant figures and binary representation in computers
- Truncation errors from using approximations instead of exact mathematical representations
- Error propagation when combining results with arithmetic operations
It also covers topics like accuracy vs precision, definitions of relative and absolute errors, floating point representation standards, and techniques to estimate errors like Taylor series expansions and machine epsilon values. The goal is to understand the sources and magnitudes of different errors to improve the reliability of numerical analysis methods.
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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.
Bisection method
The document discusses the bisection method for finding roots of equations. It begins by defining the bisection method as a root finding technique that repeatedly bisects an interval and selects a subinterval containing the root. It notes that while simple and robust, the bisection method converges slowly. The document then provides the step-by-step algorithm for implementing the bisection method and works through an example of finding the root of f(x) = x^2 - 2 between 1 and 2. It concludes by presenting the bisection method code in C++.
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This document discusses methods for finding the roots of equations. It begins by explaining the importance of determining roots and how it relates to other mathematical problems. It then outlines different types of methods including graphical methods, which provide initial estimates but lack precision, and closed methods, which limit the search domain. Specific closed methods discussed include bisection, false position, fixed point, Newton-Raphson, and secant methods. Graphics are provided to illustrate each method.
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The document discusses various numerical methods for finding the roots or zeros of equations, including closed and open methods. Closed methods like bisection and false position trap the root within a closed interval by repeatedly dividing the interval in half. Open methods like Newton-Raphson and secant methods use information about the nonlinear function to iteratively refine the estimated root without being restricted to an interval. The document also covers methods for equations with multiple roots like Muller's method.
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The document discusses the bisection method for finding real roots of equations. It provides the step-by-step algorithm for applying the bisection method. The key steps are: (1) find two values a and b where the function has opposite signs, (2) compute the midpoint x0 between a and b and evaluate the function there, (3) replace either a or b with x0 depending on whether f(x0) is positive or negative, and (4) repeat until the desired accuracy is reached. The document includes an example of applying the bisection method to find the root of the equation f(x) = x^3 - 2x - 5 between 2 and 3.
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The document discusses numerical computing and various interpolation techniques. Numerical computing involves solving complex mathematical problems using simple arithmetic operations by formulating models that can be solved numerically. The document then discusses nonlinear equations and various iterative methods to solve them, including bracketing methods like bisection and regula falsi, and open-end methods like Newton-Raphson and secant. It also discusses fixed point iteration. Finally, it covers interpolation techniques like Lagrange interpolation and Newton interpolation to estimate values of a function at intermediate points.
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This document discusses the secant method for finding roots of equations numerically. It provides an overview of the secant method graphically and analytically based on the Newton-Raphson method formula. It then gives an example problem of using the secant method to find a real root of an equation accurate to five significant figures, showing the calculations in a tabular form. The root is calculated to be 3.1004.
The newton raphson method
Newton's method is an iterative method used to find successively better approximations to the roots, or zeroes, of a real-valued function. It works by taking an initial guess, calculating the tangent line to the function at that point, and using the x-intercept of the tangent line as the next guess. The process is repeated by calculating new tangent lines until an accurate value is reached. Newton's method can also be extended to complex functions and systems of equations.
Numerical Methods
The document summarizes numerical integration methods for solving equations of motion directly in the time domain, including explicit and implicit methods. It describes Newmark's β method, the central difference method, and Wilson-θ method. Key steps involve discretizing the equations of motion and relating response parameters at different time steps using finite difference approximations. Stability, accuracy, and error considerations are also discussed.
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.
Newton raphson
The Newton-Raphson method is used to find the root of a function by iteratively guessing values that get closer to the root. It involves writing the function, taking its derivative, and using the previous guess to calculate the next guess until the guesses converge on the root. The document provides examples of using Newton-Raphson as well as the fixed point and secant methods, which similarly find roots through iterative guessing.
Newton-Raphson Method
The document describes the Newton-Raphson method for finding the roots of nonlinear equations. It provides the derivation of the method, outlines the algorithm as a 3-step process, and gives an example of applying it to find the depth a floating ball submerges in water. The advantages are that it converges fast if it converges and requires only one initial guess. Drawbacks include potential issues with division by zero, root jumping, and oscillations near local extrema.
APPLICATION OF NUMERICAL METHODS IN SMALL SIZE
This document summarizes numerical methods used in various fields including engineering, crime detection, scientific computing, finding roots, and solving heat equations. It discusses how numerical methods are widely used in engineering to model systems using mathematical equations when analytical solutions are not possible. Examples of applying numerical methods include structural analysis, fluid dynamics, image processing to deblur photos, and algorithms for finding roots of equations and solving differential equations.
Application of Numerical method in Real Life
This presentation discusses the application of numerical methods in real-life scenarios. It provides examples such as estimating ocean currents, modeling combustion flow in coal power plants, and simulating airflow over airplane bodies. The presentation also examines modeling electromagnetics, shuttle/tank separation, and other applications involving differential equations, programming, control systems, and data fitting. In total, 16 real-world uses of numerical methods are outlined.
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.
Roots equation
The document discusses different methods for finding the roots or zeros of equations, where the root is the value of x that makes the equation f(x) equal to 0. It describes graphical methods, incremental search methods like bisection which divides the interval in half each iteration, and open methods like Newton-Raphson that use calculus to home in on the root. It also addresses challenges in finding multiple roots where the function is tangent to the x-axis at the root.
Secante
The document discusses the secant method, a root-finding algorithm that approximates the derivative using two points. It derives the secant method from Newton's method and geometry. The document also provides an example of using the secant method to find the maximum deflection of a loaded bookshelf, running through three iterations and calculating the absolute relative approximate error and number of significant digits after each iteration.
Newton-Raphson Method
The document discusses the Newton-Raphson method for solving nonlinear equations. It involves starting with an initial guess for the root and iteratively improving the estimate using the function value and derivative until reaching an acceptable approximation of the true root. The method relies on linearizing the function using its tangent line to generate improved estimates in each step.
Numerical analysis (Bisectional method) application
Monsur Ahmed Shafiq's presentation discusses the application of numerical methods. It focuses on the bisection method, a simple root-finding technique that approximates solutions to equations. The presentation provides examples of how numerical methods can be used in scientific programming, modeling airflow over airplanes, estimating ocean currents, modeling combustion flow in power plants, and electromagnetics. It also mentions two projects Shafiq completed, making a calendar and a Bangla dictionary.
Newton raphson method
The document discusses the Newton Raphson method for finding roots of equations. It describes how Isaac Newton and Joseph Raphson discovered the method in the 17th century. The method works by taking the derivative of the function and using it to calculate successive approximations that converge on a root. The document provides an example of using the method to find the root of a function and discusses advantages like fast convergence and requiring only an initial guess, as well as potential drawbacks such as failure to converge or slow convergence for roots with multiplicity greater than one.
Applied numerical methods lec4
This document discusses the bracketing method for finding the roots of equations. It begins by introducing bracketing methods and noting that they require initial guesses that bracket a root. It then provides an example of using a graphical method to find the root of an equation by plotting the function and observing where it crosses the x-axis. Finally, it describes the bisection method in more detail, noting that it iteratively narrows the range containing the root until the desired accuracy is reached.
Presentation on Solution to non linear equations
The document provides information about the bisection method for finding roots of non-linear equations. It defines the bisection method, outlines its basis and key steps, and provides an example of using the method to find the depth at which a floating ball is submerged in water. Over 10 iterations, the bisection method converges on an estimated root of 0.06241 for the example equation, with 2 significant digits found to be correct after the final iteration. The document also discusses an application of using the bisection method to find resistance of a thermistor at a given temperature.
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Bisection
- 1. Bisection Method http://numericalmethods.eng.usf.edu Transforming Numerical Methods Education for STEM Undergraduates 07/28/10 http://numericalmethods.eng.usf.edu
- 2. Bisection Method http://numericalmethods.eng.usf.edu
- 6. Basis of Bisection Method http://numericalmethods.eng.usf.edu Figure 4 If the function changes sign between two points, more than one root for the equation may exist between the two points.
- 7. Algorithm for Bisection Method http://numericalmethods.eng.usf.edu
- 11. Step 4 http://numericalmethods.eng.usf.edu Find the new estimate of the root Find the absolute relative approximate error where
- 12. Step 5 http://numericalmethods.eng.usf.edu Is ? Yes No Go to Step 2 using new upper and lower guesses. Stop the algorithm Compare the absolute relative approximate error with the pre-specified error tolerance . Note one should also check whether the number of iterations is more than the maximum number of iterations allowed. If so, one needs to terminate the algorithm and notify the user about it.
- 16. Example 1 Cont. To aid in the understanding of how this method works to find the root of an equation, the graph of f(x) is shown to the right, where http://numericalmethods.eng.usf.edu Figure 7 Graph of the function f(x) Solution
- 17. Example 1 Cont. http://numericalmethods.eng.usf.edu Let us assume Check if the function changes sign between x and x u . Hence So there is at least on root between x and x u, that is between 0 and 0.11
- 18. Example 1 Cont. http://numericalmethods.eng.usf.edu Figure 8 Graph demonstrating sign change between initial limits
- 19. Example 1 Cont. http://numericalmethods.eng.usf.edu Iteration 1 The estimate of the root is Hence the root is bracketed between x m and x u , that is, between 0.055 and 0.11. So, the lower and upper limits of the new bracket are At this point, the absolute relative approximate error cannot be calculated as we do not have a previous approximation.
- 20. Example 1 Cont. http://numericalmethods.eng.usf.edu Figure 9 Estimate of the root for Iteration 1
- 21. Example 1 Cont. http://numericalmethods.eng.usf.edu Iteration 2 The estimate of the root is Hence the root is bracketed between x and x m , that is, between 0.055 and 0.0825. So, the lower and upper limits of the new bracket are
- 22. Example 1 Cont. http://numericalmethods.eng.usf.edu Figure 10 Estimate of the root for Iteration 2
- 23. Example 1 Cont. http://numericalmethods.eng.usf.edu The absolute relative approximate error at the end of Iteration 2 is None of the significant digits are at least correct in the estimate root of x m = 0.0825 because the absolute relative approximate error is greater than 5%.
- 24. Example 1 Cont. http://numericalmethods.eng.usf.edu Iteration 3 The estimate of the root is Hence the root is bracketed between x and x m , that is, between 0.055 and 0.06875. So, the lower and upper limits of the new bracket are
- 25. Example 1 Cont. http://numericalmethods.eng.usf.edu Figure 11 Estimate of the root for Iteration 3
- 26. Example 1 Cont. http://numericalmethods.eng.usf.edu The absolute relative approximate error at the end of Iteration 3 is Still none of the significant digits are at least correct in the estimated root of the equation as the absolute relative approximate error is greater than 5%. Seven more iterations were conducted and these iterations are shown in Table 1.
- 27. Table 1 Cont. http://numericalmethods.eng.usf.edu Table 1 Root of f(x)=0 as function of number of iterations for bisection method.
- 28. Table 1 Cont. http://numericalmethods.eng.usf.edu Hence the number of significant digits at least correct is given by the largest value or m for which So The number of significant digits at least correct in the estimated root of 0.06241 at the end of the 10 th iteration is 2.