NUMERICAL METHODS
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The Bairstow method is an iterative technique for finding the roots of polynomials by using synthetic division to divide the polynomial by quadratic factors of the form (x2 - rx - s). It aims to find values of r and s that make the quadratic factor exactly divide the polynomial, resulting in a remainder of zero. Starting with an initial approximation of r0 and s0, it generates better approximations rk and sk using a Taylor series expansion to develop equations relating changes in r and s to the remainder values b1 and b0. This allows calculating updated rk+1 and sk+1 values that bring b1 and b0 closer to zero, iterating until a desired tolerance is reached. Once tolerance is met, the estimated
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Roots of polynomials
The Bairstow method is an iterative technique for finding the roots of polynomials by calculating the coefficients of a quadratic factor. It works by taking an initial approximation of the quadratic factor and generating better approximations using partial derivatives until the remainder of dividing the polynomial by the quadratic factor is zero, giving the roots. The method calculates the partial derivatives to update the approximations in a way that avoids having to perform calculations with complex numbers directly. When applied repeatedly, it can find all roots of polynomials of third order or higher.
NUMERICAL METHODS
The Bairstow method is an iterative technique for finding the roots of polynomials by using synthetic division to divide the polynomial by quadratic factors of the form (x2 - rx - s). It aims to find values of r and s that make the quadratic factor exactly divide the polynomial, resulting in a remainder of zero. Starting with an initial approximation of r0 and s0, it generates better approximations rk and sk using a Taylor series expansion to develop equations relating changes in r and s to the remainder values b1 and b0. This allows calculating updated rk+1 and sk+1 values that bring b1 and b0 closer to zero, iterating until a desired tolerance is reached. Once tolerance is met, the estimated
Roots of polynomials
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
METHOD
The Bairstow method is an iterative technique for finding the roots of polynomials by using synthetic division. It calculates both real and complex roots without using complex arithmetic. The method works by taking an initial approximation of r and s values and generating better approximations iteratively until the remainder of dividing the polynomial by (x^2 - rx - s) is zero. It develops functions b1 and b0 in a Taylor series to obtain equations relating changes in r and s, allowing new approximations to be calculated. The partial derivatives required are obtained through a secondary synthetic division. Once tolerance is reached, the estimated r and s values are used to calculate the roots.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
Both methods use iterative techniques to refine initial guesses and isolate the roots of polynomials without using
Chapter 3
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.
ROOTS OF EQUATIONS
1. There are conventional methods for finding multiple roots of equations, such as the Muller method and Bairstow method. The Muller method uses three initial values to obtain the coefficients of a parabola, which are then substituted into the quadratic formula to estimate the root where the parabola intersects the x-axis. The Bairstow method iteratively calculates the factors of a polynomial to determine its roots.
2. Complex roots of equations can be calculated using Euler's formula, which states that every complex number has exactly n complex roots. The formula generates each root by substituting different values for k between 0 and n-1. This allows polar forms of complex numbers to be substituted to find their roots
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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Roots of polynomials
The Bairstow method is an iterative technique for finding the roots of polynomials by calculating the coefficients of a quadratic factor. It works by taking an initial approximation of the quadratic factor and generating better approximations using partial derivatives until the remainder of dividing the polynomial by the quadratic factor is zero, giving the roots. The method calculates the partial derivatives to update the approximations in a way that avoids having to perform calculations with complex numbers directly. When applied repeatedly, it can find all roots of polynomials of third order or higher.
NUMERICAL METHODS
The Bairstow method is an iterative technique for finding the roots of polynomials by using synthetic division to divide the polynomial by quadratic factors of the form (x2 - rx - s). It aims to find values of r and s that make the quadratic factor exactly divide the polynomial, resulting in a remainder of zero. Starting with an initial approximation of r0 and s0, it generates better approximations rk and sk using a Taylor series expansion to develop equations relating changes in r and s to the remainder values b1 and b0. This allows calculating updated rk+1 and sk+1 values that bring b1 and b0 closer to zero, iterating until a desired tolerance is reached. Once tolerance is met, the estimated
Roots of polynomials
The document discusses methods for finding the roots of polynomial equations, including Muller's method and Bairstow's method. Muller's method uses three points to derive the coefficients of a parabola and find an approximated root. Bairstow's method involves synthetically dividing a polynomial by a quadratic factor to find values of r and s that make the coefficients b1 and b0 equal to zero, through an iterative process. It provides an example of applying Bairstow's method to find the roots of a 5th order polynomial.
METHOD
The Bairstow method is an iterative technique for finding the roots of polynomials by using synthetic division. It calculates both real and complex roots without using complex arithmetic. The method works by taking an initial approximation of r and s values and generating better approximations iteratively until the remainder of dividing the polynomial by (x^2 - rx - s) is zero. It develops functions b1 and b0 in a Taylor series to obtain equations relating changes in r and s, allowing new approximations to be calculated. The partial derivatives required are obtained through a secondary synthetic division. Once tolerance is reached, the estimated r and s values are used to calculate the roots.
Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain coefficients a, b, c. It then uses the quadratic formula on the parabola to find the root, and iterates with new guesses until converging on a solution.
Both methods use iterative techniques to refine initial guesses and isolate the roots of polynomials without using
Chapter 3
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.
ROOTS OF EQUATIONS
1. There are conventional methods for finding multiple roots of equations, such as the Muller method and Bairstow method. The Muller method uses three initial values to obtain the coefficients of a parabola, which are then substituted into the quadratic formula to estimate the root where the parabola intersects the x-axis. The Bairstow method iteratively calculates the factors of a polynomial to determine its roots.
2. Complex roots of equations can be calculated using Euler's formula, which states that every complex number has exactly n complex roots. The formula generates each root by substituting different values for k between 0 and n-1. This allows polar forms of complex numbers to be substituted to find their roots
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.
Chapter 3
The document discusses solving linear equations by using addition, subtraction, multiplication, and division properties. It provides examples of solving single-step and multi-step equations, including equations with decimals that require rounding. It also covers rewriting formulas to make one variable a function of another and solving real-world application problems involving formulas.
Calculus multiple integral
1. The document discusses multiple integral calculus topics including double integrals over rectangles and general regions, double integrals in polar coordinates, triple integrals in rectangular, cylindrical and spherical coordinates, and change of variables in multiple integrals.
2. It provides definitions, properties, examples and solutions for calculating double and triple integrals using rectangular, polar, cylindrical and spherical coordinate systems.
3. Formulas are given for calculating double and triple integrals in polar, cylindrical and spherical coordinates by representing the volume element and limits of integration in terms of the respective coordinate systems.
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Logarithms are exponents that indicate the power to which a base number must be raised to equal the original number. John Napier introduced logarithms in the early 17th century as a way to simplify calculations. Logarithms have many applications in fields like science, engineering, mathematics, psychology, and music. They allow complex calculations to be performed more easily.
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This document discusses several numerical methods for finding roots of equations, including the bisection method, false position method, fixed point method, Newton-Raphson method, and secant method. It provides an example of using the Newton-Raphson method to find the maximum deflection of a bookshelf beam. It also asks which function could be used with the fixed point method to find a root between 10 and 15 for the equation x3 - 30x2 + 2400 = 0. Finally, it provides an example of using the bisection method to find the root of the equation f(x)=3x+sinx-ex.
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This document provides a tutorial on solving one-variable equations with fractions through 13 examples of different types of fractional equations. The examples cover equations of the form a/bx = c, -a/bx = c, a/bx = -c, a/bx = cx + d, a/bx = c/dx + e, and other variations involving addition, subtraction, and combinations of fractions and variables.
Matlab lecture 7 – regula falsi or false position method@taj
Matlab lecture 7 – regula falsi or false position 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
Formulas for calculating surface area and volume
This document provides formulas for calculating the surface area and volume of various 3D shapes. It defines key terms like perimeter, base, lateral surface area, and total surface area. Formulas are provided for calculating the surface area and volume of cubes, rectangular prisms, triangular prisms, cylinders, cones, square pyramids, and spheres. An example problem is worked through to find the volume and total surface area of a sphere with a radius of 30 cm.
Chapter 3: Roots of Equations
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.
Polar area
1) The area enclosed by a polar curve can be found by integrating the area of sectors swept out by the radius as the angle varies over the interval.
2) The area of each sector is 1/2*r^2*Δθ. Integrating this expression from α to β gives the total area.
3) Special care must be taken when curves intersect or complete multiple cycles over the interval to break the problem into multiple integrals if needed. Graphing and inspecting the curves is important to set up the integrals correctly.
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2. The three main laws are: (1) When expressions with the same base are multiplied, the indices are added. (2) When expressions with the same base are divided, the indices are subtracted. (3) When an expression with an exponent is raised to another power, the exponents are multiplied.
3. Examples are provided to illustrate each law, such as 76 × 74 = 710 by the first law, and (a3)4 = a12 by the third law.
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December 19
The document provides instructions on graphing linear equations from points and determining the slope and situation described by a line passing through two points. It asks the reader to match a graph to an equation, graph an equation using at least three points, and find the slope of a line passing through (1,7) and (6,4). It also provides examples of forming a linear equation given the slope and y-intercept, a point and slope, or two points. Finally, it asks the reader to find the perimeters of two triangles with side lengths provided.
Roots of equations
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.
Matlab analytical solution of cubic equation
In this paper code to find real roots of cubic equation with brief of explanation about it and snapshot of program
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This document discusses techniques for integrating trigonometric functions using u-substitution and trigonometric identities. It provides examples of integrating tan(x), cot(x), sec(x), csc(x) using the log rule with u-substitutions of u = ln(x), u = cos(x), and u = sec(x) + tan(x). It also evaluates integrals using Pythagorean identities and finds the average value of functions over an interval.
NUMERICAL & STATISTICAL METHODS FOR COMPUTER ENGINEERING
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.
xx
Eray SALTIK has a computer engineering degree from Bogazici University in Istanbul. He graduated from Istanbul Erkek Lisesi high school in 2006 with honors. Saltik has advanced experience in programming languages like C, C++, and Java, as well as basic experience in C# and ASP.NET. He has worked internships and part-time jobs involving information technology. Saltik is fluent in Turkish, English, and German and has been involved with educational and extracurricular organizations. His career interests include software design, internet technologies, mobile solutions, and information technology.
Bairstow's methods
Este documento describe el método de Bairstow para determinar las raíces de un polinomio. Explica cómo tomar valores iniciales para r y s, realizar iteraciones para corregir estos valores, y calcular las raíces del polinomio original una vez se alcanza la convergencia deseada. Incluye un ejemplo numérico para ilustrar el proceso.
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Calculus multiple integral
1. The document discusses multiple integral calculus topics including double integrals over rectangles and general regions, double integrals in polar coordinates, triple integrals in rectangular, cylindrical and spherical coordinates, and change of variables in multiple integrals.
2. It provides definitions, properties, examples and solutions for calculating double and triple integrals using rectangular, polar, cylindrical and spherical coordinate systems.
3. Formulas are given for calculating double and triple integrals in polar, cylindrical and spherical coordinates by representing the volume element and limits of integration in terms of the respective coordinate systems.
logarithms
Logarithms are exponents that indicate the power to which a base number must be raised to equal the original number. John Napier introduced logarithms in the early 17th century as a way to simplify calculations. Logarithms have many applications in fields like science, engineering, mathematics, psychology, and music. They allow complex calculations to be performed more easily.
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The document discusses topological dimension and fractal dimension. It provides examples of objects with different dimensions, such as a line being 1-dimensional, a plane being 2-dimensional, and the Cantor set having a fractal dimension of 0.6309. Fractals are self-similar objects that can have non-integer dimensions calculated using the logarithmic formula D = log(N)/log(1/r), where N is the number of parts and r is the scaling ratio. The Koch snowflake is used to illustrate this, with N=4, r=1/3, giving a fractal dimension of 1.26.
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The document discusses key concepts in geometry including segments, measures of segments, and the Pythagorean theorem. It defines segments and their measures using the ruler postulate and distance formula. It also explains how to use the distance formula, segment addition postulate, and Pythagorean theorem to measure and relate segments on a coordinate plane. Students are assigned example problems and a homework assignment involving these geometric concepts.
1557 logarithm
Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics
Exercise roots of equations
This document discusses several numerical methods for finding roots of equations, including the bisection method, false position method, fixed point method, Newton-Raphson method, and secant method. It provides an example of using the Newton-Raphson method to find the maximum deflection of a bookshelf beam. It also asks which function could be used with the fixed point method to find a root between 10 and 15 for the equation x3 - 30x2 + 2400 = 0. Finally, it provides an example of using the bisection method to find the root of the equation f(x)=3x+sinx-ex.
Tutorials--Equations with Fractions
This document provides a tutorial on solving one-variable equations with fractions through 13 examples of different types of fractional equations. The examples cover equations of the form a/bx = c, -a/bx = c, a/bx = -c, a/bx = cx + d, a/bx = c/dx + e, and other variations involving addition, subtraction, and combinations of fractions and variables.
Matlab lecture 7 – regula falsi or false position method@taj
Matlab lecture 7 – regula falsi or false position 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
Formulas for calculating surface area and volume
This document provides formulas for calculating the surface area and volume of various 3D shapes. It defines key terms like perimeter, base, lateral surface area, and total surface area. Formulas are provided for calculating the surface area and volume of cubes, rectangular prisms, triangular prisms, cylinders, cones, square pyramids, and spheres. An example problem is worked through to find the volume and total surface area of a sphere with a radius of 30 cm.
Chapter 3: Roots of Equations
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.
Polar area
1) The area enclosed by a polar curve can be found by integrating the area of sectors swept out by the radius as the angle varies over the interval.
2) The area of each sector is 1/2*r^2*Δθ. Integrating this expression from α to β gives the total area.
3) Special care must be taken when curves intersect or complete multiple cycles over the interval to break the problem into multiple integrals if needed. Graphing and inspecting the curves is important to set up the integrals correctly.
Media,265106,en
1. The document discusses the laws of indices, which provide rules for simplifying expressions involving exponents or indices.
2. The three main laws are: (1) When expressions with the same base are multiplied, the indices are added. (2) When expressions with the same base are divided, the indices are subtracted. (3) When an expression with an exponent is raised to another power, the exponents are multiplied.
3. Examples are provided to illustrate each law, such as 76 × 74 = 710 by the first law, and (a3)4 = a12 by the third law.
Soal Olimpiade
The document contains 9 multiple choice questions about geometry, algebra, and arithmetic concepts. Specifically, question 1 asks about finding the equation of a line given 3 points, question 2 involves solving an equation for x+y, and question 3 deals with calculating the length of a diagonal space of a beam given other measurements.
Intro To Formula Project
The document provides examples of working with literal equations, which are formulas written as equations. It shows how to transform a formula by solving it for a different variable. For example, rewriting the volume formula V=lwh to solve for width by isolating w on one side. Students are assigned a project to research two formulas, write them, describe the variables, show a real-life application by substituting values, and present their findings.
December 19
The document provides instructions on graphing linear equations from points and determining the slope and situation described by a line passing through two points. It asks the reader to match a graph to an equation, graph an equation using at least three points, and find the slope of a line passing through (1,7) and (6,4). It also provides examples of forming a linear equation given the slope and y-intercept, a point and slope, or two points. Finally, it asks the reader to find the perimeters of two triangles with side lengths provided.
Roots of equations
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.
Matlab analytical solution of cubic equation
In this paper code to find real roots of cubic equation with brief of explanation about it and snapshot of program
Calc 5.2b
This document discusses techniques for integrating trigonometric functions using u-substitution and trigonometric identities. It provides examples of integrating tan(x), cot(x), sec(x), csc(x) using the log rule with u-substitutions of u = ln(x), u = cos(x), and u = sec(x) + tan(x). It also evaluates integrals using Pythagorean identities and finds the average value of functions over an interval.
What's hot (19)
Matlab lecture 7 – regula falsi or false position method@taj
Matlab lecture 7 – regula falsi or false position method@taj
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NUMERICAL & STATISTICAL METHODS FOR COMPUTER ENGINEERING
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.
xx
Eray SALTIK has a computer engineering degree from Bogazici University in Istanbul. He graduated from Istanbul Erkek Lisesi high school in 2006 with honors. Saltik has advanced experience in programming languages like C, C++, and Java, as well as basic experience in C# and ASP.NET. He has worked internships and part-time jobs involving information technology. Saltik is fluent in Turkish, English, and German and has been involved with educational and extracurricular organizations. His career interests include software design, internet technologies, mobile solutions, and information technology.
Bairstow's methods
Este documento describe el método de Bairstow para determinar las raíces de un polinomio. Explica cómo tomar valores iniciales para r y s, realizar iteraciones para corregir estos valores, y calcular las raíces del polinomio original una vez se alcanza la convergencia deseada. Incluye un ejemplo numérico para ilustrar el proceso.
Fixedpoint
This document discusses fixed point iteration, a numerical method for finding the zeros of a function. It uses the example of finding the fixed point of the function g(x)=mx/(m-1) for different slope values of m. For m<1, the fixed point iteration converges monotonically to the fixed point, while for m>1 it diverges monotonically away from the fixed point. This simple linear example illustrates the general behavior for continuous mapping functions - convergence when the slope of g(x) is less than 1, and divergence when it is greater than 1.
Es272 ch3b
- Müller's method and Bairstow's method are conventional methods for finding both real and complex roots of polynomials.
- Müller's method fits a parabola to three initial guesses to estimate roots, then iteratively refines the estimate.
- Bairstow's method divides the polynomial by a quadratic factor to estimate roots, then iteratively adjusts the factor's coefficients to minimize the remainder using a process similar to Newton-Raphson.
- Both methods can find all roots of a polynomial by sequentially applying the process after removing already located roots from the polynomial.
Gauss
Gaussian Elimination is a variation of the Gauss elimination method that can solve up to 15-20 simultaneous equations with 8-10 significant digits of precision on a computer. It differs from Gaussian elimination by normalizing all rows when using them as the pivot equation, resulting in an identity matrix rather than a triangular matrix. This avoids needing to perform back substitution. The method is demonstrated through solving a system of 3 equations with 3 unknowns via Gaussian elimination, resulting in values for the 3 unknowns. Advantages of the Gaussian-Jordan method include requiring approximately 50% fewer operations than Gaussian elimination and providing a direct method for obtaining the inverse matrix.
Newton's forward difference
This document provides Newton's formula for forward difference interpolation and an example of using it to find the value of tan(0.12).
- Newton's formula uses forward difference interpolation to find the value of a polynomial of degree n that fits a set of (n+1) equally spaced (x,y) points.
- The coefficients of the polynomial are determined using forward differences of the y-values.
- In the example, the value of tan(0.12) is found by applying Newton's formula to a table of tan(x) values from 0.10 to 0.30 using forward differences up to degree 4.
Es272 ch3a
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.
Gauss Elimination Method With Partial Pivoting
Gauss Elimination Method with Partial Pivoting:
Goal and purposes:
Gauss Elimination involves combining equations to eliminate unknowns. Although it is one of the earliest methods for solving simultaneous equations, it remains among the most important algorithms in use now a days and is the basis for linear equation solving on many popular software packages.
Description:
In the method of Gauss Elimination the fundamental idea is to add multiples of one equation to the others in order to eliminate a variable and to continue this process until only one variable is left. Once this final variable is determined, its value is substituted back into the other equations in order to evaluate the remaining unknowns. This method, characterized by step‐by‐step elimination of the variables.
Gauss Seidel Method:
Goal and purposes:
The main goal and purpose of the program is to solve a system of n linear simultaneous equation using Gauss Seidel method.
This Slides includes:
Goal and purpose, Description, Algorithm, C-code, Screenshot etc.
Regulafalsi_bydinesh
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.
Gauss elimination method
Gaussian elimination is a method to solve systems of linear equations by transforming the matrix of coefficients into upper triangular form using elementary row operations. It has two steps: forward elimination reduces the matrix to row echelon form, and back substitution solves the system. The method is equivalent to computing an LU matrix decomposition that writes the original matrix as the product of an invertible matrix and an upper triangular matrix. Examples demonstrate solving systems by Gaussian elimination.
Gauss elimination
Gaussian elimination is a method for solving systems of linear equations consisting of two steps:
1) Forward elimination transforms the coefficient matrix into an upper triangular matrix by eliminating variables from lower-numbered equations. This is done by subtracting appropriate multiples of pivot equations from other equations.
2) Back substitution solves the upper triangular system of equations by substituting the solution of higher-numbered equations into lower ones and solving for each variable sequentially, starting from the last equation.
Gauss y gauss jordan
The document discusses Gauss elimination and Gauss-Jordan elimination, which are methods for solving systems of linear equations. Gauss elimination uses elementary row operations to systematically eliminate variables from equations until a solution is reached. Gauss-Jordan elimination further manipulates the matrix of coefficients to put it into row echelon form, allowing direct inspection of the solution. Potential problems that can arise include parallel lines with no solution, division by zero errors, and accumulation of rounding errors in large matrices.
Gauss jordan and Guass elimination method
This ppt is based on engineering maths.
the topis is Gauss jordan and gauss elimination method.
This ppt having one example of both method and having algorithm.
NUMERICAL METHODS -Iterative methods(indirect method)
The document discusses two iterative methods for solving systems of linear equations: Gauss-Jacobi and Gauss-Seidel. Gauss-Jacobi solves each equation separately using the most recent approximations for the other variables. Gauss-Seidel updates each variable with the most recent values available. The document provides an example applying both methods to solve a system of three equations. Gauss-Seidel converges faster, requiring fewer iterations than Gauss-Jacobi to achieve the same accuracy. Both methods are useful alternatives to direct methods like Gaussian elimination when round-off errors are a concern.
Numerical method
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.
Newton’s Forward & backward interpolation
This document discusses Newton's forward and backward interpolation formulas. Newton's forward interpolation formula is used to find the value of tan(0.12) given values of tan(x) at other x values between 0.1 and 0.3. Newton's backward interpolation formula is then introduced and an example shows how to use it to determine the value of y(300) given y-values at x-values between 50 and 250.
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Roots of polynomials
The Bairstow method and Muller method are techniques for calculating the roots of polynomials.
The Bairstow method uses synthetic division to iteratively calculate the roots of a polynomial by dividing it by quadratic factors (x^2 - rx - s) until the remainder is zero. It obtains better approximations of r and s at each iteration to isolate the roots.
The Muller method fits a parabola through three initial guesses to obtain the coefficients a, b, c of the quadratic formula. It then uses the quadratic formula to find the root, and iterates with new guesses to converge on more accurate roots.
Both methods use iterative techniques to gradually converge on the real and complex roots of polynomials without
matrixMultiplication
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Class 14 3D HermiteInterpolation.pptx
1) Interpolation is a process that defines a function to connect data points by determining a unique polynomial curve that passes through the specified points.
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3) Hermite interpolation also uses piecewise cubics but specifies the tangent at each control point, allowing local control of the curve sections. Boundary conditions define the curves to match positions and specified tangent derivatives at endpoints.
real numbers
This document provides an introduction to rational numbers. It defines integers, whole numbers, counting numbers, and rational numbers. Rational numbers can be expressed as the quotient of two integers, such as fractions. Some numbers, like pi, cannot be expressed as quotients of integers and are called irrational numbers. The combination of rational and irrational numbers make up the set of real numbers. The document then provides examples and theorems regarding properties of rational numbers, such as Euclid's division algorithm for finding the highest common factor of two numbers and the fundamental theorem of arithmetic regarding unique prime factorizations.
Quadratic equations
This document discusses methods for solving quadratic equations. It describes that a quadratic equation is an equation of the form ax^2 + bx + c = 0, with a, b, and c being real numbers. A quadratic equation has at most two real or imaginary roots. The methods covered for solving quadratic equations are: square root property, factoring, completing the square, and the quadratic formula. Examples are provided for each method.
Quadratic equation
The document provides an overview of quadratic equations, including:
1) Quadratic equations take the form of ax^2 + bx + c = 0 and involve an unknown (x) and known coefficients (a, b, c).
2) Quadratic equations can be solved through factoring, completing the square, using the quadratic formula, or graphing.
3) The quadratic formula provides the general solution for quadratic equations as x = (-b ± √(b^2 - 4ac))/2a.
Measurement of the angle θ .docx
Measurement
of
the
angle
θ
For
better
understanding
I
am
showing
you
a
different
particle
track
diagram
bellow.
Where
at
point
C
particle
𝜋! 𝑎𝑛𝑑 Σ!
are
created
and
the
Σ!
decays
into
𝜋∓ 𝑎𝑛𝑑 K!
particles
The
angle
θ
between
the
π−
and
Σ−
momentum
vectors
can
be
determined
by
drawing
tangents
to
the
π−
and
Σ−
tracks
at
the
point
of
the
Σ−
decay.
We
can
then
measure
the
angle
between
the
tangents
using
a
protractor.
Alternative
method
which
does
not
require
a
protractor
is
also
possible.
Let
AC
and
BC
be
the
tangents
to
the
π−
and
Σ−
tracks
respectively.
Drop
a
perpendicular
(AB)
and
measure
the
distances
AB
and
BC.
The
ratio
AB/BC
gives
the
tangent
of
the
angle180◦−θ.
It
should
be
noted
that
only
some
of
the
time
will
the
angle
θ
exceed
90◦
as
shown
here.
Determining
the
uncertainty
of
Measurements
In
part
B,
It
is
asked
to
estimate
the
uncertainty
of
your
measurements
of
𝜃
and
r.
Uncertainty
of
measurement
is
the
doubt
that
exists
about
the
result
of
any
measurement.
You
might
think
that
well-‐made
rulers,
clocks
and
thermometers
should
be
trustworthy,
and
give
the
right
answers.
But
for
every
measurement
-‐
even
the
most
careful
-‐
there
is
always
a
margin
of
doubt.
It
is
important
not
to
confuse
the
terms
‘error’
and
‘uncertainty’.
Error
is
the
difference
between
the
measured
value
and
the
‘true
value’
of
the
thing
being
measured.
Uncertainty
is
a
quantification
of
the
doubt
about
the
measurement
result
Since
there
is
always
a
margin
of
doubt
about
any
measurement,
we
need
to
ask
‘How
big
is
the
margin?’
and
‘How
bad
is
the
doubt?’
Thus,
two
numbers
are
really
needed
in
order
to
quantify
an
uncertainty.
One
is
the
width
of
the
margin,
or
interval.
The
other
is
a
confidence
level,
and
states
how
sure
we
are
that
the
‘true
value’
is
within
that
margin.
You
can
increase
the
amount
of
information
you
get
from
your
measurements
by
taking
a
number
of
readings
and
carrying
out
Graphics6 bresenham circlesandpolygons
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The document provides information on correlation and linear regression. It defines correlation as the association between two variables and discusses how the correlation coefficient r measures the strength of this linear association. It then discusses:
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NUMERICAL METHODS
- 1. Bairstow Method Jorge Eduardo Celis ROOTS OF POLYNOMIALS
- 2. Bairstow Method A method for calculating roots of polynomials can calculate peer (conjugated in the case of complex roots). Unlike Newton, calculate complex roots without having to make calculations with complex numbers. It is based on the synthetic division of the polynomial Pn (x) by the quadratic (x2 - rx - s).
- 3. Bairstow Method The synthetic division can be extended to quadratic factors: and even by multiplying the coefficients is obtained:
- 4. Bairstow Method We want to find the values of r and s that make b1 and b0 equal to zero since, in this case, the factor divided exactly quadratic polynomial. The first method works by taking an initial approximation (r0, s0) and generate approximations (rk, sk) getting better using an iterative procedure until the remainder of division by the quadratic polynomial (x2 - rkx - sk) is zero. The iterative procedure of calculation is based on the fact that both b1 and b0 are functions of r and s.
- 5. Bairstow Method In developing b1 (rk, sk) and b0 (rk, sk) in Taylor series around the point (r *, s *), we obtain: It takes (r *, s *) as the point where the residue is zero and Δr = r * - rk, Δs = s * - sk. Then:
- 6. Bairstow Method Bairstow showed that the required partial derivatives can be obtained from the bi by a second synthetic division between factor (x2 - r0x - s0) in the same way that the bi are obtained from the ai. The calculation is:
- 7. Bairstow Method Thus, the system of equations can be written
- 8. Bairstow Method Calculation of approximate error:When tolerance is reached estimated coefficientsrand s is used to calculate the roots:
- 9. Bairstow Method Then: When the resulting polynomial is of third order or more, the Bairstow method should be applied to obtain a resultant function of order 2. When the result is quadratic polynomial, defines two of the roots using the quadratic equation. When the final function is first order root is determined from the clearance of the equation.
- 10. Bibliography CHAPRA, Steven C. y CANALE, Raymond P.: Métodos Numéricos para Ingenieros. McGraw Hill 2002. http://ocw.mit.edu/OcwWeb/Mathematics PPTX EDUARDO CARRILLO, PHD.