Muller’s Method Dan Heflin Sarah Hare. Muller’s Method What problem does it solve? How is it represented? Where does it come from?

Slides:

Advertisements

Similar presentations

TSW#3 Solving Compound Inequalities

Advertisements

This algorithm is used for dimension reduction. Input: a set of vectors {Xn є }, and dimension d,d

Complex Numbers.

Numerical Computation Lecture 4: Root Finding Methods - II United International College.

Chapter 5 Section 4: Complex Numbers. VOCABULARY Not all quadratics have real- number solutions. For instance, x 2 = -1 has no real-number solutions because.

MA/CS 375 Fall MA/CS 375 Fall 2002 Lecture 29.

AB 11 22 33 44 55 66 77 88 99 10  20  19  18  17  16  15  14  13  12  11  21  22  23  24  25  26  27  28.

MATH 577http://amadeus.math.iit.edu/~fass1 3.2 The Secant Method Recall Newton’s method Main drawbacks: requires coding of the derivative requires evaluation.

Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. 7.1 – Completing the Square x 2 = 20 5x =

7.1 – Completing the Square

Secant Method Another Recursive Method. Secant Method The secant method is a recursive method used to find the solution to an equation like Newton’s Method.

Newton’s Method: Homework Create a Google Spreadsheet – that uses Newton’s Method – to find roots of x^2 + x = 1 Link to this page from your homepage Send.

Chapter 4 Roots of Polynomials.

5.6 Complex Numbers. Solve the following quadratic: x = 0 Is this quadratic factorable? What does its graph look like? But I thought that you could.

Warm-Up Exercises ANSWER ANSWER x =

5.7.1 – Complex Numbers. We used the method of square roots to solve specific types of quadratics Only used when we had problems such as x 2 – 16 = 0.

5.4 Complex Numbers Until now, you have always been told that you can’t take the square root of a negative number. If you use imaginary units, you can!

DO NOW: FACTOR EACH EXPRESSION COMPLETELY 1) 1) 2) 3)

Solving Quadratic Equations

5.4 Complex Numbers. Let’s see… Can you find the square root of a number? A. E.D. C.B.

Factor: Factor: 1. s 2 r 2 – 4s 4 1. s 2 r 2 – 4s b b 3 c + 18b 2 c b b 3 c + 18b 2 c 2 3. xy + 3x – 2y xy + 3x – 2y -

Lecture 3 Bisection method Download bisection02.m And ftest2.m From math.unm.edu/~plushnik/375.

Pre-Calculus Section 1.5 Equations Objectives: To solve quadratics by factoring, completing the square, and using the quadratic formula. To use the discriminant.

4.6 Perform Operations With Complex Numbers. Vocabulary: Imaginary unit “i”: defined as i = √-1 : i 2 = -1 Imaginary unit is used to solve problems that.

The Islamic University of Gaza Faculty of Engineering Civil Engineering Department Numerical Analysis ECIV 3306 Chapter 7 Roots of Polynomials.

The Quadratic Formula Students will be able to solve quadratic equations by using the quadratic formula.

ALGEBRA 1 SECTION 10.4 Use Square Roots to Solve Quadratic Equations Big Idea: Solve quadratic equations Essential Question: How do you solve a quadratic.

Solving Polynomial Functions involving Complex Numbers.

4.2 Quadratic Functions Objective: Solve quadratic equations. Use the discriminant to describe the roots of a quadratic equation.

Solve by factoring. x² = - 4 – 5x 2,. Solve by factoring. n² = -30 – 11n -4 and -1.

5-7: COMPLEX NUMBERS Goal: Understand and use complex numbers.

How do I use the imaginary unit i to write complex numbers?

160 as a product of its prime factors is 2 5 x 5 Use this information to show that 160 has 12 factors.

Section 3.2 Mean Value Theorem Math 1231: Single-Variable Calculus.

 Complex Numbers  Square Root- For any real numbers a and b, if a 2 =b, then a is the square root of b.  Imaginary Unit- I, or the principal square.

Lesson 6.5: The Quadratic Formula and the Discriminant, pg. 313 Goals: To solve quadratic equations by using the Quadratic Formula. To use the discriminant.

9.4 Solving Quadratic Equations Standard Form: How do we solve this for x?

Factor: 1. 2x 2 – 3x x 2 - 8x x 2 – 10x – 20.

5-5 Quadratic Equations Hubarth Algebra II. Zero Product Property For every real number a, b, if ab = 0, then a = 0 or b = 0. EXAMPLEIf (x + 3)(x + 2)

WARM UP What are the solutions of each equation? 1.) x = 4 2.) x = 0 3.) x 2 – 49 = 0.

Chapter 6 Polynomials Review

4.6 Quadratic formula.

Section 6.5 Theorems about Roots of Polynomial Equations Objective: Students will be able to solve equations using Theorems involving roots. Rational Root.

Finding the square root of a complex number

Find the equation of the tangent line to the curve y = 1 / x that is parallel to the secant line which runs through the points on the curve with x - coordinates.

Solve by Factoring:.

Complex integers? Here a and b are integers.

Ex. Factor a) x2 + 5x + 6 b) x2 + 3x – 40 c) 5x2 – 17x + 6 d) 9x2 – 25.

4.6 Quadratic formula.

Rational Exponents Section 6.1

بعض المشاكل المناعتية عند الإنسان

Solve x2 + 2x + 24 = 0 by completing the square.

Numerical Analysis Lecture 7.

Sec Math II Performing Operations with Complex Numbers

3.2 Complex Numbers.

Objectives Student will learn how to define and use imaginary and complex numbers.

Rational Exponents and Nth Roots Objectives:

Day 2 Write in Vertex form Completing the Square Imaginary Numbers Complex Roots.

5.4 – Complex Numbers.

Solving Polynomial Equations in Factored Form

Day 5  Happy Friday!!.

Segment Lengths in Circles

Exercise Every positive number has how many real square roots? 2.

9-5 Factoring to Solve Quadratic Equations

Presentation transcript:

Muller’s Method Dan Heflin Sarah Hare

Muller’s Method What problem does it solve? How is it represented? Where does it come from?

What does Muller’s Method Solve? It finds the Roots of functions including Complex. This method allows the user to find roots for functions that do not have real roots.

How is Muller’s Method Represented?

Where does Muller’s Method Come From? x n-1 xnxn f(x n-1 ) f(x n ) Secant Method Muller’s Method

Algorithm for Muller’s Method For[i=3,(i<=maxit)&&(Abs[xn1-xn2]<=err),i++, t=(xn-xn1)*(xn-xn2); u=(xn1-xn)*(xn1-xn2); v=(xn2-xn)*(xn2-xn1); If[Abs[t]>0 &&Abs[u]>0&&Abs[v]>0, a= (Y0/t)+(Y1/u)+(Y2/v); b=(-Y0(xn1+xn2)/t)+(-Y1(xn+xn2)/u)+ (-Y2(xn+xn1)/v); c=(Y0 (xn1*xn2)/t)+(Y1 (xn*xn2)/u)+ (Y2 (xn*xn1)/v); r1=(-b+Sqrt[b^2 -4a*c])/(2a); r2=(-b-Sqrt[b^2 -4a*c])/(2a); If[Abs[xn2-r1]<Abs[xn2-r2], xn = xn1; xn1 = xn2; xn2 = r1; Y0 = f[xn]; Y1= f[xn1]; Y2= f[xn2];,(*else*) xn = xn1; xn1 = xn2; xn2 = r2; Y0 = f[xn]; Y1= f[xn1]; Y2= f[xn2]; ];

Representation Again

Example

Continued

Since r1 is closer to the actual root of the function, we choose r1 instead of r2. The program continues until the limit of iterations has been met, or the actual root has been found. This is the first iteration of Muller’s Method.