3.Get a real root using Newton's Method Newton's Method for

Polynomial Equations GE1319 Interdisciplinary Research

for Smart Professionals Conclusion Newton's Method

for Polynomial Equations Motivation Motivation & Prospect Steps & Methods Examples FEI Teng

MA Hoi Wan Wendy

SHEN Jiahui Three parts in this presentation Motivation & prospect by Wendy Steps & Methods by FEI Teng Examples by SHEN Jiahui 1.Plot 2.Initial guess 4.Synthetic division 5.Redo 1-4 FOR REAL ROOTS FOR COMPLEX ROOTS 1.Use m+in substitute x in the polynomial and get a system of nonlinear equations 4.Using Newton's Method to solve the equation system 'PINS' Positive influence on our major study Applications for Business To apply what we have learnt An opportunity to gain research experiences Newton's Method &

other numerical methods f '(Xn) Modified Newton's Method equation

Step1:

Step2:

Step3:

Step4: example 2: example 4: example 1: example 3: FEATURE: Conducting Newton's Method on a new function g(x)=f(x)/f'(x), which can simplify the multiple roots and have the same but simple root. Formula: Newton's Method for Systems of Non-linear Equations FEATURE: Two unknown values.

Used when solving complex roots.

The aim is to find intersection points of two non-linear equations. Formula: FEATURE: Rewrite equation p(x) = 0 as x = g(x), and find a intersection point of function y=x and y = g(x). Fixed Point Iteration Formula: In geometry: Formula: Newton's Method: The main method used in this project The Secant Method FEATURE: Instead of a tangent line, we use a secant line to do the iteration. So it needs two initial guesses. Formula: Synthetic Division After finding a root, synthetic division can be used to get a new lower-degree polynomial. This method uses the coefficients of the higher-degree polynomial and the found root to get the coefficients of the new lower-degree polynomial. The remainder should be close to zero. function w = fn(input1,input2)

x(1)=input1;

y(1)=input2;

for i = 1:20

u= x(i).^4-6*x(i).^2*y(i).^2+2*x(i).^2+y(i).^4-x(i)-2*y(i).^2+2;

v=4*x(i).^3-4*x(i)*y(i).^2+4*x(i)-1;

u1=4*x(i).^3-12*x(i)*y(i).^2+4*x(i)-1;

u2=-12*x(i).^2*y(i)+4*y(i).^3-4*y(i);

v1=12*x(i).^2-4*y(i).^2+4;

v2=-8*x(i)*y(i);

x(i+1)= x(i)-(u*v2-v*u2)/(u1*v2-u2*v1);

y(i+1)= y(i)-(v*u1-u*v1)/(u1*v2-u2*v1);

if abs(x(i+1) - x(i) )< 1e-10 && abs(y(i+1) - y(i) )< 1e-10

break;

end

end

x

y

end Laguerre polynomials

Equation

plot

Input

[1 -49 882 -7350 29400 -52920 35280 -5040], 20

X1=19.3957278

Input

[1 -49 882 -7350 29400 -52920 35280 -5040],19.3957278

Input

[1 -29.60427220000 307.80359469169 -1379.92526149840 2635.34524363320 -1805.56099546575 259.83040564916],9

redo, find all roots situation when newton's method fails

function:

When n goes to ∞, x(n) keep on increasing.

No matter what the initial guess is, Newton’s method will fail. real situation Plot & Initial Guess To plot the graph of a polynomial, we may use MATLAB or Grapher. Using zoom in, we may get a initial guess and use this number to start iteration. A possible initial guess initial function substituting y and d

function y = em(x)

y = 1./((x.^2+0.01).^2)+1./((x.^2-2*x+1.01).^2)+2*(x-x.^2-0.01)./(((x.^2+0.01).^1.5).*((x.^2-2*x+1.01).^1.5))-100;

end

derivative function

function w=dem(x)

w=-4*x./((x.^2+0.01).^3)-4*(x-1)./((x.^2-2*x+1.01).^3)+2*(((1-2*x).*((x.^2)+0.01).^1.5)*((x.^2-2*x+1.01).^1.5)-(4*x.^3-6*x.^2+2.04*x-0.02).*(x-x.^2-0.01))./((x.^4-2*x.^3+1.02*x.^2-0.02*x+0.0101) .^3);

end

using newton’s method

function q=dipole(guess)

x(1)=guess;

for i = 1:500

x(i+1)=x(i)- em(x(i))/dem(x(i));

if abs(x(i+1) - x(i) )< 1e-5

break;

end

end

x'

end The coefficients of derivative function

function y = deri( coef )

y = coef(1:end-1).*(length(coef)-1:-1:1)

end

To get the sum of a function with inputting the coefficients and the sum of unknown number

function d = pop(c,x)

d= sum(c.*(x.^(length(c)-1:-1:0)));

end

Newton’s method

function y = newton(input1,input2)

coef =input1;

x(1) = input2;

for i = 1:200

x(i+1)=x(i)- (pop(coef,x(i)))/pop(deri(coef), x(i)),

if abs(x(i+1) - x(i) )< 1e-6

break; end

end

x'

end

To calculate the remainder and the coefficients by synthetic division

function [y,remainder]=syn(input1,input2)

coef=input1;

root=input2;

y(1)=coef(1);

for i=2:length(coef)

y(i) = y(i-1).*root+coef(i);

end

y=y(1:end)

remaider=y(length(y))

end Prospect to describe the input-output relationship for a radio to turn electromagnetic waves into music to broadcast to model the markets Varied applications

in various fields to predict the trends in the stock market Applications

in Physics Describe the trajectory of projectiles Express the change of some physical quantity by polynomial models FORENSICS:

An interdisciplinary example to create code-breaking programs to gain access to confiscated computers. (Computer Science) to work with chemical equations in determining what a chemical composition or reaction is. (Chemistry) to determine the speed, impact strength etc. of the firearm (Physics) Applications in

Electric Engineering Use math model to obtain the contours regarding an electric or magnetic dipole field strength 2.Plot the graph of the system of nonlinear equations 3.Initial guess 'SPIN' Solving the roots of polynomial is widely used in colorful field, engineering, science, business, and forensics.

'PINS' and 'SPIN' are the steps for finding the roots, whose core part is using numerical methods including Newton's Method.

Newton's Method can be divergent in some cases, we may use other methods as alternative methods.

Some examples are given to illustrate the steps and methods. Analytical Methods For quadratic equation : Quadratic Formula: For cubic equation : Cardano’s method For quartic equation = 0: Ferrari's method Suppose that we have an approximation to a simple real root of the equation . According to Taylor's Theorem: Neglecting the second and higher order terms in h: So a better approximation to x* is equal to x+h', which is = h' Newton's Method for System of Nonlinear Equations Suppose is a system of two nonlinear equations. Suppose that is an approximation to an exact solution of the mentioned system. Then we have Using Taylor series expansions about : Neglecting the higher order terms, the next approximation to Applications

in Chemistry Widely used to

describe chemical equations Nurture the academic habits O We use this method when there are multiple roots.

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# GE1319

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