chapter 3

CHAPTER 3: Roots of Equations

By Erika Villarreal

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Roots of Equations

The determination of the roots of an equation is one of the oldest problems in mathematics and there have been many efforts in this regard. Its importance is that if we can determine the roots of an equation we can also determine the maximum and minimum values of matrices, solving systems of linear differential equations.


CLOSED METHODS

Closed domain methods are methods that start with two values of x between which is the root of the equation, x = α, the interval is reduced consistently keeping the root within the interval. The two methods that fall into this category are:

1. Graphical method 2. Bisection method (assuming the range in two) 3. Method of false position

These methods will be developed in the following sections. Closed methods are robust in the sense that ensure that a solution will be obtained because the root is? Trapped? on a closed interval. The counterpart of them is they have a slow convergence.


CLOSED METHODS

Graphic methods consist of plotting the function f (x) and observe where the function crosses the x-axis

Example 1:Find the following:

1. Graphical Methods

040138.667 146843.0 xex

xf

-15

-10

-5

0

5

10

15

20

25

30

35

40

0 5 10 15 20 25

x f(x)

4 34.114889388 17.65345264

12 6.06694996316 -2.26875420820 -8.400624408


CLOSED METHODS

Example 2:Find the following:

1. Graphical Methods

f(x) = sen 10x + cos 3x

-2.50

-2.00

-1.50

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0.00 1.00 2.00 3.00 4.00 5.00 6.00

x f(x)

0.00 1.000.25 1.330.50 -0.890.75 0.311.00 -1.531.25 -0.891.50 0.441.75 -0.462.00 1.872.25 0.412.50 0.212.75 0.313.00 -1.903.25 -0.063.50 -0.903.75 0.054.00 1.594.25 -0.014.50 1.454.75 -0.485.00 -1.02


CLOSED METHODS

It's about finding the zeros of

f (x) = 0

Where f is a continuous function on [a, b] with f (a) f (b) with different signs

According to the mean value theorem, there is p [a, b] such that f (p) = 0.The method involves dividing the interval in half and trace half containing p.The process is repeated to achieve the desired accuracy.

2. Bisection method


CLOSED METHODS

2. Bisection method

Half of the interval containing p

First iteration of the algorithm


CLOSED METHODS

2. Bisection method

Accession: ends a, b, number of iterations or, tolerance tol

Bisection algorithm

1. p=a; i=1; eps=1;2. mientras f(p)0 y i ni eps>tol 2.1. pa = p; 2.2. p = (a+b)/2 2.3. si f(p)*f(a)>0 entonces a=p; 2.4. sino 2.5. si f(p)*f(b)>0 entonces b=p; 2.6. i = i + 1; eps = |p-pa|/p;


CLOSED METHODS

2. Bisection method

double biseccion(double a, double b, double error, int ni){ double p,pa,eps; int i; p = a; i = 1; eps = 1; while(f(p) != 0 && i<ni && eps > error){ pa = p; p = (a+b)/2; if(f(p)*f(a)>0) a = p; else if(f(p)*f(b)>0) b = p; i++; eps = fabs(p-pa)/p; } return p;}


Methods closed

2. Bisection method


CLOSED METHODS

2. Error in the bisection methodFor the bisection method is known that the root is within the range, the result must be within Dx / 2, where Dx = xb - xa.The solution in this case is equal to the midpoint of the intervalxr = (xb + xa) / 2Should be expressed by

xr = (xb + xa) / 2 Dx / 2

Approximate Error

replacing


CLOSED METHODS

This method considers that the range limit is closer to the root.

From Figure

clearing

3. Method of false position


CLOSED METHODS

Example 1:

Find the following:

3. Method of false position

040138.667 146843.0 xex

xf

xl xu xr f(xl) f(xu) f(xr)12.0000000 16.0000000 14.9113077 6.0669500 -2.2687542 -0.254277512.0000000 14.9113077 14.7941976 6.0669500 -0.2542775 -0.027257212.0000000 14.7941976 14.7817001 6.0669500 -0.0272572 -0.002907612.0000000 14.7817001 14.7803676 6.0669500 -0.0029076 -0.000310012.0000000 14.7803676 14.7802255 6.0669500 -0.0003100 -0.0000330


CLOSED METHODS

False position in C


OPEN METHODS

Open domain methods are not restricted to an interval root. Consequently,these methods are not as robust as methods dimensional and can diverge. However, these methods use information coming from the nonlinear function to refine the estimated result.

Thus, these methods are more efficient than dimensional methods. The most representative methods are:

1. Fixed-point iterative method.2. Newton-Raphson Method3. Secant method4. Muller Method


OPEN METHODS

A fixed point of a function g (x) is a number p such that g (p) = p.Given a problem f (x) = 0, we can define a function g (x) with a fixed point p in different ways.For example g (x) = x - f (x).

Theorem

If g C [a, b] and g (x) C [a, b] for all x C [a, b], then g has a fixed point in [a, b].If in addition g '(x) exists in (a, b) and a positive constant k <1 exists| G '(x) | <= k,

For all x (a, b)

Then the fixed point in [a, b] is unique

1. Fixed-Point Iteration


OPEN METHODS

1. Graph of fixed-point algorithm


OPEN METHODS

Consider the function: x3 + 4x2 -10 = 0 has a root in [1, 2]

You can unwind in:

a. x = g1(x) = x – x3 – 4x2 +10

b. x = g2(x) = ½(10 – x3)½

c. x = g3(x) = (10/(4 + x))½

d. x = g4(x) = x – (x3 + 4x2 – 10)/(3x2 + 8x)


OPEN METHODS

1. Fixed point iterations


OPEN METHODS

1. Functions plotted in Mathlab


OPEN METHODS

1. Cases of non-convergence


OPEN METHODS

2. Newton Method

The equation of the tangent line is

y – f(xn) = f ’ (xn)(x – xn)

When y = 0, x = xn+1 and

0 – f(xn) = f ’ (xn)(xn+1– xn)

.

x xf x

f xn nn

n 1

( )

'( )


OPEN METHODS

2. Newton Method

Example

f(x) = x – cos(x) f’(x) = 1 + sen(x)

.

pn+1 = pn – (pn – cos(pn))/(1 + sen(pn))

Taking p0 = 0, finding

pn f(pn) f’(pn) pn+1

0 -1 1 11 0.459698 1.8414 0.75036390.7503639 0.0189 1.6819 0.73911280.7391128 0.00005 1.6736 0.7390851 0.7390851 3E-10 1.6736 0.7390851


OPEN METHODS

3. Alternative method to evaluate the derivative (secant method)

.

The secant method starts at two points (no one like Newton's method) and estimates the tangent by an approach according to the expression:

The expression of the secant method gives us the next iteration point:


OPEN METHODS

3. Alternative method to evaluate the derivative (secant method)

.

:

In the next iteration, we use the points x1 and x2para estimate a new point closer to the root of Eq. The figure represents geometric method.


OPEN METHODS

Multiple roots

In the event that a polynomial has multiple roots, the function will have zero slope when crossing the x-axis

Such cases can not be detected in the bisection method if the multiplicity is even.

In Newton's method the derivative is zero at the root.Usually the function value tends to zero faster than the derivative and can be used Newton's method

.

:


OPEN METHODS

1. Muller Method

.

This method used to find roots of equations with multiple roots, and is to obtain the coefficients of the parabola passing through three selected points. These coefficients are substituted in the quadratic formula to get the value where the parabola intersects the X axis, the estimated result. The approach can be facilitated if we write the equation of the parabola in a convenient way.

One of the biggest advantages of this method is that by working with the quadratic formula is therefore possible to locate real estate, and complex roots.

Formula

The three initial values are denoted as needed xk, xk-1 xk-2. The parabola passes through the points (xk, f (xk)) (xk-1, f (xk-1)) and (xk-2, f (xk-2)), if written in the form Newton, then:

where f [xk, xk-1] f [xk, xk-1, xk-2] denote subtraction divided. This can be written as:

where

The next iteration is given by the root that gives the equation y = 0


OPEN METHODS

2. Lin-Bairstow

.

The Lin-Bairstow method finds all the roots (real and complex) of a polinomioP (x). Given initial values of r and s, made a synthetic divide P (x) by (x2 - rx - s). Use Newton's method to find r and s values that make the waste is zero, ie, find the roots of the system of equations.

bn(r, s) = 0, (55)bn−1(r, s) = 0. (56)

Using the recursive ruler ← r+Δr (57)s ← s+Δs

Where

Once you find a quadratic factor of P (x) is solved with the formula

and work continues to take Q (x) as the new polynomial P (x).


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• MÉTODOS NUMÉRICOS Maestría en Ingeniería de Petróleos Escuela de Ingeniería de Petróleos David Fuentes Díaz Escuela de Ingeniería de Mecánica

. Métodos Numéricos (SC–854) Solución de ecuaciones no lineales

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BIBLIOGRAPY

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