ROOTS OF EQUATIONS

PRESENTED BY: KATHERINE SILVA

NUMERICAL METHODS IN PETROLEUM ENGINEERING

ROOTS OF EQUATIONS

MULTIPLE ROOTS

MULLER METHOD

BAIRSTOW METHOD

COMPLEX ROOT

1. CALCULATION OF MULTIPLE ROOTS

A multiple root of a polynomial is a root that occurs more than once and corresponds to a point where a function is tangent to the x axis

for example the polynomial:

The factor occurs twice, so it is a multiple root. Since it occurs twice, also called a double root.

Source: http://www.lifeisastoryproblem.org/vocab/es/m/multipleroot.html

1. 1 CONVENTIONAL METHODS

• MULLER METHOD• BAIRSTOW METHOD

used to find roots of equations with multiple roots

this is to obtain the coefficients of the parabola through three points selected. These coefficients are substituted in the quadratic formula to get the value where the parabola intersects the X axis, so get the estimated result.

MULLER METHOD

Source: www.embedded.com/story/OEG20020321S0075

• The three initial values are denoted as xk, xk-1 y xk-2.

• The parabola passes through the points: (xk, f(xk)), (xk-1, f(xk-

1)) y (xk-2, f(xk-2)), if it is written in the form of Newton, then:

where [xk, xk-1] y  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.

• Bairstow's method is an iterative method, based on the method of Müller and Newton Raphson. Given a polinonio fn(x) are two factors, a quadratic polynomial f2(x) = x2 – rx – s y fn-2(x).

• The general procedure for the Bairstow method is: 

1. Since fn(x) and r0 and s0

2. Using the method of NR calculate f2(x) = x2 – r0x – s0 and fn-2(x) , such that the residue of fn(x)/ f2(x) is zero.

3. Determine the roots f2(x), using the general formula.

4. It is estimated fn-2(x)= fn(x)/ f2(x).

5. We fn(x)= fn-2(x)

6. If the degree of the polynomial is greater than three back to step 2If we do not finish 

BAIRSTOW METHOD

2. COMPLEX ROOT CALCULATION

• To calculate the result of a complex number, or N-ESIMA root, we use a variant of Euler, this formula states that every complex number has exactly n complex roots.Where: k = 0,1,2 ,...... n-1 that is generating each of the roots.

www.katjaas.nl/rootsofunity/rootsofunity.html

EXAMPLE: calculate the square root of:

first polar graph mode is solved in polar form first and then being able

to replace in the formula above written.

Re

i1

Im

2

45)1(tan1

)45sin)45(cos(2 iz

Re

this is the polar form now follows the substitution in the Euler formula ..

EXAMPLE: 1ªroot when k = 0.

ii

ii

ii

ii

4551.00987.11

)3827.09239.0(1892.11

))2

45sin()

2

45(cos(1892.11

)]2

360*045sin()

2

360*045[cos()2(1 2/1

ii

ii

ii

ii

ii

4551.00987.11

)3827.09239.0(1892.11

))2

405sin()

2

405(cos(1892.11

))2

36045sin()

2

36045(cos(1892.11

))2

)360*1(45sin()

2

)360*1(45(cos(1892.11

2ª root when k = 1

• http://lc.fie.umich.mx/~calderon/programacion/Mnumericos/RMult.html

• http://www.lifeisastoryproblem.org/vocab/es/m/multipleroot.html

• http://illuminatus.bizhat.com/metodos/Muller.htm

• http://lc.fie.umich.mx/~calderon/programacion/Mnumericos/Bairstow.html

• http://temasmatematicos.uniandes.edu.co/Complejos/paginas/intro.htm#_Toc86777347

• notannhelita.blogspot.com 

BIBLIOGRAPHY