Roots of Algebraic Equations

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Chapter one: Roots of algebraic andtranscendental equations

1.1 equations

In this chapter methods of finding roots to various equations are

explored. Roots of an equation are defined as values of x where

the solution of an equation is true.

Equations are generally grouped into two main categories,

algebraic equations and transcendental equations.

The first type of equation, algebraic, is defined as an equation that

involves only powers of x. The following are examples of

algebraic equations:

!" # $ %x x x + =

&! %x

x+ =

&.!# %x =

'n the other hand, transcendental equations are non(algebraic

equations or functions that transcend, or cannot be expressed in

terms of algebra. Examples of such are exponential functions,

trigonometric functions, and the inverses of each. The following

are examples of transcendental equations:

cos sin %x x+ =

%x

e + =

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Transcendental functions may have an infinite number of roots or

may not have any roots at all. *or example, the function sinx has

an infinite number of roots with x k= , and %,&,!,...,etc.k=

The solution of algebraic equations is rarely carried out from thebeginning to the end by one method. The roots of the equation are

generally determined by one method with some small accuracy,

and then made more accurate by other methods. *or the intent and

purpose of this text, only a handful of the available methods are

discussed. They include: +escartes Rule, -ynthetic +ivision,

Incremental -earch, Refined Incremental -earch, isection, *alse

/osition, -ecant, 0ewton(Raphson, 0ewtons -econd 'rder,

1raeffes Root -quaring, and airstows methods.

1.2 Polynomials

) polynomial is defined as an algebraic equation involving only

positive integer powers of x. /olynomials are generally expressed

in the following form:

& ! &

& ! &.... %n n n n

n nx a x a x a x a x a

+ + + + + + =

*or these polynomials, the following apply:

2 The order or degree of the polynomial is equal to the highest

power of x and the number of roots is directly equal to the

degree or 3n4, where na is not equal to %. *or example, a

sixth degree polynomial, or a polynomial with $n = has six

roots.

2 The value of 3n4 must be a non(negative integer. In other

words, it must be whole number that is equal to 5ero or a

positive integer.

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2 The coefficients 6 % & ! &, , ,.... ,n na a a a a 7 are real numbers.

2 There will be at least one real root if 3n4 is an odd integer.

2 It is possible that equal roots exist.

2 8hen complex roots exist, they occur in con9ugate pairs, for

example:

&== vuviux

1.3 Descartes Rule

+escartes Rule is a method of determining the maximum number

of positive and negative real roots of a polynomial. This rule states

that the number of positive real roots is equal to the number of sign

changes of the coefficients or is less than this number by an even

integer. *or positive roots, start with the sign of the coefficient of

the lowest 6or highest7 power and count the number of sign

changes from the lowest to the highest power 6ignore powers

which do not appear7. The number of sign changes proves to be the

number of positive roots. sing &x= in evaluating 6 7 %f x = is theeasiest way to loo; at the coefficients.

*or negative roots, begin by transforming the polynomial to6 7 %f x = . The signs of all the odd powers are reversed while the

even powers remain unchanged. 'nce again, the sign changes can

be counted from either the highest to lowest power, or vice versa.

The number of negative real roots is equal to the number of sign

changes of the coefficients, or less than by an even integer. sing&x= in evaluating 6 7 %f x = is the easiest way to loo; at the

coefficients.

8hen considering either positive or negative roots, the statement

3less than by an even integer4 is included. This statement accounts

Roots of )lgebraic and Transcendental Equations &(


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for complex con9ugate pairs that could exist. +escartes rule is

valid as long as there are no 5ero coefficients. If 5ero coefficients

exist, they are ignored in the count. )lso, one could find a root

and divide it out to form a new polynomial of degree 3n(&4 and

apply +escartes rule again.

Example &.(&

Example &.(!

Example &.(

Example &.( , second interval contains the root

& 6 7 6 7 %f x f x = , x is the root

If the first interval contains the root, let the following be assigned:

&x and &6 7f x remain unchanged

! x x=

! 6 7 6 7f x f x=

If the second interval contains the root, let the following be

assigned:

!x and !6 7f x remain unchanged

& x x=& 6 7 6 7f x f x=

Bontinue with the process until the desired accuracy is obtained.

Example &."(&

Roots of )lgebraic and Transcendental Equations &(F
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1., !ecant %ethod

This method is similar to the *alse /osition Aethod except that the

two most recent values ofx

6 !x

and x

7 and their correspondingfunction values 6 !6 7f x and 6 7f x 7 are used to obtain a new

approximation to the root instead of chec;ing values that bound the

root. In the renaming process for iteration, use the following:

& ! ! ,x x x x= =

& ! ! 6 7 6 7, 6 7 6 7f x f x f x f x= =

In some instances interpolation occurs, while in others,extrapolation occurs.

x!

f6x&7

f6x!7

x&x

*igure &.F

Example &.F(&

1.1- e/ton0Raphson %ethod e/tonsangent

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'nce an approximate root nx has been found, the slope of the

function at that point, G6 7nf x , is incorporated to converge to the

root more rapidly. The slope intersects the x(axis at &nx + .

xn

f6xn7

fG6xn7

xn@&

f6x7

*igure &.&%

&

6 7G6 7 nn

n n

f xf x

x x+

= 'R &

6 7

G6 7

nn n

n

f xx x

f x+ =

Repeat the process using a new value until convergence occurs.

Bonvergence may not occur in the following two cases:

2 GG6 7f x 6curvature7 changes the sign near a root.

2 Initial approximation is not sufficiently close to the true root

and the slope at that point has a small value.

Example &.&%(&

1.11 e/tons !econd 4rder %ethod

Roots of )lgebraic and Transcendental Equations &(&&
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0ewtons -econd 'rder Aethod is often a preferred method to

determine the value of root due to its rapid convergence and

extremely close approximation.

The equation 6 7 %f x = is considered once again.

f6x7x

xnxn@& x

*igure &.&&.&

The following is a Taylor series expansion of 6 7f x about nx x= :

!

&

GG6 76 7 GGG6 76 76 7 6 7 G6 76 7 ...

!? ?

n nn n n

f x x f x xf x f x f x x

+

= + + + +

*or a means of determining a value of x that will ma;e theTaylor series expansion go to 5ero, the first three terms of the right

hand side of the equation is set equal to 5ero to obtain an

approximate value:

GG6 76 76 7 G6 7 %

!

nn n

f x xf x x f x

+ + =

The exact value of x can not be determined from this equationsince only the first three terms of the infinite series were used in

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the calculation. Dowever, a close approximation of the root is a

result.

8hen using this equation to calculate x , a quadratic must besolved yielding two possible roots. In order to avoid this problem,

6 7 C G6 7n nx f x f x = from 0ewtons Tangent may be substitutedinto the brac;eted term only:

GG6 7 6 76 7 G6 7 %

! G6 7

n nn n

n

f x f xf x x f x

f x

+ =

-olving for x we obtain the following:

6 7

GG6 7 6 7G6 7

! G6 7

n

n nn

n

f xx

f x f xf x

f x

=

*rom *igure &.&&, &n nx x x+ = . -ubstituting into the previousequation, the formula can be written as follows:

&

6 7

GG6 7 6 7G6 7

! G6 7

nn n

n nn

n

f xx x

f x f xf x

f x

+

=

If the first derivative is small, use 9ust the second derivative termas follows:

G6 7 %nf x

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GG6 76 76 7 %

!

nn

f x xf x x

+ =

!

GG6 76 7!

nn x f xf x =

! !6 7 6 7 %GG6 7 GG6 7

! !

n n

n n

f x f xx x

f x f x

= + =

-olving by the quadratic equation where,

6 7&, %,

GG6 7

!

n

n

f xa b c

f x= = =

!

6 7 6 7


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+ouble roots occur when the first derivative is 5ero. Triple roots

occur when the first and second derivatives are 5ero, etcH

*igure &.&&.!

-ynthetic division may be used with all the previously discussed

methods.

Example &.&&(&

1.12 5raeffes Root !quaring %ethod

1raeffes Root -quaring Aethod is a root(finding method which

was among the most popular methods for finding roots of

polynomials in the &Fth and !%th centuries. The 1raeffes Root

-quaring Aethod is especially effective if all roots are real. The

method proceeds by multiplying a polynomial 6 7f x by 6 7f x using

the following equations:

& !6 7 6 76 7......6 7nf x x a x a x a=

& !6 7 6 &76 76 7......6 7nf x x a x a x a = + + +

so the result is:

Roots of )lgebraic and Transcendental Equations &(
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! ! ! !

&6 7 6 7 6 &7 6 7.....6 7n

nf x f x x a x a =

*or example, use a

rd

degree polynomial with root & !,x x

and x

asfollows:

!

& ! 6 7 %f x x a x a x a= = + + +

) polynomial with roots & !,x x and x would be as follows:

!

& ! 6 7 %f x x a x a x a = = + +

Aultiplying the two equations together yields the following:

$ ! < ! ! !

& ! ! & 6 7 6 7 % 6 ! 7 6 ! 7f x f x x a a x a a a x a = = + + + +

etting!y x= , this equation may be written as follows:

! ! ! !

& ! ! & % 6 ! 7 6 ! 7y a a y a a a y a

= + + +

This equation has roots of! !

& !x x and!

x . If the procedure was

applied again, another polynomial would be derived with roots of

Roots of Algebraic Equations

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