beyond the quadratic equations and the n-d newton-raphson...

IOSR Journal of Mathematics (IOSR-JM)

e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 4 Ver. IV (Jul. - Aug.2016), PP 44-76

www.iosrjournals.org

DOI: 10.9790/5728-1204044476 www.iosrjournals.org 44 | Page

Beyond the Quadratic Equations and the N-D Newton-Raphson

Method

Winai Ouypornprasert (Department of Civil Engineering, College of Engineering/ Rangsit University, Thailand)

Abstract: The objective of this technical paper was to propose an approach for solving polynomials of degree

higher than two. The main concepts were the decomposition of a polynomial of a higher degree to the product of

two polynomials of lower degrees and the n-D Newton-Raphson method for a system of nonlinear equations.

The coefficient of each term in an original polynomial of order m will be equated to the corresponding term

from the collected-expanded product of the two polynomials of the lower degrees based on the concept of

undetermined coefficients. Consequently a system of nonlinear equations was formed. Then the unknown

coefficients of the decomposed polynomial of the lower degree of the two decomposed polynomials would be

eliminated from the system of the nonlinear equations. After that the unknown coefficients in the decomposed

polynomial of the higher degree would be obtained by the n-D Newton-Raphson. Finally the unknown

coefficients for the decomposed polynomial of the lower degree would be obtained by back substitutions. In this

technical paper the formulations for the decomposed polynomials would be derived for the polynomials degree

three to nine. Several numerical examples were also given to verify the applicability of the proposed approach.

Keywords: Roots of a Polynomial of a High Degree, the n-D Newton-Raphson Method, Undetermined

Coefficients, Jacobian of the Functions, Matrix Inversion

I. Introduction

Finding solutions to a polynomial of order higher than two has been unavoidable in engineering works.

Mostly only real roots were required. In these cases the graphical method could give good initial guesses for

some efficient numerical methods such as the Newton-Raphson method. However the determination of all

possible roots has been very challenging. There are general solutions for cubic- and quartic polynomials [1].

Beyond the quartic polynomials some special forms and sufficient conditions for solvable polynomials have

been studied [1-10].

The purpose of this technical paper was to propose a mathematical tool for solving for all possible roots

of a polynomial of degree higher than two. It included the decomposition technique and the Newton-Raphson

method for a system of nonlinear equations. The decomposition technique was applied for rewriting the original

polynomial into the form of product of two polynomials of lower degrees. Based on the concept of

undetermined coefficients each coefficient of an x power in the original polynomial would be equated to the

corresponding collected-expanded one of the product of the two decomposed polynomials. The unknown

coefficients in decomposed polynomials of the lower degree would be eliminated. Based on this a system of

nonlinear equations of unknown coefficients in the decomposed polynomial of the higher degree was obtained.

Then the n-D Newton-Raphson method was used to solve for the unknown coefficients from the system of

nonlinear equations. The eliminated coefficients were obtained by back substitutions. The formulations and the

concepts would be discussed in Section 2. In Section 3 the applicability of the proposed mathematical tool

would be demonstrated in several numerical examples. From which critical conclusion could be drawn in

Section 4.

II. Decomposition of the Original Polynomial Equation 2.1 Decomposition of a Polynomial Equation of Order m

A polynomial equation of order m may be generally expressed in form of (1):

001

22

...11

axaxamxm

amxm

a (1)

Where miai ,...,1,0, are the coefficients of ix and 0ma . Without losing generality 1ma may be used

throughout this technical paper. Thus:

001

22

...11

axaxamxm

amx (2)

Given miri ,...,1,0, are all possible roots of the polynomial equation. The polynomial equation of (2)

can be rewritten as:

0321 mrxrxrxrx (3)

Beyond the Quadratic Equations and the N-D Newton-Raphson Method


In the case that a root rx is found, rxx will be one factor of the original polynomial equation. The

deflated equation may be obtained by the direct long division of the original equation of (2) by rxx as:

0......2

32132

122

11

mrrrrrmm

mrm

m xxaxaaxxxaaxxax (4)

Further deflations can be done as soon as additional roots will be found. Real roots of polynomial

equations could be obtained in Bernstein form by numerical analysis [11]. The graphical method can be served

as a powerful tool for determining the number of real roots, approximate values of real roots and nature of the

real roots e.g. double real roots or triple real roots etc. Usually any value of a real root from the graphical

method can be used as an initial guess value of a numerical method e.g. the Newton-Raphson method.

The rest of this technical paper will be devoted for the decomposition of an original polynomial

equation into two decomposed polynomial equations of lower degrees.

2.2 Decomposition of a Polynomial Equation

2.2.1 Proposed Decomposition

An original polynomial equation of order m as in the form of (2) can always be decomposed into two

polynomial equations of lower degrees. For an original polynomial equation of an odd degree the equation will

be decomposed into two decomposed equations of lower degrees i.e. one decomposed polynomial equation with

an odd degree and the other decomposed polynomial equation with an even degree. For an original polynomial

equation of even degree, however, the function will be decomposed to two decomposed equations of even

degrees. The reason behind is that a polynomial equation of odd degree will always have at least one real root.

However no real root is guaranteed for a polynomial equation of an even degree. Therefore for the case of an

original polynomial equation of even degree it is conservative to assume both two decomposed polynomial

equations of even degrees, since a quadratic equation can always be solved in a closed form formula. Our

discussions will be focused on the polynomial equations of degree higher than two i.e. degree three onwards.

The orders of the two decomposed equations are summarized in Table 1.

Table 1 Degrees of Two Decomposed Equations for Original Equations of Degree from 3-9 Degree of Equation

Original Equation 1st Decomposed Equation 2nd Decomposed Equation

3 1 2

4 2 2

5 2 3

6 2 4

7 3 4

8 4 4

9 4 5

2.2.2 Bairstow’s Decomposition

Bairstow [12] proposed decomposing a polynomial of order m in form of (5) into a product of two

lower degrees, i.e. a quadratic function and a polynomial of degree 2m , plus a remainder term in form of (6).

0

... 12

21

1 axaxaxaxaxP mm

mm

(5)

dcxbxbxbxbxbvxuxxP mm

mm

012

23

32

22 ... (6)

where 2,...,1,0, mibi are the coefficients of ix dc, are coefficients and 02 mb . The latter polynomial can

be obtained by long dividing xP by vxux 2 and the term dcx is the remainder.

Once the values of u and v are assumed, All ib ’s as well as c and d can be determined. By iterative

procedures, the actual u and v as well as all ib ’s can be obtained as soon as dcx approaches zero. Thus, (6)

is reduced to (7).

0... 012

23

32

22

bxbxbxbxbvxux mm

mm (7)

For the sake of further discussions ma and 2mb can be set to 1 without losing any generality. Thus, (7)

may be rewritten as:

0... 012

23

32

012

exexexexdxdx m

mm

(8)

Based on (8) a polynomial equation can be decomposed to the product of two polynomial equations,

i.e. a quadratic equation and a polynomial equation of order 2m . There can be several pairs of the

decomposed polynomials depending on the initial guess of the iterative procedures. Since any quadratic

equation and linear equation is always solved, all possible roots of a polynomial equation can be determined by

the Bairstow’s method. The convergence of the method is quadratic, only if the zeros are complex conjugate



pairs of multiplicity one, or are real of multiplicity at most two. For higher multiplicities it is impractically slow

or subject to failure. The modifications of the Bairstow’s method were proposed [13-14], but the details are out

of the scope of this technical paper.

2.2.3 Complete Bairstow’s Decomposition

The decomposed polynomial equation of (8) can be further decomposed until the degree of the last

decomposed equation is two or one. For an original polynomial equation of an even degree or an odd degree, the

complete Bairstow’s solution can be rewritten in form of (9) or (10), respectively.

0...012

012 xQexexdxdx (9)

0...012

012

0 xQexexdxdxcx (10)

where xQ is a quadratic function.

2.3 Decomposition of a Cubic Function

Consider a general cubic equation:

0012

23 axaxax (11)

The cubic equation may be decomposed into a product of two equations i.e. one linear equation and

one quadratic equation as shown below:

0012

0 exexdx (12)

where 10 ,ed and 0e are the unknown coefficients of the equations. Expanding the product in (12) yields:

0001002

103 edxedexedx (13)

Equating each coefficient in (13) to the corresponding term in the original equation in (11) leads to 3 equations:

210 aed (14.a)

1100 aede (14.b)

000 aed (14.c)

0d in (14.c) can be rewritten in term of 0a and 0e as:

0

00

e

ad (15)

Thus, 0d can be eliminated from (14.a) and (14.b) such that a system of two nonlinear equations in

two simultaneous unknowns 1e and 0e is formed.

210

0 aee

a (16.a)

10

100 a

e

eae

(16.b)

Once 1e and 0e are obtained from a numerical method i.e. the Newton-Raphson method in two

dimensions, 0d can be obtained by back substitution via (15).

The proposed decomposition for a cubic equation has exactly the same form as the Bairstow’s method

and it is already complete.

2.4 Decomposition of a Quartic Function

Consider a general quartic equation:

0012

23

34 axaxaxax (17)

The quartic equation may be decomposed into a product of two equations i.e. two quadratic equations

as shown below:

0012

012 exexdxdx (18)

where 101 ,, edd and 0e are the unknown coefficients of the equations. Expanding the product in (18) yields:

00001102

11003

114 edxededxededxedx (19)

Equating each coefficient in (19) to the corresponding term in the original equation in (17) leads to 4

equations:

311 aed (20.a)



21100 aeded (20.b)

10110 aeded (20.c)

000 aed (20.d)

1d in (20.a) and 0d in (20.d), respectively, can be rewritten in term of the other terms as:

131 ead (21.a)

0

00

e

ad (21.b)

Thus, 1d and 0d can be eliminated from (20.b) and (20.c) such that a system of two simultaneous

nonlinear equations in two unknowns 1e and 0e is formed.

211300

0 aeeaee

a (22.a)

10130

10 aeeae

ea

(22.b)

Once 1e and 0e are obtained by the Newton-Raphson method in two dimensions, 1d and 0d can be

obtained by back substitution via (21.a) and (21.b), respectively.

The proposed decomposition for a quartic equation has exactly the same form as the Bairstow’s method

and it is already complete.

2.5 Decomposition of a Quintic Function

2.5.1 The Proposed Decomposition

Consider a general quintic equation:

0012

23

34

45 axaxaxaxax (23)

The quintic equation may be decomposed into a product of two equations i.e. one quadratic equation

and one cubic equation as shown below:

0012

23

012 exexexdxdx (24)

where 1201 ,,, eedd and 0e are the unknown coefficients of the equations. Expanding the product in (24) yields:

00001102

112003

21104

215 edxededxededexededxedx (25)


equations:

421 aed (26.a)

32110 aeded (26.b)

211200 aedede (26.c)

10110 aeded (26.d)

000 aed (26.e)

1d in (26.a) and 0d in (26.e), respectively, can be rewritten in term of the other terms as:

241 ead (27.a)

0

00

e

ad (27.b)

Thus, 1d and 0d can be eliminated from (26.b), (26.c) and (26.d) such that a system of three

simultaneous nonlinear equations in three unknowns 12 ,ee and 0e is formed.

322410

0 aeeaee

a (28.a)

21240

200 aeea

e

eae

(28.b)

10240

10 aeeae

ea

(28.c)

Once 12 ,ee and 0e are obtained by the Newton-Raphson method in three dimensions, 1d and 0d can be

obtained by back substitution via (27.a) and (27.b), respectively.




The proposed decomposition for a quintic equation has exactly the same form as the Bairstow’s method

and the cubic equation obtained can be further decomposed by using the proposed decomposition as discussed

in Section 2.3.


The quintic equation of (23) can be rewritten in form of a complete Bairstow’s decomposition as:

0012

012

0 exexdxdxcx (29)

Expanding the product in (29) yields:

0000

010100002

11001100000

311101000

4110

5

edc

xedcedcedxedcededecdc

xedecdcedxedcx

(30)


4110 aedc (31.a)

311101000 aedecdced (31.b)

211001100000 aedcededecdc (31.c)

101010000 aedcedced (31.d)

0000 aedc (31.e)

Thus, a system of five simultaneous nonlinear equations in five unknowns 1010 ,,, eddc and 0e is formed.

2.6 Decomposition of a Sextic Function


Consider a general sextic equation: (32)

0012

23

34

45

56 axaxaxaxaxax

The sextic equation may be decomposed into a product of two equations i.e. one quadratic equation and

one quartic equation as shown below:

0012

23

34

012 exexexexdxdx (33)

where 12301 ,,,, eeedd and 0e are the unknown coefficients of the equations. Expanding the product in (33)

yields:

000

01102

112003

213014

31205

316

ed

xededxededxededexededxedx (34)


equations:

531 aed (35.a)

43120 aeded (35.b)

321301 aedede (35.c)

211200 aedede (35.d)

10110 aeded (35.e)

000 aed (35.f)

1d in (35.a) and 0d in (35.f), respectively, can be rewritten in term of the other terms as:

351 ead (36.a)

0

00

e

ad (36.b)

Thus, 1d and 0d can be eliminated from (35.b), (35.c), (35.d) and (35.e) such that a system of four

simultaneous nonlinear equations in three unknowns 123 ,, eee and 0e is formed.

433520

0 aeeaee

a (37.a)



30

301235 a

e

eaeeea

(37.b)

21350

200 aeea

e

eae

(37.c)

10350

10 aeeae

ea

(37.d)

Once 123 ,, eee and 0e are obtained by the Newton-Raphson method in four dimensions, 1d and 0d can

be obtained by back substitution via (36.a) and (36.b), respectively.


The proposed decomposition for a sextic equation has exactly the same form as the Bairstow’s method

and the quartic equation obtained can be further decomposed by using the proposed decomposition as discussed

in Section 2.4.


The sextic equation of (32) can be rewritten in form of a complete Bairstow’s decomposition as:

0012

012

012 fxfxexexdxdx (38)


0000001010100

2011101110000000

3111011001100110

4111111000

5111

6

fedxfedfedfed

xfedfedfedfefded

xfedfefefdfdeded

xfefdedfedxfedx

(39)


5111 afed (40.a)

4111111000 afefdedfed (40.b)

3111011001100110 afedfefefdfdeded (40.c)

2011101110000000 afedfedfedfefded (40.d)

1001010100 afedfedfed (40.e)

0000 afed (40.f)

Thus, a system of six simultaneous nonlinear equations in six unknowns 10101 ,,,, feedd and 0f is formed.

2.7 Decomposition of a Septic Function


Consider a general septic equation:

0012

23

34

45

56

67 axaxaxaxaxaxax (41)

The septic equation may be decomposed into a product of two equations i.e. one cubic equation and

one quartic equation as shown below:

0012

23

34

012

23 exexexexdxdxdx (42)

where 123012 ,,,,, eeeddd and 0e are the unknown coefficients of the equations. Expanding the product in (42)

yields:

00001102

021120

31221300

4223110

53221

632

7

edxededxededed

xedededexedededxededxedx (43)


632 aed (44.a)

53221 aeded (44.b)

4223110 aededed (44.c)

31221300 aededede (44.d)



2021120 aededed (44.e)

10110 aeded (44.f)

000 aed (44.g)

2d in (44.a), 1d in (44.b) and 0d in (44.g), respectively, can be rewritten in term of the other terms as:

362 ead (45.a)

336251 eeaead (45.b)

0

00

e

ad (45.c)

Thus, 12 ,dd and 0d can be eliminated from (44.c), (44.d), (44.e) and (44.f) such that a system of four

simultaneous nonlinear equations in four unknowns 123 ,, eee and 0e is formed.

423633362510

0 aeeaeeeaeaee

a (46.a)

313623362530

00 aeeaeeeaeae

e

ae (46.b)

20361336250

20 aeeaeeeaeae

ea

(46.c)

10336250

10 aeeeaeae

ea

(46.d)

Once 123 ,, eee and 0e are obtained by the Newton-Raphson method in four dimensions, 12 ,dd and

0d can be obtained by back substitution via (45.a), (45.b) and (45.c), respectively.


The septic equation of (41) can be rewritten in form of the Bairstow’s decomposition as:

0012

23

34

45

012 exexexexexdxdx (47)


00001102

11200

321301

431402

54130

641

7

edxededxedede

xededexededexededxedx (48)


641 aed (49.a)

54130 aeded (49.b)

431402 aedede (49.c)

321301 aedede (49.d)

211200 aedede (49.e)

10110 aeded (49.f)

000 aed (49.g)

Thus, a system of seven simultaneous nonlinear equations in seven unknowns 101010 ,,,,, feeddc and

0f is formed.


The septic equation of (41) can be rewritten in form of a complete Bairstow’s decomposition as:

0012

012

012

0 fxfxexexdxdxcx (50)

Expanding the product in (50) and equating each coefficient to the corresponding term in the original

equation in (41) leads to 7 equations:

61110 afedc (51.a)

5111111101010000 afefdedfcecdcfed (51.b)

4111110110011001100110000000 afedfdcedcfefefdfdededfcecdc (51.c)



31110011101110

100010100010100000000

afedcfedfedfed

fecfdcfdcedcedcfefded

(51.d)

2011011101010001010100000000000 afedcfedcfedcfedfedfedfecfdcedc (51.e)

1001001001000000 afedcfedcfedcfed (51.f)

00000 afedc (51.h)

Thus, a system of seven simultaneous nonlinear equations in seven unknowns 101010 ,,,,, feeddc and

0f is formed.

2.8 Decomposition of an Octic Function


Consider a general octic equation:

0012

23

34

45

56

67

78 axaxaxaxaxaxaxax (52)

The octic equation may be decomposed into a product of two equations i.e. two quartic equations as

shown below:

0012

23

34

012

23

34 exexexexdxdxdxdx (53)

where 1230123 ,,,,,, eeedddd and 0e are the unknown coefficients of the equations. Expanding the product in

(54) and equating each coefficient to the corresponding term in the original equation in (52) leads to 8

equations:

733 aed (54.a)

63322 aeded (54.b)

5233211 aededed (54.c)

413223100 aedededed (54.d)

303122130 aedededed (54.e)

2021120 aededed (54.f)

10110 aeded (54.g)

000 aed (54.h)

3d in (54.a), 2d in (54.b), 1d in (54.c) and 0d in (54.h), respectively, can be rewritten in term of the

other terms as:

373 ead (55.a)

337262 eeaead (55.b)

23733372615151 eeaeeeaeaeaead (55.c)

0

00

e

ad (55.d)

Thus, 123 ,, ddd and 0d can be eliminated from (54.d), (54.e), (54.f) and (54.g) such that a system of

four simultaneous nonlinear equations in four unknowns 123 ,, eee and 0e is formed.

4137233726

32373337261500

0

aeeaeeeaea

eeeaeeeaeaeaee

a

(56.a)

3133726037

2237333726150

30

aeeeaeaeea

eeeaeeeaeaeae

ea

(56.b)

20337261237333726150

20 aeeeaeaeeeaeeeaeaeae

ea

(56.c)

10237333726150

10 aeeeaeeeaeaeae

ea

(56.d)

Once 123 ,, eee and 0e are obtained by the Newton-Raphson method in four dimensions, 123 ,, ddd and

0d can be obtained by back substitution via (55.a), (55.b), (55.c) and (55.d), respectively.




The octic equation of (52) can be rewritten in form of the Bairstow’s decomposition as:

0012

23

34

45

56

012 exexexexexexdxdx (57)



751 aed (58.a)

65140 aeded (58.b)

541503 aedede (58.c)

431402 aedede (58.d)

321301 aedede (58.e)

211200 aedede (58.f)

10110 aeded (58.g)

000 aed (58.h)

Thus, a system of eight simultaneous nonlinear equations in eight unknowns 1234501 ,,,,,, eeeeedd and

0e is formed.


The octic equation of (52) can be rewritten in form of a complete Bairstow’s decomposition as:

0012

012

012

012 gxgxfxfxexexdxdx (59)



71111 agfed (60.a)

61111111111110000 agfgefegdfdedgfed (60.b)

5111111111111

011001100110011001100110

agfegfdgedfed

gfgfgegefefegdgdfdfdeded

(60.c)

41111011101110011101110011

101110011101110000000000000

agfedgfegfegfegfdgfdgfdged

gedgedfedfedfedgfgefegdfded

(60.d)

30111101111011110001010

100001010100001010100001010100

agfedgfedgfedgfedgfegfe

gfdgfdgfdgfdgedgedgedfedfedfed

(60.e)

21010001101011001

100100101100000000000000

agfedgfedgfedgfed

gfedgfedgfedgfegfdgedfed

(60.f)

10001001001001000 agfedgfedgfedgfed (60.g)

00000 agfed (60.h)

Thus, a system of eight simultaneous nonlinear equations in eight unknowns 1010101 ,,,,,, gffeedd and

0g is formed.

2.9 Decomposition of a Nonic Function


Consider a general nonic equation:

0012

23

34

45

56

67

78

89 axaxaxaxaxaxaxaxax (61)

The nonic equation may be decomposed into a product of two equations i.e. one quartic equations and

one quintic equation as shown below:

0012

23

34

45

012

23

34 exexexexexdxdxdxdx (62)

where 12340123 ,,,,,,, eeeedddd and 0e are the unknown coefficients of the equations. Expanding the product in

(62) and equating each coefficient to the corresponding term in the original equation in (61) leads to 9

equations:

843 aed (63.a)

74332 aeded (63.b)

6334221 aededed (63.c)



523324110 aedededed (63.d)

4132231400 aedededede (63.e)

303122130 aedededed (63.f)

2021120 aededed (63.g)

10110 aeded (63.h)

000 aed (63.i)

3d in (63.a), 2d in (63.b), 1d in (63.c) and 0d in (63.i), respectively, can be rewritten in term of the

other terms as:

483 ead (64.a)

448372 eeaead (64.b)

348444837261 eeaeeeaeaead (64.c)

0

00

e

ad (64.d)

Thus, 123 ,, ddd and 0d can be eliminated from (63.d), (63.e), (63.f), (63.g) and (63.h) such that a

system of five simultaneous nonlinear equations in five unknowns 1234 ,,, eeee and 0e is formed.

5248344837

43484448372610

0

aeeaeeeaea

eeeaeeeaeaeaee

a

(65.a)

4148244837

3348444837260

400

aeeaeeeaea

eeeaeeeaeaeae

eae

(65.b)

3048144837

2348444837260

30

aeeaeeeaea

eeeaeeeaeaeae

ea

(65.c)

20448371348444837260

20 aeeeaeaeeeaeeeaeaeae

ea

(65.d)

10348444837260

10 aeeeaeeeaeaeae

ea

(65.e)

Once 1234 ,,, eeee and 0e are obtained by the Newton-Raphson method in five dimensions,

123 ,, ddd and 0d can be obtained by back substitution via (64.a), (64.b), (64.c) and (64.d), respectively.


The nonic equation of (61) can be rewritten in form of the Bairstow’s decomposition as:

0012

23

34

45

56

67

012 exexexexexexexdxdx (66)



861 aed (67.a)

76150 aeded (67.b)

651604 aedede (67.c)

541503 aedede (67.d)

431402 aedede (67.e)

321301 aedede (67.f)

211200 aedede (67.g)

10110 aeded (67.h)

000 aed (67.i)

Thus, a system of nine simultaneous nonlinear equations in nine unknowns 12345601 ,,,,,,, eeeeeedd and 0e is

formed.




The nonic equation of (61) can be rewritten in form of a complete Bairstow’s decomposition as:

0012

012

012

012

0 gxgxfxfxexexdxdxcx (68)



811110 agfedc (69.a)

7111111111111101010100000 agfgefegdfdedgcfcecdcgfed (69.b)

6111111111111110110110110110110

0110011001100110011000000000

agfegfdgedfedgfcgecfecgdcfdcedc

gegefefegdgdfdfdededgcfcecdc

(69.c)

5111111101110

11101110011101110011101110011101

110011101110010100010100010100

010100010100010100000000000000

agfedgfdcgedc

fedcfedcgfegfegfegfdgfdgfdgedged

gedfedfedfedgfcgfcgecgecfecfec

gdcgdcfdcfdcedcedcgfgefegdfded

(69.d)

411010

0110101011000110101011000110

10101010011010101100011110111110

0010101000010101000010101101

100001010100000000000000000000

agfedc

gfecgfecgfecgfdcgfdcgfdcgedc

gedcgfdcfedcfedcfedcgfedgfedgfed

gfegfegfegfdgfdgfdgedgedgfed

gedfedfedfeddfcgecfecgdcfdcedc g

(69.e)

30111010110

110101110000100100100000100100

10000010010010000010010010001100

00110101100101101100000000000000

agfedcgfedc

gfedcgfedcgfecgfecgfecgfdcgfdc

gfdcgedcgedcgedcfedcfedcfedcgefd

gfedgfedgfedgfedgfedgfegfdgedfed

(69.f)

2001100101010010011001010011000

00010010010010000000000000000000

agfedcgfedcgfedcgfedcgfedcgfedc

gfedgfedgfedgfedgfecgfdcgedcfedc

(69.g)

1000100010001000100000000 agfedcgfedcgfedcgfedcgfed (69.h)

000000 agfedc (69.i)

Thus, a system of nine simultaneous nonlinear equations in nine unknowns 10101010 ,,,,,,, gffeeddc

and 0g is formed.

III. N-D Newton-Raphson Method and the Jacobian of the Functions 3.1 Newton-Raphson Method for a Nonlinear Function

The prediction of the Newton-Raphson method was based on a first order Taylor series expansion:

iiiii xfxxxfxf '11 (70)

where ix is the initial guess at the root or the previous estimate of the root and 1ix is the point at which the

slope intercepts the x axis. At this intercept 01 ixf by definition and (70) can be rearranged to:

i

iii

xf

xfxx

'1 (71)

which is the Newton-Raphson method for a nonlinear equation.

3.2 Newton-Raphson Method in more than one Dimension

3.2.1 Newton-Raphson Method in two Dimensions

The Newton-Raphson method for two simultaneous nonlinear equations can be derived in the similar

fashion. However, a multivariate Taylor series has to be taken into account for the fact of more than one

independent variables contributing to the determination of the root. For the two-variable case, a first order

Taylor series can be written for each nonlinear equation as:

y

uyy

x

uxxuu i

iii

iiii

111 (72.a)

y

vyy

x

vxxvv i

iii

iiii

111 (72.b)



Just as for the single-equation case, the root estimate corresponds to the points which 01 iu

and 01 iv . (72.a) and (72.b) can be rearranged to:

y

uy

x

uxuy

y

ux

x

u ii

iiii

ii

i

11 (73.a)

y

vy

x

vxvy

y

vx

x

v ii

iiii

ii

i

11 (73.b)

In (73) only 1ix and 1iy are unknown. Thus (73) is a set of two simultaneous linear equations with

two unknowns. Consequently, with simple algebraic manipulations, e.g. Cramer’s rule, 1ix and 1iy can be

solved as:

x

v

y

u

y

v

x

u

y

uv

y

vu

xxiiii

ii

ii

ii

1 (74.a)

x

v

y

u

y

v

x

ux

uv

x

vu

yyiiii

ii

ii

ii

1 (74.b)

The denominator of each of (74) is the determinant of the Jacobian of the system. (73) is the equation

for the Newton-Raphson method in two dimensions.

For the benefits of further discussions on the method for more than two dimensions, (74) should be

rewritten in term of the matrix notation i.e. the Jacobian of the function – Z .

ii

ii

i

i

i

i

yxv

yxuZ

y

x

y

x

,

,1

1

1 (75)

y

yxv

y

yxux

yxv

x

yxu

Ziiii

iiii

,,

,,

(76)

where Z is the Jacobian of the function and 1Z is the inverse of Z .

3.2.2 Newton-Raphson Method in More Than Two Dimensions

Consider a system of n simultaneous nonlinear equations:

0,...,,

0,...,,2

0,...,,1

21

21

21

n

n

n

xxxfn

xxxf

xxxf

(77)

The solution of this system consists of a set of x values that simultaneously result in all the equations

equaling zero. Just for the case of two nonlinear equations a Taylor series expansion is written for each equation

n

ilinin

ilii

iliiilil

x

fxx

x

fxx

x

fxxff

,,1,

2

,,21,2

1

,,11,1,1, (78)

where the subscript, l , represents the equation or unknown and the second subscript denotes whether the value

of function under consideration is at the present value ( i ) or at the next vale ( 1i ).

Equations in the form of (78) are written for each of the original nonlinear equations. All 1, ilf terms

are set to zero and (78) can be rewritten as:

n

ilin

ili

ili

n

ilin

ili

iliil

x

fx

x

fx

x

fx

x

fx

x

fx

x

fxf

,1,

2

,1,2

1

,1,1

,,

2

,,2

1

,,1, (79)

Notice that only the njx ij ,...,2,1,1, terms on the right-hand side are unknowns. As a result, a set of

n linear simultaneous equation is obtained.



The partial derivatives can be expressed in term of matrix notation as:

n

ininin

n

iii

iii

x

f

x

f

x

f

x

f

x

f

x

f

xn

f

x

f

x

f

Z

,

2

,

1

,

,2

2

,2

1

,2

,1

2

,1

1

,1

(80)

The present and the next values can be expressed in the vector form as

iniiT

i xxxX ,,2,1 (81)

1,1,21,11 iniiT

i xxxX (82)

The function values at i can be expressed as

iniiT

i fffF ,,2,1 (83)

Using these relationships, (79) can be rewritten as

ii WXZ 1 (84)

iii XZFW (85)

Assumed that the inverse of Z can be obtained. Then 1iX in (84) can be solved.

iiiiii XZZFZXZFZWZX

1111

1 (86)

In (86) IZZ 1

is a unit matrix. Thus, (86) can be rewritten as:

iii FZXX

1

1 (87)

3.3 Functions from the Decomposed Equations and the Jacobian of the Functions

For the benefits of applications the system of simultaneous nonlinear equations derived for the

decomposed equations discussed in Section 2.3 – Section 2.9 are summarized in the following subsections.

3.3.1 Original Cubic Equations

10

100

210

0

ae

eae

aee

a

F (88)

0

0

20

10

20

0

1

1

e

a

e

ea

e

a

Z (89)

3.3.2 Original Quartic Equations

10130

10

211300

0

aeeae

ea

aeeaee

a

F (90)

00

0

20

1013

1320

0 21

ee

a

e

eaea

eae

a

Z (91)



3.3.3 Original Quintic Equations

3.3.3.1 The Proposed Decomposition

10240

10

21240

200

322410

0

aeeae

ea

aeeae

eae

aeeaee

a

F (92)

00

0242

0

10

10

0242

0

20

2420

0

1

21

ee

aea

e

ea

ee

aea

e

ea

eae

a

Z (93)

3.3.3.2 Bairstow’s Decomposition

000

11010

211200

32110

411

aed

aeded

aedede

aeded

aed

F (94)

000

0

1

101

10010

00

0101

0112

12

de

ddee

ddee

de

Z (95)

3.3.3.3 Complete Bairstow’s Decomposition

0000

101010000

211001100000

311101000

4110

aedc

aedcedced

aedcededecdc

aedecdced

aedc

F (96)

00

11

10101

000000

00100001000110

10010100101100

101011

dceced

dcdcdececeeded

dcddceceeceded

dceced

Z (97)



3.3.4 Original Sextic Equations

3.3.4.1 The Proposed Decomposition and Bairstow’s Decomposition

10350

10

21350

200

30

301235

433520

0

aeeae

ea

aeeae

eae

ae

eaeeea

aeeaee

a

F (98)

00

0352

0

10

10

0352

0

20

0

0352

0

30

320

0

0

1

1

210

ee

aea

e

ea

ee

aea

e

ea

e

aea

e

ea

ee

a

Z

(99)


0000

1001010100

2011101110000000

3111011001100110

4111111000

5111

afed

afedfedfed

afedfedfedfefded

afedfefefdfdeded

afefdedfed

afed

F (100)

000

111

101010

000000

000110000110000110

011011000110110001101100

110011110011110011

111111

edfdfe

edededfdfdfdfefefe

ededededfdfdfdfdfefefefe

edededfdfdfdfefefe

edfdfe

Z (101)

3.3.5 Original Septic Equations


10336250

10

20361336250

20

31362336250

300

423633362510

0

aeeeaeae

ea

aeeaeeeaeae

ea

aeeaeeeaeae

eae

aeeaeeeaeaee

a

F (102)



300600

0336252

0

10

3116010

033625362

0

20

322610

033625362

0

30

1336253620

0

2

2

221

2221

eeeaee

aeeaea

e

ea

eeeaeee

aeeaeaea

e

ea

eeeaee

aeeaeaea

e

ea

eeeaeaeae

a

Z

(103)


000

10110

211200

321301

431402

54130

641

aed

aeded

aedede

aedede

aedede

aeded

aed

F (104)

00000

000

001

010

100

10001

1000010

00

0101

0112

0123

0134

14

de

ddee

ddee

ddee

ddee

de

Z (105)


00000

1001001001000000

2011010101100001010100000000000

31110

011101110010100010100010100000000

4111110110110011001100110000000

5111111101010000

61110

afedc

afedcfedcfedcfed

afedcfedcfedcfedfedfedfecfdcedc

afedc

fedfedfedfecfecfdcfdcedcedcfefded

afedfecfdcedcfefefdfdededfcecdc

afefdedfcecdcfed

afedc

F (106)

000

111

1010101

000000000000

0006,60004,60002,61,6

7,56,55,54,53,52,51,5

7,46,45,44,43,42,41,4

7,31105,31103,31101,3

110110110111

edcfdcfecfed

edczfdczfeczz

zzzzzzz

zzzzzzz

zedczfdczfecz

edcedcfecfed

Z (107)

1111110001,3 fefdedfedz (108.a)

111010003,3 fefcecfez (108.b)

111010005,3 fdfcdcfdz (108.c)

111010007,3 fefdedfez (108.d)

1110110011001101,4 fedfefefdfdededz (108.e)

111010002,4 fefcecfez (108.f)

110011000003,4 fecfefefcecz (108.g)



111010004,4 fdfcdcfdz (108.h)

110011000005,4 fdcfdfdfcdcz (108.i)

111010006,4 fefdedfez (108.j)

110011000007,4 fedfefefdedz (108.k)

0111011100000001,5 fedfedfedfefdedz (108.l)

110011000002,5 fecfefefcecz (108.m)

010100003,5 fecfecfez (108.n)

110011000004,5 fdcfdfdfcdcz (108.o)

110011000006,5 fedfefefdedz (108.q)

010100007,5 fedfedfez (108.r)

0010101001,6 fedfedfedz (108.s)

010100002,6 fecfecfez (108.t)

010100004,6 fdcfdcfdz (108.u)

010100006,6 fedfedfez (108.v)

3.3.6 Original Octic Equations


10237333726150

10

20337261237333726150

20

3133726

0372237333726150

30

4137

23372632373337261500

0

aeeeaeeeaeaeae

ea

aeeeaeaeeeaeeeaeaeae

ea

aeeeaea

eeaeeeaeeeaeaeae

ea

aeea

eeeaeaeeeaeeeaeaeaee

a

F

(109)

4,4300700

01,4

4,3311700

02,3

2337262

0

20

4,23,22

3372620

3037

4,16372

323720

0

2

2

2

23221

zeeeaee

az

zeeeaee

azeeaea

e

ea

zzeeaeae

eaea

zaeaeeeae

a

Z (110)

323

32

373627154,1 643222 eeeeaeaeaeaz (111.a)

3332

2373627153,2 422 eeeeaeaeaeaz (111.b)

23232731

2226170

0

04,2 3222 eeeeaeeeeaeae

e

az (111.c)

3332

2373627152,3 22 eeeeaeaeaeaz (111.d)

231317213016074,3 3222 eeeeaeeeeeaeaz (111.e)

323

32

3736271520

101,4 2 eeeeaeaeaea

e

eaz

(111.f)

23030720064,4 322 eeeeaeeeaz (111.g)




000

10110

211200

321301

431402

541503

65140

751

aed

aeded

aedede

aedede

aedede

aedede

aeded

aed

F (112)

000000

0000

0001

0010

0100

1000

100001

10000010

00

0101

0112

0123

0134

0145

15

de

ddee

ddee

ddee

ddee

ddee

de

Z (113)


00000

10001001001001000

21010001101011001

100100101100000000000000

30111101111011110001010

100001010100001010100001010100

41111011101110011101110011

101110011101110000000000000

5111111111111

011001100110011001100110

61111111111110000

71111

agfed

agfedgfedgfedgfed

agfedgfedgfedgfed

gfedgfedgfedgfegfdgedfed

agfedgfedgfedgfedgfegfe

gfdgfdgfdgfdgedgedgedfedfedfed

agfedgfegfegfegfdgfdgfdged

gedgedfedfedfedgfgefegdfded

agfegfdgedfed

gfgfgegefefegdgdfdfdeded

agfgefegdfdedgfed

agfed

F (114)

0000

1111

10101010

000000000000

0007,70005,70003,70001,7

8,67,66,65,64,63,62,61,6

8,57,56,55,54,53,52,51,5

8,47,46,45,44,43,42,41,4

8,31116,31114,31112,3111

111111111111

fedgedgfdgfe

fedzgedzgfdzgfez

zzzzzzzz

zzzzzzzz

zzzzzzzz

zfedzgedzgfdzgfe

fedgedgfdgfe

Z

(115)

1111110002,3 gfgefegfez (116.a)

1111110004,3 gfgdfdgfdz (116.b)

1111110006,3 gegdedgedz (116.c)

1111110008,3 fefdedfedz (116.d)

1111110001,4 gfgefegfez (116.e)

1110110011001102,4 gfegfgfgegefefez (116.f)

1111110003,4 gfgdfdgfdz (116.g)

1110110011001104,4 gfdgfgfgdgdfdfdz (116.h)

1111110005,4 gegdedgedz (116.i)



1110110011001106,4 gedgegegdgdededz (116.j)

1111110007,4 fefdedfedz (116.k)

1110110011001108,4 fedfefefdfdededz (116.l)

1110110011001101,5 gfegfgfgegefefez (116.m)

0111011100000002,5 gfegfegfegfgefez (116.n)

1110110011001103,5 gfdgfgfgdgdfdfdz (116.o)

0111011100000004,5 gfdgfdgfdgfgdfdz (116.p)

1110110011001105,5 gedgegegdgdededz (116.q)

0111011100000006,5 gedgedgedgegdedz (116.r)

1110110011001107,5 fedfefefdfdededz (116.s)

0111011100000008,5 fedfedfedfefdedz (116.u)

1010111100000001,6 gfegfegfegfgefez (116.v)

001101002,6 gfegfegfez (116.w)

0111011100000003,6 gfdgfdgfdgfgdfdz (116.x)

1000010104,6 gfdgfdgfdz (116.z)

1100111010000005,6 gedgedgedgegdedz (116.aa)

0010101006,6 gedgedgedz (116.ab)

0111011100000007,6 fedfedfedfefdedz (116.ac)

0100011008,6 fedfedfedz (116.ad)

0010101001,7 gfegfegfez (116.ae)

0010101003,7 gfdgfdgfdz (116.af)

0010101005,7 gedgedgedz (116.ag)

0010101007,7 fedfedfedz (116.ah)

3.3.7 Original Nonic Equations


10348444837260

10

20448371348444837260

20

3048

1448372348444837260

30

4148

2448373348444837260

400

5248

34483743484448372610

0

aeeeaeeeaeaeae

ea

aeeeaeaeeeaeeeaeaeae

ea

aeea

eeeaeaeeeaeeeaeaeae

ea

aeea

eeeaeaeeeaeeeaeaeae

eae

aeea

eeeaeaeeeaeeeaeaeaee

a

F

(117)



5,5400800

01,5

5,44,410

02,41,4

5,34,33,32

448374820

30

5,24,22

448374820

40

5,14,14820

0

2

21

21

zeeeaee

az

zzee

azz

zzzeeaeaeae

ea

zzeeaeaeae

ea

zzeae

a

Z (118)

2448374,1 322 eeaeaz (119.a)

3443

2484738265,1 463222 eeeeaeaeaeaz (119.b)

3443

248438264,2 4722 eeeeaeeaeaz (119.c)

23

2434384237281

0

05,2 2322 eeeeeaeeeaeae

e

az (119.d)

3443

2484738263,3 22 eeeeaeaeaeaz (119.e)

422810

04,3 2 eeeae

e

az (119.f)

2424283241271805,3 3222 eeeeaeeeeeaeaez (119.g)

244872

0

201,4 eeaa

e

eaz

(119.h)

3443

2484738262,4 2 eeeeaeaeaeaz (119.i)

411804,4 2 eeeaez (119.j)

241418314017085,4 3222 eeeeaeeeeeaeaz (119.k)

3443

2484738262

0

101,5 2 eeeeaeaeaea

e

eaz

(119.l)

24040830075,5 322 eeeeaeeeaz (119.m)


000

10110

211200

321301

431402

541503

651604

76150

861

aed

aeded

aedede

aedede

aedede

aedede

aedede

aeded

aed

F (120)



0000000

00000

00001

00010

00100

01000

10000

1000001

100000010

00

0101

0112

0123

0134

0145

0156

16

de

ddee

ddee

ddee

ddee

ddee

ddee

de

Z (121)


000000

1000100010001000100000000

2001100101010010011001010011000

00010010010010000000000000000000

30111010110

110101110000100100100000100100

10000010010010000010010010001100

00110101100101101100000000000000

411010

0110101011000110101011000110

10101010011010101100011110111110

0010101000010101000010101101

100001010100000000000000000000

5111111101110

11101110011101110011101110011101

110011101110010100010100010100

010100010100010100000000000000

6111111111111110110110110110110

0110011001100110011000000000

7111111111111101010100000

811110

agfedc

agfedcgfedcgfedcgfedcgfed

agfedcgfedcgfedcgfedcgfedcgfedc

gfedgfedgfedgfedgfecgfdcgedcfedc

agfedcgfedc

gfedcgfedcgfecgfecgfecgfdcgfdc

gfdcgedcgedcgedcfedcfedcfedcgefd

gfedgfedgfedgfedgfedgfegfdgedfed

agfedc

gfecgfecgfecgfdcgfdcgfdcgedc

gedcgfdcfedcfedcfedcgfedgfedgfed

gfegfegfegfdgfdgfdgedgedgfed

gedfedfedfeddfcgecfecgdcfdcedc

agfedgfdcgedc

fedcfedcgfegfegfegfdgfdgfdgedged

gedfedfedfedgfcgfcgecgecfecfec

gdcgdcfdcfdcedcedcgfgefegdfded

agfegfdgedfedgfcgecfecgdcfdcedc

gegefefegdgdfdfdededgcfcecdc

agfgefegdfdedgcfcecdcgfed

agfedc

F

g

(122)

9,96,94,92,91,9

9,88,87,86,85,84,83,82,81,8

9,78,77,76,75,74,73,72,71,7

9,68,67,66,65,64,63,62,61,6

9,58,57,56,55,54,53,52,51,5

9,48,47,46,45,44,43,42,41,4

9,38,37,36,35,34,33,32,31,3

9,27,25,23,21,2

0000

1111

101010101

zzzzz

zzzzzzzzz

zzzzzzzzz

zzzzzzzzz

zzzzzzzzz

zzzzzzzzz

zzzzzzzzz

zzzzz

Z (123)

11111,2 gfedz (124.a)

11103,2 gfecz (124.b)

11105,2 gfdcz (124.c)

11107,2 gedcz (124.d)

11109,2 fedcz (124.e)

11111111111100001,3 gfgefegdfdedgfedz (124.f)



11102,3 gfecz (124.g)

1111111010100003,3 gfgefegcfcecgfez (124.h)

11104,3 gfdcz (124.i)

1111111010100005,3 gfgdfdgcfcdcgfdz (124.j)

11106,3 gedcz (124.k)

1111111010100007,3 gegdedgcecdcgedz (124.l)

11108,3 fedcz (124.m)

1111111010100009,3 fefdedfcecdcfedz (124.n)

111111111111

0110011001100110011001101,4

gfegfdgedfed

gfgfgegefefegdgdfdfdededz

(124.o)

1111111010100002,4 gfgefegcfcecgfez (124.p)

111

1101101100110011001100000003,4

gfe

gfcgecfecgfgfgegefefegcfcecz

(124.q)

1111111010100004,4 gfgdfdgcfcdcgfdz (124.r)

111

1101101100110011001100000005,4

gfd

gfcgdcfdcgfgfgdgdfdfdgcfcdcz

(124.s)

1111111010100006,4 gegdedgcecdcgedz (124.t)

111

1101101100110011001100000007,4

ged

gecgdcedcgegegdgdededgcecdcz

(124.u)

1111111010100008,4 fefdedfcecdcfedz (124.v)

111

1101101100110011001100000009,4

fed

fecfdcedcfefefdfdededfcecdcz

(124.w)

1111011101110011101110

01110111001110111000000000001,5

gfedgfegfegfegfdgfdgfd

gedgedgedfedfedfedgfgegdfdedz

(124.x)

111110

1101100110011001100000002,5

gfegfc

gecfecgfgfgegefefegcfcecz

(124.y)

1110011011

1100101000101000101000000003,5

gfecggfefe

gfegfcgfcgecgecfecfecgfgefez

(124.z)

111

1101101100110011001100000004,5

gfd

gfcgdcfdcgfgfgdgdfdfdgcfcdcz

(124.aa)

1110011101

1100101000101000101000000005,5

gfdcgfdgfd

gfdgfcgfcgdcgdcfdcfdcgfgdfdz

(124.ab)

111

1101101100110011001100000006,5

ged

gecgdcedcgegegdgdededgcecdcz

(124.ac)

1110011101

1100101000101000101000000007,5

gedcgedged

gedgecgecgdcgdcedcedcgegdedz

(124.ad)

111

1101101100110011001100000008,5

fed

fecfdcedcfefefdfdededfcecdcz

(124.ae)

1110011

1011100101000101000101000000009,5

fedcfed

fedfedfecfecfdcfdcedcedcfefdedz

(124.af)

0111101111011110001010100

0010101000010101000010101001,6

gfedgfedgfedgfedgfegfegfe

gfdgfdgfdgedgedgedfedfedfedz

(124.ag)



1110011

1011100101000101000101000000002,6

gfecgfe

gfegfegfcgfcgecgecfecfecgfgefez

(124.ah)

0110101011000010101000000000003,6 gfecgfecgfecgfegfegfegfcgecfecz (124.ai)

1110011101

1100101000101000101000000004,6

gfdcgfdgfd

gfdgfcgfcgdcgdcfdcfdcgfgdfdz

(124.aj)

0110101011000010101000000000005,6 gfdcgfdcgfdcgfdgfdgfdgfcgdcfdcz (124.ak)

1110011

1011100101000101000101000000006,6

gedcged

gedgedgecgecgdcgdcedcedcgegdedz

(124.al)

0110101011000010101000000000007,6 gedcgedcgedcgedgedgedgecgdcedcz (124.am)

1110011

1011100101000101000101000000008,6

fedcfed

gedfedfecfecfdcfdcedcedcfefdedz

(124.an)

0110101011000010101000000000009,6 fedcfedcfedcfedfedfedfecfdcedcz (124.ao)

00110101

10010110101011000000000000001,7

gfedgfed

gfedgfedgfedgfedgfegfdgedfedz

(124.ap)

0110101011000010101000000000002,7 gfecgfecgfecgfegfegfegfcgecfecz (124.aq)

0010010010000003,7 gfecgfecgfecgfez (124.ar)

0110101011000010101000000000004,7 gfdcgfdcgfdcgfdgfdgfdgfcgdcfdcz (124.as)

0010010010000005,7 gfdcgfdcgfdcgfdz (124.at)

0110101011000010101000000000006,7 gedcgedcgedcgedgedgedgecgdcedcz (124.au)

0010010010000007,7 gedcgedcgedcgedz (124.av)

0110101011000010101000000000008,7 fedcfedcfedcfedfedfedfecfdcedcz (124.aw)

0010010010000009,7 fedcfedcfedcfedz (124.ax)

00010010010010001,8 gfedgfedgfedgfedz (124.ay)

0001010010000002,8 gfedgfecgfecgfez (124.az)

00003,8 gfecz (124.ba)

0010010010000004,8 gfdcgfdcgfdcgfdz (124.bb)

00005,8 gfdcz (124.bc)

0010010010000006,8 gedcgedcgedcgedz (124.bd)

00007,8 gedcz (124.be)

0010010010000008,8 fedcfedcfedcfedz (124.bf)

00009,8 fedcz (124.bg)

00001,9 gfedz (124.bh)

00002,9 gfecz (124.bi)

00004,9 gfdcz (124.bj)

00006,9 gedcz (124.bk)

00009,9 fedcz (124.bl)



IV. Numerical Examples Seven numerical examples will be demonstrated in this section to verify the applicability of the

proposed procedures. The first four examples are selected from real applications in civil engineering. They are

the cubic-, quartic-, quintic- and sextic equations, respectively. The last three examples are beyond the sextic

equation demonstrated in technical papers of the other author [7-9]. The intention is just to serve the researchers

to exercise some challenging problems and the formulas given herein should be useful for some existing

applications, if any. All calculations in this technical paper were done in the environments of Software Mathcad

Prime 3.0. Only results from the proposed decomposition will be shown rather in details. The results from the

other two alternative decompositions were also calculated to verify the correctness of the equations and to

compare the efficiency with the proposed method, but the results from the alternative methods will be excluded

because of the space limitation.

Example 1: Roots of a Cubic Equation

The required depth of a square timber section could be determined by solving the a cubic equation,

0103.55872101.40368 -3-23 xx . This equation can be decomposed to the product of a linear equation

and a quadratic equation as shown in (12). Firstly the two unknown 1e and 0e may be obtained by the Newton-

Raphson method in 2 dimensions via (75). In this case F and Z are calculated via (88) and (89),

respectively. The results of calculation are summarized as shown below.

Initial guess

2.0

01.0

1

0

e

e

Iteration 1:

2.0

01.0

1

0

e

e,

2-

-1

104.71376

101.55872F ,

1-103.558728.11744

00000.1103.55872Z

1.71241103.90601

104.81188101.712411-

-2-21

Z

1-

-2

2-

-1

1-

-2-2

1

0

101.80165

101.49374

104.71376

101.55872

1.71241103.90601

104.81188101.71241

2.0

01.0

e

e

Iteration 2:

1-

-2

1

0

101.80165

101.49374

e

e,

2-

-2

101.39488

105.80779F ,

1-102.382433.87353

00000.1101.59494Z

2.07855105.04802

101.30321103.104801-

-1-21

Z

1-

-2

2-

-2

1-

-1-2

1-

-2

1

0

101.80489

101.85584

101.39488

105.80779

2.07855105.04802

101.30321103.10480

101.80165

101.49374

e

e

Iteration 3:

1-

-2

1

0

101.80489

101.85584

e

e,

3-

-2

102.01510

101.12685F ,

1-101.917582.86494

00000.1101.03327Z

2.13207105.91159

102.06342103.956781-

-2-21

Z

1-

-2

3-

-2

1-

-2-2

1-

-2

1

0

101.82855

101.94201

102.01510

101.12685

2.13207105.91159

102.06342103.95678

101.80489

101.85584

e

e

Iteration 4:

1-

-2

1

0

101.82855

101.94201

e

e,

5-

-4

105.11779

103.95045F ,

1-101.832502.72543

00000.19.43609Z

2.11828106.11825

102.24487104.113721-

-1-21

Z

1-

-2

5-

-4

1-

-1-2

1-

-2

1

0

101.82988

101.94478

105.11779

103.95045

2.11828106.11825

102.24487104.11372

101.82855

101.94201

e

e



Iteration 5:

1-

-2

1

0

101.82988

101.94478

e

e,

8-

-7

103.34315

103.73362F ,

1-101.829882.72177

00000.19.40920Z

2.11750106.12523

102.25046104.118071-

-1-21

Z

1-

-2

2-

-1

1-

-1-2

1-

-2

1

0

101.82988

101.94478

104.71376

101.55872

2.11750106.12523

102.25046104.11807

101.82988

101.94478

e

e

Then the estimates of 1e and 0e are 0.019447 and 0.182998, respectively. Then 0d can be obtained

from (9).

0.18299101.94478

103.55872-2-

-3

0

00

e

ad . The decomposed equation becomes,

0101.94478101.829880.18299 -2-12 xxx

ix 105.0091.0,183.0 . However, for the design purpose only the positive real root 183.0x m

is taken.

Example 2: Roots of a Quartic Equation

The minimum anchored length of a sheet pile can be obtained from the equilibrium of the lateral earth

pressure acting on a sheet pile in form of a quartic equation 0496.109925.87132.12971.5 234 xxxx .

This equation can be decomposed into the product of two quadratic equations as shown in (18). Firstly the two

unknown 1e and 0e may be obtained by the Newton-Raphson method in 2 dimensions via (75). In this case

F and Z are calculated via (90) and (91), respectively. The results of calculation are summarized as shown

below.

Initial guess

4

4

1

0

e

e

Iteration 1:

4

4

1

0

e

e,

13.68500

3.35750F ,

31.3735029.34450

2.029007.84338Z

0.042050.15731

0.010880.168191Z

3.95276

4.41585

13.68500

3.35750

0.042050.15731

0.010880.16819

4

4

1

0

e

e

Iteration 2:

3.95276

4.41585

1

0

e

e,

1.17411

0.27022F ,

29.2115424.21361

1.934516.61516Z

0.045190.16540

0.013210.199541Z

3.94440

4.45425

1.17411

0.27022

0.045190.16540

0.013210.19954

3.95276

4.41585

1

0

e

e

Iteration 3:

3.94440

4.45425

1

0

e

e,

0.00882

0.00193F ,

29.0361623.79475

1.917806.51875Z

0.045380.16565

0.013350.202141Z

3.94432

4.45452

0.00882

0.00193

0.045380.16565

0.013350.20214

3.94440

4.45425

1

0

e

e

Iteration 4:

3.94432

4.45452

1

0

e

e,

7-

-8

104.61274

109.83190F ,

29.0349323.79173

1.917636.51808Z



0.045380.16565

0.013350.202151Z

3.94432

4.45452

104.61274

109.83190

0.045380.16565

0.013350.20215

3.94432

4.45452

7-

-8

1

0

e

e

Thus the estimates of 1e and 0e are 3.94432 and 4.45452, respectively. Then 1d and 0d from can be

obtained from (15.a) and (15.b), respectively.

02668.21 d and 24.580430 d . The decomposed equation becomes,

045452.43.9443224.5804302668.2 22 xxxx

ix 752.0972.1,047.4,074.6 . However, for the design purpose only the positive real root

497.4x m is taken.

Example 3: Roots of a Quintic Equation

The diameter of a circular steel column subjected to an axial force may be determined by solving a

quintic equation 010850.910951.410866.5 825335 xxx . This equation can be decomposed into

the product of two equations i.e. one quadratic equation and one cubic equation as shown in (24). Firstly the

three unknown 12 ,ee and 0e may be obtained by the Newton-Raphson method in 3 dimensions via (87). In this

case F and Z are calculated via (92) and (93), respectively. The results of calculation are summarized as

shown below.

Initial guess

1

1

1

2

1

0

e

e

e

Iteration 1:

1

1

1

2

1

0

e

e

e

,

1-

5-

-3

1010.00000

104.94163

105.86581

F ,

1.00000109.849841.00000

1010.000001.000001010.00000

00000.21.00000109.84984

8-

1-1-

-8

Z

1-1-1-

1-1-1-

-1-1-1

1

102.50000102.50000102.50000

105.00000105.00000105.00000

107.50000102.50000102.50000

Z

1-

1-

-1

1-

5-

-3

1-1-1-

1-1-1-

-1-1-1

2

1

0

107.51454

104.97042

102.48546

1010.00000

104.94163

105.86581

102.50000102.50000102.50000

105.00000105.00000105.00000

107.50000102.50000102.50000

1

1

1

e

e

e

Iteration 2:

1-

1-

-1

2

1

0

107.51454

104.97042

102.48546

e

e

e

,

1-

1-

-2

101.86771

101.25008

106.17748-

F ,

1-7-1-

1-1-1-

-6

102.48546103.96299107.51455

104.97042107.51454109.99999

1.502911.00000101.59447

Z

1-1-1-

1-1-

-1-1

1

106.79939105.10944103.83951

1.02189107.67902104.22957

1.10586101.68996101.26993

Z

1-

1-

-2

1-

1-

-2

1-1-1-

1-1-

-1-1

1-

1-

-1

2

1

0

105.36871

102.36318

107.09744

101.86771

101.25008

106.17748-

106.79939105.10944103.83951

1.02189107.67902104.22957

1.10586101.68996101.26993

107.51454

104.97042

102.48546

e

e

e



Iteration 3:

1-

1-

-2

2

1

0

105.36871

102.36318

107.09744

e

e

e

,

2-

2-

-2

103.81038

105.59467

104.60452-

F ,

2-6-1-

1-1-1-

-5

107.09744101.38780105.36876

102.36317105.36871109.99990

1.073741.00000101.95536

Z

1.971051.05823105.68132

2.116441.13627103.89974

1.60206101.39894107.51075

1-

1-

-1-2

1Z

1-

1-

-2

2-

2-

-2

1-

1-

-1-2

1-

1-

-2

1

0

103.76402

101.10060

102.12147

103.81038

105.59467

104.60452-

1.971051.05823105.68132

2.116441.13627103.89974

1.60206101.39894107.51075

105.36871

102.36318

107.09744

e

e

After 13 cycles of iteration the estimates of 12 ,ee and 0e are 1.0075110-1

, 2.1228310-3

and

4.5556410-5

, respectively. Then -1

1 101.00751 d and -3

0 102.16212 d can be obtained from (27.a) and

(27.b), respectively.

The decomposed equation becomes,

0104.55564102.122831000751.11016212.2101.00751 -5-32133-12 xxxxx

ix 23222 10160.21059.9,10975.6,10100.3,10157.8 . However, for the design

purpose the larger value of the positive real roots 06975.0x m is taken.

Example 4: Roots of a Sextic Equation

The critical water height (unit in m) in an open channel of a trapezoidal section may be considered by

solving a sextic equation 010982.310964.7100030030 543456 xxxxx . This equation can be

decomposed into the product of one quadratic equation and one quartic equation as shown in (33). Firstly the

four unknown 123 ,, eee and 0e may be obtained by the Newton-Raphson method in 4 dimensions via (87). In

this case F and Z are calculated via (98) and (99), respectively. The results of calculation are summarized

as shown below.

Initial guess

1

20

100

2000

3

2

1

0

e

e

e

e

Iteration 1:

1

20

100

2000

3

2

1

0

e

e

e

e

,

3101.73000

918.00000

320.90000

51.90000-

F ,

3 10 2.000000199.1000019.04500

100.00000199.1000029.000002.99100

179.1000029.0000010.09955

1100.09955

Z

0.000080.000740.005060.00143

0.000660.004350.004980.00967

0.003640.010940.041220.98312

0.005780.03618-0.1008510.12801

1Z

1.87166

26.22340

160.49717

1054993.1

101.73000

918.00000

320.90000

51.90000-

0.000080.000740.005060.00143

0.000660.004350.004980.00967

0.003640.010940.041220.98312

0.005780.03618-0.1008510.12801

1

20

100

2000 3

33

2

1

0

e

e

e

e



Iteration 2:

1.87166

26.22340

160.49717

1054993.1 3

3

2

1

0

e

e

e

e

,

3105.19106-

672.75595

57.98137

35.78562

F ,

3 101.549930256.915621.52433

160.49717256.9156228.128345.34679

230.6922228.1283410.31025

1100.16576

Z

0.000040.000510.004390.00793

0.000480.004340.000790.13705

0.003640.003230.026330.08827

0.002650.02309-0.031276.81178

1Z

1.87966

25.68058

139.07837

101.76259

105.19106-

672.75595

57.98137

35.78562

0.000040.000510.004390.00793

0.000480.004340.000790.13705

0.003640.003230.026330.08827

0.002650.02309-0.031276.81178

1.87166

26.22340

160.49717

1054993.1 3

3

3

3

2

1

0

e

e

e

e

After 8 cycles of iteration the estimates of 3e , 2e , 1e and 0e are 1.96104, 24.95821, -131.34117 and

-1.80954103, respectively. Then 28.038961 d and 220.056270 d can be obtained from (36.a) and (36.b),

respectively.


01076259.107837.13968058.12587966.105627.22003896.28 32342 xxxxxx

Again the quartic equation can be decomposed further to two quadratic equations as discussed earlier in

Example 2.

iix 505.7375.1,949.4019.14,983.5,198.5 . However, for the design purpose the larger value

of the positive real roots 983.5x m is taken.

Example 5: Roots of a Septic Equation

Let’s consider a septic equation 0352835142814 234567 xxxxxxx . This equation can

be decomposed into the product of one cubic equation and one quartic equation as shown in (42). Firstly the

four unknown 123 ,, eee and 0e may be obtained by the Newton-Raphson method in 4 dimensions via (87). In

this case F and Z are calculated via (102) and (103), respectively. The results of calculation are

summarized as shown below.

Initial guess

1

1

1

1

3

2

1

0

e

e

e

e

Iteration 1:

1

1

1

1

3

2

1

0

e

e

e

e

,

48.00000-

13.00000

35.00000-

7.00000

F ,

3.000001.0000035.00000-50.00000

2.0000036.00000-15.0000033.00000

33.00000-14.000002.0000036.00000

17.000003.00000135.00000

Z



0.00199-0.008720.01655-0.02808

0.010710.02527-0.011540.02725

0.02725-0.002820.010710.02527

0.000830.001990.008720.01655

1Z

0.24219

1.0274

0.14689

1.20319

48.00000-

13.00000

35.00000-

7.00000

0.00199-0.008720.01655-0.02808

0.010710.02527-0.011540.02725

0.02725-0.002820.010710.02527

0.000830.001990.008720.01655

1

1

1

1

3

2

1

0

e

e

e

e

Iteration 2:

0.24219

1.0274

0.14689

1.20319

3

2

1

0

e

e

e

e

,

7.75658-

1.66935

6.02241-

0.70239

F ,

1.786011.2031929.08923-9.72217

1.42123-28.94235-13.2734523.59696

27.41728-12.246051.242196.85544

12.605561.48439124.17667

Z

0.00369-0.012130.02650-0.02084

0.013020.02944-0.014420.02988

0.03323-0.001960.002650.01070

0.002500.004440.014590.03189

1Z

0.05959

1.04138

0.39944

1.28065

7.75658-

1.66935

6.02241-

0.70239

0.00369-0.012130.02650-0.02084

0.013020.02944-0.014420.02988

0.03323-0.001960.002650.01070

0.002500.004440.014590.03189

0.24219

1.0274

0.14689

1.20319

3

2

1

0

e

e

e

e

After 5 cycles of iteration the estimates of 123 ,, eee and 0e are 0.04747, 1.04274, -0.41342 and

1.29095, respectively. Then 13.00698,1.047472 1 dd and 27.111720 d can be obtained from (45.a),

(45.b) and (45.c), respectively.


01.290950.41342-1.042740.0474727.1117213.006981.04747 23423 xxxxxxx

Again the cubic equation can be decomposed further to one linear equation and one quadratic equation

as discussed earlier in Example1. Whereas the quartic equation can be decomposed further to two quadratic

equations as discussed earlier in Example 2.

iiix 80211.055115.0,01612.157489.0,79022.340902.0,86552.1 .

Example 6: Roots of an Octic Equation

Let’s consider an octic equation 020352830142510 2345678 xxxxxxxx . This

equation can be decomposed into the product of two quartic equations as shown in (52). Firstly the four

unknown 123 ,, eee and 0e may be obtained by the Newton-Raphson method in 4 dimensions via (87). In this

case F and Z are calculated via (109) and (110), respectively. The results of calculation are summarized as

shown below.

Initial guess

0

0

0

1

3

2

1

0

e

e

e

e



Iteration 1:

0

0

0

1

3

2

1

0

e

e

e

e

,

10.00000

18.00000

29.00000

33.00000-

F ,

10.000001.0000021.00000-25.00000-

1.0000021.00000-25.00000-10.00000

21.00000-25.00000-10.000001.00000-

25.0000010.000001.0000021.00000

Z

0.01449-0.007030.01501-0.02132-

0.007030.01501-0.02132-0.01449

0.01501-0.02132-0.014490.00703

0.02132-0.014490.007030.01501

1Z

0.00320

0.75608

0.88578

2.17302

10.00000

18.00000

29.00000

33.00000-

0.01449-0.007030.01501-0.02132-

0.007030.01501-0.02132-0.01449

0.01501-0.02132-0.014490.00703

0.02132-0.014490.007030.01501

0

0

0

1

3

2

1

0

e

e

e

e

Iteration 2:

0.00320

0.75608

0.88578

2.17302

3

2

1

0

e

e

e

e

,

7.66411-

5.84943

1.91850

13.22536-

F ,

18.45816-2.1869111.37681-27.13699-

12.2682512.26825-22.49952-12.44951

18.69062-21.73860-8.491040.98965

21.76576-8.494261.006405.23550

Z

0.00567-0.000420.01234-0.03072-

0.000430.01234-0.03076-0.02128

0.01663-0.03108-0.011950.01003

0.02606-0.012320.000910.02682

1Z

0.42048-

1.16540

0.98694

2.25417

7.66411-

5.84943

1.91850

13.22536-

0.00567-0.000420.01234-0.03072-

0.000430.01234-0.03076-0.02128

0.01663-0.03108-0.011950.01003

0.02606-0.012320.000910.02682

1.87166

26.22340

160.49717

1054993.1 3

3

2

1

0

e

e

e

e

After 5 cycles of iteration the estimates of 123 ,, eee and 0e are -0.47893, 1.26342, -1.02269 and

2.28307, respectively. Then -0.521073 d , 8.487022 d , 19.254281 d and 8.760120 d can be obtained

from (55.a), (55.b), (55.c) and (55.d), respectively.


02.283071.02269-1.263420.47893-8.7601219.254288.487020.52107x 234234 xxxxxxx

Again each of the quartic equations can be decomposed further to two quadratic equations as discussed

earlier in Example 2.

iix 0.879190.79499,1.14734i0.55552,3.288870.56419,2.03588,0.38643 .

Example 7: Roots of a Nonic Equation

Let’s consider a nonic equation 020352830142510 23456789 xxxxxxxxx . This

equation can be decomposed into the product of one quartic equation and one quintic equation as shown in (62).

Firstly the five unknown 1234 ,,, eeee and 0e may be obtained by the Newton-Raphson method in 5 dimensions



via (87). In this case F and Z are calculated via (117) and (118), respectively. The results of calculation are

summarized as shown below.

Initial guess

0

0

0

0

1

4

3

2

1

0

e

e

e

e

e

Iteration 1:

0

0

0

0

1

4

3

2

1

0

e

e

e

e

e

,

4

4

0

3

4

101.88607

101.00561

10254.00000-

101.88700-

101.46880

F ,

56.000002.000001.00000101.34400172.70100

2.000001.00000101.34400172.7010056.00000

1.00000101.34400172.7010056.000002.00000

101.34400172.7010056.000002.000001.00000

172.7010056.000002.000001101.34400-

4

4

4

4

4

Z

9-7-7-9-

7-7-9-9-

7-9-9-7-

9-9-7-7-

-9-9-7-7

1

102.94706-103.22306109.561290.00007104.37329

103.22306109.561290.00007104.37329102.94706

109.561290.00007104.37329102.94706103.22306-

0.00007104.37329102.94706103.22306-109.56129

104.37329102.94706103.22306-109.56129-0.00007

Z

0.13728

0.01533

0.72545

1.41672

2.09103

101.88607

101.00561

10254.00000-

101.88700-

101.46880

102.94706-103.22306109.561290.00007104.37329

103.22306109.561290.00007104.37329102.94706

109.561290.00007104.37329102.94706103.22306-

0.00007104.37329102.94706103.22306-109.56129

104.37329102.94706103.22306-109.56129-0.00007

0

0

0

0

1

4

4

0

3

4

9-7-7-9-

7-7-9-9-

7-9-9-7-

9-9-7-7-

-9-9-7-7

4

3

2

1

0

e

e

e

e

e

Iteration 2:

0.13728

0.01533

0.72545

1.41672

2.09103

4

3

2

1

0

e

e

e

e

e

,

3

3

3

109.92559

105.09890

114.13033-

962.57836-

107.65213

F ,

118.191683.607932.09103106.42746104.51904

76.469794.53548106.42604164.27898102.17364

45.54030106.42730163.5535356.2710545.25232-

106.42816163.5799856.286381.86272420.98764-

155.8202456.523251.725431103.07383-

33

33

3

3

3

Z

8-6-6-

6-6-7-6-

6-6-7-

8-6-6-

-8-8-6-6

1

102.85265101.45978103.769440.000160.00002-

101.46370103.969850.00016109.85295101.77027-

103.970290.00016101.04307106.067940.00010

0.00016101.59255102.12775-108.39269-0.00023

102.44886105.96497-103.05244-107.88201-0.00032

Z



0.43521

0.04187

2.27552

4.69446

4.56508

109.92559

105.09890

114.13033-

962.57836-

107.65213

102.85265101.45978103.769440.000160.00002-

101.46370103.969850.00016109.85295101.77027-

103.970290.00016101.04307106.067940.00010

0.00016101.59255102.12775-108.39269-0.00023

102.44886105.96497-103.05244-107.88201-0.00032

0.13728

0.01533

0.72545

1.41672

2.09103

3

3

3

8-6-6-

6-6-7-6-

6-6-7-

8-6-6-

-8-8-6-6

4

3

2

1

0

e

e

e

e

e

After 10 cycles of iteration the estimates of 1234 ,,, eeee and 0e are 1.87305, 20.41547, -20.31628,

71.49813 and 49.90072, respectively. Then 0.126953 d , -76.653262 d -11.401341 d and 269.33480 d

can be obtained from (64.a), (64.a), (64.c) and (64.d), respectively.


049.9007271.4981320.31628-20.415471.87305

269.334811.40134-76.653260.12695

2345

234

xxxxx

xxxx

Again the quartic equation 0269.334811.40134-76.653260.12695 234 xxxx can be

decomposed further to two quadratic equations as discussed earlier in Example 2. Whereas the quintic equation

049.9007271.4981320.31628-20.415471.87305 2345 xxxxx can be decomposed further to one

quadratic equation and one cubic equation as discussed earlier in Example 3. Finally the cubic equation can be

decomposed to one linear equation and one quadratic equation as discussed earlier in Example1.

iix 1.730630.99909,4.422891.65546,556921.84408,8.,0.56031,2.00192,8.52604 .

V. Conclusion 1) An approach for solving polynomial equations of degree higher than two was proposed.

2) The main concepts were decomposition of a polynomial of higher degrees to the product of two

polynomials of lower degrees and the n-D Newton-Raphson method for a system of nonlinear equations.

3) The coefficient of each term in an original polynomial of order m will be equated to the corresponding term

from the collected-expanded product of the two polynomials of the lower degrees based on the concept of

undetermined coefficients. Consequently a system of m nonlinear equations was formed. Then the unknown

coefficients of the decomposed polynomial of the lower degree of the two decomposed polynomials would

be eliminated from the system of nonlinear equations. Therefore the number of nonlinear equations would

be reduced to the number of unknown coefficients in the decomposed polynomial equation of higher

degree.

4) The unknown coefficients in the decomposed polynomial of the higher degree would be obtained by the

Newton-Raphson method for simultaneous nonlinear equations. Then the unknown coefficients for the

decomposed polynomial of the lower degree would be obtained by back substitutions.

5) The formulations for the decomposed polynomials would be derived for the original polynomials of degree

from three to nine.

6) The system of nonlinear equations and supplementary equations for determining the unknown coefficients

of the decomposed polynomials were also summarized for the original polynomial for degree from three to

nine.

7) For the case of an original polynomial equation of an odd degree the original polynomial equation will be

decomposed to two polynomial equations i.e. one equation of an odd degree and the other equation of an

even degree. Whereas for the case of an original polynomial equation of an even degree, two decomposed

polynomial equations of even degrees were proposed to guarantee obtaining all possible roots i.e. complex

conjugates, distinct real roots, double real root, triple real root etc.

8) Two alternative forms of decomposed polynomial equations were also given i.e. Bairstow’s decomposition

and complete Bairstow’s decomposition. For the polynomial equation of degree five or higher the vector of

nonlinear equations and the corresponding Jacobian matrix from the Bairstow’s decomposition were

simpler than the proposed decomposition. Whereas those from the complete Bairstow’s decomposition

were more complex than the proposed decomposition. Both alternative forms involved larger systems of

simultaneous nonlinear equations.

9) Seven numerical examples were also given to verify the applicability of the proposed approach. Four

numerical examples for polynomial equations of degree three, four, five and six were demonstrated. These

problems were selected from the real applications in civil engineering. The other three problems for

polynomial equations of degree seven, eight and nine were given to challenge to the researchers. Numerical

results were given rather in details so that the readers can keep track for all steps of calculations.

10) For a given polynomial equation there exist several possible pairs of decomposed polynomial equations, but

any pair of decomposed equations will always give the same final results.

11) The Newton-Raphson method in two and more dimensions were proved to be a very efficient tool for



solving a system of simultaneous nonlinear equations at least in the extent of numerical examples shown in

this technical paper and author’s experience on finding roots of a polynomial equation.

12) The decomposed polynomial equations can always decomposed further to the equations of lower degrees.

Finally the original polynomial equations can be rewritten in form of a product of linear equations and

quadratic equations. Therefore all possible roots can always be determined.

13) The method proposed can be extended for a polynomial equation of any degree beyond nine, but with

longer equations and a larger system of simultaneous nonlinear equations. Further extension was not shown

in this technical paper because of the limitation of the paper space, but it can be done systematically in form

of matrix notations and computer programming.

Acknowledgements

Full financial support and Granting a Professional License of Software Mathcad Prime 3.0 for

Calculations throughout the author’ research projects by Asia Group (1999) Company Limited was cordially

acknowledged.

References [1] R.B. King, Beyond the quartic equation (Boston-Basel-Berlin: Birkhäuser, 1996).

[2] R.G. Kulkarni, Unified methods for solving general polynomial equations of degree less than fives, Alabama Journal of

Mathematics, (Spring/Fall), 2006, 1-17.

[3] R.G. Kulkarni, Solving certain quintics, Annales Mathematicae et Informaticae, 37(10), 2010, 193-197. [4] M. A. Faggal and D. Lazard, Solving quintics and septics by radicals, International Journal of Scientific and Research

Publications, 4(12), 2014, 1-10.

[5] C. Boswell and M.L. Glasser, Solvable sextic equations, Cornel University Library, Mathematical Physics, arXiv:math-ph/0504001v1,1 April, 2005, 1-7.

[6] S.Y. Huang and S.S. Cheng, Absence of positive roots of sextic polynomials, Taiwanese Journal of Mathematics, 15(6), 2011, 2609-

2646. [7] R.G. Kulkarni, Extracting the roots of septics by polynomia decomposition, Lecturas Matemáticas, 29, 2008, 5-12.

[8] R.G. Kulkarni, On the solution to octic equations, The Montana Mathematics Enthusiast, 4(2), 2007, 193-209.

[9] R.G. Kulkarni, On the solution to nonic equations, Alabama Journal of Mathematics, (Spring/Fall), 2008, 1-17. [10] S.C. Chapra and R.P. Canale, Numerical Methods for Engineers, 6th Ed. (New York, NY: McGraw-Hall, 2010).

[11] M.R. Spencer, Polynomial real root finding in Bernstein form, doctoral diss., the Department of Civil Engineering of Brigham

Young University, Provo, UT, 1994. [12] L. Bairstow, Investigations relating to the stability of the aeroplane, Reports and Memoranda #154, Advisory Committee for

Aeronautics, October 1914, 51-64.

[13] F.M. Carrano, A modified Bairstow method for multiple zeros of a polynomial, Mathematics of Computation, 27(124), OCTOBER 1973, 781-792.

[14] L.B. Rall, Convergence of the Newton process to multiple solutions, Numer. Math., 9(MR 35 #1209), 1966, 23-37.

http://arxiv.org/abs/math-ph/0504001v1

http://arxiv.org/abs/math-ph/0504001v1

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