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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
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 45 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 46 | Page
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 47 | Page
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)
Equating each coefficient in (25) to the corresponding term in the original equation in (23) leads to 5
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.
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 48 | Page
2.5.2 Bairstow’s Decomposition
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.
2.5.3 Complete Bairstow’s Decomposition
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)
Equating each coefficient in (30) to the corresponding term in the original equation in (23) leads to 5 equations:
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
2.6.1 The Proposed Decomposition
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)
Equating each coefficient in (34) to the corresponding term in the original equation in (32) leads to 6
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 49 | Page
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.
2.6.2 Bairstow’s Decomposition
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.
2.6.3 Complete Bairstow’s Decomposition
The sextic equation of (32) can be rewritten in form of a complete Bairstow’s decomposition as:
0012
012
012 fxfxexexdxdx (38)
Expanding the product in (38) yields:
0000001010100
2011101110000000
3111011001100110
4111111000
5111
6
fedxfedfedfed
xfedfedfedfefded
xfedfefefdfdeded
xfefdedfedxfedx
(39)
Equating each coefficient in (39) to the corresponding term in the original equation in (32) leads to 6 equations:
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
2.7.1 The Proposed Decomposition
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)
Equating each coefficient in (43) to the corresponding term in the original equation in (41) leads to 7 equations:
632 aed (44.a)
53221 aeded (44.b)
4223110 aededed (44.c)
31221300 aededede (44.d)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
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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.
2.7.2 Bairstow’s Decomposition
The septic equation of (41) can be rewritten in form of the Bairstow’s decomposition as:
0012
23
34
45
012 exexexexexdxdx (47)
Expanding the product in (47) yields:
00001102
11200
321301
431402
54130
641
7
edxededxedede
xededexededexededxedx (48)
Equating each coefficient in (48) to the corresponding term in the original equation in (41) leads to 7 equations:
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.
2.7.3 Complete Bairstow’s Decomposition
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 51 | Page
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
2.8.1 The Proposed Decomposition
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.
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DOI: 10.9790/5728-1204044476 www.iosrjournals.org 52 | Page
2.8.2 Bairstow’s Decomposition
The octic equation of (52) can be rewritten in form of the Bairstow’s decomposition as:
0012
23
34
45
56
012 exexexexexexdxdx (57)
Expanding the product in (57) and equating each coefficient to the corresponding term in the original
equation in (52) leads to 8 equations:
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.
2.8.3 Complete Bairstow’s Decomposition
The octic equation of (52) can be rewritten in form of a complete Bairstow’s decomposition as:
0012
012
012
012 gxgxfxfxexexdxdx (59)
Expanding the product in (59) and equating each coefficient to the corresponding term in the original
equation in (52) leads to 8 equations:
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
2.9.1 The Proposed Decomposition
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)
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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.
2.9.2 Bairstow’s Decomposition
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)
Expanding the product in (66) and equating each coefficient to the corresponding term in the original
equation in (61) leads to 9 equations:
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.
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2.9.3 Complete Bairstow’s Decomposition
The nonic equation of (61) can be rewritten in form of a complete Bairstow’s decomposition as:
0012
012
012
012
0 gxgxfxfxexexdxdxcx (68)
Expanding the product in (68) and equating each coefficient to the corresponding term in the original
equation in (61) leads to 9 equations:
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)
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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.
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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)
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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)
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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)
3.3.4.2 Complete Bairstow’s Decomposition
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
3.3.5.1 The Proposed Decomposition
10336250
10
20361336250
20
31362336250
300
423633362510
0
aeeeaeae
ea
aeeaeeeaeae
ea
aeeaeeeaeae
eae
aeeaeeeaeaee
a
F (102)
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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)
3.3.5.2 Bairstow’s Decomposition
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)
3.3.5.3 Complete Bairstow’s Decomposition
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)
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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
3.3.6.1 The Proposed Decomposition
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)
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3.3.6.2 Bairstow’s Decomposition
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)
3.3.6.3 Complete Bairstow’s Decomposition
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)
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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
3.3.7.1 The Proposed Decomposition
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)
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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)
3.3.7.2 Bairstow’s Decomposition
000
10110
211200
321301
431402
541503
651604
76150
861
aed
aeded
aedede
aedede
aedede
aedede
aedede
aeded
aed
F (120)
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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)
3.3.7.3 Complete Bairstow’s Decomposition
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 65 | Page
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 66 | Page
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)
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 67 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 68 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 69 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 70 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 71 | Page
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.
The decomposed equation becomes,
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 72 | Page
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.
The decomposed equation becomes,
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 73 | Page
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.
The decomposed equation becomes,
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 74 | Page
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 75 | Page
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.
The decomposed equation becomes,
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
Beyond the Quadratic Equations and the N-D Newton-Raphson Method
DOI: 10.9790/5728-1204044476 www.iosrjournals.org 76 | Page
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.
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