I
nte
rna
t
io
na
l J
o
urna
l o
f
Adv
a
nces in Applie
d Science
s
(
I
J
AAS)
Vo
l.
5
,
No
.
3
,
Sep
tem
b
er
2
0
1
6
,
p
p
.
1
1
8
~1
2
7
I
SS
N:
2252
-
8814
118
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a
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:
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ttp
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AAS
A Si
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ple
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tions
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.
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a
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p
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o
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n
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d
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t
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ial
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n
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n
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ti
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term
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ti
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Der
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te
m
s
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li
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r
eq
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Un
co
n
s
tr
ain
ed
o
p
ti
m
iza
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Co
p
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rig
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t
©
201
6
In
s
t
it
u
te o
f
A
d
v
a
n
c
e
d
E
n
g
i
n
e
e
rin
g
a
n
d
S
c
ien
c
e
.
Al
l
rig
h
ts re
se
rv
e
d
.
C
o
r
r
e
s
p
o
nd
ing
A
uth
o
r
:
M.
Y.
W
az
ir
i,
Dep
ar
t
m
en
t o
f
Ma
th
e
m
at
ical
Scien
ce
s
Facu
lt
y
o
f
Scie
n
ce
,
B
ay
er
o
Un
iv
er
s
it
y
Ka
n
o
,
Kan
o
,
Nig
er
ia
.
E
m
ail:
m
y
w
az
ir
i@
g
m
ail.
co
m
1.
I
NT
RO
D
UCT
I
O
N
I
n
r
ea
l
lif
e
p
r
o
b
lem
s
,
m
an
y
p
r
o
b
lem
s
ar
e
in
lar
g
e
-
s
ca
le
s
y
s
te
m
s
o
f
n
o
n
l
in
ea
r
eq
u
atio
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s
s
u
c
h
as
co
n
ce
n
tr
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n
o
f
c
h
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m
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s
p
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cr
o
s
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tio
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p
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ties
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ct
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t
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m
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s
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m
ec
h
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ical
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k
a
g
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t.c
.
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ce
it
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x
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m
el
y
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m
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p
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as f
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w
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w
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is
ca
lled
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n
j
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ate
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r
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d
ien
t p
ar
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m
eter
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
-
8814
A
S
imp
le
Th
r
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-
term C
o
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ju
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Gra
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A
lg
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r
ith
m
fo
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lv
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(
M.
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a
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ir
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)
119
I
t
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e
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ar
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Ne
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wto
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[
1
]
w
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tai
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co
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p
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J
ac
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,
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ased
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[
5
-
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,
t
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et
h
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[
1
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s
[
1
3
-
14
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th
e
ten
s
o
r
m
eth
o
d
s
[
1
5
]
,
th
e
d
er
iv
a
tiv
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f
r
ee
m
et
h
o
d
s
[
1
6
-
18
]
an
d
th
e
s
u
b
s
p
ac
e
m
et
h
o
d
s
[
1
9
].
On
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f
t
h
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m
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t
cr
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f
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p
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d
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ls
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,
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o
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m
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.
R
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Ha
g
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h
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[
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r
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m
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ch
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6
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1
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o
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.
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ased
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ate
g
r
ad
ien
t
m
et
h
o
d
p
r
o
p
o
s
ed
b
y
[
2
1
]
f
o
r
u
n
co
n
s
tr
ai
n
ed
o
p
tim
izatio
n
.
I
n
o
t
h
er
w
o
r
d
s
o
u
r
a
lg
o
r
ith
m
ca
n
b
e
t
h
o
u
g
h
t
as
an
ex
ten
s
io
n
to
t
h
r
ee
-
ter
m
co
n
j
u
g
ate
g
r
ad
ien
t
m
et
h
o
d
to
a
g
e
n
er
al
s
y
s
te
m
s
o
f
n
o
n
l
in
ea
r
eq
u
a
tio
n
s
.
W
e
p
r
esen
t
e
x
p
er
i
m
e
n
ta
l
n
u
m
er
ical
r
es
u
lts
a
n
d
p
er
f
o
r
m
an
ce
co
m
p
ar
is
m
w
i
th
t
h
r
ee
-
ter
m
DF−
SD
C
G
co
n
j
u
g
ate
g
r
ad
i
en
t
m
et
h
o
d
b
y
[
2
0
]
w
h
ic
h
i
ll
u
s
tr
ated
th
a
t
t
h
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
is
e
f
f
icie
n
t
a
n
d
p
r
o
m
is
in
g
.
T
h
e
r
est
o
f
th
e
p
ap
er
is
o
r
g
an
ized
as
f
o
llo
w
s
:
I
n
s
ec
tio
n
2
,
w
e
d
esc
r
ib
e
th
e
p
r
o
p
o
s
ed
alg
o
r
ith
m
i
n
d
etail
s
.
S
u
b
s
eq
u
e
n
tl
y
,
C
o
n
v
er
g
en
ce
r
e
s
u
l
ts
ar
e
p
r
esen
ted
in
Sectio
n
3
.
So
m
e
n
u
m
er
ical
r
esu
lt
s
ar
e
r
ep
o
r
te
d
in
Sectio
n
4
to
s
h
o
w
its
p
r
ac
tical
p
er
f
o
r
m
a
n
ce
.
Fin
all
y
,
co
n
cl
u
s
io
n
s
ar
e
m
ad
e
in
Sectio
n
5
.
2.
AL
G
O
RI
T
H
M
T
h
is
s
ec
tio
n
,
p
r
esen
t
s
a
s
i
m
p
l
e
th
r
ee
ter
m
C
G
m
et
h
o
d
f
o
r
s
o
lv
in
g
lar
g
e
-
s
ca
le
s
y
s
te
m
s
o
f
n
o
n
li
n
ea
r
eq
u
atio
n
s
v
ia
m
e
m
o
r
y
les
s
B
FGS
u
p
d
ate.
I
n
g
e
n
er
al,
q
u
a
s
i
-
Ne
w
to
n
m
e
th
o
d
i
s
a
n
iter
ativ
e
m
et
h
o
d
t
h
at
g
en
er
ate
s
a
s
eq
u
e
n
ce
o
f
p
o
in
t
s
{x
k
}
f
r
o
m
a
g
i
v
en
i
n
itial
g
u
ess
x
0
v
ia
th
e
f
o
llo
w
in
g
f
o
r
m
:
x
k
+1
=
x
k
−
α
k
B
k
−
1
∇
f
(
x
k
)
k
=
0
,
1
,
2
. . . ,
(
5
)
w
h
er
e
B
k
is
an
ap
p
r
o
x
i
m
atio
n
to
th
e
J
ac
o
b
ian
w
h
ic
h
ca
n
b
e
u
p
d
ated
at
ea
ch
i
ter
atio
n
f
o
r
k
=
0
,
1
,
2
.
.
.
,
th
e
u
p
d
ated
m
a
tr
ix
B
k
+1
is
c
h
o
s
en
in
s
u
ch
a
w
a
y
t
h
at
it sati
s
f
ies t
h
e
s
ec
a
n
t e
q
u
atio
n
,
i.e
+
1
=
,
(
6
)
w
h
er
e
s
k
=
x
k
+1
−x
k
an
d
y
k
=
∇
f
(
x
k
+
1
)
−
∇
f
(
x
k
)
Or
teg
a
a
n
d
R
h
ei
n
b
o
ld
t
in
[
2
2
]
p
r
esen
ted
ap
p
r
o
x
i
m
atio
n
t
o
th
e
g
r
ad
ien
t
∇
f
(
x
k
)
,
i
n
o
r
d
er
to
av
o
id
co
m
p
u
ti
n
g
e
x
ac
t
g
r
ad
ien
t a
s
(
7
)
I
n
o
u
r
w
o
r
k
w
e
w
ill
u
s
e
t
h
e
ir
id
ea
an
d
α
k
to
b
e
u
p
d
ated
v
ia
lin
e
s
ea
r
c
h
tec
h
n
iq
u
e.
T
h
e
u
p
d
ate
f
o
r
m
u
la
f
o
r
th
e
B
FG
S
B
k
is
g
i
v
en
a
s
(
8
)
B
y
let
tin
g
B
k
≈
θI
,
(
8
)
ca
n
b
e
r
e
w
r
ite
as
:
(
9
)
w
h
er
e,
θ
k
as in
R
a
y
d
an
[
2
3
]
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.
5
,
No
.
3
,
Sep
tem
b
er
2
0
1
6
:
1
1
8
–
1
2
7
120
(
10
)
W
e
f
u
r
th
er
m
u
ltip
l
y
b
o
t
h
s
id
es
o
f
(
9
)
b
y
g
(
xk
+
1)
to
o
b
tain
(
1
1
)
Ob
s
er
v
e
th
at
t
h
e
d
ir
ec
tio
n
dk
+
1
f
r
o
m
(
1
1
)
ca
n
b
e
w
r
itte
n
as
(
12
)
Hen
ce
,
o
u
r
n
e
w
d
ir
ec
tio
n
i
s
(
13
)
w
h
er
e,
(
14
)
(
15
)
Fin
all
y
,
w
e
h
a
v
e
(
16
)
T
h
er
ef
o
r
e
w
it
h
t
h
e
p
r
o
p
o
s
ed
s
ea
r
ch
d
ir
ec
tio
n
w
e
ar
e
u
s
i
n
g
t
h
e
d
er
iv
ati
v
e
f
r
ee
e
li
n
e
s
ea
r
c
h
o
f
L
i
a
n
d
L
i [
1
6
]
to
f
in
d
α
k
=
m
a
x
{
s
,
ρs,
ρ
2
s
,
.
.
.
}
s
u
c
h
th
at
(
17
)
w
h
er
e
σ,
s
>
0
an
d
ρ
∈
(0
,
1)
.
W
e
p
r
esen
t th
e
b
elo
w
al
g
o
r
ith
m
A
l
g
o
r
ith
m
2
.
1
(
STT
C
G)
Step
1
: G
iv
en
x
0
,
α
>
0
,
σ
∈
(
0
,
1
)
,
ϵ
=
10
−4
an
d
co
m
p
u
te
d
0
=
−g
0
,
s
et
k
=
0
.
Step
2
: I
f
|
|
g
k
|
|
<
ϵ
.
th
e
n
s
to
p
; o
th
er
w
is
e
co
n
ti
n
u
e
w
it
h
Step
3
.
Step
3
: D
eter
m
in
e
t
h
e
s
tep
s
iz
e
α
k
b
y
u
s
i
n
g
a
li
n
e
s
ea
r
c
h
co
n
d
itio
n
s
i
n
(
1
7
)
,
Step
4
: D
eter
m
in
e
δ
k
an
d
η
k
b
y
(
1
4
)
an
d
(
1
5
)
r
esp
ec
tiv
el
y
.
Step
5
: Fin
d
th
e
s
ea
r
c
h
d
ir
ec
tio
n
b
y
(
1
3
)
.
Step
6
: P
o
w
el
r
estar
t c
r
iter
io
n
.
I
f
|
+
1
|
2
>
0
.
2
|
|
+
1
|
|
2
,
th
en
s
et
+
1
=
−
9
+
1
Step
7
: Co
n
s
id
er
k
=
k
+ 1
an
d
g
o
to
s
tep
2
.
3.
CO
NVER
G
E
NC
E
RE
S
UL
T
I
n
th
is
Sectio
n
,
w
e
w
ill
p
r
esen
t
th
e
g
lo
b
al
co
n
v
er
g
en
ce
o
f
t
h
e
s
i
m
p
le
th
r
ee
ter
m
s
co
n
j
u
g
ate
g
r
ad
ien
t
m
et
h
o
d
.
Def
i
n
itio
n
1
L
et
Ω
b
e
th
e
le
v
el
s
et
d
ef
in
ed
b
y
w
h
er
e
τ
is
a
p
o
s
itiv
e
co
n
s
ta
n
t.
T
h
e
f
o
llo
w
i
n
g
A
s
s
u
m
p
tio
n
s
a
r
e
n
ee
d
ed
o
n
th
e
n
o
n
lin
ea
r
s
y
s
te
m
s
F
in
o
r
d
er
t
o
estab
lis
h
th
e
g
lo
b
al
co
n
v
er
g
e
n
ce
o
f
o
u
r
m
et
h
o
d
Ass
u
m
p
tio
n
A
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
-
8814
A
S
imp
le
Th
r
ee
-
term C
o
n
ju
g
a
te
Gra
d
ien
t
A
lg
o
r
ith
m
fo
r
S
o
lv
in
g
S
ymm
etri
c
S
ystems
o
f
…
(
M.
Y.
W
a
z
ir
i
)
121
(
i)
T
h
e
lev
el
s
et
is
b
o
u
n
d
ed
.
(
ii)
I
n
s
o
m
e
n
ei
g
h
b
o
r
h
o
o
d
N
o
f
Ω
,
th
e
J
ac
o
b
ian
is
lip
s
ch
i
tz
co
n
tin
u
o
u
s
,
i.e
th
er
e
ex
i
s
t
a
co
n
s
ta
n
t
L
>
0
s
.
t
f
o
r
all
x,
y
∈
N
(
1
8)
(
iii)
T
h
er
e
ex
is
ts
x
∗
∈
Ω
s
u
ch
t
h
at
F
(
x
∗
)
=
0
an
d
F′
(
x
)
is
co
n
tin
o
u
s
f
o
r
all
x
.
Ass
s
u
m
p
tio
n
A
(
ii
)
an
d
A
(
iii
)
i
m
p
lies
t
h
at
t
h
er
e
ex
i
s
t p
o
s
itiv
e
co
n
s
tan
ts
κ
1
, κ
2
an
d
L
1
s
u
ch
th
at
(
19
)
(
20
)
T
h
e
f
o
llo
w
i
n
g
le
m
m
a
s
h
o
w
s
t
h
at
th
e
d
ir
ec
tio
n
d
k
d
eter
m
i
n
e
d
b
y
(
1
3
)
is
in
ter
esti
n
g
L
e
mm
a
1
Su
p
p
o
s
e
t
h
at
F
is
u
n
i
f
o
r
m
l
y
co
n
v
e
x
t
h
en
d
k
is
d
e
f
in
ed
b
y
(
1
3
)
,
th
en
w
e
h
a
v
e
(
21
)
an
d
(
22
)
P
ro
o
f
.
w
h
e
n
k
=
0
(
2
1
)
an
d
(
2
2
)
h
o
ld
s
in
ce
d
0
=
−g
0
.
Fro
m
th
e
d
e
f
in
a
ti
o
n
o
f
d
k
in
(
1
3
)
w
e
h
a
v
e
T
h
u
s
(
2
1
)
h
o
ld
f
o
r
all
k
≥
1
a
n
d
B
y
L
ip
ch
i
tz
co
n
ti
n
u
i
t
y
,
we
k
n
o
w
th
at
||y
k
||
≤
L||s
k
||.
On
th
e
o
th
er
h
an
d
b
y
u
n
if
o
r
m
co
n
v
e
x
it
y
,
it
y
ield
s
(
23
)
T
h
u
s
,
(
24
)
(
25
)
(
26
)
Sin
ce
(
27
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.
5
,
No
.
3
,
Sep
tem
b
er
2
0
1
6
:
1
1
8
–
1
2
7
122
Fin
all
y
(
2
8)
Hen
ce
,
d
k
+
1
is
b
o
u
n
d
ed
.
T
h
e
co
m
m
i
n
g
le
m
m
a
s
h
o
w
s
t
h
at
t
h
e
li
n
e
s
ea
r
ch
in
s
tep
3
o
f
ST
T
C
G
Alg
o
r
it
h
m
is
r
ea
s
o
n
ab
le,
th
en
th
e
p
r
esen
ted
al
g
o
r
ith
m
is
w
el
l d
ef
in
ed
.
L
e
mm
a
2
L
et
t
h
e
Ass
u
m
p
t
io
n
A
h
o
ld
.
T
h
en
ST
T
C
G
A
lg
o
r
it
h
m
p
r
o
d
u
ce
s
an
i
ter
ate
o
f
z
k
=
x
k
+
α
k
d
k
,
in
a
f
i
n
it
e
n
u
m
b
er
o
f
b
ac
k
tr
a
k
in
g
s
tep
s
.
P
r
o
o
f
:
W
e
s
u
p
p
o
s
e
t
h
at
||g
k
||
→
0
d
o
es
n
o
t
h
o
ld
,
o
r
th
e
alg
o
r
it
h
m
i
s
s
to
p
ed
.
T
h
en
th
er
e
e
x
i
s
t
s
a
co
n
s
ta
n
t
ϵ
0
>
0
s
u
ch
t
h
at
(
29
)
W
e
w
ill
g
et
t
h
is
b
y
co
n
tr
ad
icti
o
n
.
Su
p
p
o
s
e
th
at
f
o
r
s
o
m
e
iter
ate
in
d
ex
es
s
u
c
h
as
k
∗
th
e
co
n
d
itio
n
(
1
7
)
is
n
o
t tr
u
e.
T
h
en
b
y
le
tti
n
g
∗
=
,
it c
an
b
e
co
n
clu
d
ed
th
at
co
m
b
i
n
i
n
g
w
i
th
a
s
s
u
m
p
t
io
n
A
(
ii)
an
d
(
2
1
)
,
w
e
h
av
e
(
30
)
B
y
(
1
9
)
an
d
(
2
8
)
T
h
u
s
,
w
e
o
b
tain
T
h
u
s
,
it
co
n
tr
ad
icts
w
it
h
t
h
e
d
ef
in
at
io
n
o
f
∗
.
C
o
n
s
eq
u
en
tl
y
,
th
e
lin
e
s
ea
r
c
h
p
r
o
ce
d
u
r
e
(
1
7
)
ca
n
attain
a
p
o
s
iti
v
e
s
tep
le
n
g
th
α
k
in
a
f
in
ite
n
u
m
b
er
o
f
b
ac
k
tr
ac
k
in
g
s
tep
s
.
He
n
ce
i
t
t
u
r
n
s
o
u
t
th
e
r
e
s
u
l
t
o
f
th
i
s
le
m
m
a.
T
h
e
p
r
o
o
f
is
co
m
p
lete.
No
w
w
e
estab
li
s
h
th
e
g
lo
b
al
co
n
v
er
g
e
n
ce
t
h
eo
r
e
m
T
h
eo
r
em
L
et
t
h
e
p
r
o
p
er
ties
o
f
ass
u
m
p
tio
n
A
h
o
ld
.
T
h
en
th
e
s
eq
u
en
ce
{xk}
b
e
g
en
er
ated
b
y
STT
C
G
alg
o
r
ith
m
co
n
v
er
g
e
s
g
lo
b
all
y
,
th
at
is
,
(
31
)
P
r
o
o
f
.
W
e
p
r
o
v
e
th
i
s
t
h
eo
r
em
b
y
co
n
tr
ad
ictio
n
.
S
u
p
p
o
s
e
th
at
(
3
1
)
is
n
o
t
tr
u
e,
t
h
e
n
t
h
e
r
e
ex
is
t
s
a
p
o
s
itiv
e
co
n
s
tan
t
τ
s
u
c
h
th
a
t
(
32
)
Sin
ce
∇
f
(
x
k
)=
J
k
F
k
,
(
3
2
)
im
p
lie
s
th
at
t
h
er
e
ex
i
s
ts
a
p
o
s
iti
v
e
co
n
s
ta
n
t
τ
1
s
ati
s
f
y
i
n
g
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
-
8814
A
S
imp
le
Th
r
ee
-
term C
o
n
ju
g
a
te
Gra
d
ien
t
A
lg
o
r
ith
m
fo
r
S
o
lv
in
g
S
ymm
etri
c
S
ystems
o
f
…
(
M.
Y.
W
a
z
ir
i
)
123
(
33
)
C
ase
(
i)
:
li
m
s
u
p
k→∞
α
k
>
0
.
t
h
en
b
y
(
2
2
)
,
w
e
h
a
v
e
li
m
i
n
f
k→
∞
||F
k
||
=
0
.
T
h
is
an
d
L
e
m
m
a
1
s
h
o
w
t
h
at
li
m
k→∞
||F
k
||
=
0
,
w
h
ic
h
co
n
tr
a
d
icts
(
3
2
)
.
C
ase
(
ii):
li
m
s
u
p
k→∞
α
k
=
0
.
Sin
ce
α
k
≥
0
,
th
is
ca
s
e
i
m
p
l
ies t
h
at
(
34
)
b
y
d
e
f
in
itio
n
o
f
g
k
i
n
(
7
)
,
w
e
h
av
e
(
35
)
(
36
)
(
37
)
w
h
er
e
w
e
u
s
e
(
1
9
)
an
d
(
2
0
)
in
th
e
last
i
n
eq
u
a
lit
y
.
(
1
7
)
an
d
(
3
2
)
s
h
o
w
t
h
at
t
h
er
e
ex
i
s
ts
a
co
n
s
tan
t
τ
2
>
0
s
u
ch
t
h
at
(
3
8)
B
y
(
7
)
an
d
(
1
9
)
,
w
e
g
e
t
(
39
)
Fro
m
(
2
0
)
an
d
(
3
9
)
,
w
e
o
b
tain
(
40
)
(
41
)
(
42
)
T
h
is
to
g
et
h
er
w
it
h
(
3
4
)
an
d
l
e
m
m
a
2
s
h
o
w
th
at
li
m
k→∞
|
|y
k|
|
=
0
.
Fro
m
(
3
8
)
,
(
3
9
)
,
(
4
0
)
an
d
(
4
1
)
,
w
e
h
av
e
(
43
)
m
ea
n
in
g
t
h
er
e
ex
is
ts
a
co
n
s
tan
t
λϵ
(0
,
1
)
s
u
ch
th
at
f
o
r
s
u
f
f
icie
n
tl
y
lar
g
e
k
(
44
)
Sin
ce
li
m
k→∞
α
k
=
0
,
th
e
n
′
=
d
o
es n
o
t satis
f
y
(
1
7
)
n
a
m
el
y
,
Sin
ce
{x
k
}
⊂
Ω
is
b
o
u
n
d
ed
an
d
(
2
8
)
,
w
i
th
o
u
t lo
s
s
o
f
g
e
n
er
al
it
y
,
w
e
as
s
u
m
e
x
k
→
x
∗
.
B
y
(
7
)
,
w
e
h
av
e
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
2
5
2
-
8814
IJ
AA
S
Vo
l.
5
,
No
.
3
,
Sep
tem
b
er
2
0
1
6
:
1
1
8
–
1
2
7
124
(
45
)
th
e
f
ac
t th
at
t
h
e
s
eq
u
en
ce
{d
k
}
is
b
o
u
n
d
ed
.
o
n
th
e
o
th
er
h
an
d
(
46
)
Hen
ce
,
f
r
o
m
(
4
5
)
an
d
(
4
6
)
,
we
o
b
tain
−
∇
f
(
x
∗
)
T
∇
f
(
x
∗
)
≥
0
,
wh
ich
m
ea
n
s
||
∇
f
(
x
∗
)
||
=0
.
T
h
is
co
n
tr
ad
icts
w
it
h
(
3
2
)
.
T
h
e
p
r
o
o
f
is
th
en
co
m
p
leted
.
4.
NUM
E
RICAL
R
E
SU
L
T
S
I
n
th
is
s
ec
tio
n
,
w
e
test
ed
a
s
i
m
p
le
th
r
ee
ter
m
co
n
j
u
g
at
e
g
r
ad
ien
t
alg
o
r
ith
m
a
n
d
co
m
p
ar
e
it
’
s
p
er
f
o
r
m
a
n
ce
w
i
th
a
f
a
m
il
y
o
f
d
er
iv
ativ
e
f
r
ee
co
n
j
u
g
ate
g
r
a
d
ien
t
m
eth
o
d
f
o
r
lar
g
esca
le
n
o
n
lin
ea
r
s
y
s
te
m
s
o
f
eq
u
atio
n
s
[
2
4
]
:
P
r
o
b
lem
3
,
5,
8
,
an
d
1
0
ar
e
co
n
s
tr
u
cted
b
y
u
s
w
h
er
e
a
s
t
h
e
r
e
m
ai
n
in
g
ar
e
th
e
r
ef
er
en
ce
t
h
er
ein
.
T
h
e
test
f
u
n
ctio
n
s
ar
e
lis
ted
as
f
o
ll
o
w
s
P
r
o
b
lem
1
: see
[
1
8
]
F
i
(
x
)
=
x
i
(
co
s
x
i
−
n
1
)
− x
n
[
s
in
x
i
−
1
−
(
x
i
−
1)
2
−
1
∑
=
1
i
=
1
,
2
,
.
.
.
,
n
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
2
: [
2
5
]
F
i
(
x
)
=
e
x
i −
1
i
=
1
,
2
,
.
.
.
,
n
.
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
3
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
F
i
(
x
)
=
1
− x
2
i
+
x
i
+
x
i
x
n−
2
x
n−
1
x
n
−
2;
i
=
2
,
3
,
.
.
.
,
n
.
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
4
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
[
1
8
]
F
i
(
x
)
=
x
i
−
3
s
in
(
x
3
i
−
0
.
6
6
)
+
2
,
i
=
2
,
3
,
.
.
.
,
n
−
1
.
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
5
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
F
i
(
x
)
=
co
s
x
1
−
9
+
3
x
1
+ 8
e
x
2
,
F
i
(
x
)
=
co
s
x
i
−
9
+
3
x
i
+ 8
e
xi
1
,
i
=
1
,
2
,
.
.
.
,
n
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
6
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
[
2
5
]
F
(
x
)
=
1
2
+
(
−
3
)
l
og
(
+
3
)
−
9
+ (
x
−
3)
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
7
: S
y
s
te
m
o
f
n
n
o
n
l
i
n
ea
r
eq
u
atio
n
s
[
1
8
]
F
i
(
x
)
=
1
2
−
−
c
os
(
1
−
)
,
i
=
1
,
2
,
.
.
.
,
n
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
8
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
F
i
(
x
)
=
(0
.
5
-
x
i
)
2
+
(
n
+
1
− i
)
2
−
0
.
25
x
i
−
1
,
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
-
8814
A
S
imp
le
Th
r
ee
-
term C
o
n
ju
g
a
te
Gra
d
ien
t
A
lg
o
r
ith
m
fo
r
S
o
lv
in
g
S
ymm
etri
c
S
ystems
o
f
…
(
M.
Y.
W
a
z
ir
i
)
125
F
n
(
x
)
=
10
1
−
−
2
, i
=
1
,
2
,
.
.
.
,
n
.
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
9
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
[
25
]
F
i
(
x
)
=
4
x
i
+
x
i
+1
−
2
x
i
−
+
1
3
F
n
(
x
)
=
4
x
n
+
x
n−
1
−
2
x
n
−
−
+
1
3
i
=
1
,
2
,
.
.
.
,
n
−
1
.
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
P
r
o
b
lem
1
0
: S
y
s
te
m
o
f
n
n
o
n
li
n
ea
r
eq
u
atio
n
s
F
i
(
x
)
=
2
−
4
,
x
0
=
(0
.
5
,
0
.
5
,
0
.
5
,
.
.
.
,
0
.
5)
T
T
ab
le
1
.
Nu
m
er
ical
R
es
u
lt
s
S
TT
C
G
A
l
g
o
r
i
t
h
m
DF
-
S
D
C
G
A
l
g
o
r
i
t
h
m
F
D
i
m
NI
NF
C
P
U
t
i
me
NI
NF
C
P
U
t
i
me
10
0
0
8
7
.
1
3
E
-
04
0
.
3
0
9
1
6
2
27
5
.
3
5
E
-
07
0
.
4
8
3
5
8
8
1
50
0
0
9
1
.
6
0
E
-
04
0
.
2
2
5
9
7
5
27
1
.
5
9
E
-
06
2
.
0
1
5
2
5
9
10000
9
2
.
2
6
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-
04
0
.
3
3
9
4
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5
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3
3
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3
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127
[4
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.
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[5
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Yu
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[6
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Hu
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4
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