I
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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
.
1
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Ma
r
ch
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0
1
6
,
p
p
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5
0
~
5
7
I
SS
N:
2252
-
8814
50
J
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h
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Enha
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ra
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m
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ua
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a
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R
ec
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No
v
29
,
2
0
1
5
R
ev
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s
ed
Feb
2
,
2
0
1
6
A
cc
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ted
Feb
1
6
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2
0
1
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In
th
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t
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m
p
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n
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th
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c
o
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b
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m
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tri
x
.
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a
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6
In
s
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ar
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at
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th
w
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s
t
Un
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o
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ail:
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a
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(
x)
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(
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s
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1
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t
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A
4
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e
J
ac
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ian
F
′
(
x)
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y
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m
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n
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n
s
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x
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x,
y
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h
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p
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t
m
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d
f
o
r
f
i
n
d
in
g
th
e
s
o
l
u
tio
n
o
f
(
1
)
,
is
th
e
class
ical
Ne
w
to
n
’
s
m
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t
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w
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2
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w
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Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
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2252
-
8814
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1
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m
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f
o
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l
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o
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li
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ates
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o
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r
o
m
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ial
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v
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x
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s
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s
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k
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as c
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ar
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ai
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izatio
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p
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w
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o
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p
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o
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n
d
ar
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v
al
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e
p
r
o
b
lem
,
a
n
d
etc.
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q
u
atio
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(
1
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is
th
e
f
ir
s
t
-
o
r
d
er
n
ec
es
s
ar
y
co
n
d
itio
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f
o
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th
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u
n
co
n
s
tr
ain
ed
o
p
ti
m
iza
tio
n
p
r
o
b
le
m
w
h
e
n
F is
th
e
g
r
ad
ien
t
m
ap
p
in
g
o
f
s
o
m
e
f
u
n
c
tio
n
f
:
R
n
−→
R
,
(
6
)
Fo
r
th
e
eq
u
alit
y
co
n
s
tr
ai
n
ed
p
r
o
b
lem
min
f
(
x
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,
s
.t
h
(
z
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=
0
,
(
7
)
w
h
er
e
h
is
a
v
ec
to
r
-
v
alu
ed
f
u
n
ctio
n
,
th
e
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n
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itio
n
s
ca
n
b
e
r
ep
r
esen
ted
as
th
e
s
y
s
te
m
(
1
)
w
it
h
x
=
(
z
,
v
)
,
an
d
(
8
)
w
h
er
e
v
is
t
h
e
v
ec
to
r
o
f
L
ag
r
an
g
e
m
u
ltip
lier
s
.
No
tice
th
a
t
th
e
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ac
o
b
ian
∇
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(
z
,
v
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s
y
m
m
etr
ic
f
o
r
all
(
z
,
v
)
(
s
ee
,
e.
g
.
,
[
?
]).
P
r
o
b
lem
(
1
)
ca
n
b
e
co
n
v
er
ted
to
th
e
f
o
llo
w
i
n
g
g
lo
b
al
o
p
tim
iza
tio
n
p
r
o
b
lem
(
5)
w
i
th
o
u
r
f
u
n
ctio
n
f
d
ef
i
n
ed
b
y
(
9
)
A
lar
g
e
n
u
m
b
er
o
f
e
f
f
icie
n
t
s
o
lv
er
s
f
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r
lar
g
e
-
s
ca
le
s
y
m
m
etr
ic
n
o
n
li
n
ea
r
eq
u
atio
n
s
h
av
e
b
ee
n
p
r
o
p
o
s
ed
,
an
al
y
ze
d
,
an
d
test
ed
in
th
e
last
d
ec
ad
e.
Am
o
n
g
th
e
m
ar
e
[
4
,
2
,
9
]
.
Sti
ll
th
e
m
atr
i
x
s
to
r
ag
e
an
d
s
o
lv
i
n
g
o
f
n
-
l
in
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r
s
y
s
te
m
ar
e
r
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ir
ed
in
t
h
e
B
FG
S
t
y
p
e
m
et
h
o
d
s
p
r
ese
n
ted
i
n
th
e
lit
er
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r
e.
T
h
e
r
ec
en
t
d
esig
n
ed
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o
n
m
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n
o
to
n
e
s
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tr
al
g
r
ad
ien
t a
l
g
o
r
ith
m
[
1
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f
alls
w
it
h
i
n
t
h
e
f
r
a
m
e
w
o
r
k
o
f
m
atr
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-
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.
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ad
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o
d
s
f
o
r
s
y
m
m
etr
ic
n
o
n
li
n
ea
r
eq
u
ati
o
n
s
h
as
r
ec
ei
v
ed
a
g
o
o
d
atten
s
io
n
an
d
ta
k
e
a
n
ap
p
r
o
p
r
iate
p
r
o
g
r
ess
.
Ho
w
ev
er
,
L
i
an
d
W
a
n
g
[
5
]
p
r
o
p
o
s
ed
a
m
o
d
i
f
ied
Flectc
h
er
-
R
ee
v
e
s
co
n
j
u
g
ate
g
r
ad
ien
t
m
et
h
o
d
w
h
ich
i
s
b
ased
o
n
t
h
e
w
o
r
k
o
f
Z
h
an
g
et
a
l.
[
3
]
,
an
d
th
e
r
es
u
lt
s
illu
s
tr
ate
t
h
at
t
h
eir
p
r
o
p
o
s
ed
co
n
j
u
g
ate
g
r
ad
ien
t
m
et
h
o
d
is
p
r
o
m
is
i
n
g
.
I
n
lin
e
w
it
h
t
h
is
d
ev
elo
p
m
e
n
t,
f
u
r
th
er
s
tu
d
ies
o
n
co
n
j
u
g
at
e
g
r
ad
ien
t
ar
e
[
7
,
1
0
,
8
,
1
3
]
.
E
x
ten
s
i
v
e
n
u
m
er
ical
e
x
p
er
i
m
en
ts
s
h
o
w
ed
t
h
at
ea
c
h
o
v
er
m
en
tio
n
ed
m
et
h
o
d
p
er
f
o
r
m
s
q
u
ite
w
el
l.
W
e
o
r
g
an
ized
t
h
e
p
ap
er
as
f
o
llo
w
s
:
I
n
t
h
e
n
ex
t
s
ec
t
io
n
,
w
e
p
r
esen
t
th
e
d
etails
o
f
t
h
e
p
r
o
p
o
s
ed
m
et
h
o
d
.
C
o
n
v
er
g
e
n
ce
r
es
u
lts
ar
e
p
r
esen
ted
in
Sectio
n
3
.
S
o
m
e
n
u
m
er
ical
r
es
u
lt
s
ar
e
r
ep
o
r
ted
in
Sectio
n
4
.
Fin
all
y
,
co
n
cl
u
s
io
n
s
ar
e
m
ad
e
in
Sectio
n
5
.
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
.
1
,
Ma
r
ch
201
6
:
5
0
–
5
7
52
2.
E
NH
ANC
E
D
D
E
RIV
AT
I
V
E
-
F
RE
E
(
E
DF
)
Giv
e
n
an
in
i
tial
p
o
in
t
x
0
,
an
iter
ativ
e
s
c
h
e
m
e
f
o
r
p
r
o
b
lem
(
1
)
g
en
er
all
y
g
en
er
ate
s
a
s
eq
u
en
ce
o
f
iter
ates
(
1
0
)
w
h
ic
h
e
m
p
lo
y
s
a
li
n
e
s
ea
r
c
h
p
r
o
ce
d
u
r
e
alo
n
g
th
e
d
ir
ec
ti
o
n
d
k
to
co
m
p
u
te
t
h
e
s
tep
s
i
ze
αk
.
T
y
p
ical
li
n
e
s
ea
r
ch
es
i
n
cl
u
d
e
B
ac
k
tr
ac
k
i
n
g
,
A
r
m
ij
o
o
r
W
o
lf
e
lin
e
s
ea
r
ch
e
s
.
L
et
z
k
=
x
k
+
α
k
d
k
−
1
,
th
e
h
y
p
e
r
p
lan
e
(
1
1
)
s
tr
ictl
y
s
ep
ar
ates
x
k
f
r
o
m
t
h
e
s
o
lu
tio
n
s
et
o
f
(
1
)
.
T
h
er
ef
o
r
e,
f
r
o
m
th
i
s
f
ac
ts
So
lo
d
o
v
an
d
Sv
aiter
[
6
]
ad
v
is
ed
to
let
th
e
n
e
x
t iter
ate
x
k
+
1
b
e
th
e
p
r
o
j
ec
tio
n
o
f
x
k
o
n
to
th
is
h
y
p
er
p
lan
e
H
k
.
T
h
at
is
,
x
k
+
1
is
d
ef
i
n
e
d
b
y
(
1
2
)
I
n
th
i
s
p
ap
er
,
th
e
d
ir
ec
tio
n
d
k
is
b
ase
o
n
[
1
3
]
,
s
p
ec
if
icall
y
(
1
3
)
w
h
er
e
F
k
m
ea
n
s
F
(
x
k
)
an
d
β
k
d
ef
i
n
ed
as
(
1
4
)
(
1
5
)
T
h
r
o
u
g
h
o
u
t th
i
s
p
ap
er
,
||.||
is
t
h
e
E
u
clid
ea
n
n
o
r
m
,
s
k
=
x
k
−
x
k
−
1
an
d
y
k
=
F
k
−
F
k
−
1
Ho
w
e
v
er
,
to
co
m
p
u
te
th
e
s
tep
s
ize
α
k
,
n
o
n
m
o
n
o
to
n
e
lin
e
s
ea
r
ch
p
r
o
p
o
s
ed
b
y
[
4
]
is
an
in
ter
esti
n
g
id
ea
th
at
av
o
id
s
t
h
e
n
ec
es
s
it
y
o
f
d
escen
t
d
ir
ec
tio
n
s
to
g
u
ar
an
tee
t
h
at
ea
ch
iter
atio
n
i
s
w
e
ll
d
ef
i
n
ed
.
L
et
ω
1
>
0
,
ω
2
>
0
,
r
∈
(0
,
1
)
b
e
co
n
s
tan
ts
a
n
d
{η
k
}
b
e
a
g
iv
e
n
p
o
s
iti
v
e
s
eq
u
en
ce
s
u
ch
t
h
at
(
1
6
)
L
et
α
k
=
ma
x
{1
, r
k
}th
at
s
at
is
f
y
(
1
7
)
E
DF
Alg
o
rit
h
m
Ste
p
1
: G
iv
e
n
x
0
,
α
>
0
,
ω
∈
(0
,
1
)
,
r
∈
(0
,
1
)
an
d
a
p
o
s
itiv
e
s
eq
u
e
n
ce
ηk
s
ati
s
f
y
i
n
g
(
1
6
)
,
an
d
s
et
k
=
0
.
Ste
p
2
:
T
est a
s
to
p
p
in
g
cr
iter
io
n
.
I
f
y
es,
th
e
n
s
to
p
; o
th
er
w
is
e
co
n
tin
u
e
w
it
h
Step
3
.
Ste
p
3
: Co
m
p
u
te
d
k
b
y
(
1
3
)
.
Ste
p
4
: Co
m
p
u
te
α
k
b
y
t
h
e
li
n
e
s
ea
r
c
h
(
1
7
)
.
Ste
p
5
: Co
m
p
u
te
Ste
p
6
: Co
n
s
id
er
k
=
k
+
1
an
d
g
o
to
s
tep
2
.
3.
G
L
O
B
A
L
CO
NVE
RG
E
NC
E
T
h
is
s
ec
tio
n
p
r
ese
n
t
s
g
lo
b
al
c
o
n
v
er
g
e
n
ce
r
es
u
lt
s
o
f
th
e
E
n
h
an
ce
d
d
er
iv
ati
v
ef
r
ee
co
n
j
u
g
at
e
g
r
ad
ien
t
m
et
h
o
d
.
I
n
o
r
d
er
to
an
aly
ze
t
h
e
co
n
v
er
g
en
ce
o
f
o
u
r
m
et
h
o
d
,
w
e
w
i
ll
m
a
k
e
t
h
e
f
o
llo
w
i
n
g
ass
u
m
p
tio
n
s
o
n
n
o
n
li
n
ea
r
s
y
s
te
m
s
F
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IJ
AA
S
I
SS
N:
2252
-
8814
E
n
h
a
n
ce
d
Deriva
tive
-
F
r
ee
C
o
n
ju
g
a
te
Gra
d
ien
t Meth
o
d
F
o
r
S
o
lvin
g
S
ymm
etri
c
… (
Ja
milu
S
a
b
i'u
)
53
Ass
u
m
ptio
n
(
i)
T
h
e
lev
el
s
et
Ω
=
{
x|F
(
x
)
≤
e
n
F
(
x
0
)
}
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
,
F
(
x
)
is
L
ip
s
ch
itz
co
n
ti
n
o
u
s
i.e
th
er
e
ex
is
t
s
a
p
o
s
itiv
e
co
n
s
t
an
t
L
>
0
s
u
c
h
th
at
(
1
8
)
∀
x,
y
∈
N
.
P
r
o
p
er
ties
(
i)
an
d
(
i
i)
i
m
p
lies
t
h
at
t
h
er
e
ex
is
ts
p
o
s
i
tiv
e
co
n
s
tan
ts
M
1
,
M
2
s
u
c
h
t
h
a
t
(
1
9
)
L
e
mm
a
3
.
1
[
4
]
Let
th
e
s
eq
u
en
ce
{x
k
}
b
e
g
en
era
ted
b
y
t
h
e
a
lg
o
r
ith
ms
a
b
o
ve
.
Th
en
t
h
e
s
eq
u
en
ce
{|
|
F
k
|
|
}
co
n
ve
r
g
es a
n
d
x
k
∈
N
fo
r
a
ll k
≥
0
.
L
e
mm
a
3
.
2
Let th
e
p
r
o
p
erti
es
o
f
(
1
)
a
b
o
v
e
h
o
l
d
.
Th
en
w
e
h
a
ve
(
2
0
)
(
2
1
)
P
ro
o
f
.
b
y
(
1
6
)
an
d
(
1
7
)
w
e
h
a
v
e
f
o
r
all
k
>
0,
(
2
2
)
b
y
s
u
m
m
i
n
g
t
h
e
ab
o
v
e
k
in
eq
u
alit
y
,
t
h
en
w
e
o
b
tain
:
(
2
3
)
s
o
,
f
r
o
m
(
1
9
)
an
d
th
e
f
ac
t
t
h
at
{η
k
}
s
a
tis
f
ies
(
1
6
)
th
e
s
er
ie
s
∑
is
co
n
v
er
g
e
n
t.
T
h
is
i
m
p
lies
(
2
4
)
.
B
y
a
s
i
m
i
lar
w
a
y
,
w
e
ca
n
p
r
o
v
e
t
h
at(
2
1
)
h
o
ld
s
.
L
e
mm
a
3
.
1
S
u
p
p
o
s
e
th
a
t {x
k
} is
g
en
era
ted
b
y
E
DF
a
lg
o
r
ith
m.
Let s
k
=
x
k
−
x
k
−
1
.
Th
en
,
w
e
h
a
ve
(
2
4
)
P
ro
o
f
.
(
2
5
)
(
2
6
)
T
h
e
f
o
llo
w
i
n
g
t
h
eo
r
e
m
e
s
tab
li
s
h
e
s
th
e
g
lo
b
al
co
n
v
er
g
en
ce
o
f
th
e
E
D
F a
lg
o
r
it
h
m
.
T
heo
re
m
3
.
3
Let {x
k
} b
e
g
en
era
ted
b
y
E
DF
a
lg
o
r
ith
m.
Th
en
,
w
e
h
a
ve
(
2
7
)
P
ro
o
f
.
W
e
p
r
o
v
e
th
is
th
eo
r
e
m
b
y
co
n
tr
ad
ictio
n
.
S
u
p
p
o
s
e
th
at
(
2
7
)
d
o
es
n
o
t
h
o
ld
,
th
en
th
e
r
e
ex
is
t
s
a
p
o
s
itiv
e
co
n
s
ta
n
t
τ
s
u
c
h
t
h
at
(
2
8
)
C
lear
l
y
,
∥
F
k
∥
≤
∥
d
k
∥
,
w
h
ic
h
i
m
p
lies
(
2
9
)
Ob
s
er
v
e
th
at,
(
3
0
)
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
.
1
,
Ma
r
ch
201
6
:
5
0
–
5
7
54
(3
1
)
m
ea
n
in
g
t
h
er
e
ex
is
ts
a
co
n
s
tan
t
λ
∈
(0
,
1
)
s
u
ch
th
a
t f
o
r
s
u
f
f
ic
ien
tl
y
lar
g
e
k
(
3
2
)
Ag
ai
n
f
r
o
m
t
h
e
d
ef
in
i
tio
n
o
f
o
u
r
∗
w
e
o
b
tain
(
3
3
)
w
h
ic
h
i
m
p
lie
s
t
h
er
e
ex
is
ts
a
c
o
n
s
ta
n
t
ρ
∈
(0
,
1
)
s
u
ch
th
at
f
o
r
s
u
f
f
icien
tl
y
lar
g
e
k
(
3
4
)
W
ith
o
u
t lo
s
t o
f
g
e
n
er
alit
y
,
(
3
5
)
w
h
ic
h
s
h
o
w
s
t
h
at
th
e
s
eq
u
e
n
ce
{d
k
}
is
b
o
u
n
d
ed
.
T
h
is
to
g
eth
er
w
it
h
(
2
8
)
an
d
(
2
9
)
y
iel
d
s
a
co
n
tr
ad
ictio
n
.
Hen
ce
th
e
p
r
o
o
f
is
co
m
p
le
te.
4.
NUM
E
RICAL
E
XP
E
R
I
M
E
NT
I
n
th
i
s
s
ec
t
io
n
,
w
e
co
m
p
ar
e
t
h
e
p
er
f
o
r
m
a
n
ce
o
f
o
u
r
m
e
th
o
d
f
o
r
s
o
lv
i
n
g
n
o
n
-
lin
ea
r
eq
u
a
tio
n
(
1
)
w
it
h
A
d
er
iv
ati
v
e
-
f
r
ee
co
n
j
u
g
ate
g
r
ad
ien
t
m
et
h
o
d
an
d
its
g
lo
b
al
co
n
v
er
g
e
n
ce
f
o
r
s
o
lv
i
n
g
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IJ
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2252
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8814
E
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57
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[1
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Ch
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D.H.,
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[7
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[8
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Yu
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[9
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Yu
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G
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tru
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Do
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D.,
M
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J.J.,
Ben
c
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Op
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Per
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