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I
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I
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6
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9
:
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5
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9
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er
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n
t
f
r
o
m
t
h
e
s
o
m
e
cla
s
s
ical
tech
n
iq
u
e
s
w
h
er
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y
n
o
n
l
in
ea
r
eq
u
atio
n
s
ar
e
s
o
lv
ed
ea
s
i
l
y
a
n
d
m
o
r
e
ac
c
u
r
a
tel
y
.
VI
M
h
as
r
ec
en
tl
y
b
ee
n
a
p
p
lied
to
m
a
n
y
f
ield
s
i
n
p
h
y
s
i
cs
an
d
e
n
g
in
ee
r
i
n
g
p
r
o
b
lem
s
[
20
-
22
].
T
h
is
m
et
h
o
d
is
v
er
y
u
s
e
f
u
l
f
o
r
s
o
lv
i
n
g
h
i
g
h
o
r
d
er
lin
ea
r
an
d
n
o
n
lin
e
ar
in
it
ial
v
al
u
e
p
r
o
b
lem
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ir
ec
tl
y
w
i
th
o
u
t
r
e
d
u
cin
g
to
a
f
ir
s
t
s
y
s
te
m
a
n
d
r
eq
u
ir
e
m
en
t
o
r
r
estricti
v
e
ass
u
m
p
tio
n
s
f
o
r
th
e
n
o
n
li
n
ea
r
ter
m
s
[
1
7
]
co
m
p
ar
ed
to
th
e
HP
M
an
d
A
DM
,
wh
er
e
co
m
p
u
tatio
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al
alg
o
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it
h
m
s
ar
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n
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r
m
all
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u
s
ed
to
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ec
o
m
p
o
s
e
th
e
n
o
n
li
n
ea
r
te
r
m
s
.
I
n
th
i
s
s
t
u
d
y
,
w
e
i
n
tr
o
d
u
ce
th
e
B
r
atu
eq
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n
in
f
u
zz
y
f
o
r
m
f
o
r
th
e
f
ir
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t
ti
m
e
an
d
p
r
o
p
o
s
ed
th
e
ap
p
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x
im
ate
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al
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tica
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s
o
lu
t
io
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f
o
r
s
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co
n
d
o
r
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er
n
o
n
lin
ea
r
FIV
P
u
s
i
n
g
VI
M.
Mo
d
if
ied
VI
M
w
il
l
b
e
th
e
f
ir
s
t
ti
m
e
u
s
e
d
f
o
r
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li
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r
B
r
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a
n
d
is
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al
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ze
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ase
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o
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th
e
f
r
a
m
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f
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h
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et
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h
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y
.
2.
B
ASI
C
F
U
Z
Z
Y
CO
NC
E
P
T
S
Def
ini
t
io
n
2
.
1
[
2
3
]
:
T
h
e
r
-
lev
el
(
o
r
r
-
cu
t)
s
et
o
f
a
f
u
zz
y
s
et
̃
,
lab
eled
as
̃
,
is
th
e
cr
is
p
s
et
o
f
all
∈
s
u
c
h
th
at
̃
≥
i.e
.
̃
=
{
∈
|
̃
>
,
∈
[
,
]
}
.
T
h
e
r
-
lev
el
s
et
i
s
th
e
li
n
k
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et
w
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n
th
e
f
u
zz
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d
o
m
ai
n
an
d
th
e
cr
is
p
d
o
m
ain
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y
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s
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n
g
th
e
ad
v
an
ta
g
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of
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e
th
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r
ies i
n
cr
is
p
d
o
m
ai
n
in
th
e
f
u
zz
y
d
o
m
ain
.
Fig
u
r
e
1
Sh
o
w
s
t
h
e
d
eg
r
ee
o
f
f
u
zz
y
m
e
m
b
er
s
h
ip
f
u
n
ct
io
n
[
2
4
]
c
o
n
n
ec
ted
w
ith
t
h
e
v
a
lu
e
s
o
f
r
-
le
v
e
l
s
ets
f
o
r
all
∈
[
,
]
.
W
h
en
th
e
v
al
u
e
o
f
r
-
lev
el
s
et
is
ap
p
r
o
ac
h
to
1
,
m
ea
n
s
t
h
at
t
h
e
f
u
zz
y
ar
ea
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o
m
e
s
m
al
ler
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d
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e
to
cr
is
p
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o
m
ain
.
Fig
u
r
e
1
.
T
h
e
f
u
zz
y
r
-
le
v
el
s
e
t
s
Def
ini
t
io
n
2
.
1
:
Fu
zz
y
n
u
m
b
er
s
ar
e
a
s
u
b
s
et
o
f
t
h
e
r
ea
l
n
u
m
b
er
s
s
et,
an
d
r
ep
r
esen
t
u
n
ce
r
ta
in
v
a
lu
e
s
.
Fu
zz
y
n
u
m
b
er
s
ar
e
lin
k
ed
to
d
eg
r
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s
o
f
m
e
m
b
er
s
h
ip
th
a
t
s
t
ate
h
o
w
tr
u
e
it
is
to
s
a
y
i
f
s
o
m
et
h
in
g
b
elo
n
g
s
o
r
n
o
t
to
a
d
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m
i
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ed
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A
f
u
zz
y
n
u
m
b
er
µ
is
ca
lled
a
tr
ian
g
u
lar
f
u
zz
y
n
u
m
b
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[
2
4
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as
s
h
o
w
n
in
F
ig
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r
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2
i
f
d
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in
ed
b
y
t
h
r
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m
b
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s
<
<
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th
e
g
r
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µ
(
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is
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ase
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[
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ip
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(
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{
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W
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[
]
=
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+
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−
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1
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I
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e
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[
2
5
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-
(
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al,
i.e
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0
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w
it
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(
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s
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(
+
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1
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{
(
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(
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}
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,
[
0
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1
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,
-
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p
p
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Def
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I
n
th
e
s
in
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ar
a
m
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ic
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o
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m
[
2
6
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,
a
f
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n
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m
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s
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1
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w
h
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s
atis
f
ies
:
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(
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a
b
o
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n
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ed
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t c
o
n
ti
n
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o
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s
n
o
n
-
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g
f
u
n
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n
o
v
er
[
0
,
1
]
.
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(
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is
a
b
o
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n
d
ed
r
ig
h
t c
o
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ti
n
u
o
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s
n
o
n
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n
cr
ea
s
i
n
g
f
u
n
ctio
n
o
v
er
[
0
,
1
]
.
-
(
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≤
(
)
,
∈
[
0
,
1
]
.
A
cr
is
p
n
u
m
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er
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s
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m
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y
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ep
r
esen
ted
b
y
(
)
=
(
)
=
,
∈
[
0
,
1
]
.
Def
ini
t
io
n
2
.
2
[
2
5
]:
I
f
̃
b
e
th
e
s
et
o
f
all
f
u
zz
y
n
u
m
b
er
s
,
we
s
a
y
th
a
t
(
)
is
a
f
u
zz
y
f
u
n
ctio
n
i
f
:
ℝ
→
̃
Def
ini
t
io
n
2
.
3
[
2
7
]:
A
m
ap
p
in
g
:
→
̃
f
o
r
s
o
m
e
i
n
ter
v
al
⊆
̃
is
ca
lled
a
f
u
zz
y
f
u
n
ctio
n
p
r
o
ce
s
s
an
d
w
e
d
e
n
o
te
r
-
le
v
el
s
et
b
y
:
[
̃
(
)
]
=
[
(
;
)
,
(
;
)
]
,
∈
,
∈
[
0
,
1
]
T
h
e
r
-
lev
el
s
e
ts
o
f
a
f
u
zz
y
n
u
m
b
er
ar
e
m
u
ch
m
o
r
e
ef
f
ec
tiv
e
as
r
ep
r
esen
tatio
n
f
o
r
m
s
o
f
f
u
z
z
y
s
et
th
a
n
th
e
ab
o
v
e.
Fu
zz
y
s
ets
ca
n
b
e
d
ef
i
n
ed
b
y
t
h
e
f
a
m
ilie
s
o
f
t
h
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r
-
le
v
el
s
ets
b
ased
o
n
th
e
r
eso
lu
tio
n
id
en
tit
y
th
eo
r
e
m
.
T
h
is
is
ca
l
led
th
e
Z
ad
eh
ex
te
n
s
io
n
p
r
in
cip
le.
Def
ini
t
io
n
2
.
4
[
2
5
]:
C
o
n
s
id
er
̃
,
̃
∈
̃
.
I
f
th
er
e
e
x
i
s
ts
̃
∈
̃
s
u
ch
th
a
t
̃
=
̃
+
̃
,
th
en
̃
is
ca
lled
th
e
H
-
d
if
f
er
en
ce
(
H
u
k
u
h
ar
a
d
if
f
er
en
ce
)
o
f
x
an
d
y
an
d
is
d
en
o
ted
b
y
̃
=
̃
⊝
̃
.
Def
ini
t
io
n
2
.
5
[
2
6
]
:
If
̃
:
→
̃
an
d
0
∈
,
w
h
er
e
∈
[
0
,
]
.
W
e
s
ay
t
h
at
̃
Hu
k
u
h
ar
a
d
if
f
er
e
n
tiab
le
at
0
,
if
t
h
er
e
ex
i
s
t
s
an
ele
m
en
t
[
′
̃
]
∈
̃
s
u
c
h
t
h
at
f
o
r
all
ℎ
>
0
s
u
f
f
icie
n
tl
y
s
m
al
l
(
n
ea
r
to
0
)
,
ex
is
t
s
̃
(
0
+
ℎ
;
)
⊝
̃
(
0
;
)
,
̃
(
0
;
)
⊝
̃
(
0
−
ℎ
;
)
an
d
th
e
li
m
i
ts
ar
e
tak
e
n
in
t
h
e
m
etr
ic
(
̃
,
)
:
ℎ
→
0
+
̃
(
0
+
ℎ
;
)
⊝
̃
(
0
;
)
ℎ
=
ℎ
→
0
+
̃
(
0
;
)
⊝
̃
(
0
−
ℎ
;
)
ℎ
T
h
e
f
u
zz
y
s
et
[
′
̃
(
0
)
]
is
ca
lled
th
e
Hu
k
u
h
ar
a
d
er
iv
ati
v
e
o
f
[
′
̃
]
at
0
.
T
h
ese
lim
its
ar
e
tak
en
in
t
h
e
s
p
ac
e
(
̃
,
)
if
0
o
r
K
,
th
en
w
e
co
n
s
i
d
er
th
e
co
r
r
esp
o
n
d
in
g
o
n
e
-
s
id
e
d
er
iv
atio
n
.
R
ec
all
t
h
at
̃
⊝
̃
=
̃
∈
̃
ar
e
d
ef
in
ed
o
n
r
-
le
v
el
s
et,
w
h
er
e
[
̃
]
⊝
[
̃
]
=
[
̃
]
,
∀
∈
[
0
,
1
]
.
B
y
co
n
s
id
er
atio
n
o
f
d
ef
i
n
it
io
n
o
f
t
h
e
m
etr
ic
D
a
ll
t
h
e
r
-
le
v
el
s
et
[
̃
(
0
)
]
ar
e
Hu
k
u
h
ar
a
d
i
f
f
er
en
tiab
le
at
0
,
w
it
h
Hu
k
u
h
ar
a
d
er
i
v
ati
v
es
[
′
̃
(
0
)
]
,
w
h
e
n
̃
:
→
̃
is
Hu
k
u
h
ar
a
d
i
f
f
er
en
tiab
le
at
0
w
it
h
Hu
k
u
h
ar
a
d
er
iv
ativ
e
[
′
̃
(
0
)
]
it’
lead
to
th
at
̃
is
Hu
k
u
h
ar
a
d
if
f
er
en
t
iab
le
f
o
r
al
l
∈
[
0
,
1
]
w
h
ic
h
s
ati
s
f
ies
t
h
e
ab
o
v
e
li
m
it
s
i.e
.
if
f
is
d
if
f
er
en
tiab
le
at
0
∈
[
0
+
,
]
th
en
all
its
r
-
le
v
el
s
[
′
̃
(
)
]
ar
e
Hu
k
u
h
ar
a
d
if
f
er
e
n
tiab
le
at
0
.
x
(
x)
1
0
0.5
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
6
,
Dec
em
b
er
2
0
1
9
:
5
0
7
5
-
508
3
5078
Def
ini
t
io
n
2
.
8
[
2
8
]
:
Def
in
e
th
e
m
ap
p
in
g
′
̃
:
→
̃
an
d
0
∈
I
,
w
h
er
e
∈
[
0
,
]
.
W
e
s
ay
t
h
at
′
̃
Hu
k
u
h
ar
a
d
if
f
er
e
n
tiab
le
∈
̃
,
if
t
h
er
e
ex
is
t
s
an
ele
m
e
n
t
[
̃
(
)
]
∈
̃
s
u
ch
th
at
f
o
r
all
ℎ
>
0
s
u
f
f
icie
n
tl
y
s
m
al
l
(
n
ea
r
to
0
)
,
ex
is
ts
̃
(
−
1
)
(
0
+
ℎ
;
)
⊝
̃
(
−
1
)
(
0
;
)
,
̃
(
−
1
)
(
0
;
)
⊝
̃
(
−
1
)
(
0
−
ℎ
;
)
an
d
t
h
e
li
m
its
ar
e
tak
en
i
n
th
e
m
etr
ic
(
̃
,
)
l
im
ℎ
→
0
+
̃
(
−
1
)
(
0
+
ℎ
;
)
⊝
̃
(
−
1
)
(
0
;
)
ℎ
=
l
im
ℎ
→
0
+
̃
(
−
1
)
(
0
;
)
⊝
̃
(
−
1
)
(
0
−
ℎ
;
)
ℎ
E
x
is
t
s
an
d
eq
u
al
to
̃
(
)
an
d
f
o
r
=
2
w
e
h
av
e
s
ec
o
n
d
o
r
d
er
Hu
k
u
h
ar
a
d
er
iv
ativ
e
.
T
heo
re
m
2
.
2
[
2
6
]
:
L
et
̃
:
[
0
+
,
]
→
̃
b
e
Hu
k
u
h
ar
a
d
if
f
er
e
n
tiab
le
a
n
d
d
en
o
te
[
′
̃
(
)
]
=
[
′
(
)
,
′
(
)
]
=
[
′
(
;
)
,
′
(
;
)
]
.
T
h
en
th
e
b
o
u
n
d
ar
y
f
u
n
ctio
n
s
′
(
;
)
,
′
(
;
)
ar
e
d
if
f
er
en
tiab
le
w
e
ca
n
w
r
ite
f
o
r
s
ec
o
n
d
o
r
d
er
f
u
zz
y
d
er
iv
ati
v
e
[
̃
′′
(
)
]
=
[
(
′′
(
;
)
)
′
,
(
′′
(
;
)
)
′
]
,
∀
∈
[
,
]
.
Def
ini
t
io
n
2
.
5
[
2
9
,
3
0
]:
E
ac
h
f
u
n
ctio
n
:
→
in
d
u
ce
s
an
o
t
h
er
f
u
n
ctio
n
̃
:
(
)
→
(
)
d
ef
in
e
d
f
o
r
ea
ch
f
u
zz
y
in
ter
v
al
in
b
y
:
̃
(
)
(
)
=
{
Sup
∈
−
1
(
)
(
)
,
∈
(
)
0
,
∉
(
)
T
h
is
is
also
k
n
o
w
n
a
s
th
e
Z
ad
eh
ex
te
n
s
io
n
p
r
in
cip
le.
3.
G
E
NE
R
AL
F
U
Z
Z
Y
V
I
M
T
h
e
f
o
r
m
u
latio
n
o
f
VI
M
f
o
r
o
f
th
e
ap
p
r
o
x
i
m
ate
s
o
lu
tio
n
o
f
cr
is
p
g
e
n
er
al
d
if
f
er
en
tial
eq
u
atio
n
w
as
g
iv
e
n
i
n
[
1
5
]
.
I
n
th
is
s
ec
tio
n
,
th
e
n
e
w
f
o
r
m
u
latio
n
o
f
VI
M
f
o
r
th
e
g
en
er
al
f
u
zz
y
d
i
f
f
er
en
tia
l
eq
u
atio
n
b
y
u
s
i
n
g
th
e
p
r
o
p
er
ties
o
f
th
e
f
u
zz
y
d
ef
i
n
itio
n
s
in
s
ec
tio
n
2
w
ill
b
e
p
r
esen
ted
.
C
o
n
s
id
er
g
e
n
er
al
f
u
zz
y
d
if
f
er
e
n
tial
eq
u
atio
n
̃
(
)
−
(
̃
(
)
)
=
ℎ
̃
(
)
(
1
)
W
h
er
e
is
a
lin
ea
r
o
p
e
r
ato
r
,
a
n
o
n
li
n
ea
r
o
p
er
ato
r
o
f
th
e
f
u
z
z
y
f
u
n
ct
io
n
̃
(
)
o
f
cr
is
p
v
ar
iab
le
an
d
ℎ
̃
(
)
is
th
e
i
n
h
o
m
o
g
e
n
eo
u
s
ter
m
s
u
ch
th
a
t
̃
(
)
=
̃
(
;
)
=
[
(
;
)
,
(
;
)
]
,
ℎ
̃
(
)
=
ℎ
̃
(
;
)
=
[
ℎ
(
;
)
,
ℎ
(
;
)
]
,
f
o
r
all
lev
el
s
et
∈
[
0
,
1
]
.
A
cc
o
r
d
in
g
to
VI
M
[
1
5
]
,
a
co
r
r
ec
tio
n
f
u
zz
y
f
u
n
c
tio
n
i
s
co
n
s
tr
u
cted
as f
o
llo
w
s
{
+
1
(
;
)
=
(
;
)
+
∫
{
(
;
)
−
(
(
;
)
⏞
)
−
ℎ
(
;
)
}
=
0
+
1
(
;
)
=
(
;
)
+
∫
{
(
;
)
−
(
(
;
)
⏞
)
−
ℎ
(
;
)
}
=
0
(
2
)
f
o
r
=
0
,
1
,
2
,
…
.
.
Her
e
in
(
2
)
is
a
L
a
g
r
an
g
e
m
u
ltip
lier
th
a
t c
an
b
e
id
en
ti
f
ied
o
p
tim
al
l
y
v
ia
v
ar
iatio
n
al
it
er
atio
n
m
et
h
o
d
[
1
8
]
.
A
cc
o
r
d
in
g
to
[
1
5
]
̃
(
;
)
⏞
is
co
n
s
id
er
ed
as
r
estricte
d
v
ar
iatio
n
s
u
c
h
th
a
t
̃
(
;
)
⏞
=
0
.
E
x
p
an
d
(
2
)
s
i
m
ilar
to
VI
M
in
cr
is
p
to
g
et
th
e
f
o
llo
w
in
g
eq
u
a
tio
n
s
:
L
o
w
er
b
o
u
n
d
s
o
lu
tio
n
1
(
;
)
=
0
(
;
)
+
∫
{
0
(
;
)
−
(
0
(
;
)
)
−
ℎ
(
;
)
}
=
0
2
(
;
)
=
1
(
;
)
+
∫
{
1
(
;
)
−
(
1
(
;
)
)
−
ℎ
(
;
)
}
=
0
3
(
;
)
=
2
(
;
)
+
∫
{
2
(
;
)
−
(
2
(
;
)
)
−
ℎ
(
;
)
}
=
0
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Dir
ec
t so
lu
tio
n
o
f u
n
ce
r
ta
in
b
r
a
tu
in
itia
l v
a
lu
e
p
r
o
b
lem
(
N
.
R
.
A
n
a
kira
)
5079
Up
p
er
b
o
u
n
d
s
o
lu
t
io
n
1
(
;
)
=
0
(
;
)
+
∫
{
0
(
;
)
−
(
0
(
;
)
)
−
ℎ
(
;
)
}
=
0
2
(
;
)
=
1
(
;
)
+
∫
{
1
(
;
)
−
(
1
(
;
)
)
−
ℎ
(
;
)
}
=
0
3
(
;
)
=
2
(
;
)
+
∫
{
2
(
;
)
−
(
2
(
;
)
)
−
ℎ
(
;
)
}
=
0
W
h
er
e
̃
0
(
;
)
is
an
in
itial
g
u
es
s
f
u
n
ctio
n
w
it
h
p
o
s
s
ib
le
u
n
k
n
o
wn
s
th
a
t
s
ati
s
f
ie
s
th
e
in
itial
co
n
d
itio
n
s
o
f
th
e
g
i
v
e
n
FIV
P
an
d
is
th
e
L
a
g
r
an
g
e
m
u
ltip
lier
o
p
ti
m
all
y
.
T
h
e
s
u
cc
ess
i
v
e
ap
p
r
o
x
i
m
a
tio
n
,
f
o
r
i
≥
0
o
f
th
e
s
o
lu
t
io
n
̃
(
;
)
w
i
ll
b
e
r
ea
d
ily
o
b
tain
e
d
u
p
o
n
u
s
i
n
g
th
e
d
eter
m
i
n
ed
th
e
s
u
itab
le
L
ag
r
a
n
g
e
m
u
ltip
lier
an
d
s
elec
ti
v
e
o
f
th
e
i
n
it
ial
ap
p
r
o
x
im
atio
n
̃
0
(
;
)
co
n
s
eq
u
e
n
tl
y
,
th
e
n
t
h
e
ex
ac
t f
u
zz
y
s
o
lu
t
io
n
is
g
i
v
en
b
y
̃
(
;
)
=
l
im
→
∞
(
;
)
(
3
)
4.
ANALY
SI
S O
F
T
H
E
F
U
Z
Z
Y
B
RATU E
Q
UA
T
I
O
N
A
cc
o
r
d
in
g
to
[
1
]
th
e
g
en
er
al
F
I
VP
B
r
atu
eq
u
atio
n
i
s
g
i
v
en
b
y
:
{
̃
′′
(
)
=
̃
̃
(
)
,
>
,
̃
>
̃
̃
(
)
=
̃
,
̃
′
(
)
=
̃
(
4
)
Fo
llo
w
i
n
g
Sectio
n
2
to
d
ef
u
zz
if
y
(
4
)
f
o
r
all
∈
[
,
]
as f
o
llo
w
s
:
̃
(
)
: is a
f
u
zz
y
f
u
n
ct
io
n
o
f
t
h
e
cr
is
p
v
ar
iab
le
s
u
ch
t
h
at
[
̃
(
)
]
=
[
(
;
)
,
(
;
)
]
,
[
̃
]
=
[
,
]
,
[
̃
]
=
[
,
]
,
[
̃
]
=
[
,
]
ar
e
tr
ian
g
u
lar
f
u
zz
y
n
u
m
b
e
r
s
,
̃
′′
(
)
: is th
e
s
ec
o
n
d
o
r
d
er
f
u
zz
y
H
-
d
er
iv
ativ
e
o
f
s
u
c
h
t
h
at
[
̃
′′
(
)
]
=
[
′′
(
;
)
,
′′
(
;
)
]
,
L
et
̃
(
,
̃
(
;
)
)
=
̃
̃
(
)
s
u
ch
t
h
at
[
̃
(
,
̃
(
;
)
)
]
=
[
(
,
̃
(
;
)
)
,
(
,
̃
(
;
)
)
]
th
en
t
h
e
f
o
llo
w
i
n
g
{
(
,
̃
(
;
)
)
=
1
[
,
,
]
(
,
̃
(
;
)
)
=
2
[
,
,
]
(
5
)
Sin
ce
̃
′′
(
)
=
̃
(
,
̃
(
;
)
)
b
y
u
s
i
n
g
th
e
Z
ad
e
h
e
x
ten
s
io
n
p
r
in
ci
p
le
i
n
Sectio
n
2
,
w
e
ca
n
d
ef
i
n
e
th
e
f
o
llo
w
i
n
g
m
e
m
b
er
s
h
ip
f
u
n
ctio
n
{
(
,
̃
(
;
)
)
=
min
{
̃
(
,
̃
(
)
)
|
̃
(
)
∈
̃
(
;
)
}
(
,
̃
(
;
)
)
=
ma
x
{
̃
(
,
̃
(
)
)
|
̃
(
)
∈
̃
(
;
)
}
(
6
)
W
h
er
e
{
(
,
̃
(
;
)
)
=
1
(
,
(
;
)
,
(
;
)
)
=
1
(
,
̃
(
;
)
)
=
(
)
(
;
)
(
,
̃
(
;
)
)
=
2
(
,
(
;
)
,
(
;
)
)
=
2
(
,
̃
(
;
)
)
=
(
)
(
;
)
(
7
)
Fin
all
y
(
4
)
ca
n
b
e
w
r
itte
n
as
{
′′
(
;
)
=
(
)
(
;
)
,
>
0
,
̃
>
0
̃
(
0
;
)
=
(
)
,
′
(
0
;
)
=
(
)
(
8
)
{
′′
(
;
)
=
(
)
(
;
)
,
>
,
̃
>
̃
(
;
)
=
(
)
,
′
(
;
)
=
(
)
(
9
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
6
,
Dec
em
b
er
2
0
1
9
:
5
0
7
5
-
508
3
5080
I
n
th
i
s
s
ec
t
io
n
,
w
e
p
r
esen
t
th
e
s
o
lu
tio
n
o
f
(
8
-
9
)
b
y
m
ea
n
s
o
f
VI
M
in
Sectio
n
3
an
d
[
1
]
.
W
e
fi
r
s
t
co
n
s
tr
u
ct
a
co
r
r
ec
tio
n
f
u
n
ctio
n
al
+
(
;
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=
(
;
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+
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(
)
{
′′
(
;
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−
(
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(
;
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(
1
0
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w
h
er
e
(
;
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⏞
ar
e
co
n
s
id
er
ed
as
r
estr
icted
v
ar
iatio
n
s
,
w
h
ich
m
ea
n
s
(
;
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⏞
=
.
T
o
o
b
tain
th
e
s
u
itab
le
v
a
lu
e
of
(
)
,
w
e
p
r
o
ce
ed
as f
o
llo
w
s
:
+
(
;
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=
(
;
)
+
∫
(
)
{
′′
(
;
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−
(
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(
;
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}
=
(
1
1
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an
d
co
n
s
eq
u
e
n
tl
y
+
(
;
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=
(
;
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+
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(
)
{
′′
(
;
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}
=
(
1
2
)
I
n
teg
r
ati
n
g
(
1
2
)
b
y
p
ar
t
w
h
ich
r
esu
lts
t
h
e
f
o
llo
w
in
g
+
(
;
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(
;
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(
−
′
(
)
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+
(
;
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(
)
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∫
(
;
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=
′′
(
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=
(
1
3
)
T
h
e
s
tatio
n
ar
y
co
n
d
itio
n
s
ca
n
b
e
o
b
tain
ed
as f
o
llo
w
s
:
{
−
′
(
)
=
}
=
(
)
=
}
=
′′
(
)
=
}
=
(
1
4
)
Fro
m
(
1
3
)
,
th
e
L
ag
r
a
n
g
e
m
u
lti
p
lier
s
,
th
er
ef
o
r
e,
ca
n
b
e
id
en
ti
fi
ed
as:
(
)
=
−
(
1
5
)
an
d
th
e
iter
atio
n
VI
M
o
f
th
e
lo
w
er
b
o
u
n
d
f
o
r
m
u
la
f
o
r
s
o
l
u
tio
n
o
f
(
4
)
is
g
i
v
en
a
s
+
(
;
)
=
(
;
)
+
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(
−
)
{
′′
(
;
)
−
(
)
(
;
)
}
=
(
1
6
)
Si
m
i
lar
l
y
f
o
r
th
e
u
p
p
er
b
o
u
n
d
s
o
lu
tio
n
o
f
(
4
)
:
+
(
;
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(
;
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+
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(
−
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{
′′
(
;
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(
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(
;
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(
1
7
)
5.
AP
P
L
I
CA
T
I
O
N
I
n
th
i
s
s
ec
tio
n
,
w
e
e
m
p
lo
y
t
h
e
m
o
d
i
f
ied
VI
M
i
n
s
ec
tio
n
to
o
n
FIV
P
B
r
atu
eq
u
atio
n
a
n
d
d
is
p
la
y
th
e
r
esu
lt
in
t
h
e
f
o
r
m
o
f
tab
les
an
d
f
ig
u
r
es.
A
cc
o
r
d
in
g
to
Sectio
n
2
,
w
e
co
n
s
id
er
th
e
f
u
zz
y
v
er
s
io
n
o
f
th
e
cr
is
p
in
itial
v
al
u
e
B
r
atu
eq
u
atio
n
[
2
]
f
o
r
all
∈
[
,
]
as f
o
llo
w
s
{
̃
′′
(
)
=
̃
(
)
,
>
̃
(
)
=
[
.
−
.
,
.
−
.
]
,
̃
′
(
)
=
[
.
−
.
,
.
−
.
]
(
1
8
)
Fro
m
th
e
f
u
zz
y
a
n
al
y
s
is
i
n
s
ec
tio
n
4
,
(
1
6
)
ca
n
b
e
w
r
itte
n
as
{
′′
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;
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=
2
(
;
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,
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0
(
0
;
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=
0
.
1
−
0
.
1
,
′
(
0
;
)
=
0
.
1
−
0
.
1
(
1
9
)
{
′′
(
;
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=
(
;
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,
>
(
;
)
=
.
−
.
,
′
(
;
)
=
.
−
.
(
2
0
)
Evaluation Warning : The document was created with Spire.PDF for Python.
I
n
t J
E
lec
&
C
o
m
p
E
n
g
I
SS
N:
2
0
8
8
-
8708
Dir
ec
t so
lu
tio
n
o
f u
n
ce
r
ta
in
b
r
a
tu
in
itia
l v
a
lu
e
p
r
o
b
lem
(
N
.
R
.
A
n
a
kira
)
5081
A
cc
o
r
d
in
g
to
Sectio
n
4
th
e
VI
M
f
o
r
m
u
la
o
f
t
h
e
lo
w
er
b
o
u
n
d
s
o
lu
tio
n
o
f
(
1
8
)
is
:
+
(
;
)
=
(
;
)
+
∫
(
−
)
{
′′
(
;
)
−
(
;
)
}
=
(
2
1
)
W
h
er
e
in
itial
f
u
n
ctio
n
o
f
(
2
0
)
is
(
;
)
=
.
−
.
.
Si
m
ilar
l
y
f
o
r
th
e
u
p
p
er
b
o
u
n
d
s
o
lu
t
io
n
of
(
2
1
)
th
e
VI
M
f
o
r
m
u
la
i
s
+
(
;
)
=
(
;
)
+
∫
(
−
)
{
′′
(
;
)
−
(
;
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}
=
(
2
2
)
W
h
er
e
in
itia
l
f
u
n
ctio
n
o
f
(
2
0
)
is
(
;
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=
.
−
.
.
Sin
ce
VI
M
o
b
tai
n
th
e
s
o
lu
tio
n
o
f
i
n
itia
l
v
al
u
e
p
r
o
b
lem
s
in
th
e
f
o
r
m
o
f
s
er
io
u
s
s
o
lu
tio
n
t
h
at
co
n
v
er
g
e
to
th
e
e
x
ac
t
s
o
l
u
tio
n
[
1
6
]
,
in
o
r
d
er
to
o
b
tain
th
e
co
n
v
er
g
e
n
ce
s
er
ies
w
e
n
ee
d
to
ex
p
an
d
th
e
n
o
n
li
n
ea
r
ter
m
̃
(
;
)
in
(
2
1
)
an
d
(
2
2
)
in
to
Ma
cl
au
r
in
s
er
ie
s
o
f
o
r
d
er
th
r
ee
s
u
ch
th
a
t
{
(
;
)
=
+
(
;
)
+
(
;
)
+
(
;
)
(
;
)
=
+
(
;
)
+
(
;
)
+
(
;
)
(2
3
)
Su
b
s
ti
tu
te
(
2
3
)
in
(
2
1
-
22
)
to
o
b
tain
th
e
f
o
llo
w
i
n
g
VI
M
f
o
r
m
u
la:
+
(
;
)
=
(
;
)
+
∫
(
−
)
{
′′
(
;
)
−
(
+
(
;
)
+
(
;
)
+
(
;
)
)
}
=
(
2
4
)
+
(
;
)
=
(
;
)
+
∫
(
−
)
{
′′
(
;
)
−
(
+
(
;
)
+
(
;
)
+
(
;
)
)
}
=
(
2
5
)
No
te
th
at
in
cr
ea
s
in
g
t
h
e
o
r
d
er
o
f
th
e
s
er
ies
co
r
r
esp
o
n
d
in
g
w
i
th
t
h
e
in
cr
ea
s
in
g
o
f
it
er
atio
n
o
f
th
e
ap
p
r
o
x
i
m
ate
m
e
th
o
d
th
e
ac
cu
r
ac
y
o
f
t
h
e
s
o
lu
tio
n
w
ill
b
e
in
cr
ea
s
e.
A
s
m
e
n
tio
n
ed
in
Sectio
n
1
lar
g
e
n
u
m
b
er
o
f
n
o
n
li
n
ea
r
i
n
itial
v
alu
e
p
r
o
b
lem
u
n
d
er
u
n
ce
r
ta
in
t
y
ca
n
n
o
t
o
b
tain
t
h
e
e
x
ac
t
s
o
lu
tio
n
f
r
o
m
it
s
eq
u
atio
n
o
r
v
er
y
d
i
f
f
icu
lt
to
g
et
t
h
e
ex
ac
t
s
o
lu
tio
n
.
Si
n
ce
th
e
ex
ac
t
s
o
lu
tio
n
o
f
(
1
7
)
ca
n
n
o
t
b
e
o
b
tain
ed
an
al
y
ticall
y
,
i
n
o
r
d
e
r
to
d
etec
t
th
e
ac
c
u
r
ac
y
o
f
VI
M
ap
p
r
o
x
i
m
ate
s
o
l
u
tio
n
w
e
d
e
f
in
e
th
e
f
o
llo
w
i
n
g
r
esid
u
a
l
er
r
o
r
[
1
1
]
,
let
VI
M
s
o
lu
tio
n
d
en
o
ted
b
y
̃
(
;
)
s
u
c
h
th
a
t
̃
(
;
)
=
|
̃
(
;
)
−
̃
(
;
)
|
(
2
6
)
Fro
m
(
2
4
)
an
d
(
2
5
)
th
e
th
ir
d
o
r
d
er
VI
M
ap
p
r
o
x
im
ate
s
o
lu
tio
n
o
f
(
1
7
)
f
o
r
all
∈
[
,
]
is
g
iv
e
n
b
y
̃
(
;
)
=
∑
̃
(
;
)
=
(
2
7
)
T
ab
le
1
d
is
p
lay
ed
th
e
t
h
ir
d
o
r
d
er
VI
M
ap
p
r
o
x
im
ate
s
o
lu
t
i
o
n
an
d
th
e
ac
c
u
r
ac
y
u
n
d
er
th
e
r
esid
u
al
er
r
o
r
(
2
5
)
o
f
(
1
8
)
w
h
en
=
.
f
o
r
al
l
f
u
zz
y
le
v
el
s
et
s
∈
[
,
]
.
I
n
T
a
b
le
1
,
also
th
e
VI
M
s
o
lu
tio
n
o
f
(
1
8
)
s
atis
f
ies
th
e
p
r
o
p
er
ties
o
f
f
u
zz
y
s
et
t
h
eo
r
y
as
m
en
tio
n
i
n
s
ec
tio
n
2
f
o
llo
w
in
g
t
h
e
p
r
o
p
er
ties
o
f
s
in
g
le
p
ar
am
etr
ic
f
o
r
m
o
f
f
u
zz
y
n
u
m
b
er
s
.
T
ab
le
1
.
T
h
ir
d
o
r
d
er
V
I
M
s
o
lu
tio
n
o
f
(
1
7
)
f
o
r
all
∈
[
,
]
at
=
.
Evaluation Warning : The document was created with Spire.PDF for Python.
I
SS
N
:
2
0
8
8
-
8708
I
n
t J
E
lec
&
C
o
m
p
E
n
g
,
Vo
l.
9
,
No
.
6
,
Dec
em
b
er
2
0
1
9
:
5
0
7
5
-
508
3
5082
T
h
e
th
r
ee
d
i
m
en
s
io
n
al
Fi
g
u
r
e
3
,
s
h
o
w
s
t
h
at
t
h
e
th
ir
d
o
r
d
er
VI
M
ap
p
r
o
x
i
m
ate
s
o
lu
tio
n
f
o
r
ea
ch
v
al
u
e
o
f
∈
[
,
.
]
an
d
f
u
zz
y
le
v
el
s
et
s
∈
[
,
]
f
o
llo
w
s
t
h
e
p
r
o
p
er
ties
o
f
f
u
zz
y
n
u
m
b
er
s
as
m
e
n
tio
n
ed
in
s
ec
tio
n
2
in
t
h
e
f
o
r
m
o
f
t
h
e
tr
i
an
g
u
lar
f
u
zz
y
n
u
m
b
er
.
Fig
u
r
e
3
.
T
h
ir
d
o
r
d
er
VI
M
s
o
lu
tio
n
o
f
(
1
7
)
in
th
e
f
o
r
m
o
f
tr
i
an
g
u
lar
f
u
zz
y
n
u
m
b
er
f
o
r
all
∈
[
,
]
at
∈
[
,
.
]
Sin
ce
t
h
e
ex
ac
t
s
o
lu
tio
n
ca
n
n
o
t
o
b
tain
in
(
1
8
)
,
Fig
u
r
e
4
s
h
o
w
s
s
u
f
f
icien
t
ac
cu
r
ac
y
w
i
th
o
n
l
y
t
h
ir
d
o
r
d
er
VI
M
ap
p
r
o
x
im
a
te
lo
w
er
an
d
u
p
p
er
s
o
lu
tio
n
o
f
(
18
)
in
ter
m
s
o
f
r
esid
u
a
l
er
r
o
r
(
2
5
)
w
h
er
e
th
e
m
a
x
i
m
u
m
er
r
o
r
−
f
o
r
th
e
lo
w
er
s
o
l
u
tio
n
a
n
d
−
f
o
r
th
e
u
p
p
er
s
o
lu
t
io
n
f
o
r
ea
ch
v
al
u
e
o
f
∈
[
,
.
]
an
d
f
u
z
z
y
lev
el
s
et
s
∈
[
,
]
.
Fig
u
r
e
4
.
A
cc
u
r
ac
y
o
f
t
h
ir
d
o
r
d
er
VI
M
s
o
lu
tio
n
o
f
(
1
7
)
f
o
r
a
ll
∈
[
,
]
at
∈
[
,
.
]
6.
CO
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O
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I
n
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5083
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NC
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S
[1
]
B.
Ba
ti
h
a
,
"
Nu
m
e
ri
c
a
l
S
o
lu
ti
o
n
o
f
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tu
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Ty
p
e
Eq
u
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ti
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s
b
y
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e
Va
riatio
n
a
l
Itera
ti
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n
M
e
th
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d
,
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c
e
tt
e
p
e
J
o
u
rn
a
l
o
f
M
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ti
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n
d
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t
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ti
stics
,
v
o
l
.
3
9
(1
),
p
p
.
23
–
29
,
2
0
1
0
.
[2
]
B.
S
.
H.
Ka
sh
k
a
ri
a
n
d
S
.
S.
Z.
A
b
b
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s,
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o
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ti
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In
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lu
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Bra
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p
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Eq
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Ho
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e
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ter
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ti
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l
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o
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rn
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o
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ter
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ti
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p
.
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2
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[3
]
A
.
O
m
e
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O.
O
m
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"
A P
ra
y
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n
d
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o
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c
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th
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t
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ta
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41
(
3
),
p
p
.
3
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0
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[4
]
T
.
S
m
it
a
,
S
.
Ch
a
k
ra
v
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y
,
"
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m
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rica
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ti
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p
.
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4
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7
3
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2
0
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3
.
[5
]
M
.
S
.
El
Na
sc
h
ie,
"
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ro
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tal
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.
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p
.
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7
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[6
]
M
.
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b
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.
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.
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Ba
b
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1
4
9
,
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p
.
5
4
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–
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5
7
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0
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[8
]
M
.
G
h
a
n
b
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ri,
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L
.
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6
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0
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M
.
G
h
a
n
b
a
ri,
"
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l
so
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M
,
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n
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M
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ti
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s
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l.
1
(1
),
p
p
.
19
–
39
,
2
0
0
9
.
[1
1
]
A.
F
.
Ja
m
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e
l,
A
.
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M
d
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m
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s,
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Un
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m
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l.
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p
.
2
2
1
-
2
2
9
,
2
0
1
5
.
[1
2
]
T
.
S
m
it
a
,
S
.
Ch
a
k
ra
v
e
rt
y
,
"
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m
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rica
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ti
o
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r
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Ho
m
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,"
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p
.
0
9
7
5
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8
8
7
,
2
0
1
3
.
[1
3
]
T
.
A
ll
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lo
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A
.
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a
n
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n
d
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o
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a
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a
r,
"
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ti
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An
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a
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a
ti
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a
,
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l.
1
7
(1
),
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p
.
5
–
14
,
2
0
0
9
.
[1
4
]
A.
F
.
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e
e
l,
A
.
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a
a
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n
,
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.
A
.
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.
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A
h
m
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d
,
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r
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r
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e
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s
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l.
1
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,
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p
.
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64
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0
1
8
.
[1
5
]
J.
H.
He
,
"
V
a
riatio
n
a
l
Itera
ti
o
n
M
e
th
o
d
f
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r
A
u
to
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m
o
u
s
Ord
in
a
ry
Diff
e
r
e
n
ti
a
l
S
y
ste
m
s,
"
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p
li
e
d
M
a
th
e
ma
t
ics
a
n
d
Co
mp
u
t
a
ti
o
n
,
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o
l.
1
1
4
,
p
p
.
1
1
5
–
1
2
3
,
2
0
0
0
.
[1
6
]
J.
H.
He
,
"
V
a
riatio
n
a
l
Itera
ti
o
n
M
e
th
o
d
-
A
Kin
d
o
f
No
n
-
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in
e
a
r
A
n
a
l
y
ti
c
a
l
T
e
c
h
n
iq
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e
:
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o
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e
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m
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les
,
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t
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J
o
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rn
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r M
e
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n
ics
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3
4
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p
.
6
9
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[1
7
]
J.
H.
He
,
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.
[1
8
]
J.
H.
He
,
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M
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.
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[1
9
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A
.
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a
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.
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ra
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d
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ter
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[2
0
]
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Am
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Biaz
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r,
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Ne
w
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.
[2
1
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K.
Ha
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.
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ter
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o
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trica
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e
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tro
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d
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[2
2
]
A
.
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.
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[2
3
]
S
.
Bo
d
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v
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,
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lp
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.
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.
[2
4
]
D.
Du
b
o
is,
H.
P
ra
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e
,
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s
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z
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rt
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l.
8
,
p
p
.
2
2
5
-
233
,
1
9
8
2
.
[2
5
]
O.
S
.
F
a
rd
,
"
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n
it
e
ra
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s
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s J
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5
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9
.
[2
6
]
R.
F
a
ra
n
a
k
,
I.
F
u
d
z
iah
,
A
.
A
li
,
S
.
S
o
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il
,
"
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m
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rica
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so
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iffere
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sin
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ro
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Ru
n
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Pro
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s in
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.
[2
7
]
O.
Ka
lev
a
,
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u
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7
,
1
9
8
7
.
[2
8
]
S
.
S
.
M
a
n
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o
u
r,
N.
A
h
m
a
d
y
,
"
A
n
u
m
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rica
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m
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[2
9
]
L
.
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.
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.
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l.
1
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(2
),
p
p
.
1
–
40
,
2
0
0
5
.
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