Indonesi
an
Journa
l
of El
ect
ri
cal Engineer
ing
an
d
Comp
ut
er
Scie
nce
Vo
l.
13
,
No.
3
,
Ma
rch
201
9
, p
p.
9
4
5
~
9
5
3
IS
S
N: 25
02
-
4752, DO
I: 10
.11
591/ijeecs
.v1
3
.i
3
.pp
9
4
5
-
9
5
3
945
Journ
al h
om
e
page
:
http:
//
ia
es
core.c
om/j
ourn
als/i
ndex.
ph
p/ij
eecs
A
ne
w class of B
FGS up
dating fo
rmula b
ased on th
e new quasi
-
newton e
qu
atio
n
Basim A
. Has
sa
n,
Hu
ssein
K.
K
ha
l
o
Depa
rtment
o
f M
at
hematics,
C
oll
eg
e
of
Com
pute
rs Sci
ences a
n
d
Mathe
m
atics,
Univer
sit
y
of
Mos
ul,
Ira
q
.
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Sep
15
, 201
8
Re
vised
N
ov
28
, 2
018
Accepte
d
Dec
17
, 201
8
Quasi
-
Newton
m
et
hods”
are
a
m
ongst
the
m
a
inly
usefu
l
and
compete
nt
it
er
at
iv
e
proc
ess
for
solving
un
restr
ai
n
ed
m
ini
m
iz
at
ion
func
tions
.
In
thi
s
pape
r
we
der
iv
e
a
new
quasi
-
Newton
equa
ti
on
w
it
h
on
the
Hess
ia
n
esti
m
at
e
updat
es
and
a
ltera
t
ions
in
te
nd
e
d
a
t
d
evelopin
g
th
ei
r
per
for
m
anc
e
.
Th
e
“
Nu
m
eri
ca
l
res
ult
s”
il
lustr
at
e
t
hat
the
propose
d
te
chni
qu
e
useful
for
the
known t
est
fun
ctions.
Ke
yw
or
d
s
:
Conver
ge
nce
pro
per
ti
es
Qu
asi
-
Ne
wton
equ
at
io
ns
Qu
asi
-
Ne
wton
m
et
ho
ds
Copyright
©
201
9
Instit
ut
e
o
f Ad
vanc
ed
Engi
n
ee
r
ing
and
S
cienc
e
.
Al
l
rights re
serv
ed
.
Corres
pond
in
g
Aut
h
or
:
Ba
si
m
A
. H
a
ss
an
,
Dep
a
rtm
ent o
f M
at
hem
a
ti
cs,
Coll
ege
of
C
om
pu
te
rs
Scie
nc
es an
d
Ma
them
at
ic
s,
Un
i
ver
sit
y o
f M
os
ul,
Iraq
.
Em
a
il
:
basi
m
a
bas
39@g
m
ai
l.
com
, h
us
sei
nz
om
ar5
4@gm
ail.co
m
1.
INTROD
U
CTION
Un
c
overi
ng a s
olu
ti
on to
a
ge
ner
al
broa
d de
gr
ee
no
nlinear
op
ti
m
iz
ation
P
roblem
:
n
R
x
,
)
x
(
M
in
(1)
wh
е
re wher
e
i
s a
“
s
m
oo
th funct
ion
”
of n
vari
ables, b
y q
uasi
-
Ne
wton m
et
h
od
s is p
ai
ns
ta
ki
ng
. Quasi
-
Ne
wton
m
et
ho
ds
a
re
awfull
y
us
ef
ul
utensils
f
or
s
olv
in
g
unr
est
rain
ed
opti
m
iz
at
io
n
pro
blem
s
[
1
]
.
At
the
k
th
it
erati
on
of
the
quasi
-
N
ewto
n
m
et
ho
d,
a
sy
m
m
et
ric
a
n
d
non
neg
at
iv
e
def
init
e
k
B
is
kn
ow
n,
a
nd
a
se
arch
directi
on
is
com
pu
te
d by:
,
1
k
k
k
J
B
d
(2)
wh
e
re
k
J
is t
he g
rad
ie
nt of
eval
uated
at
t
he
c
urren
t i
te
rate
k
x
. O
ne
the
n
c
om
pute
s the
nex
t i
te
r
at
e b
y:
k
k
k
k
d
x
x
1
(3)
It co
m
pu
te
d a s
te
p
le
ngth
k
that
m
ake su
re t
he a
ppr
ov
al
of the
Wo
lfe c
onditi
on
s:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
13
, N
o.
3
,
Ma
rc
h 201
9
:
9
4
5
–
9
5
3
946
k
T
k
k
k
k
k
k
d
J
x
d
x
)
(
)
(
(4)
k
T
k
k
k
k
T
k
J
d
d
x
J
d
)
(
”
(5)
wh
e
re
1
0
.
F
or
m
or
e
detai
ls
can
be
f
ound
in
[
2
]
.
By
tradit
io
n,
k
B
sat
isfie
s
the
“
qu
a
si
-
Ne
wton
equ
at
io
n:
“
,
1
k
k
k
p
B
(6)
wh
e
re
k
k
k
x
x
1
and
k
k
k
J
J
p
1
.
L
et
k
H
be
the
in
verse
of
k
B
.
T
he
fam
ou
s
in
ver
se
up
date
k
H
is t
he
sta
nda
rd
BFGS f
or
m
ula:
k
T
k
T
k
k
k
T
k
k
k
T
k
k
T
k
k
T
k
k
T
k
k
k
k
B
F
G
S
k
p
p
p
H
p
p
H
p
p
H
H
H
1
1
(7)
Ce
rtai
nly,
BFGS
m
et
ho
d
is
on
e
of
t
he
m
os
t
excell
ent
m
et
hods
an
d
do
i
ng
t
o
now
f
or
so
lvi
ng
(1).
F
or
m
or
e
detai
ls can b
e
f
ound in [
3
]
. A
lo
t of
a
ppr
oaches ha
ve
bee
n
s
uggeste
d
to f
i
nd b
et
te
r
t
he
qu
asi
-
“
Ne
wton Hessi
an
est
i
m
at
e u
pd
at
es. In
this f
ra
gm
ent w
e sk
et
c
h
a fe
w
la
te
st suggeste
d
up
dates ta
ke
by m
odify
ing
the
vect
or
k
p
,
a
s sho
wn in T
a
ble 1.
Table
1.
M
od
i
f
yi
ng
T
he Vect
or
Na
m
e
m
e
th
o
d
s
Dif
f
erence in g
radien
ts
Ref
erences
P
k
k
k
k
k
k
B
p
p
)
1
(
*
[
4
]
LF
6
*
10
,
k
k
k
k
k
p
p
[
5
]
W
L
Q
k
k
k
T
k
k
k
k
k
k
J
J
p
p
2
1
1
*
)
(
)
(
2
[
6
]
ZDC
k
k
k
T
k
k
k
k
k
k
J
J
p
p
2
1
1
*
)
(
3
)
(
6
[
7
]
The
i
dea
of
va
r
ia
nt QN m
et
ho
ds
ha
d been
stu
died by m
any r
esearche
rs f
or
exam
ple
, [
8]
, [
9]
.
Now we
will
derive
ne
w qu
as
i
-
Ne
wton e
qu
a
ti
on
s a
nd
a
naly
ze it
s conv
e
rge
nce.
2.
DERIVI
NG N
EW Q
UA
S
I
-
N
EWT
ON
E
Q
UA
TI
ON
A
N
D A
ALG
ORI
THM
In
this
f
rag
m
ent
we
de
rive
th
e
new
quasi
-
N
ewto
n
eq
uatio
ns
.
T
her
e
fore
we
can
a
pp
ly
it
to
fu
nctio
ns
m
or
e g
ene
ral t
han
qua
dr
at
ic
a
s f
r
om
:
k
k
T
k
k
k
T
k
k
T
k
k
k
T
U
J
1
1
1
1
6
1
2
1
(8)
wh
e
re
1
k
T
is
the
t
ens
or
of
at
the
po
i
nt
k
x
.
we
at
ta
in,
by
re
voki
ng
t
he
co
ndit
ion
s
w
hich
c
om
pr
ise
the
te
ns
or:
k
T
k
k
k
k
k
T
k
k
k
T
k
J
J
p
U
)
(
3
)
(
6
1
1
1
(9)
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A n
ew
cl
as
s
of
BFG
S upd
atin
g
fo
r
m
ula b
as
e
d on the
new
quas
i
-
newt
on equa
ti
on (
B
as
im
A.
Ha
ss
an
)
947
Fo
r
m
or
e
detai
ls
can
be
f
ound
in
[
10
].
So,
would
li
ke
deri
vative
of
m
anag
es
to
form
ul
at
e
gen
e
ral
qu
adr
at
ic
,
we have:
k
T
k
k
k
k
k
T
k
k
k
T
k
J
J
p
U
)
(
2
3
)
(
3
2
1
2
1
1
1
1
(10)
The
ste
p
siz
e s
cal
ar
,
k
wh
ic
h
m
ini
m
iz
es
),
(
k
x
is ap
pro
xim
a
te
d
by:
k
T
k
k
T
k
k
Ud
d
d
J
(11)
Af
te
r
s
om
e alge
br
ai
c m
anipu
l
at
ion
s
one
ob
ta
ins
:
k
T
k
k
T
k
k
k
k
T
k
k
k
T
k
J
J
p
U
1
1
1
2
3
)
(
3
2
1
(12)
Since
k
k
B
1
is nee
d
t
o
est
im
at
ed
,
1
k
k
U
it
is reas
on
a
ble t
o need
:
k
T
k
k
T
k
k
k
k
T
k
k
k
T
k
J
J
p
B
1
1
1
2
3
)
(
3
2
1
(13)
A goo
d
c
ho
ic
e
to esti
m
a
te
k
k
B
1
is kno
wn b
y
:
k
k
T
k
k
T
k
k
T
k
k
k
k
k
k
k
k
J
J
p
p
p
B
)
(
2
/
3
)
(
3
2
1
,
1
1
*
*
1
(14)
wh
e
re
k
u
is any
ve
ct
or
s
uc
h
that
0
k
T
k
.
Var
ie
ti
es
of
th
is
qu
asi
-
Ne
wt
on
e
quat
io
ns
di
ff
er
i
n
the
wa
y
of
sel
ect
in
g
the
vect
or
k
in
(14)
we
h
ave
the
f
or
m
s:
1.
First
case
k
k
p
giv
es:
k
k
T
k
k
T
k
k
T
k
k
k
k
k
k
k
k
p
p
J
J
p
p
p
B
)
(
2
/
3
)
(
3
2
1
,
1
1
*
*
1
(15)
2.
Seco
nd case
1
k
k
J
gi
ves
:
1
1
1
1
*
*
1
)
(
2
/
3
)
(
3
2
1
,
k
k
T
k
k
T
k
k
T
k
k
k
k
k
k
k
k
J
J
J
J
p
p
p
s
B
(16)
Diff
e
re
nt
change
gradie
nt
us
e
d
in
quasi
-
Ne
wton
eq
uatio
n
for
yi
el
d
diff
e
r
ent
quasi
-
New
t
on
m
et
ho
ds
.
The ne
w
CB
F
GS
al
gorithm
can be stat
ed
form
al
l
y as fo
ll
ows.
Step
1
:
Data
n
R
x
0
a
nd
I
H
0
. S
et
0
k
.
Step
2:
Sto
p
i
f
0
k
J
.
Step
3:
Ca
lc
ula
te
k
d
by
:
k
k
k
J
H
d
Step
4:
Fin
ds a
k
w
hich
sati
sfie
s the
(4)
a
nd (5
).
Step
5:
Iterati
ve
pro
ces
s
be
as
k
k
k
k
d
x
x
1
.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
13
, N
o.
3
,
Ma
rc
h 201
9
:
9
4
5
–
9
5
3
948
Step
6:
U
pdat
e
0
H
f
or tim
es to f
i
nd
1
k
H
by (7) an
d (
14)
.
Step
7:
P
ut
1
k
k
. Go
to
step
2.
A prope
rty
pos
it
ive d
efi
nite o
f
k
H
is aw
fu
ll
y i
m
portant,
can
b
e
v
e
rify in t
he n
ext the
or
em
.
Theorem
2.1.
Let
“
)
,
,
,
(
1
1
1
k
k
k
k
d
J
x
”
be
gen
e
r
at
ed
by
the
ne
w
al
gorithm
.
The
n
1
k
B
is
po
sit
ive
def
i
nite
for
al
l k
pr
ov
i
de
d
that
0
*
k
T
k
p
.
Pro
of
.
Now,
we
eval
ua
te
the
qu
a
ntit
y
*
k
T
k
y
.
I
f
the
ste
p
le
ng
t
h
k
sat
isfie
s
the
Wo
l
fe
c
onditi
on
s
(
4)
an
d
(5),
t
hen w
e
h
a
ve:
k
T
k
k
T
k
k
T
k
k
k
k
T
k
k
T
k
k
k
k
T
k
k
T
k
J
J
J
J
J
p
p
3
2
1
2
1
)
(
3
2
3
)
(
3
2
1
1
1
1
1
*
(17)
To
at
ta
in
this
intenti
on,
pr
e
fe
r
the
values
of
and
with
3
/
1
2
/
1
an
d
0
3
2
/
1
.
No
ti
ng
the
0
k
T
k
k
k
T
k
J
d
J
,
we k
now
t
hat the
re e
xists a co
ns
ta
nt
0
m
su
c
h
that:
0
*
k
T
k
k
T
k
J
d
m
p
(18)
The pr
oof
is co
m
ple
te
.
3.
GLOB
AL P
R
OPERT
Y
We
rea
dy
the
“
l
ocal
co
nver
ge
nc
e
pro
per
ty
”
of the
m
od
ifie
d
B
FG
S
m
et
hod.
Th
e fo
ll
owin
g
assum
ption
is require
d.
Assu
m
pt
i
on
(i)
T
he
le
vel se
t
)
(
)
(
0
x
x
R
x
S
n
is b
ounde
d.
(ii)
“
T
he
f
unct
ion
f
is
“
twic
e
con
ti
nu
ously
di
ff
ere
ntiable
”
on
S
an
d
the
re
e
xists
a
c
onsta
nt
”
0
L
su
c
h
that:
y
x
L
y
J
x
J
)
(
)
(
(19)
Since
k
is
a
di
m
inishin
g
se
ries,
it
is
ob
vi
ous
t
hat
the
series
k
x
gen
e
rated b
y
ne
w
Al
gorithm
i
s
enclose
d
in
S
, and
the
re e
xists a c
onsta
nt
*
suc
h t
hat:
*
lim
k
k
(20)
(iii
)
“
The
fu
nction
is
“
un
if
or
m
l
y
”
con
vex, i.
e.,
the
re e
xist
posit
ive c
on
sta
nt
s
M
an
d
m
su
c
h
t
ha
t:
2
2
)
(
d
M
d
x
U
d
d
m
T
(21)
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Ind
on
esi
a
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J
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c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A n
ew
cl
as
s
of
BFG
S upd
atin
g
fo
r
m
ula b
as
e
d on the
new
quas
i
-
newt
on equa
ti
on (
B
as
im
A.
Ha
ss
an
)
949
ho
l
ds
for
al
l
S
x
an
d
,
n
R
d
w
her
e
)
(
)
(
2
x
x
U
. Th
es
e assu
m
ption
s
are the
sam
e as those in
[
11
].
Theorem
2.2.
“
Let
k
x
be
g
e
ne
ra
te
d
by t
he new
al
gorithm
. Th
e
n we
hav
e:
,
,
*
2
*
2
k
k
k
k
T
k
k
M
L
p
M
p
m
(22)
and
1
k
k
T
k
k
d
J
(23)
Pro
of
.
Fo
ll
owin
g
the
def
i
niti
on
of
*
k
p
and the
Tay
lor'
s se
ries,
w
e
g
et
:
)
(
6
2
4
)
(
3
)
(
6
2
3
)
(
3
2
1
1
1
1
1
1
1
*
k
k
k
T
k
k
T
k
k
T
k
k
k
k
k
T
k
k
T
k
k
T
k
k
k
k
T
k
k
T
k
J
J
J
J
p
J
J
p
p
(24)
By
u
sin
g
Tay
l
or’s
se
ries, a
nd
m
ean v
al
ue
t
he
or
em
, w
e
get:
k
k
T
k
k
T
k
k
k
U
J
)
(
2
1
1
1
(25)
Fr
om
)
24
(
an
d
)
25
(
we
get
:
)
(
2
2
)
(
2
)
(
6
2
)]
(
)
(
2
1
[
4
)
(
3
)
(
6
2
3
)
(
3
2
1
1
1
1
1
1
1
1
*
k
k
k
T
k
k
k
T
k
k
k
k
T
k
k
k
k
k
T
k
k
T
k
k
k
k
k
T
k
k
T
k
k
T
k
k
k
k
T
k
k
T
k
J
U
J
U
J
J
p
J
J
p
p
s
(26)
By
u
sin
g qu
a
dr
at
ic
f
unct
ion, a
n
d m
ean v
al
ue
theo
rem
, w
e h
a
ve:
k
k
T
k
k
T
k
k
k
U
J
)
(
2
1
1
(27)
Ther
e
f
or
e,
it
fo
ll
ow
s
from
)
26
(
an
d
)
27
(
that
:
k
k
T
k
k
k
T
k
k
k
T
k
k
k
k
T
k
k
k
T
k
k
T
k
U
U
U
J
U
p
)
(
)
(
)
(
2
)
(
2
2
)
(
2
1
*
(28)
wh
e
re
)
(
1
k
k
k
k
x
x
x
an
d
)
1
,
0
(
(29)
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S
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:
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Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
13
, N
o.
3
,
Ma
rc
h 201
9
:
9
4
5
–
9
5
3
950
Mi
ng
li
ng
with
Assum
pt
ion
(
ii
i), it i
s sim
ple t
o
cat
ch:
2
*
2
k
k
T
k
k
M
p
m
(30)
Using
def
i
niti
on
of
*
k
p
and the T
ay
lor'
s ser
ie
s
once m
or
e,
we g
et
:
k
k
k
k
k
k
T
k
k
k
k
k
k
T
k
k
k
T
k
k
k
k
k
k
T
k
k
T
k
k
k
k
k
k
T
k
k
T
k
k
T
k
k
k
k
k
M
L
M
L
U
p
p
U
p
J
J
p
J
J
p
p
]
)
(
[
)]
(
2
/
1
)
(
[
2
1
)
(
2
/
3
)
(
3
2
1
)
(
2
/
3
)
(
3
2
1
1
1
1
1
*
(31)
Now we t
urn
t
o
the
pr
oof of (
23). By
the
W
WP r
ule (4)
and
Assum
ption
(ii) w
e
obta
in:
2
1
)
(
)
1
(
k
k
k
T
k
k
k
T
k
d
L
d
J
J
d
J
(32)
This yi
el
ds
t
ha
t
:
~
2
2
)
1
(
)
1
(
)
1
(
k
k
k
k
T
k
k
k
T
k
k
L
m
d
L
d
B
d
d
L
d
J
(33
)
Conver
sel
y, f
r
om
(
1)
,
w
e
obt
ai
n
:
*
1
1
1
1
1
1
1
)
(
lim
)
(
lim
)
(
k
N
k
k
k
N
k
k
k
(34
)
Th
us
,
,
)
(
1
1
k
k
k
(35
)
wh
ic
h
m
ixed wit
h
the
WW
P
law
(4)
that i
s:
1
k
k
T
k
k
d
J
(36
)
Currentl
y we
e
sta
blishe
d t
he
world
wide
con
verge
nce
of
ne
w Alg
or
it
hm
.
Theorem
2.3.
Let
k
x
be
create
d
by n
ew
Algo
rithm
an
d
le
t
s
at
isfie
s A
ssu
m
ption
s i and
ii
.
S
uppose to f
aci
li
ta
te
there e
xists c
onsta
nts
1
an
d
2
s
uc
h
the
s
ub
se
qu
ent r
el
at
io
n:
Evaluation Warning : The document was created with Spire.PDF for Python.
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci
IS
S
N:
25
02
-
4752
A n
ew
cl
as
s
of
BFG
S upd
atin
g
fo
r
m
ula b
as
e
d on the
new
quas
i
-
newt
on equa
ti
on (
B
as
im
A.
Ha
ss
an
)
951
k
k
k
B
1
an
d
2
2
2
k
k
T
k
B
(37
)
ho
l
ds
,
Th
e
n we
h
a
ve
:
0
i
n
f
lim
k
k
J
(38
)
Pro
of
.
We
pro
ceed b
y
con
tr
adict
io
n,
w
e
ass
um
e
k
J
for
al
l
k
with
s
om
e
po
sit
ive
c
onsta
nt
.
Taki
ng
into acc
ount
k
k
k
k
k
k
k
J
d
B
B
, t
hen
2
2
2
)
/
1
(
k
k
k
k
B
J
, w
e
get
from
(
36
)
that
:
k
k
k
k
k
T
k
k
k
k
k
k
k
k
T
k
k
k
k
T
k
k
k
T
k
B
B
J
B
d
J
d
J
2
1
2
2
~
2
2
1
1
)
(
)
(
(39
)
wh
e
re
the
in
eq
ualit
y
fo
ll
ow
s
from
~
k
k
ineq
ualit
ie
s
k
k
k
B
1
and
2
2
2
k
k
T
k
B
,
w
hic
h
con
cl
ud
e
s the
pro
of
.
4.
NUMER
IC
A
L
RES
ULTS
AND
DISC
USSION
Now,
we
detai
ls
the
nu
m
erical
resu
lt
s
f
or
Algorithm
s
CB
FG
S
an
d
BF
GS
.
T
he
prob
l
e
m
s
that
w
e
te
ste
d
are
f
rom
[
12
,
13]
.
H
i
m
m
eblau
us
e
d
the
nex
t
st
op
la
w
see
in
[
14
]
“I
f
,
10
)
(
5
k
x
le
t
;
)
(
/
)
(
)
(
1
1
k
k
k
x
x
x
s
t
o
p
Otherwise,
le
t
)
(
)
(
1
1
k
k
x
x
s
t
o
p
.
For
each
pro
blem
,
if
k
J
or
5
10
1
s
t
o
p
was
sat
isfie
d,
t
he
pr
ogram
will
b
e sto
pped
. A
ll
codes
w
e
r
e w
r
it
te
n
i
n
M
ATL
AB 4.
4
and
W
i
ndows
XP
op
e
rati
on
s
yst
e
m
.
The
pa
r
a
m
et
ers
are
ch
os
e
n
as:
5
10
,
9
.
0
,
1
.
0
and
t
he
init
ia
l
m
at
rix
I
B
0
is
the
un
it
m
at
rix”.
Table
1
e
xp
la
i
ns
t
he
re
su
lt
s,
wh
e
re
t
he
c
olu
m
ns
co
ntain
the
f
ollo
wing
i
m
plyi
ng
:
Pr
oble
m
: the na
m
e o
f
t
he
te
st
pro
blem
in
M
ATL
AB;
Dim
: t
he
dim
e
ns
io
n of t
he p
r
ob
le
m
;
NI
:
the
num
ber
of
it
erati
ons;
NF
: t
he nu
m
ber
of
f
un
ct
io
n
e
valuati
ons.
Their
num
eric
al
exp
e
rience
sign
ify
t
hat
nu
m
ero
us
up
dates
from
this
idea
la
bored
well
in
ap
plied
,
s
pecial
ly
the
m
od
i
fied
update
1
k
k
J
but
the
m
od
ifi
ed
update
k
k
p
give
a
sli
ght
im
pr
ov
em
ent
ove
r
the
ori
gi
nal
BFGS
m
et
ho
d.
Com
par
ison
of
se
ver
al
am
ou
nts
betw
een
diff
e
re
nt
quasi
-
Ne
wton
m
et
ho
ds
a
s
sh
ow
n
in
Ta
ble 1
.
Gen
e
rall
y,
we
can
c
om
pu
te
the
per
ce
ntage
perform
ance
of
the
ne
w
pro
pose
d
al
gorithm
s
com
par
ed
against
th
e
sta
nd
a
r
d
BF
GS
a
lgorit
hm
fo
r
th
e
ge
ner
al
to
ols
NI
an
d
NF
as
fo
ll
ows
.
Re
la
ti
ve
ef
fici
en
cy
of
t
he
new a
lg
or
it
hm
s
as s
how
n
in
Table
2.
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2502
-
4752
Ind
on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
13
, N
o.
3
,
Ma
rc
h 201
9
:
9
4
5
–
9
5
3
952
Table
1.
C
om
par
iso
n
o
f
Se
veral
A
m
ou
nts Be
tween
Dif
fer
e
nt
Q
ua
si
-
Ne
wton
Me
th
ods
BFGS
a
lg
o
rith
m
CB
FGS
with
k
k
p
CB
FGS
with
1
k
k
J
P
.
No
.
n
N
I
NF
NI
NF
NI
N
F
‘
48
’
‘
8
’
‘
146
’
‘
37
’
‘
140
’
‘
35
’
‘
2
’
Ro
se
‘
16
’
‘
5
’
‘
23
’
‘
8
’
‘
26
’
‘
9
’
‘
2
’
Froth
‘
11
’
‘
3
’
‘
142
’
‘
39
’
‘
166
’
‘
43
’
‘
2
’
Bad
scp
‘
30
’
‘
3
’
‘
30
’
‘
3
’’
‘
30
’
‘
3
’
‘
2
’
Bad
sc
‘
21
’
‘
6
’
‘
43
’
‘
13
’
‘
50
’
‘
15
’
‘
2
’
Beale
‘
27
’
‘
2
’
‘
27
’’
‘
2
’
‘
27
’
‘
2
’
‘
2
’
Jen
sa
m
‘
20
’
7
‘
90
’
‘
28
’
‘
113
’
‘
34
’
‘
3
’
Helix
‘
19
’
‘
6
’
‘
55
’
‘
17
’
‘
54
’
‘
16
’
‘
3
’
Bard
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
3
’
Gau
ss
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
3
’
Gu
lf
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
3
’
Bo
x
‘
17
’
‘
5
’
‘
35
’
‘
11
’
‘
60
’
‘
20
’
‘
4
’
Sin
g
‘
13
’
‘
4
’
‘
60
’
‘
19
’
‘
61
’
‘
19
’
‘
4
’
W
o
o
d
‘
13
’
‘
5
’
‘
117
’
‘
21
’
‘
65
’
‘
21
’
‘
4
’
Ko
wo
sb
‘
17
’
‘
5
’
‘
46
’
‘
15
’
‘
54
’
‘
17
’
‘
4
’
Bd
‘
27
’’
‘
2
’
‘
27
’
‘
2
’
‘
2
7
’
‘
2
’
‘
5
’
Osb
1
‘
12
’
‘
4
’
‘
48
’
‘
8
’
‘
72
’
‘
25
’
‘
6
’
Big
g
s
‘
31
’
‘
3
’
‘
13
’
‘
3
’
‘
31
’
‘
3
’
‘
11
’
Osb
2
‘
13
’
‘
4
’
‘
97
’
‘
31
’
‘
102
’
‘
31
’
‘
20
’
W
atso
n
‘
17
’
‘
5
’
109
‘
35
’
‘
209
’
‘
64
’
‘
400
’
Sin
g
x
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
27
’
‘
2
’
‘
400
’
Pen
1
‘
5
’
‘
2
’
‘
5
’
‘
2
’
‘
5
’
‘
2
’
‘
200
’
Pen
2
‘
27
’
‘
2
’
27
‘
2
’
‘
27
’
‘
2
’
‘
100
’
Vardi
m
‘
28
’
‘
8
’
‘
32
’
‘
9
’
‘
33
’
‘
9
’
‘
500
’
Tr
ig
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
500
’
Bv
‘
16
’
‘
6
’
‘
19
’
7
‘
61
’
‘
6
’
‘
500
’
Ie
‘
16
’
‘
5
’
‘
82
’
‘
15
’
‘
281
’
‘
57
’
‘
500
’
Ban
d
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
4
’
‘
2
’
‘
500
’
Lin
‘
7
’
‘
3
’
‘
7
’
‘
3
’
‘
7
’
‘
3
’
‘
500
’
Lin1
‘
7
’
‘
3
’
‘
7
’
‘
3
’
‘
7
’
‘
3
’
‘
500
’
Lino
551
118
1380
345
1756
453
Total
Table
2.
Re
la
ti
ve
E
ff
ic
ie
ncy
of t
he
N
e
w
Algorithm
s
BF
GS algo
ri
th
m
CB
FGS w
ith
k
k
p
CB
FGS
with
1
k
k
J
2
6
.04
%
7
6
.15
%
100%
NI
4
3
.37
%
7
8
.58
%
100%
NF
5.
CONCL
US
I
O
N
S
In
t
his
pa
pe
r,
supporte
d
th
e
ne
w
Q
N
-
e
quat
ion,
we'
ve
go
t
pro
j
ect
ed
so
m
e
new
quasi
-
New
t
on
strat
egies.
T
he
co
nv
e
rg
e
nce
r
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