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.
954
-
961
IS
S
N: 25
02
-
4752, DO
I: 10
.11
591/ijeecs
.v1
3
.i
3
.pp
954
-
961
954
Journ
al h
om
e
page
:
http:
//
ia
es
core.c
om/j
ourn
als/i
ndex.
ph
p/ij
eecs
A
ne
w fo
rmula fo
r conju
gat
e para
meter c
om
pu
tatio
n bas
ed
on
the quad
ratic m
od
el
Basim A
. Has
sa
n
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,
I
raq
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Sep
13, 201
8
Re
vised
Dec
4
,
2018
Accepte
d
Dec
15
, 201
8
The
conj
ug
ancy
coe
fficie
nt
is
th
e
ver
y
b
asis
of
a
dive
rsit
y
of
th
e
conj
uga
te
gra
die
n
t
m
et
hods.
In
thi
s
rese
arch,
we
der
ivation
a
new
form
ula
of
conj
ugate
gra
die
n
t
m
et
hod
s
base
d
on
the
quadr
atic
m
odel.
Our
ari
thmet
ical
findi
ngs
have
r
eveal
ed
t
hat
,
our
n
ew
m
et
hod
h
as
the
m
ost
exc
ellent
per
form
anc
e
cont
rast
to
th
e
ot
her
standa
rd
CG
m
et
hods.
Also
give
proof
vi
ewi
ng
tha
t
thi
s
m
et
hod
converg
es
globall
y
.
Ke
yw
or
ds:
Conj
ug
at
e
grad
ie
nt m
et
ho
d
Global
con
vergen
ce
Su
f
fici
ent
desc
ent pr
op
e
rty
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
of
Ma
them
a
ti
cs,
Coll
ege
of
C
om
p
uters
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
1.
INTROD
U
CTION
Me
thods
of
c
onjug
at
e
gra
dient
are
par
ti
c
ularly
i
m
po
rtant
cl
ass
becau
se
of
their
conve
rg
e
nc
e
featur
e
s,
a
ve
ry
si
m
ple
appl
ic
at
ion
endea
vor
in
c
om
pu
te
r
perf
or
m
ances
and
ve
ry
good
in
s
olv
i
ng
big
pro
blem
s
[
1
]
.
We
are
co
nce
rn
e
d
with
c
onj
ugat
e
gr
a
dien
t
m
et
ho
ds
for
fin
ding
a
loc
al
m
ini
m
u
m
o
f
th
e
functi
on:
n
R
x
)
x
(
f
m
i
n
(1)
wh
е
re
1
R
R
:
f
n
is a inc
essantl
y dif
fеr
e
ntiable
fu
nctio
n.
Conj
ug
at
e
grad
ie
nt m
et
ho
ds
f
or so
l
ving
(1)
a
re itе
rati
ve
m
еth
ods
of thе
form
:
k
k
k
k
n
h
x
x
,
R
x
1
0
(2)
wh
е
re
0
k
is a stе
p
siz
e
a
nd
k
h
is t
he
sеa
rch di
rеc
ti
on
gen
e
rated
by:
k
k
k
k
h
q
h
,
q
h
1
1
0
0
(3)
wh
e
re
1
k
q
denotes
g
ra
dient
of
)
x
(
f
k
1
at
the point
1
k
x
an
d
k
is a scal
ar r
e
presenti
ng d
i
ff
e
r
ent m
et
ho
ds.
The best
-
know
n param
et
er f
or
k
is :
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
f
ormul
a f
or
c
on
j
ugate
paramet
er c
omp
uta
ti
on
based
on the
quadr
at
ic
m
odel
(
Ba
sim A
. Hassa
n
)
955
k
T
k
k
T
k
FR
k
q
q
q
q
1
1
(4)
introd
uced
by
Flet
cher
an
d
Re
eve,
FR
[
2
]
.
For
ot
her
re
view
s
of
CG
-
cl
assic
m
et
ho
ds
se
e
fo
r
instance
[
3
-
7
].
“
As
we
ca
n
se
e
in
the
pa
ram
et
er
”
for
k
the
di
ff
e
ren
ce
“
1
k
k
f
f
”
is
not
us
ed
at
”
al
l. To
o
btai
n
bette
r
co
njuga
te
gr
a
dient
m
e
thods,
”
m
any
m
od
ifie
d
m
e
tho
ds
us
i
ng
val
ue
of
ob
j
ect
iv
e
functi
on
ha
ve
been
pr
ese
nted
s
ome
of it
:
Hidea
ki and
Y
asus
hi [
8
]
m
ade
“
a m
od
ific
at
ion o
n
t
he
C
G p
aram
et
er as
fo
l
lows
”
:
“
)
f
f
)(
/
(
q
q
k
k
k
k
T
k
HY
k
1
1
1
2
”
(5)
Lat
el
y, Ba
si
m
and H
a
nee
n [
9
],
“
us
in
g qu
a
dr
at
ic
f
unct
ion,
m
od
ifie
d
CG
pa
ram
et
er as
fo
l
lows
”
:
“
))
f
f
(
(
/
)
h
q
(
q
q
k
k
k
T
k
k
k
T
k
B
H
Q
k
1
2
1
1
2
”
(6)
Ther
e
a
lot
of
are
ot
her
m
od
ifie
d
C
G
-
m
et
hods
that
we
did
not
co
ver
up
i
n
this
pa
per.
C
omm
on
ly
,
ste
plen
gth
k
in
(
2)
“
is
sel
ect
ed
t
o
sat
isfy t
he W
olfe
li
ne
searc
h
”
sta
te
s:
k
T
k
k
k
k
k
k
h
q
)
h
x
(
f
)
x
(
f
(7)
k
T
k
k
T
k
k
k
h
q
h
)
h
x
(
q
(8)
wh
e
re
1
0
. In
a
ddit
ion
, t
he
s
uffici
ent d
e
scent c
onditi
on :
2
1
1
1
k
k
T
k
q
c
h
q
(9)
All
from
the
Wo
l
fe
sta
te
s
and
desce
nt
co
ndit
ion
a
re
go
od
pro
per
ty
to
prov
e
c
оnve
r
gence.
More
detai
ls
can
be
fou
nd in [
10
,
1
1
].
A
key
facto
r
of
co
njugate
gr
a
dient
m
et
ho
ds
is
ho
w
to
sel
ec
t
the
con
ju
ga
nc
y
coef
fici
ent
k
.
Be
low
base
d
on
the
quad
rati
c
m
od
el
will
intr
oduce
d
the
ne
w
co
njugate
gr
a
dient
m
et
ho
ds.
T
he
resu
l
ti
ng
m
odifie
d
CG
m
et
ho
d
re
ta
ins
gl
ob
al
c
onve
r
gen
ce
,
a
nd
pe
r
form
s
slig
htly
bette
r
th
an
the
FR
-
CG
m
e
tho
d
on
som
e
te
st
pro
blem
s.
2.
NEW F
ORM
ULA FO
R C
ONJU
GATE
PARA
METE
R CO
MP
UTA
TION
A
ND AL
GOR
IT
HM
In co
njugate
gradient
(CG
)
m
et
hods
t
he
f
or
m
ula f
or
the
ne
w
ste
p becom
es:
k
k
k
k
h
q
h
1
1
(10)
wh
e
re
k
is f
ound
by im
po
sing
the con
diti
on that
k
T
k
v
h
and is
giv
e
n
as:
k
T
k
k
T
k
k
v
h
v
q
1
(11)
wh
e
re
n
n
R
is
a
no
n
ne
gative
def
i
nite
an
d
i
f
k
is
t
he
e
xact
(
0
1
k
T
k
d
q
)
on
e
dim
ension
al
m
ini
m
iz
er
giv
e
n by:
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
2019
:
954
–
961
956
h
h
h
q
k
T
k
k
T
k
k
(12)
More det
ai
ls a
bout the
con
ju
gate gra
dient
m
et
ho
d
ca
n be
fou
nd in [
8,
12
].
Now,
we
de
riv
e
the
ne
w
for
m
ulas
fo
r
c
onjug
at
e
par
am
et
er
com
pu
ta
ti
on
.
W
e
sh
al
l
thi
nk
a
diff
e
re
nt
look
of the
de
nom
inator
k
T
k
v
h
. Based on q
ua
dr
at
i
c m
od
el
an
d
usi
ng
,
h
q
k
T
k
0
1
we
g
et
:
k
T
k
k
k
k
T
k
k
T
k
k
k
h
h
)
x
(
f
)
v
(
)
v
(
)
v
(
q
)
x
(
f
)
x
(
f
2
1
1
1
2
1
2
1
(13)
wh
ic
h
im
plies t
hat:
2
2
1
1
2
/
h
q
/
))
x
(
f
)
x
(
f
(
v
h
/
h
q
)
x
(
f
)
x
(
f
h
h
k
T
k
k
k
k
k
T
k
k
T
k
k
k
k
k
T
k
k
(14)
Fr
om
this w
e
define
form
ula:
2
1
1
/
h
q
/
))
x
(
f
)
x
(
f
(
y
q
k
T
k
k
k
k
k
T
k
k
(15)
Qu
a
drat
ic
f
un
c
ti
on
is
optim
al
so
l
ution
to
t
he
open
pr
ob
le
m
kn
own
a
nd
is
ta
ken
f
ro
m
Yu
a
n
[
13
]
.
More det
ai
ls c
an be
fou
nd in [
1
]
.
“
For
quad
rati
c
functi
on
s
a
nd
und
er
e
xact
li
ne
search
es,
al
l
the
gr
adien
ts
of
f
at
the
diff
e
re
nt
it
erates are m
utu
al
ly
o
rth
ogon
al
”
. Th
at
is
0
1
k
T
k
q
q
. For
m
ula (
15
)
f
urt
her re
duces t
o:
2
1
1
1
/
h
q
/
))
x
(
f
)
x
(
f
(
q
q
k
T
k
k
k
k
k
T
k
B
k
(16)
In ord
e
r
t
o we
adjust
or exte
nsi
on the
over
f
or
m
ula as foll
ow:
)
/
h
q
/
))
x
(
f
)
x
(
f
(
,
y
h
m
a
x(
q
q
k
T
k
k
k
k
k
T
k
k
T
k
MB
k
2
1
1
1
(17)
W
it
h
this
n
e
w,
we prese
ntin
g
al
gorithm
as f
ol
lows
.
New Al
go
ri
thm
:
1.Givе
n
init
ia
l
n
R
x
1
an
d
est
im
at
e t
he
1
q
an
d
1
1
q
d
.
2.
If
6
1
10
k
q
, th
e
n
sto
p.
3.
C
ub
ic
searc
h
to
est
i
m
at
e
k
a
nd
w
hic
h
sat
isfyi
ng
the
Wo
lf
e
conditi
on
s
(
7)
-
(
8)
a
nd
up
da
te
the
var
ia
bles
k
k
k
k
h
x
x
1
.
4.
Esti
m
at
e
k
w
hi
ch
de
fine
d
in
(
16)
a
nd
(
17)
.
5.
Sеt
1
k
k
an
d re
pe
at
stеp
2
to
ste
p 5.
3.
GLOB
AL
CONVE
RGE
NCE
A
N
AL
YS
IS
BY
SE
VER
A
L L
INE SE
A
RCHES F
OR
B
k
METHO
D
The
as
pire
of t
his sесti
on is t
о
st
ud
y t
hе worl
dwide c
onve
r
gen
ce
acti
viti
es of
new Al
gоr
it
h
m
.
Dеscen
t
c
оn
di
tion
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
f
ormul
a f
or
c
on
j
ugate
paramet
er c
omp
uta
ti
on
based
on the
quadr
at
ic
m
odel
(
Ba
sim A
. Hassa
n
)
957
Fо
r
the
suff
ic
iе
nt
sta
te
tо h
оld, the
n:
0
2
1
1
1
c
,
q
c
h
q
k
k
T
k
(18)
If
k
is
vied by th
e
Wo
lfe stat
es
(7)
a
nd
(
8),
a
fterw
a
rd the
sear
ch direct
io
n
B
k
sa
ti
sfies
(16
)
.
Theorem
1.
Con
si
der the
ne
w
B
k
m
e
tho
d.
If
k
is vie
d by the
Wo
l
fe s
ta
te
s
(7)
a
nd
(
8),
afte
r
ward:
2
1
1
1
k
k
T
k
q
h
q
(19)
More
ov
e
r
0
B
k
.
Pro
of:
The
pr
oof
is
by
ind
uctio
n.
F
or
0
k
then
2
0
2
0
0
0
q
q
d
q
T
.
Sup
po
s
e
that
(
16)
is
sat
isfie
s
for
k
. No
w we
pro
ve
t
hat
(
18)
ho
l
ds
for
1
k
.
By
m
ul
ti
plyi
ng
T
k
q
1
on
bo
t
h
si
des of
(3),
we o
btain:
k
T
k
k
T
k
k
T
k
k
k
k
k
k
k
T
k
k
T
k
k
k
k
k
k
k
T
k
B
k
k
T
k
k
T
k
h
q
h
q
]
/
h
q
/
)
f
f
[(
q
q
h
q
)
/
h
q
(
/
)
f
f
(
q
q
)
h
q
q
(
q
h
q
1
1
2
1
2
1
1
1
2
1
2
1
1
1
1
1
1
2
1
2
(20)
Fr
om
Wo
lfe
s
ta
te
s
we
get,
k
T
k
k
k
k
h
q
)
f
f
(
1
an
d
k
T
k
k
T
k
h
q
d
g
1
.
P
ut
thi
s
val
ue
in
the
above e
quat
io
n t
o get:
2
1
2
1
2
1
1
2
1
2
1
1
1
2
1
2
1
k
k
T
k
k
T
k
k
k
k
T
k
k
T
k
k
T
k
k
k
T
k
k
k
k
k
T
k
q
h
q
h
q
]
/
[
q
q
h
q
h
q
]
/
h
q
/
)
h
q
[(
q
q
h
q
(21)
wh
e
re
2
1
1
/
.
The
refоr
e
,
(
18)
is
f
ul
fill
ed
fоr
k
.
A
ddit
ion
al
,
frоm
(21)
a
naly
sis,
we
to
o
obta
in
0
B
k
. B
y m
at
he
m
a
tical
inducti
оn
m
et
ho
d, we
оb
ta
in the desi
re
d res
ult.
Glob
al conver
gence
of the
B
k
meth
od
To
st
ud
y t
he w
or
l
dw
i
de
c
onve
rg
e
nce
of
B
k
-
m
et
hod,
t
he
a
nu
m
ber
o
f basic
a
ssu
m
ption
.
B1.
“
T
he
le
vel
sеt
)
x
(
f
)
x
(
f
R
x
n
1
is
”
boun
de
d.
B2.
T
hе
re е
xists a c
оn
sta
nt
0
L
suc
h
that
f
or
a
ny
:
U
y
,
x
,
y
x
L
)
y
(
q
)
x
(
q
W
it
hin
this
su
bd
ivisi
on
, w
e m
ake th
e con
ver
gen
ce of
the
B
k
m
et
ho
d.
I
niti
al
ly
, th
e research
er
m
anifests
that
B
k
has
the
sa
m
e
featur
es
to
the
DY
k
m
et
ho
d
and
Z
ou
te
nd
i
jk
sta
te
,
that
is
ver
y
m
uch
e
m
plo
ye
d
t
o d
e
m
on
strat
e
world
wide
c
onve
r
gen
ce
.
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
2019
:
954
–
961
958
The
B
k
m
et
ho
d
ha
s
the
sam
e
featur
es
of
the
DY
k
m
et
ho
d,
t
hat
is
con
si
der
e
d
of
i
m
po
rtance
t
o
the
world
wide
con
verge
nce
of
t
he
foll
ow
i
ng in
ve
sti
gations
.
Theorem
2
Con
si
der
k
x
gen
e
rated
by
ne
w
m
et
ho
d.
T
he
n
for
ever
y
,
k
the
relat
ion
s
k
T
k
k
T
k
B
k
h
q
h
q
1
1
0
al
ways
ho
l
d.
Pro
of:
Fr
om
Th
e
or
em
1
,
w
e
kn
ow
B
k
0
. Mu
lt
iply
ing
(3)
by
T
k
q
1
with
new
form
ula w
e obt
ai
n:
,
)
h
q
)]
/
h
q
(
/
)
f
f
[(
(
h
q
q
q
h
q
k
k
T
k
k
T
k
k
k
k
k
T
k
k
k
T
k
k
T
k
1
1
1
1
1
1
1
2
(22)
More
ov
e
r, by
The
or
em
1
, we
h
a
ve
that:
0
2
1
1
1
1
1
k
T
k
k
T
k
k
T
k
k
T
k
k
T
k
k
T
k
k
T
k
k
T
k
k
k
k
h
q
h
q
h
q
h
q
h
q
y
h
h
q
)]
/
h
q
(
/
)
f
f
[(
(23)
This als
o
s
how
s that
k
T
k
k
T
k
B
k
h
q
h
q
1
1
. Th
e
refo
re th
e
proo
f
is
com
plete
.
The
le
m
m
a b
elo
w
is cal
le
d
Zo
uten
dijk stat
e
[
14
]
.
Le
mma 1
Supposit
io
n
(B
1)
-
(B
2)
ho
l
ds
.
Let
the
m
et
ho
ds
in
the
f
or
m
of
(
2)
-
(
3),
w
hе
re
k
d
is
sat
isfy
(18)
a
nd
k
sat
isfie
s thе
(7)
-
(8)
sta
te
s. Th
en we:
1
2
1
2
1
1
k
k
k
T
k
h
)
h
q
(
(24)
We asce
rtai
n
t
he worl
dwide
conve
rg
e
nce
of the
B
k
m
e
thod.
Theorem
3 .
Supposit
io
n
(
B1)
-
(B
2)
holds.
C
onside
r
k
x
be
ge
nerat
ed
by
Ne
w
Al
gorithm
.
The
n
0
i
nf
lim
1
k
k
q
.
Pro
of
.
We
ca
rr
y
on
by
disa
gr
eem
ent.
P
res
um
e
t
hat
2
1
k
q
for
0
.
By
(3),
it
f
оllo
ws
so
a
s
to
k
B
k
k
k
h
q
h
1
1
. T
his
jointl
y
by
theorem
2
,
im
pl
y:
2
1
1
1
2
2
1
1
2
1
1
1
2
2
2
1
2
2
k
k
T
k
k
k
T
k
k
T
k
k
k
T
k
k
B
k
k
q
q
h
d
h
q
h
q
q
q
h
h
)
(
h
(25)
Divid
i
ng bоth
sides
of
(
25)
by
2
1
1
)
q
h
(
k
T
k
,
we
obtai
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
f
ormul
a f
or
c
on
j
ugate
paramet
er c
omp
uta
ti
on
based
on the
quadr
at
ic
m
odel
(
Ba
sim A
. Hassa
n
)
959
2
1
2
2
2
1
2
1
1
1
1
2
2
2
1
1
2
1
1
1
2
2
2
1
1
2
1
1
1
1
2
k
k
T
k
k
k
k
k
T
k
k
k
T
k
k
k
T
k
k
k
T
k
k
T
k
k
k
T
k
k
q
)
q
h
(
h
q
q
q
h
q
)
q
h
(
h
)
q
h
(
q
)
q
h
(
)
q
h
(
h
)
q
h
(
h
(26)
No
ti
ng t
hat
,
q
)
q
h
(
h
T
2
1
2
1
1
2
1
1
by r
ecu
rr
e
nce
for
m
ula ab
ov
e
(
26)
,
w
e
hav
e:
k
q
.
.
.
.
.
.
.
q
q
)
q
h
(
h
q
)
q
h
(
h
)
q
h
(
h
k
i
i
k
k
k
T
k
k
k
k
T
k
k
k
T
k
k
1
1
2
1
2
1
2
2
1
1
2
1
2
1
2
2
2
1
1
2
1
1
1
1
1
(27)
Th
us
,
k
h
)
q
h
(
k
k
T
k
2
1
2
1
1
an
d
this
i
m
plies t
hat:
1
2
1
2
1
1
k
k
k
T
k
h
)
h
q
(
(28)
it
co
ntra
dicts L
e
m
m
a 1
. Th
e
re
fore,
t
he desire
d resu
lt
ho
l
ds
.
4.
AR
IT
HMETI
CA
L
FIN
DI
N
GS
AND DIS
CUSSIO
N
In
t
his
pa
rt,
a
rithm
etical
find
in
gs
a
re
re
po
rted.
We
te
st
and
com
par
e
t
he
new
m
et
hods
with
FR
m
et
ho
d
w
hose
resu
lt
s
be give
n by [
2
]
.
Using
F
ort
ra
n
90
to
co
de
t
hi
s
m
e
tho
ds.
In
our
a
ppli
cat
ion
,
we
sel
ect
t
he
fo
ll
owin
g
par
am
et
ers
:
001
0
.
and
9
0
.
.
The
e
xa
m
inati
on
pro
bl
e
m
s
are
sel
ected
f
ro
m
ref
.
[
15,
16
]
.
O
ptim
iz
at
ion
pro
ble
m
s
exist i
n
m
any areas
[17]. T
he st
opping stat
e i
s :
6
1
10
k
q
(29)
The
arit
hm
et
ical
find
in
gs
are
cat
al
og
ed
in
t
able
1,wh
e
re
the
colum
n
“
Prob
le
m
”
sta
nd
s
fo
r
the
la
bel
of
th
e
exam
ined
pro
bl
e
m
.
“
Dim
”
refe
rs
to
the d
im
e
ns
io
n
of
t
he
te
s
t
p
r
ob
le
m
s.
Th
e
res
ults
are d
e
no
te
d
by
N
I
a
nd N
F
ref
e
r
to
the ta
ble of it
erati
ons
and f
unct
ion es
tim
a
ti
on
s s
ucc
essivel
y
”
.
In
su
m
m
ary,
the
a
rithm
et
ic
a
l
fin
dings
s
ho
w
that
Ne
w
m
et
ho
ds
are
m
or
e
eff
ic
ie
nt
than
the
FR
m
et
ho
d
a
nd
prov
i
des
a
n
e
ff
ic
ie
nt m
et
ho
d
f
or s
olv
in
g u
nconstraine
d o
ptim
iz
at
ion
p
r
oblem
s.
Fail
: The algo
rithm
f
ai
l t
o
con
ver
ge.
Pr
ob
le
m
s n
um
ber
s ind
ic
ant f
or
1.
is t
he
Extend
ed
“
Rosenb
ro
ck,
2.
is t
he
Extend
ed
”
W
hite & Ho
lst
,
3.
is t
he
Extend
ed
“
Be
al
e,
4.
is t
he
Extend
ed
”
Tridiago
nal 1
,
5.
is t
he
Extend
ed
“
Thr
ee Exp
o
Term
s,
6.
is t
he
Gen
erali
zed
”
Tridiago
nal 2
,
7.
is t
he
Extend
ed
“
Po
well
,
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on
esi
a
n
J
E
le
c Eng &
Co
m
p
Sci,
Vo
l.
13
, N
o.
3
,
Ma
rc
h
2019
:
954
–
961
960
8.
is t
he
Qu
adr
at
ic
D
ia
go
nal Per
turb
ed,
9.
is t
he
Extend
ed
”
W
oo
d,
10.
is t
he
“
Qu
adr
at
ic
Q
F2
,
11.
is t
he
NO
ND
IA
(
CUTE),
12.
is t
he
DI
XMAA
NE
(CUTE)
,
13.
is t
he
Partia
l
”
Pertur
bed
Q
uad
rati
c,
14.
is t
he
“
Extend
ed
Bl
ock
-
Diago
nal”B
D2
,
15.
is
the
LIA
RW
HD
(CUTE)
”.
Com
m
on
ly
,
app
raisal
of
the
aver
ages
of
sever
al
qu
antit
ie
s
between
d
iffer
ent co
nj
ug
at
e g
rad
ie
nt m
et
ho
ds
as f
ollow
s
Table 1
and
2
.
Table
1.
Num
erical
Result
s
of n
e
w Alg
or
it
hm
s
and FR
-
C
G
al
go
rithm
FR
alg
o
rithm
B alg
o
rithm
MB
alg
o
rithm
P.
No
.
n
NI
NF
NI
NF
NI
NF
75
37
79
39
93
47
100
‘
1
’
81
38
78
37
131
78
1000
84
37
84
37
88
43
100
‘
2
’
62
28
79
35
92
46
1000
26
13
32
17
52
32
100
‘
3
’
24
12
24
13
42
22
1000
26
10
23
11
64
32
100
‘
4
’
31
16
26
13
129
77
1000
25
17
13
8
25
15
100
‘
5
’
474
25
342
28
Fail
Fail
1000
63
41
62
40
67
37
100
‘
6
’
101
64
98
60
115
73
1000
115
81
115
61
313
180
100
‘
7
’
159
83
149
78
Fail
Fail
1000
96
55
81
47
231
124
100
‘
8
’
281
160
313
181
711
445
1000
59
31
50
26
110
71
100
‘
9
’
51
26
52
26
84
47
1000
174
111
167
111
196
130
100
‘
10
’
753
471
Fail
Fail
593
364
1000
26
13
26
13
25
13
100
‘
11
’
25
12
29
14
29
15
1000
133
84
120
80
218
121
100
‘
12
’
388
249
347
219
634
345
1000
125
81
134
89
123
74
100
‘
13
’
410
244
454
273
616
370
1000
23
12
23
12
156
122
100
‘
14
’
23
12
23
14
166
130
1000
33
17
34
19
45
23
100
‘
15
’
53
24
45
20
55
27
1000
2613
1541
2611
1515
4610
2739
Total
Fail
:
T
h
e
algo
rithm
f
ail to co
n
v
erge.
Table
2.
A
ver
a
ges
E
ff
ic
ie
nc
y
of the
Ne
w
Algorithm
s
MB
alg
o
rith
m
B alg
o
rith
m
FR alg
o
rith
m
5
6
.26
%
5
5
.32
%
100
%
NI
5
6
.68
%
5
6
.64
%
100
%
NF
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
f
ormul
a f
or
c
on
j
ugate
paramet
er c
omp
uta
ti
on
based
on the
quadr
at
ic
m
odel
(
Ba
sim A
. Hassa
n
)
961
5.
CONCL
US
I
O
N
S
In
t
his
re
searc
h,
we
ha
ve
de
rive
d
a
nеw
C
G
-
m
et
ho
ds
ba
sed
on
the
qua
dr
at
ic
m
od
el
.
Ar
it
hm
et
ic
a
l
fin
dings
hav
e
been acco
unte
d,
w
hich
e
xpla
ined
the
us
e
fu
l
ne
ss of
our
m
et
ho
d.
REFERE
NCE
S
[1]
Andrei
N.
Open
proble
m
s
in
nonli
ne
ar
conj
ug
ate
gra
di
ent
a
lgorithm
s
for
uncons
tra
in
ed
opti
m
iza
ti
on.
Bul
le
t
in
of
the
Bu
ll
.
Mala
ysian
Math
emati
ca
l
Sc
ie
nc
es
Soc
iet
y
,
2011
;
34
:
319
-
330.
[2]
Flet
ch
er
R
,
Re
ev
es
C.
Functi
on
m
ini
m
iz
at
ion
b
y
conj
ug
at
e
gra
d
i
ent
s.
Comput
er
J
,
1964;
7,
149
-
1
54.
[3]
Dai
H,
Yuan
Y.
A
nonli
ne
ar
c
onjuga
t
e
gra
di
e
nt
m
et
hod
with
a
strong
global
conve
rge
n
ce
pr
oper
t
y
.
SI
AM
J.
O
pti
mization
,
19
99:
177
-
182.
[4]
Flet
ch
er
R
.
Pra
ctical
Method
of O
pti
m
iz
at
io
n.
2n
d
Edition, John
W
il
e
y
and
Sons
.
New York. 198
9.
[5]
Heste
nes
R,
Sti
e
fel
L
.
Method
of
conj
uga
te
gr
adients
for
Solving linear
s
y
st
ems
.
J
ournal
Nati
onal Standards
,
1952;
49:
409
-
436.
[6]
Polak
E,
Rib
iere
G.
Note
fo
r
Converge
nc
e
Di
rec
ti
on
Co
njuga
t
e.
Revu
e
Franc
ai
se
Inf
orm
ant
,
Reser
c
he.
Opertione
lle
,
19
69:
35
-
43.
[7]
Li
u
Y,
Store
y
C
.
Eff
ic
i
ent
g
ene
r
a
li
z
ed
conj
ug
at
e
gra
die
n
ts
al
gori
t
hm
s.
Part
1:
Theor
y
.
J.
Optimization
The
or
y
and
Applic
a
ti
ons,
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