Int
ern
at
i
onal
Journ
al of
P
ower E
le
ctr
on
i
cs a
n
d
Drive
S
ystem
(I
J
PE
D
S
)
Vo
l.
11
,
No.
3
,
Septem
be
r 2020
, pp.
1153
~
116
4
IS
S
N:
20
88
-
8694
,
DOI: 10
.11
591/
ij
peds
.
v11.i
3
.
pp1153
-
116
4
1153
Journ
al h
om
e
page
:
http:
//
ij
pe
ds
.i
aescore.c
om
Review
of
f
ast s
quar
e roo
t calcul
ation me
t
hods f
or
fi
xed poi
nt
microco
ntroll
er
-
based c
ont
ro
l
sys
tems of powe
r el
ec
t
ro
nics
An
t
on
Di
anov
1
,
Aleksey
Anuchi
n
2
1
Digital
Appl
iance
s Business,
Samsung Ele
c
tron
ic
s,
R
epubl
i
c
of
Korea
2
El
e
ct
ri
ca
l
Driv
es
Depa
rt
me
n
t,
Mos
cow
Pow
er
Engi
ne
eri
ng
I
nstit
ute,
Russ
ia
Art
ic
le
In
f
o
ABSTR
A
CT
Art
ic
le
history:
Re
cei
ved
Sep
23
, 201
9
Re
vised
Jan
9
, 20
20
Accepte
d
Fe
b
16
,
2020
Square
root
c
alculation
is
a
wi
del
y
used
t
ask
i
n
real
-
t
im
e
con
t
rol
sys
te
ms
espe
ciall
y
in
th
ose,
which
control
power
elec
tr
onic
s:
mot
ors
dr
ive
s,
power
conve
rt
ers,
pow
er
fa
c
tor
cor
r
ectors,
e
tc.
At
th
e
sam
e
t
im
e
c
a
lc
ul
at
ion
of
square
roots
is
a
bottle
-
n
ec
k
in
the
op
ti
m
izati
o
n
of
cod
e
ex
ecution
t
im
e
.
Ta
king
in
to
acc
ount
tha
t
for
m
a
ny
app
li
c
at
ions
appr
oximate
ca
l
cul
a
ti
on
of
a
square
roo
t
is
e
nough,
ca
l
culation
time
c
an
be
dec
re
a
sed
with
t
he
pr
ic
e
of
pre
ci
sion
of
calc
ula
ti
on
.
Th
is
pap
er
analyses
exi
st
ing
methods
for
fast
square
root
ca
l
culati
on
,
which
ca
n
be
i
mpl
ement
ed
fo
r
fixe
d
point
m
ic
r
ocont
rollers.
It
discusses
a
lgo
rit
hms’
pros
and
cons,
analyses
ca
l
cul
a
ti
on
err
or
s
and
giv
es
some
rec
o
mmen
dat
ions
on
th
ei
r
use.
Th
e
pap
er
al
so
proposes
an
origi
n
al
me
thod
for
fast
square
root
c
al
cu
la
t
ion,
whic
h
does
not
use
har
dwar
e
ac
c
el
e
ration
and
the
r
efo
re
,
is
s
uit
able
fo
r
i
mpleme
n
ta
t
ion
at
a
var
i
et
y
of
mode
rn
Digi
ta
l
Signal
Proce
ss
ors,
which
h
ave
high
-
spe
e
d
har
dware
mul
ti
p
li
ers
,
bu
t
do
not
hav
e
eff
e
ct
iv
e
d
ivi
de
rs.
T
he
ma
xi
mum
re
l
at
iv
e
err
or
of
the
proposed
me
thod
is
3.
36%
for
calc
ul
ation
without
div
ision,
and
c
an
be
dec
re
ase
d
to
0
.
055%
using
o
ne
divi
sion
op
era
t
ion.
Fina
ll
y
,
the
most
promi
sing
m
ethods
are
co
m
par
ed
and
res
ult
s
of
th
ei
r
per
forma
n
ce
com
par
isons a
re
depi
c
te
d
in ta
bl
e
s.
Ke
yw
or
d
s
:
Appro
ximate
c
ompu
ti
ng
Appro
ximati
on
algorit
hm
s
New
t
on meth
od
Numerical
met
hods
Roo
t
mean
squ
are
This
is an
open
acc
ess arti
cl
e
un
der
the
CC
BY
-
SA
l
ic
ense
.
Corres
pond
in
g
Aut
h
or
:
An
t
on D
ia
nov,
Digital
appli
an
ces b
us
ine
ss, S
amsu
ng Elec
tr
on
ic
s
,
129, Sam
sun
gro,
Y
oungto
ng
-
gu, Su
won, G
oe
nggi
-
do, 1
6677, Re
public
of
Korea.
Emai
l:
ant
on
.
dianov
@gmai
l.com
1.
INTROD
U
CTION
M
ode
rn
c
ontr
ol
sy
ste
m
s
of
powe
r
el
ect
ronics
ha
ve
e
xtend
e
d
functi
on
al
it
y
and
la
r
ge
cod
e
t
o
be
execu
te
d.
At
t
he
sa
me
ti
me
t
he
mo
st
of
ma
ss
pr
oduce
d
de
vices
a
re
s
ubje
ct
for
c
os
t
op
ti
miza
ti
on
,
the
refor
e
dev
el
op
e
rs
fr
e
qu
e
ntly
sel
ect
cheap
micr
oc
ontr
ollers,
w
hic
h
are
not
powe
rful.
T
he
bu
l
k
of
t
he
c
od
e
inc
lud
in
g
math
f
unct
ions
and
m
od
el
of
con
t
ro
l
ob
je
ct
s
are
r
un
eve
r
y
samplin
g
per
i
od
of
the
syst
em,
w
hic
h
is
ty
pical
ly
equ
al
t
o
m
odul
at
ion
per
i
od.
T
he
m
odulati
on
per
i
od
of
the
majority
of
power
el
ect
r
onic
s
co
ntro
l
sy
ste
ms
li
es
in
the
inter
val
of
4
-
20
kHz,
wh
il
e
ope
rati
ng
fr
e
quenc
y
of
t
he
cost
-
ef
f
ect
ive
microc
ontr
oller
bel
ong
to
the
range
of
40
-
80
MHz,
wh
ic
h
giv
e
s
2,0
00
-
20,
000
proce
sso
r
c
ycles
pe
r
mod
ulati
on
per
i
od.
High
powe
r
dev
ic
es
operat
e
at
lowe
r
m
odulati
on
f
re
quencies,
wh
il
e
l
ow
e
r
power
el
ect
ronics
an
d
dev
ic
es
with
l
ow
e
r
inducta
nces
,
wh
ic
h
a
re
wi
dely
use
d
in
home
a
ppli
an
ces,
a
utomoti
ve
a
nd
i
ndus
t
ry,
operate
at
hi
gh
e
r
modu
la
ti
on
f
re
qu
e
ncies.
The
pr
ic
e
of
high
powe
r
el
ect
ron
ic
s
is
al
so
high,
s
o
inc
rease
of
mic
roco
ntr
ol
le
r’
s
cost
in
1
-
2
$
do
e
s
no
t
im
pa
ct
sign
i
ficantl
y
the
t
otal
pri
ce
of
dev
ic
e,
bu
t
the
c
os
t
of
th
e
low
power
unit
s
is
low,
a
nd
the
pressu
re
form
c
ompeti
tors
ma
ke
eve
r
y
cent
sign
ific
a
nt.
S
o
dev
el
opers
of
powe
r
el
ect
ron
ic
s
pu
t
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
11
53
–
116
4
1154
cheap
Digital
Sign
al
P
r
ocess
or
s
(
DSP)
in
t
heir
so
l
ution
s
and
the
n
t
ry
to
opti
mize
s
of
t
war
e
to
be
abl
e
r
un
in
the sele
ct
ed D
SP.
O
ne of
t
he mai
n
mil
est
one
s in
t
he way
of
op
ti
miza
ti
on is
squar
e
ro
ot calc
ulati
on
r
ou
ti
ne
.
Square
r
oot
operati
on
is
fr
e
quently
us
e
d
i
n
ma
ny
c
on
t
ro
l
sy
ste
ms
of
po
wer
el
ect
ronic
s
an
d
di
gital
sign
al
processi
ng
al
gorithm
s.
Tasks
with
s
quare
r
oo
ts
ca
n
be
div
ide
d
int
o
tw
o
groups
.
Tasks
from
t
he
first
gro
up
a
re
exec
uted
e
very
sam
pling
per
i
od
an
d
proce
ss
ne
wl
y
sam
pled
data.
Tasks
from
t
he
second
gro
up
are
execu
te
d
ra
rely
(
with
a
fr
e
qu
ency
le
ss
tha
n
120
Hz
),
w
he
n
measu
reme
nt
resu
lt
s
are
nee
ded
a
nd
proces
s
data
from
m
ulti
ple
samples.
Fi
rst
gro
up
incl
ud
es
volt
age
an
d
c
urren
t
vect
or
t
ran
s
f
or
mati
ons
an
d
nor
mali
zat
ion
s,
models
of
c
ontrol
obje
ct
s,
ob
serv
e
rs,
et
c.
S
econ
d
gr
oup
inclu
des
cal
cul
at
ion
of
R
oot
M
ea
n
Square
s
(RM
S
)
and
Total
Harmo
nic
Disto
rtion
(T
HD),
le
a
st
sq
ua
res
bas
ed
al
gorith
ms,
et
c.
Squa
re
r
oo
t
operati
on
is
al
s
o
intensivel
y
us
e
d
i
n
el
ect
rical
dr
i
ves,
w
he
re
i
t
is
in
volve
d
into
m
otor
models
cal
c
ulati
on,
M
axim
um
Torq
ue
Per
Ampe
re
(
M
TP
A)
c
on
t
rol
[
1
],
M
a
ximum
T
orq
ue
Per
V
oltage
(
M
T
PV
)
c
on
t
ro
l,
f
ie
ld
wea
ke
ning
[
2
],
var
i
ou
s
obse
rvers, Po
wer Fac
tor
C
orrecto
rs (PFC)
, etc.
Au
t
hors
of
t
his
pa
per faced
w
it
h
the
same
pr
ob
le
m
, which
aro
se
,
w
hen 8
0
MHz DSP wa
s
s
ub
sti
tuted
with
64
MHz
DS
P
a
nd
s
of
t
war
e
op
e
rati
ng
at
16
kHz
f
ai
le
d
due
to
t
he
M
C
U
over
load.
An
al
ys
is
of
the
so
ft
war
e
rev
ea
le
d
seve
ral
bott
le
-
neck
s
,
one
of
wh
ic
h
was
s
qu
a
re
r
oot
cal
culat
ion
routine
cal
le
d
sever
al
ti
mes
per
sam
pling
pe
rio
d
a
nd
t
wo
ti
mes
in
the
60
Hz
inte
rrup
t.
A
t
the
same
ti
me
impleme
ntati
on o
f
t
he
s
quare
roo
t
cal
culat
io
n
r
ou
ti
ne
us
e
d
di
git
-
by
-
dig
it
met
hod,
w
hich
was
no
t
perfect
a
nd
t
ook
a
l
ot
of
process
or
c
ycles.
Ther
e
f
or
e,
aut
hors
c
oncent
rated thei
r
ef
f
or
ts
on the
opti
miza
ti
on
of ti
me used
for
s
quare
root cal
culat
io
n.
Let
's
form
ulate
a
pro
blem
f
or
t
he
disc
us
si
on.
F
or
a
gi
ve
n
num
be
r
S,
it
is
necessar
y
to
fin
d
the
numb
e
r X s
o
t
hat:
√
≈
.
(
1
)
M
ode
rn
D
SPs
for
c
hea
p
c
on
t
ro
l
s
ys
te
ms
of
powe
r
el
ect
r
onic
s
are
t
yp
ic
al
ly
16
or
32
-
bit
with
10
or
12
-
bit
AD
C.
Ther
e
f
or
e,
maj
or
it
y
of
va
riab
le
s
in
co
nt
r
ol
al
gorithms
a
re
16
-
bit
an
d
int
erme
diate
res
ul
ts
of
cal
culat
ion
s a
r
e
32
-
bit
va
riabl
es
.
Co
ns
e
quent
ly,
X
is e
xpect
ed
to
b
e
16
-
bit
and S is
expect
ed
to
b
e
32
-
bit.
The
over
w
helming
maj
or
it
y
of
D
SPs
are
de
sign
e
d
f
or
fas
t
mu
lt
ipli
cat
io
n
with
ad
diti
on,
but
has
n
o
hard
war
e
acc
el
erati
on
f
or
the
div
isi
on
opera
ti
on
.
C
onseq
ue
ntly,
im
pleme
nt
at
ion
of
t
he
div
isi
on
op
e
rati
on
i
s
slow
a
nd
ty
pical
ly
ta
kes
as
man
y
pr
ocess
or
cycles
as
nu
mb
e
r
of
bits
of
the
re
su
lt
.
F
or
a
16
-
bit
res
ult,
one
div
isi
on
operat
ion
ta
kes
a
bout
16
c
ycl
es
an
d
si
gnific
antly
slo
ws
do
wn
cal
culat
ion
s.
T
hu
s
,
it
is
desir
ed
t
o
minimi
ze the
num
ber o
f divisi
on
s
in
the s
qua
re ro
ot calc
ulat
ion
al
gorithm
.
In
s
uc
h
a
wa
y
our
pu
rpose
i
s
to
acce
le
rate
the
square
root
cal
culat
ion
with
the
pr
ic
e
of
decr
ea
sed
tolerance
.
As
sq
ua
re
r
oo
t
op
erati
on
is
t
yp
i
cal
ly
use
d
f
or
the
processi
ng
of
meas
ur
e
d
sign
al
s
co
rru
pted
by
no
ise
,
cal
c
ulati
on
e
rror
s
s
houl
d
be
simi
la
r
t
o
t
he
meas
ur
i
ng
e
rro
rs,
w
hic
h
a
re
ty
pical
ly
0.5
–
5%
.
H
oweve
r
so
me
cases
de
man
d
higher
preci
sion,
s
o
cal
culat
ion
meth
o
ds
mu
st
be
abl
e
to
decre
ase
e
rror
s
by
r
unning
one
to
tw
o
a
ddit
ion
al
it
erati
ons.
Ther
e
f
or
e,
the
cal
culat
ion
m
et
hod
m
us
t
qu
ic
kly
cal
c
ulate
the
s
quare
r
oot
with
error
s
in
the
ra
ng
e
of
0.5
–
5%
a
nd
m
us
t
ha
ve
fast
c
onve
rg
e
nce
to
be
a
ble
to
decr
ea
se
er
rors
in
one
to
tw
o
add
it
io
nal it
era
ti
on
s.
Ba
sic
al
ly,
the
al
gorithms
f
or
sq
ua
re
r
oo
t
ev
al
uation
can
be
di
vid
e
d
i
nto
two
main
groups:
it
erati
ve
methods
an
d
appr
ox
imat
io
n
by
real
f
un
ct
ion
s
.
Ite
rati
ve
meth
ods
c
omprise
t
hr
ee
cl
asses:
direct
m
et
hods
,
al
gorithms
bas
ed
on th
e
Ne
wton
-
Ra
phs
on fo
rm
ula and
nor
mali
zat
ion
tech
niques
[
3
].
Au
t
hors
of
[
4
–
13
]
repor
te
d
ha
rdwar
e
implem
entat
ion
of
no
n
-
it
erati
ve
met
hods.
T
hese
al
gorith
ms
ar
e
mainly
de
sig
ne
d
for
lo
w
-
reso
l
ution data (8
-
bi
t), b
ut oper
at
e
ex
tremel
y
fast and
ca
n
be
us
e
d
for
pre
-
proce
ssing
of
sa
mp
le
d
da
ta
Su
ch
met
hods
nee
d
ad
diti
on
al
hardware
an
d
ca
nnot
be
us
e
d
in
gen
e
ral
D
SP
-
base
d
al
gorithms.
A
ut
hors of [
14
]
use
d
par
al
le
l cal
culat
ion
i
n rep
ort
ed
s
pee
d
inc
r
ease i
n mo
re t
ha
n 38%.
The
un
c
omm
on
a
ppro
ac
h
for
square
r
oo
t
ca
lc
ulati
on
was
r
eported
i
n
[
16
]
.
A
uthor
s
pr
opos
e
d
to
us
e
it
erati
ve
exec
ut
ion
of
the
no
nlinear
I
nf
init
e
Im
pu
lse
Re
s
ponse
(
II
R)
filt
er
to
ob
ta
i
n
th
e
square
root
of
t
he
giv
e
n
num
ber
.
This
meth
od
do
e
s
no
t
in
vo
l
ve
div
isi
on
op
erati
on,
bu
t
in
te
ns
ively
use
s
m
ulti
plica
ti
on
with
add
it
io
n operat
ion
s
.
This
pap
e
r
is
div
ide
d
i
nto
fi
ve
sect
io
ns
.
I
n
Sect
ion
I
I
the
auth
ors
e
xp
la
i
n
e
xisti
ng
met
hods
f
or
t
he
fixe
d
po
i
nt
s
qu
are
root
cal
c
ulati
on
,
an
d
disc
us
s
their
pr
os
and
c
on
s
.
Sect
ion
I
II
pr
opos
e
s
a
ne
w
meth
od
a
nd
discusse
s
it
s
to
le
ran
ce.
E
xperi
mental
res
ults
and
perf
or
ma
nc
e
of
the
m
os
t
promisin
g
met
hods
are
giv
e
n
in
t
he
Sect
ion
I
V.
Se
ct
ion
V
c
on
ta
i
ns
c
oncl
us
io
ns.
2.
EXISTI
NG M
ET
HOD
S
The
a
uthors
of
[
3
]
gi
ve
cl
ass
ific
at
ion
an
d
e
xtensi
ve
re
view
of
the
gen
e
r
al
sq
ua
re
r
oo
t
cal
culat
ion
methods
,
but
not
al
l
of
t
hem
are
s
uitable
f
or
fixe
d
po
i
nt
implementa
ti
on.
Pape
r
[
3
]
can
be
e
xten
ded
wi
th
[
15
]
and
[
17
], w
hic
h revie
w
a
nd a
nalyse
only
f
ix
ed po
i
nt alg
or
it
hm
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Revi
ew
o
f f
as
t
sq
uare
root c
al
culatio
n meth
ods f
or
fi
xed
poin
t micr
oc
on
tr
ol
le
r
-
based
cont
ro
l
…
(
A.
Dian
ov)
1155
Ther
e
a
re
seve
ral
s
quare
r
oot
cal
culat
io
ns
methods
,
w
hich
can
be
co
ns
i
der
e
d
to
be
fa
st.
S
om
e
of
them
util
iz
e
ha
r
dware
featu
r
es
of
pr
ocess
or
c
or
e
a
nd
ca
n
be
im
pleme
nted
i
n
t
he
li
mit
ed
num
ber
of
DS
Ps
,
wh
il
e
oth
e
rs
a
re
gen
e
ral
met
hods
,
w
hic
h
do
no
t
de
pend
on
pr
ocess
or
c
or
e
.
T
his
pap
e
r
re
view
s
the
mo
s
t
pros
pecti
ve me
thods a
nd
discuss
es t
heir p
ros and c
ons.
2.1.
Digit
-
by
-
digit c
alcula
ti
on
This
met
hod
f
or
squa
re
r
oot
cal
culat
ion
at
fixe
d
point
D
SPs,
is
the
most
popula
r
a
s
i
t
is
easy
t
o
unde
rstan
d
an
d
impleme
nt.
Desp
it
e
relat
iv
el
y
slow
c
onve
rg
e
nce,
man
y
eng
i
neer
s
sti
ll
pay
at
te
ntio
n
to
thi
s
method,
oft
en
impleme
nting
i
t
in
hard
war
e
[
18
]
a
nd
s
oft
wa
re
[
19
].
This
m
et
hod
ca
n
be
c
on
si
der
e
d
as
a
basi
c
method a
nd ca
n be
us
e
d
f
or t
he
e
valuati
on
of
oth
er
meth
od
s.
F
or
be
tt
er
un
de
rstan
ding
th
e
al
gorithm
im
pl
ementat
io
n
in
bin
a
ry
syst
em,
le
t’s
co
ns
i
d
er
it
s
op
e
rati
on
with
decimal
num
ber
s
, whic
h i
s o
fte
n use
d
f
or ma
nual
calc
ul
at
ion
of s
quar
e r
oo
ts
with
pa
per an
d pe
ncil.
Suppose
X
k
is t
he k
-
t
h digit
of
X
,
so
:
=
⋅
1
0
+
−
1
⋅
1
0
−
1
+
.
.
.
+
1
⋅
10
+
0
(
2
)
The
m
os
t
sig
ni
ficant
di
git
X
k
is
the
first
ap
pr
ox
imat
io
n
of
t
he
s
qu
a
re
r
oot
X
an
d
ca
n
be
e
asi
ly
fou
nd
as the
highest i
ntege
r wh
ic
h
s
at
isfie
s:
2
⋅
1
0
2
≤
(
3
)
The ne
xt d
i
git
X
k
-
1
of s
qu
a
re
r
oo
t
X
ca
n be fo
und usi
ng the
f
ollow
i
ng ine
qual
it
y:
≥
(
⋅
1
0
+
−
1
⋅
1
0
−
1
)
2
(
4
)
wh
ic
h
tra
nsfo
r
ms int
o:
−
2
⋅
1
0
2
≥
−
1
⋅
1
0
2
(
−
1
)
(
20
⋅
+
−
1
)
(
5
)
Simi
la
rly,
t
he next
dig
it
ca
n be calc
ulate
d
a
s:
(
)
(
)
(
)
2
2
2
2
1
2
2
2
1
1
4
2
10
10
20
10
10
20
10
−
−
−
−
−
−
−
+
+
+
+
−
k
d
i
g
i
t
N
e
w
k
d
i
g
i
t
N
e
w
k
s
t
e
p
p
r
e
v
i
o
u
s
at
R
o
o
t
k
k
k
s
u
b
t
r
a
h
e
n
d
p
a
i
r
N
e
x
t
k
k
k
s
u
b
t
r
a
h
e
n
d
p
a
i
r
L
e
f
t
m
o
s
t
k
X
X
X
X
X
X
X
X
S
(
6
)
This
a
ppro
ac
h
can
be
us
e
d
recursivel
y
du
rin
g
the
ne
xt
ste
ps
to
obta
in
the
ne
cessar
y
num
ber
of
dig
it
s.
Since
e
very
di
git
of
th
e
re
su
lt
X
is
m
ulti
plied
by
10
2k,
it
’
s
ea
sie
r
to
sepa
rate
S
in
to
pairs
of
dig
i
ts
an
d
cal
culat
e
X
k
f
or
e
very
pai
r.
If
S
c
on
ta
in
s
odd
nu
mb
e
r
of
di
gits,
a
0
m
us
t
be
a
dd
e
d
to
t
he
le
ft.
T
he
reafter
,
every
pair
is
combine
d wit
h p
r
evio
us
ste
p re
mainde
r
a
nd
ne
w digit
sati
sf
yi
ng
:
≥
(
20
⋅
+
−
1
)
−
1
(
7
)
mu
st
be fo
und.
Let
’s
denote
r
oo
t
cal
c
ulate
d
at
the
cu
rr
e
nt
s
te
p
as
“R
”
a
nd
remai
nd
e
r
of
th
e
init
ia
l
val
ue
as
“R
em”.
R
is
init
ia
l
iz
ed
with
ze
r
o,
and
e
ve
ry
it
er
at
ion
of
the
a
lgorit
hm
cal
cu
la
te
s
the
nex
t
dig
it
of
the
root.
The flo
wch
a
rt
of the al
gorith
m is s
how
n
in
Figure
1
:
Let
’s
il
lustrate
this al
gorith
m
with the
foll
ow
ing
e
xam
ple: T
he
s
quare
r
oo
t
of 54
756
is:
So
a
nswer
is
234.
This
meth
od
c
an
be
e
xten
ded
to
bin
a
ry
num
ber
s
.
A
bin
a
ry
sy
ste
m
is
simp
le
r
for
cal
culat
ion
becau
s
e
the
great
est
num
ber
can
be
fou
nd
just
by
com
pa
rin
g
num
ber
s
.
T
he
only
dif
fer
e
nce
is
that
c
onditi
on
(
7
)
tran
s
forms
into
:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
11
53
–
116
4
1156
≥
(
10
0
⋅
+
−
1
)
−
1
7
(
8
)
because
the
r
a
dix
num
ber ch
ang
e
d from
10 t
o
2.
The ne
xt ex
a
m
ple s
hows r
oo
t
cal
culat
ion
i
n
t
he bina
ry syst
em.
So
,
res
ult i
s
101111b
=
47d
Figure
1
.
Flo
w
char
t
of
di
git
-
by
-
dig
it
met
hod
This
meth
od
does
not
use
sp
e
ci
fic
hardw
a
re
and
it
s
exec
uti
on
ti
me
do
es
not
sign
i
ficantl
y
dep
e
nd
on
the
process
or
typ
e.
T
he
a
dvantages
of
t
his
method
are
preci
se
res
ults
(
LSB
or
half
L
SB
if
remain
de
r
is
analyse
d),
sim
ple
impleme
nt
at
ion
a
nd
a
bs
e
nce
of
div
isi
on
.
It
cal
culat
es
r
oo
t
dig
it
by
di
git,
sta
rtin
g
f
rom
the
mo
st
si
gn
i
fic
ant,
s
o
cal
c
ulati
on
s
ca
n
be
sto
p
pe
d
be
fore
processin
g
of
f
ul
l
numb
e
r
S,
if
acce
ptab
le
t
oleran
ce
is reache
d.
Howe
ver,
the
mo
st
si
gn
ific
a
nt
disa
dv
a
nta
ge
of
the
e
xp
la
i
ned
meth
od
is
cal
culat
ion
ti
me.
Desp
it
e
this,
dig
it
-
by
-
di
git
method
is
f
reque
ntly
us
e
d
in
the
co
ntr
ol
sy
ste
ms,
w
her
e the
process
or
is
no
t hig
hly
lo
aded.
It
was
al
so
p
r
opose
d
f
or
impl
ementat
io
n
in h
ar
dware
a
nd
t
he
aut
hors
of
[
19
]
en
ha
nce
d
this
idea
to
cal
c
ulati
on
of o
t
her
f
un
ct
i
on
s
a
nd imple
mented
on FP
GA.
2.2.
P
olynom
ial approxi
m
ati
on
Square
r
oo
t
functi
on
is
di
ff
ic
ult
for
a
ppr
ox
i
mati
on
beca
use
it
s
der
i
vative
rap
i
dly
c
ha
nges
cl
os
e
t
o
zero,
w
hic
h
re
su
lt
s
in
high
a
ppr
ox
imat
io
n
error
s
.
T
her
e
f
ore,
s
quare
r
oo
t
can
be
s
ucces
sfu
ll
y
ap
p
roxi
mate
d
on
l
y
at
the
li
mit
ed
inter
val.
(
)
(
)
(
)
(
)
2325
1856
176
129
9
2
4
5
5
23
0
1856
4
4
23
0
1856
1856
4
4
2
0
147
3
3
2
0
129
147
3
05
04
05
56
47
05
+
+
−
+
+
−
−
2
2
be
c
a
u
s
e
4
is
di
g
i
t
t
h
i
r
d
T
h
e
2
2
be
c
a
u
s
e
3
is
di
g
i
t
s
e
c
on
d
T
h
e
2
be
c
a
u
s
e
2
is
di
g
i
t
f
i
r
s
t
T
h
e
n
u
m
be
r
G
i
v
e
n
2
(
)
(
)
(
)
(
)
(
)
(
)
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
1
0
1
1
1
0
1
1
1
1
0
1
1
1
00
1
0
1
1
1
0
1
1
0
1
1
1
0
1
1
0
0
0
1
0
0
1
1
1
0
1
1
00
0
1
0
1
1
0
1
1
0
0
0
1
0
0
1
0
0
1
1
0
1
1
101
00
0
1
0
1
0
1
1
0
0
1
1
0
1
0
0
1
0
1
1
10
00
0
1
0
0
1
1
0
0
1
0
100
1
1
1
00
000
100
10
01
01
10
01
00
10
10
00
10
1
0
1
1
1
0
1
1
0
1
1
0
1
10101
1001
101
+
−
+
−
+
−
+
−
+
−
−
1
bec
a
u
s
e
1
is
di
g
i
t
S
i
x
t
h
1
bec
a
u
s
e
1
is
di
g
i
t
F
i
f
t
h
1
bec
a
u
s
e
1
is
di
g
i
t
F
ou
r
t
h
1
bec
a
u
s
e
1
is
di
g
i
t
T
h
i
r
d
1
bec
a
u
s
e
0
is
di
g
i
t
S
e
c
on
d
bec
a
u
s
e
1
is
di
g
i
t
F
i
r
s
t
2209
n
u
m
ber
G
i
v
e
n
d
S
e
p
a
r
a
t
e
S
i
n
t
o
p
a
i
r
o
f
d
i
g
i
t
s
.
A
d
d
“
0
”
t
t
h
e
l
e
f
t
i
f
n
e
c
e
s
s
a
r
y
R
=
0
;
R
e
m
=
L
e
f
t
m
o
s
t
p
a
i
r
;
F
i
n
d
g
r
e
a
t
e
s
t
d
i
g
i
t
X
s
u
c
h
t
h
a
t
X
·
(
20
·
R
+
X
)
≤
R
e
m
A
l
l
p
a
i
r
s
p
r
o
c
e
s
s
e
d
?
R
e
m
=
R
e
m
·
1
0
0
+
N
e
x
t
p
a
i
r
;
R
=
10
·
R
+
X
;
B
e
g
i
n
E
n
d
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Revi
ew
o
f f
as
t
sq
uare
root c
al
culatio
n meth
ods f
or
fi
xed
poin
t micr
oc
on
tr
ol
le
r
-
based
cont
ro
l
…
(
A.
Dian
ov)
1157
Po
ly
nomial
a
ppr
oximat
ion
is pro
po
se
d
i
n
[
20
]
an
d
recomm
end
e
d
by
e
ng
i
ne
e
rs
from
A
nalog
De
vices
to
us
e
with
th
ei
r
D
SPs
.
T
he
y
pro
pose
to
appr
ox
imat
e
s
qu
a
re
r
oo
t
f
unct
ion
f(x)
=
x0
.5
at
the
inter
val
of
[
0.5
..
1]
,
w
it
h fi
fth orde
r po
l
ynom
ia
l:
0
.
2
0
7
5
8
0
6
1
.
4
5
4
8
9
5
x
1
.
3
4
4
9
1
x
1
.
1
0
6
8
1
2
x
0
.
5
3
6
4
9
9
-
x
0
.
1
1
2
1
2
1
6
x
2
3
4
5
+
+
−
+
x
(
9
)
Since
a
ppr
ox
i
mati
ng
poly
no
mial
giv
es
a
ppropr
ia
te
res
ults
in
t
he
li
mit
ed
interval,
the
in
it
ia
l
nu
m
ber
mu
st
be
scal
e
d
to
fall
i
ns
ide
i
t.
A
fter
cal
cula
ti
on
of
t
he
squ
are
root
of
t
he
scal
ed
num
ber,
the
re
su
lt
in
g
va
lue
mu
st
be
m
ulti
pl
ie
d
by
the
squ
are
root
of
the
scal
ing
value
to
ob
ta
in
the
c
orrec
t
a
ns
w
er.
The
a
ut
hors
of
[
20
]
publishe
d
as
se
mb
le
r
code
of t
he pr
opos
e
d m
et
hod
a
nd clai
med maxi
mum
ex
ec
ution t
i
m
e of
75 cy
cl
es.
T
his
esse
ntial
scal
ing
operati
on
is
not
sim
pl
e
for
ma
ny
D
SPs
a
nd
ta
kes
ti
me,
howe
ve
r
proces
sors
from
A
nal
og
Dev
ic
es
hav
e
hard
war
e
un
it
s
cal
le
d
an
e
xp
on
e
nt
detect
or,
w
hich
cal
c
ul
at
es
the
am
ount
of
the
redu
nd
a
nt
si
gn
bits
a
nd
s
hifts
init
ia
l
num
bers,
s
o
cal
c
ulati
on
is
acce
le
rate
d.
Mo
reover
,
s
ign
operati
on
of
the
expo
nen
t
detect
or
li
mit
s
the
r
ang
e
of
i
nput
value
by
3276
8,
w
hic
h
is
res
tric
ti
ng
in
m
ost
cases.
Evalua
ti
on
of
the
hi
gh
pow
e
r
po
l
ynomi
al
increases
cal
cula
ti
on
er
r
or
a
nd
impact
s
tolera
nce
of
t
he
res
ul
t.
So
this
method
i
s
no
t
rec
om
me
nded
for i
mp
le
mentat
io
n.
2.3.
T
ab
le
approxi
ma
tion
Lo
okup
ta
ble
-
base
d
meth
ods
are
co
mm
on
f
or
functi
on
ap
pro
ximati
on.
T
hey
a
re
relat
iv
el
y
fast,
bu
t
us
e
a
l
ot
of
me
mory
t
o
inc
rea
se
tolera
nce.
B
iparti
te
an
d
m
u
lt
iparti
te
method
s
are
e
xcell
ent
exa
mp
le
s
of
ta
ble
lookups
an
d
ar
e
desc
ribe
d
in
[18].
A
uthors
pro
po
se
inse
rting
tw
o
lo
okup
ta
bles
i
nto
the
data
pat
hs
a
nd
us
e
on
e
table
for
t
he
init
ia
l seeds,
wh
il
e a
no
t
her
on
e
s
hould be
us
e
d
f
or a
dd
i
ng a
small
corre
ct
ing
va
lue
.
The
a
ppr
ox
im
at
ion
e
rror
ca
n
al
so
be
re
du
ced
by
inter
po
la
ti
on
bet
wee
n
ta
ble
points
(
e.g
.
li
near),
bu
t
it
increases
the
cal
culat
ion
ti
me
an
d
dim
inishes
the
main
ad
va
ntage
of
the
meth
od
-
s
peed.
Th
e
flo
w
chart
of
f
un
ct
io
n
a
ppr
oximat
ion
w
it
h
va
riable
se
gm
e
nts
ta
ble
a
nd
li
near
inter
po
la
ti
on
betwe
en
po
i
nts
is
de
picte
d
in
Figure
2
.
I
t i
nd
ic
at
es that cal
culat
ion
c
onta
in
s one
div
isi
on,
wh
ic
h
si
gn
ific
a
ntly sl
ow
s
do
w
n
cal
culat
io
n.
The
cal
c
ulati
on
ti
me
of
t
he
f
un
ct
io
n
ap
pro
xi
mate
d
with
ta
ble
ca
n
be
sig
nificantl
y
decre
ased,
if
t
he
ta
ble
co
ns
ist
s
of
e
qual
se
gme
nts
a
nd
a
numb
e
r
of
th
em
is
t
o
the
powe
r
of
tw
o.
A
flo
wc
har
t
of
the
corres
p
on
ding
al
gorithm
is
de
picte
d
i
n
Fi
gur
e
3
.
It
in
dicat
es
that
cal
c
ulati
on
wa
s
sim
plif
ie
d
a
nd
di
visio
n
ha
d
been excl
u
de
d.
Table
ap
pro
xi
mati
on
perfect
ly
w
orks
for
t
he
sm
ooth
fun
ct
ion
wit
h
sm
oo
t
h
de
rivati
ve
and
outp
uts
higher e
r
rors f
or
t
he fu
nctio
n wit
h
ra
pi
dly
c
ha
ng
i
ng d
e
rivati
ves.
A
t
yp
ic
al
exam
ple of
fun
ct
ion
s, w
hich
c
an be
appr
ox
imat
e
d ea
sil
y,
is si
ne
a
nd
co
sine.
As
sta
te
d
a
bove
,
s
qu
a
re
r
oo
t
is
not
eas
y
f
or
ap
pro
ximati
on,
beca
us
e
it
s
der
i
vative
rap
i
dly
cha
nges
cl
os
e
t
o
ze
r
o.
Ther
e
f
or
e,
vari
able
ste
p
ta
bl
e
ap
pro
ximati
on
or
pr
e
-
scal
ing
m
us
t
be
use
d.
F
or
this
s
pecific
reason,
Fig
ure
3
c
on
ta
in
s tw
o sca
li
ng
blo
c
ks
.
On
t
he
one
ha
nd,
the
tole
rance
of
this
meth
od
mainly
de
pe
nds
on
the
siz
e
of
the
ta
ble
a
nd
i
n
some
cases
the
s
iz
e
is
unacce
ptabl
y
la
r
ge.
O
n
the
oth
e
r
ha
nd,
t
his
met
hod
nee
ds
pre
-
scal
ing
be
fore
ta
ble
is
r
ead
a
nd
after, w
hich di
minishes
it
s m
ai
n
ad
va
ntage
-
cal
culat
ion sp
eed.
2.4.
Newt
on
’
s method
This
met
hod
is
very
popula
r
f
or
nume
rical
cal
culat
ion
beca
us
e
it
is
sim
ple
an
d
has
fast
c
onve
rg
e
nce.
It
can
be
su
cce
ssfu
ll
y
us
e
d
f
or
the
a
pproxim
at
ion
of
var
io
us
functi
ons
a
nd
cal
culat
io
n
operati
ons
[21].
Let
’s
consi
der it
s appli
cat
ion
t
o
the
calc
ulati
on of
sq
ua
re
r
oo
ts.
Find
i
ng X,
wh
i
ch
is s
quare
ro
ot of
S is the
sa
me as s
olv
i
ng
:
2
−
=
0
(
10
)
Accor
ding to
New
t
on’s met
hod, the
n
e
xt ste
p
a
ppr
ox
imat
io
n
xi+1
of the
fun
ct
io
n r
oo
t i
s:
+
1
=
−
(
)
′
(
)
(
11
)
wh
e
re
xi
is
root
ap
pro
ximati
on
at
t
he
cu
rrent
ste
p.
The
cal
culat
ion
pro
cess
can
be
il
lustrate
d
by
drawin
g
series
of
se
ca
nt
li
nes
to
pa
r
abo
la
as
s
how
n
i
n
Fi
gure
4
.
The
it
erati
ve
cal
culat
ion
is
sto
pp
e
d,
wh
e
n
t
he
necessa
ry accu
racy has
b
ee
n r
eached:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
11
53
–
116
4
1158
Figure
2
.
Flo
w
char
t
of fu
nction ap
pro
ximati
on w
it
h
var
ia
ble segme
nts table a
nd li
near i
nter
po
la
ti
on
betwee
n po
i
nts
Figure
3
.
Flo
w
char
t
of s
qu
a
re
roo
t a
ppr
ox
im
at
ion
with e
qu
al
l
y
se
gm
e
nted
ta
ble
with size
of 2
Tb
lSize
and
li
near
inte
rpola
ti
on
betwee
n p
oin
ts
|
+
1
−
|
≤
(
12
)
Howe
ver,
f
or
s
impler
cal
c
ulati
on
,
the
it
erati
ve
process
ca
n
be
li
mit
ed
to
f
ixed
num
ber
of
it
erati
on
s
.
Th
us
, t
he fu
nct
ion
giv
e
n
i
n (
10
)
, usin
g (
11
)
tran
s
f
or
m
s int
o:
+
1
=
−
2
−
2
=
1
2
(
+
)
(
13
)
This e
qu
at
io
n
i
s also kno
wn
a
s the Bab
yloni
an
met
hod [
22
]
. I
t h
as qua
dr
at
ic
co
nver
ge
nce
and all
ows
us
to
ca
lc
ulate
square
r
oo
t
w
it
h
low
num
be
r
of
op
e
rati
ons
.
H
oweve
r,
ea
ch
it
erati
on
of
the
meth
od
c
onta
ins
div
isi
on,
whic
h
ta
ke
s a si
gn
i
f
ic
ant am
ount
of
process
or ti
me
Pr
ope
rly
sel
ect
ed
i
niti
al
value
can
sig
nifica
nt
ly
acce
le
rate
c
al
culat
ion
,
s
o
an
i
niti
al
gu
es
s
x0
is
ve
r
y
importa
nt.
W
e
su
ggest
t
o
loca
te
squar
e
r
oo
t
by f
i
nd
i
ng
n
, so that
square
roo
t
belo
ngs t
o t
he
inter
val:
2
≤
<
2
+
1
(
14
)
This
ste
p
ca
n
be
easi
ly
im
ple
mented
in
the
D
SPs
a
nd
pr
ovides
a
dv
a
ntag
e
of u
sin
g
the pow
e
r
of
tw
o,
th
eref
or
e
div
isi
on
at
the
fir
st
ste
p
ca
n
be
substi
tuted
with
bit
s
hift.
The
n
t
he
i
niti
al
valu
e
ca
n
be
sel
ect
ed
du
rin
g
t
his
interval,
u
si
ng
any crit
eria.
Typica
l i
niti
al
v
al
ues
a
re m
i
dd
le
po
i
nt of t
he
int
erv
al
:
0
=
3
⋅
2
−
1
(
15
)
wh
ic
h
is
th
e
mo
st
pro
ba
ble
num
be
r,
an
d
val
ue,
w
hich
giv
e
s
symmet
rica
l
relat
ive
er
ror
at
interval
bor
ders:
n
x
2
3
4
0
=
(
16
)
Using
(
15
)
as a
n
init
ia
l val
ue a
nd combi
ning
w
it
h (
13
) resul
ts:
1
=
1
2
(
3
⋅
2
−
1
+
3
⋅
2
−
1
)
=
3
⋅
2
−
2
+
3
⋅
2
(
17
)
wh
e
re
div
isi
on
can
be
substi
tuted
by
the
bi
t
sh
ift.
H
ow
e
ver,
the
f
ollo
wing
ste
ps
i
nvolv
e
di
vision
and
are
performe
d
acc
ordin
g
t
o
(
13
).
D
e
t
e
r
m
i
n
e
t
a
b
l
e
s
e
g
m
e
n
t
,
i
.
e
.
d
e
f
i
n
e
t
a
b
l
e
i
n
d
e
x
N
,
s
o
t
h
a
t
S
[
N
]
≤
S
<
S
[
N
+
1
]
R
e
a
d
d
a
t
a
f
r
o
m
t
a
b
l
e
:
X
[
N
]
,
X
[
N
+
1
]
,
S
[
N
]
,
S
[
N
+
1
]
B
e
g
i
n
E
n
d
P
e
r
f
o
r
m
l
i
n
e
r
a
p
p
r
o
x
i
m
a
t
i
o
n
:
(
)
(
)
N
S
N
S
N
S
S
N
X
N
X
N
X
X
−
+
−
−
+
+
=
1
1
S
c
a
l
e
S
t
o
b
e
i
n
t
h
e
r
a
n
g
e
[
0
.
5
..
1
].
K
i
s
e
x
p
o
n
e
n
t
(
n
u
m
b
e
r
o
f
s
h
i
f
t
e
d
b
i
t
s
)
D
e
f
i
n
e
t
a
b
l
e
i
n
d
e
x
N
N
=
S
>>
(
3
2
-
T
b
l
S
i
z
e
)
R
e
a
d
d
a
ta
f
r
o
m
ta
b
le
:
X
[
N
]
a
n
d
X
[
N
+
1
]
B
e
g
i
n
E
n
d
S
c
a
l
e
r
e
s
u
l
t
i
n
g
v
a
l
u
e
:
K
X
X
2
=
P
e
r
f
o
r
m
l
i
n
e
r
a
p
p
r
o
x
i
m
a
t
i
o
n
:
(
)
(
)
(
)
Tb
lS
ize
S
N
X
N
X
N
X
X
T
b
l
S
i
z
e
−
−
+
+
=
1
2
1
&
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Revi
ew
o
f f
as
t
sq
uare
root c
al
culatio
n meth
ods f
or
fi
xed
poin
t micr
oc
on
tr
ol
le
r
-
based
cont
ro
l
…
(
A.
Dian
ov)
1159
Let
’s
cal
culat
e
the
er
ror
for
every
it
erati
on
in
orde
r
to
def
i
ne
the
num
ber
of
re
quire
d
ste
ps.
The rel
at
ive er
ror
is
higher
, w
hen X l
ie
s at th
e inter
val’s
l
ower
borde
r: X=
2n.
Re
la
ti
ve
error
of init
ia
l appr
oxi
mati
on
:
0
=
0
−
1
=
3
⋅
2
−
1
2
−
1
=
1
2
(
18
)
First ste
p ap
prox
imat
io
n:
1
=
3
⋅
2
−
2
+
2
2
3
⋅
2
=
9
⋅
2
−
2
+
2
3
=
13
⋅
2
12
(
19
)
Re
la
ti
ve
error
of the
first ste
p ap
pro
ximati
on:
1
=
1
−
1
=
13
⋅
2
12
⋅
2
−
1
=
1
12
≈
8
.
3%
(
20
)
Seco
nd step
appr
oximat
ion
:
2
=
1
2
(
13
⋅
2
12
+
12
⋅
2
2
13
⋅
2
)
=
313
⋅
2
312
(
21
)
Re
la
ti
ve
error
of the sec
ond s
te
p
ap
pro
ximat
ion
:
2
=
2
−
1
=
313
⋅
2
312
⋅
2
−
1
=
1
312
≈
0
.
32%
(
22
)
Thir
d
ste
p
a
ppr
ox
imat
io
n:
3
=
1
2
(
313
⋅
2
312
+
312
⋅
2
2
31
3
⋅
2
)
=
19531
3
⋅
2
19531
2
(
23
)
Re
la
ti
ve
error
of the t
hird ste
p
a
ppr
ox
imat
io
n:
3
=
3
−
1
=
1
19531
2
≈
5
.
1
⋅
1
0
−
6
%
(
24
)
This
t
olera
nc
e
is
ty
pical
ly
suffici
e
nt,
a
nd
s
qu
a
re
r
oot
cal
culat
io
n
nee
ds
th
ree
it
erati
on
s
,
wh
ic
h
in
volve
on
l
y
tw
o divisi
on
s
.
Figure
4
.
G
e
ome
tric
al
inter
pret
at
ion
of the
New
t
on’s met
hod
2.5.
A
t
w
o
-
va
ri
ab
le
iter
at
i
ve
method
This
met
hod
was
propose
d
by
t
he
a
uthors
of
[
23
]
f
or
one
of
t
he
fi
rst
co
mputers.
It
is
app
li
cable
t
o
the s
qu
a
re
root
calc
ulati
on of
the num
be
r
S
s
at
isfying
0
<
S
<3,
s
o
it
als
o n
eeds s
cal
in
g o
f
the initi
al
num
ber.
This
meth
od
c
al
culat
es
tw
o
num
ber
s
a
a
nd
c
at
eve
r
y
it
era
ti
on
,
w
her
e
a
c
onve
rg
es
to
s
quare
r
oot
of
S and c
con
verges t
o 0.
These
num
ber
s
are
i
niti
al
iz
ed
as:
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
11
53
–
116
4
1160
0
=
;
0
=
−
1
(
25
)
and at eve
r
y
it
erati
on the
y
a
re
updated
as:
+
1
=
−
0
.
5
⋅
+
1
=
0
.
25
⋅
2
(
−
3
)
(
26
)
It sho
uld
be n
ot
ed
that
for fast
er calculat
io
n S can
b
e
scale
d t
o
sat
isf
y 0.5
<
S
<
2.
The
ma
xim
um
relat
ive
error
occurs
wh
e
n
S
=
2.
M
axi
mu
m
relat
ive
error
s
for
eac
h
cal
culat
io
n
ste
p
are:
0
=
√
2
−
1
≈
41
.
4%
;
1
=
1
√
2
−
1
≈
−
29%
2
=
5
4
√
2
−
1
≈
−
11
.
6%
;
3
≈
−
1
.
94%
4
≈
−
0
.
056%
(
27
)
This
meth
od,
a
s
New
t
on’s
me
thod,
ha
s
quad
rati
c
co
nver
ge
nce
an
d
ty
pical
ly
3
–
4
it
erati
on
s
of
(
26
)
is suffici
ent.
T
his calc
ulati
on
method
does
not co
ntain
di
vision o
per
at
io
n and t
ime d
el
ay
s
in
her
e
nt to
it
.
The
pro
blem
of
this
met
hod
is
mu
lt
ipli
cat
ion
of
32
-
bit
num
ber
s
,
w
hich
is
t
yp
ic
al
ly
sl
ow
e
r
tha
n
16
-
bit
num
ber
mu
lt
ipli
cat
ion
.
It
al
s
o
t
runcat
es
64
-
bit
num
be
rs
t
o
32
-
bit,
pro
duci
ng
cal
c
ul
at
ion
e
rror
s
,
wh
ic
h
te
nd
t
o be acc
umulat
ed.
2.6.
G
oldsc
h
mi
dt
’s
a
l
go
ri
t
hm
On
e
m
ore
i
nt
eresti
ng
a
nd
pros
pecti
ve
met
hod
is
G
old
sc
hm
idt
’s
al
gorithm.
Its
usual
descr
i
ption
ref
le
ct
s
a
ha
rdwar
e
ori
entat
io
n
with
diff
ic
ul
ti
es
of
i
mp
le
m
entat
ion
in
s
of
tware,
ho
wev
e
r
the
aut
hor
of
[
24
]
su
ggest
s
a s
of
t
war
e
-
f
rien
dly
way of c
omp
uting
,
which
is
de
scribe
d belo
w
.
Let
y0
be
a s
ui
ta
bly
good a
pproxim
at
io
n
to
S
1
, s
uc
h
as:
1
2
≤
⋅
0
2
≤
3
2
(
28
)
In
it
ia
li
ze var
ia
bles:
0
=
⋅
0
ℎ
0
=
0
.
5
⋅
0
(
29
)
And
t
hen it
erate as f
ollo
ws:
{
−
1
=
0
.
5
−
−
1
⋅
ℎ
−
1
=
−
1
+
−
1
⋅
−
1
ℎ
=
ℎ
−
1
+
ℎ
−
1
⋅
−
1
(
30
)
Ca
lc
ulati
on
ca
n
be
sto
pped
,
w
hen
ri
is
s
uffici
ently
lo
w
or
a
fter
fixed
num
ber
of
it
erati
ons
.
Ca
lc
ulate
d
val
ues
c
onve
rg
e
to:
→
∞
=
√
→
∞
2
ℎ
=
1
√
(
31
)
The ma
xim
um
relat
ive er
ror o
ccur
s
whe
n
,
0
=
1
√
2
⋅
(
32
)
And maxim
um rela
ti
ve
e
rror
s
for
eac
h
cal
c
ulati
on
ste
ps
a
re:
Evaluation Warning : The document was created with Spire.PDF for Python.
In
t J
P
ow Elec
& Dri S
ys
t
IS
S
N: 20
88
-
8
694
Revi
ew
o
f f
as
t
sq
uare
root c
al
culatio
n meth
ods f
or
fi
xed
poin
t micr
oc
on
tr
ol
le
r
-
based
cont
ro
l
…
(
A.
Dian
ov)
1161
0
=
1
√
2
−
1
≈
−
29%
;
1
=
5
4
√
2
−
1
≈
−
11
.
6%
2
=
355
256
√
2
−
1
≈
−
1
.
94%
;
3
≈
−
0
.
056%
(
33
)
The
a
uthor
of
[
24
]
pr
ov
es
that
Go
l
ds
m
it
h’
s
al
gorith
m,
as
Ne
wton’s
meth
od
ha
s
qu
a
drat
ic
conve
rg
e
nce,
bu
t
has
a
n
a
dv
antage
of
abse
nce
of
div
isi
on
operati
on.
T
he
featu
re
of
t
he
al
gorithm
i
s
that
it
simult
ane
ou
sl
y
ca
lc
ulate
s
square
r
oo
t
a
nd
reciprocal
s
qu
are
r
oot.
How
ever,
if
one
of
these
val
ues
is
not
need
e
d, the
co
r
respo
nd
i
ng equ
at
ion
in
(
30
) c
a
n be e
xclu
ded.
Eq
uations
in
(
30
)
ca
n
be
ef
fe
ct
ively
implem
ented
for
la
r
ge
num
ber
s
of
D
SPs,
w
hich
ha
v
e
hardw
a
re
un
it
s
f
or
m
ulti
plica
ti
on
with
add
it
io
n.
H
ow
ever,
it
s
houl
d
be
note
d
that
var
ia
bles
in
(
30
)
ha
ve
sig
nifi
cantl
y
diff
e
re
nt ord
e
r
s,
the
refor
e
a l
ot of att
entio
n must
be paid
to
their
represe
nt
at
ion
.
The
disad
va
ntage
of
t
his
m
et
hod
is
the
ne
cessi
ty
of
ini
ti
al
app
r
ox
i
ma
ti
on
(
28
)
,
w
hi
ch
is
quit
e
diff
ic
ult.
The
pa
per
[
24
]
s
ugge
sts
us
in
g
sm
a
ll
loo
ku
p
ta
ble
s
for
this
pur
pose,
but
it
ta
kes
ad
diti
on
al
m
emo
ry.
Anothe
r
dr
a
w
ba
ck
of
t
his
me
thod
is
pro
pagat
ion
of
er
rors
due
to
r
oundi
ng.
T
he
a
uthor
of
[
24
]
cl
ai
m
s
that
high
pr
eci
si
on
resu
lt
s
a
re
usu
al
ly
achie
ved
by
sta
rtin
g
with
Go
l
ds
c
hm
idt
’s
al
go
rithm
a
nd
the
n
s
witc
hi
ng
t
o
the
sel
f
-
co
rr
ec
ti
ng
New
t
on’s
it
erati
on
s.
H
oweve
r,
f
or
our
pur
pose
the
tolera
nce
of
ori
gin
al
al
gorith
m
i
s
su
f
fici
ent.
3.
PROP
OSE
D MET
HO
D
Af
te
r
a
nalysis
of
ad
va
ntages
and
disa
dvanta
ges
of
the
disc
us
se
d
m
et
hods,
the
aut
hors
pro
posed
a
com
bin
e
d met
hod, w
hich
is fa
r
s
up
e
rio
r
to
ot
her
al
gorithms
.
Au
t
hors
pro
po
se
to
de
fine
a
n
init
ia
l
r
oo
t
int
e
rv
al
as
desc
ribed
in
(
14
)
.
T
hen,
i
niti
al
ap
pro
ximati
on
can
be
ma
de b
y dr
a
wing the
s
ecant bet
ween
the ends
of the
root inter
val a
n
d
cal
culat
io
n o
f
seca
nt interse
ct
ion
with a
bs
ci
ssa a
xis,
Fi
gure
5
.
The
e
quat
ion o
f
the
secant li
ne
draw
n o
ver
t
wo points
[
x
1
,
y
1
]
an
d [
x
2
,
y
2
] i
s:
(
)
=
1
−
2
1
−
2
+
2
1
−
1
2
1
−
2
(
34
)
Its ro
ot can be
fou
nd as:
=
1
2
−
2
1
1
−
2
(
35
)
Assumin
g
x
1
=
2
n
,
x
2
=
2
n+1
a
nd combi
ning
(
10
) wit
h (
35
)
r
esults i
niti
al
gu
ess:
0
=
(
2
2
−
)
2
+
1
−
(
2
2
+
2
−
)
2
(
2
2
−
)
−
(
2
2
+
2
−
)
=
1
3
(
2
+
1
+
2
)
(
36
)
Let
’s
cal
culat
e
the ma
xim
um
error Δ
x
ma
x
.
F
rom
Fi
gure
6
e
rror f
un
ct
io
n f
or the a
ppr
ox
i
mati
on
with
se
cant is:
(
)
=
0
−
=
1
3
(
2
+
1
+
2
2
)
−
=
2
3
⋅
2
−
+
2
+
1
3
(
37
)
It h
as
ma
xim
um at
the
point:
1
2
3
−
=
n
x
m
a
x
(
38
)
And maxim
um abs
olu
te
e
rror
is:
n
x
2
12
1
−
=
m
a
x
(
39
)
Figure
7
il
lustr
at
es error f
unct
ion
a
nd
dep
ic
ts
it
s p
ea
k po
i
nt. Ma
ximum rela
ti
ve
erro
r
is:
%
6
.
5
18
1
2
3
2
12
1
1
m
a
x
0
−
−
=
−
=
=
−
n
n
x
x
(
40
)
Evaluation Warning : The document was created with Spire.PDF for Python.
IS
S
N
:
2088
-
8
694
In
t J
P
ow
Ele
c
&
D
ri
S
ys
t,
V
ol
.
11
, N
o.
3
,
Se
ptembe
r
2020
:
11
53
–
116
4
1162
As
ca
n be see
n from
(
40
), t
he e
rror i
s n
e
gativ
e, s
o
init
ia
l g
ue
ss
x
0
ca
n be im
pro
ved by a
dding shi
ft.
0
=
1
3
(
2
+
1
+
2
)
+
(
41
)
Figure
5
.
G
e
ome
tric
al
inter
pret
at
ion
of the
pro
po
se
d met
hod
Figure
6
.
Erro
r
of the
appr
oximat
ion
In
or
der
to
ha
ve
e
qual
po
sit
ive
a
nd
ne
gativ
e
abs
olu
te
er
rors,
offset
val
ue
A
s
hould
be
1/24·
2
n
a
nd
(
41
)
tra
ns
f
orm
s into:
+
=
+
+
=
−
+
n
n
n
n
n
S
S
x
2
2
17
3
1
24
2
2
2
3
1
3
1
0
(
42
)
If
we
ne
ed
sa
me
val
ues
of
the
relat
ive
er
rors,
offset
va
lue
A
s
hould
be
0.0
3367
35·
2
n
an
d
(
41
)
trans
forms
into
:
0
=
1
3
(
2
+
1
+
2
)
+
0
.
0
3
3
6
735
⋅
2
=
1
3
(
1
.
0
5
0
5
1
025
⋅
2
+
1
+
2
)
(
43
)
And fr
om (
40
) maxim
um
relat
ive er
ror
is:
%
.
m
a
x
36
3
1
0
0
−
=
x
x
(
44
)
This
er
r
or
is
al
mo
st
twic
e
le
ss
than
t
he
first
s
te
p
er
ror
in
the
Ne
wton’s
met
hod.
I
f
preci
sio
n
of
(
44
)
is
no
t
suffic
ie
nt,
one
m
ore
cal
culat
ion
ste
p
s
hould
be
pe
r
f
ormed
.
It
co
ul
d
be
performe
d
acc
ordin
g
t
o
(
13
),
wh
ic
h
is si
m
pl
e f
or
im
pleme
nt
at
ion
, but i
nvo
lves
on
e
d
i
vision o
pe
rati
on.
Let
’s
fi
nd the
maxim
um
er
ror,
wh
ic
h
c
orres
ponds t
o X=2
n.
In
it
ia
l ap
pro
ximati
on
:
0
=
1
3
(
1
.
0
5
0
5
1
025
⋅
2
+
1
+
2
2
2
)
=
1
.
0
3
3
6
735
⋅
2
(
45
)
First ste
p ap
prox
imat
io
n:
1
=
1
2
(
1
.
0
3
3
6
735
⋅
2
+
2
2
1
.
03
36
73
5
⋅
2
)
≈
1
.
0
0
0
5
485
⋅
2
(
46
)
Re
la
ti
ve
error
of the
first ste
p ap
pro
ximati
on:
1
=
1
−
1
=
1
.
00
05
48
5
⋅
2
2
−
1
≈
0
.
055%
(
47
)
Wh
ic
h
is s
uffic
ie
nt for
a
w
i
de ran
ge of
engin
eerin
g
a
pp
li
cat
ion
s
.
Evaluation Warning : The document was created with Spire.PDF for Python.