A
NODE
ORDERING
ALGORITHM
TO
SPEED
UP
THE
SOLUTION
OF
SPARSE
MATRIX
AND
SPARSE
VECTOR
LINEAR
EQUATION
SYSTEMS
An onio
G6mez
and
Leopoldo
G.
F anquelo
Dep o.
de
Ingen.
Elec ica,
Elec 6nica
y Au oma ica
Uni e sidad
de
Se illa,
Spain
ABSTRACT.
Recen ly,
mo
e a
e
n ion
hag
be
en
do
o ed
o
spa
ge
ec o
mo hoda
in
o d
. ~
o
e
duc
e
he
compu
a
ion
a l
bu den
when
aol in~
spa ae
sys ems
o!
linea
equa io
n
s.
Theae
me hode
exploi
he
spa si y
o
he
independen
ec o
and/o
he
deai e
o
know
only
a
subse
o
ho
unknown
ec o .
They
a
e
al
a o a
pplic
a
ble
when
e
a
c o iza ion
o •
s
ll
a
h ly
modi i
ed ma
ix
i . .
qui
e d .
This
pape
p opOS8a
a
scheme
o
o de
he
nodes
wi h
he
pu poae
o
educing
he
numbe
o
ope a ions
when
applying
spa aB
ec o
me hoda.
INTRODUCTION.
Spa ae
ma ix
equa lonu
o
he
o m:
Ax
• b ( 1 )
a
ppea
1n
ma
ny
e
n~ine
e
in~
i
e
ld
s .
Tho
s
anda d
o
olu ion
p ocod
u e
pe o ms
a
ap
a
si y
-
o ian ed
LOU
decomposi ion
o
A.
ollowed
by
!o
wa d
a
nd
ba
ck
ope a iona
On
ho
independen
ec o ,
b.
Thl
. Pa pe d
.a
l .
wi h
wo
kind
.
o
p oblemsl
a)
Repe
a
ed
solu ion
a
(1)
when
ma
ix
A
1s
sligh ly
modi!ied
(pa i
a l ma
ix
. a
c o i:e ion
p oblem)
Ill.
b)
Solu ion
o
(1)
when
ei he
VQc o
b
ha
D
only
a ew
non
-
ze o
elemen s
o
a
small
numbe
o
elemen s
in
he
unknown
ec o
x
i8
needod
(8p
a
~
e
ec o
p oblem)
[2l.
lIS
compeno~llon
~elhod
••
he~e
••
pecla
a e
8l.on~ly
ala ad
o
••
l
op. ~ lon
con ol
o
. acl.leal
powa
.ya .ma.
A
ecan
pape
by
Tinney.
.1.
(2)
emph
••
ized
he
in a ea
o
apa
••
~c o ~. hod.
1n
powe
.y. .~
analyala,
.hQwln~
a
d ama ic
.4u~ lon
1n
opa a lona
CQun~
co~p. .d
o
con enllonal
Me hod.,
In
lh~l
pape ,
lh.
~ulho .
noled
lh~l
naw
be
de elop~d
1n
o de
o
enhanca
he
nod.
o de lnl
alco l hma
had
o
apa al y
o
L-
1
<U-
1)
wi hou
In
his
p.ape ,
p .,) lJ,
..
.1
U)'
Tinney'
••
chama
J
(T-l)
o da 1ne
••
lao
known
••
mlnlmu"
de~ ~e
.l~o llh~
[))
.
BASIC CONCEPTS.
Fo
almpl
i
ci y
h _
p.
~
.
will
b.
••
ic ed
o
.~·.
..
eymmel le
ma i
c
es
o
ma .lea.
ha
a e
aymm. le
In
pa la n
o
non
ze o
Gi en
..
aymm8 ic
ma ix
A-<alJ)
an
~'8oclaled
undi ec ed
, aph
G=(
V,£
)
can
ba
da ined.
, n,j
1.
~h. e
V
ie
he
I
o(
e ice.
(Vl, 2'"
. nl
(
uno de ed
pai .
o
elamen e
o
V
(ad.ea),
also
called
noda.,
poin ~,
bu
••••
e e,
and
o he
namee
o
adl.'
al"'l
b oln
c
hes,
ol CI
o
linea,
I
(Vi'
J)E
.
£,
han
Vi
an
c.
J
a e
.aid
o
ba
.,jJ.c~n .
Tho
,j., ol
o
a
e ex
1e
he
numba
o
~ lcaa
adjacen
o
i .
11 C
h
a ~
n
e ica
••
han.
one- o-ona
map.
.
om
V
on o
he
.a
11.2,
__
.
,n)
18
called
a
numbe lnc
o
C,
d
ced
& aph
••
~ocl~
ad
wilh
lhe
educed
ma ix
I"
ob ained
( om
he
o
lg1noll
, olph
by
de
~
e ln,
e ex
and
inciden
a '
CI
and
adding
a ca
J 11
ba w
••
n
any
pal
o
e ic
••
which
. .
n.l~hbo .
o
nel,hbo .
o
.ach
olhe
.
1
bu
. .
no .
Cone1d.,
he
.ym~. lc
laclo lxa lon
o
•
apa
•••
~. lx
A
in o
UlU.
T ,g
".ph
a
••
ocla ed
o
he
m. l~
_u +u
,_
called
he
illed
,".aph
G -
.ulu lun
oC
(1),
10111
....
ollli.
b
111
apa
••
'U'
only
..
.w
_lam_nle
cd
"
. .
need.d.
ollowing
(21
.o~e
de lnl lona
al. ed
o
.pe ce
ec o .
will
be
in oduced
In
o de
o
make
he
papa
.el -Mu lclen .
A
alncl. on
1_.
ec o
wi h
only
on.
nonze o
ele~~n
.
(
••
~.
1n
loca ion
~).
A
pa h
(o
a
single on
,_
de ined
.8
an
o de ed
I &
oC
OWIl
o
U
which
. e
a ic ly
neeea.a y
{o
he
lo wa d
solu .lon
o
olgmen
o
X
Is
wan ed.
Such.
pa h
1.
•
••
ily
de e mined
om
he
ape ae
a uc u .
o
U
as
ollowa
(21
:
1)
Le
k
be
he
i a
ow
1n
he
pa h
.
2)
G.
he
numbe
o
he
10w.
A
-nu~be ed
nonze o
ele~en
! n
ow
l:.
o
U.
Replacs
k
wi h
his
numbe
and
include
i
in
he
pa
h.
)
I
k
1.
he
laa
ow,
.x1 .
O he wis.,
e u n
o
s ep
2.
Uhan
ow
k
o
A
1.
modi ied,
g aph
a
he
single on
k- h
~ua
only
.h.
ow.
belon&:i .,
b.
aken
in o
accoun
o
he
pa h.
du
in&"
he
. ac o iza ion
o
A.
Hance,
pa ial
~a ix
e ac o 1za ion
and
o wa d
.olu lon
p oca
••
a a
.bo dad
1n
he
••
ma
way.
P o.
.o~
on;
only
ha
(o ola d
and
backwa d
p occ
.....
'
on
he
ind.pendon
ec o
will
b.
cona
lde ed.
I
only
ha
owa
o
a
¥l en
pa h
a e
in ol ed
ln
he
solu ion
o
(1)
he
co esponding
p oce8S
la
called
he
Faa
Fo wa d
llupecL
l ll ly.
(F )
o
A ;
iln
example
conulde
he
Iii
node
IEEE
es
&ya e:n
(41,
who
••
Ina
lx
U
il
uhown
1n
Fl~
u
o
1
.•
.
The
o da in£,
al
c.
1 .hDl
uae(l
i !
T-2,
JI9
and
node
( he
a e ence
noda
In
he
Load
Flow
p ob18~)
1.
dla eaa dod.
A
pa h
c aph
o
hla
ne wo k.
which
compac ly
de.c ib
••
•
11
o
i
••
1n~le on
pa h.
I
..
ahown
In
Fleu .
l.b.
The
~ .ph
1
...
••
which
h
....
di ec ion
••
n
••
( om
low
o
hleh
nod.
o
and
om
hLeh
o
low
nod.
(o
S.
The a
1_
a
ala ion.hlp
be ween
his
pa h
aph
and
he
apa al y
s uc u e
o
U-1,
The
nodee
belon !n
o
he
k- h
noda
pa h
coincide
wi h
he
nonzo o
columma
o
he
k- h
ow
o
U-
1•
H. lx
U-
1
(o
ho
o mQ
example
1_
i an
In
FI,u .
l.c.
A.
may
ba
no ad,
he
numbe
o
nonze o
.lamen e
o
U-
1
1.
di ec ly
el. ed
o
he
a e .c
••
1n~1. on
pa h
l.na h.
Al~o
he
nu~b.
o
nonze o
.lemen .
o
U
10
di ec ly
ela ed
o
he
. e .~e
mul -adds
needed
o
pe o m
he
o wa d
o
back
aUba i u ion
p oce.s
on
.ach
UESCRIPTION
OF
THE
ALCORlnU1.
The
basic
idea
which
mo i . ed
he
alzo i hm
is
qui e
sImple.
La
Ua
obee e
&&81n
he
T-2
o de
inc
applied
o
ha
l~-node ey. e~
ahown
In
i~u e
1.
In
hi.
ca
••
ho
a . a~e
pa h
l.n~ h
o
he
••
1.
4.92~
Thi.
hi~h
alue
1e
due
~alnly
o
he
exi. ance
o
a
lon~
b anch
consis ing
o
e
node.
(2,3,4,6,9,11,12,13).
A88ume
now
h.
ha
s.me
ays am
1.
eo de ed
••
can
ba
aeen
In
icu e
2.
Tha
esul an
& o az.
pa h
l.n~ h
Is
3.
61
and
he
lonse.
b anch
la
compo8od
only
a
$
nodo
••
Th18
be e
••
ul
could
be
.xpec ed
by
_
.1~ple
ins
pec
ion
o
he
ee,
whe e
wo
ae .
o
nod.s
a e
.pp~ en
(enci clo
d )
who.e
p.lh.
ha e
only
wo
common
node
••
Thl.
shapa
haa
b.an
achie ed
by
uainc
8
clus a in¥
slco llhm
in
o de
o
ob ain
wo
waakly
in e connec ed
clue e
••
Th.
educ ion
in
he
a e a,a
pa h
leng h
impli..
a
~o e
epa
••
U-
I
ma ix
.
I
may
ba
obee ad,
howe e ,
ha
h .
1.
done
a
u.o
coa
o
a
ell,h ly
la ,a
Ill-in
o
U
(one
ex a
ele~en
appea s).

IN
Honce,
moll ~ ed
by
he
abo e
eauI e,
he
p ocedu e
18
p opoaadl
ollowlnc
heu is ic
.)
Pe o m
a
clus e ine
o
he
ma ix
A,
which
yields
he
ypical
Bo de ed
Block
018£00_1
Fo m
(BDO )
.
The
op imum
numba
o
elua e a
1s
no
~nown
-.
p lo i".
In
.ana e1,
80me
l i.l.
mua
be
ca ied
ou ,
becau.a
a
la C8
nu~b.
o
elua a..
may
deg ade
auba an ally
lhe
spa .1 y
o
U,
o e ldlnc
lh.
,.1n
e ained
wi h.
aho a
pa h
I.oa h
.
So~. l~.a.
he
~. lx
a uc u e
I ,.l
aUle
••
a
he
numbe
o
elua a a
o
be
adop ed.
b}
Reo de
lhe
node.
belon,!n,
o
.ach
elua e
ollow!n,
he
minimum
dec ee
.l,o llh~
(T-2).
A
ew
clus e ing
algo i hms
ha e
appea ed
In
he
lI e a u e
(a.e
lo
ln~ .nco
(5.6.7]).
All
o
h.~
ha .
he
d awback
o
••
lou.~y
deE .dln,
he
.pa . y
o
U,
a.
h_
.1~_
o
he
bo de
c ows
up
.
Recen ly
a
clus e inc
.l&o l h~
ha.
b
••
n
p oposed
(Bl
which
has
p o ed
o
~ old
h1s
p oblem
almoe
comple ely
.
Thl.
1.
he
on.
adop ed
in
he
ne ~l
sec ion.
DISCUSSION
O
EXPERIKENTAL RESULTS.
Th.
adml .nc.
~a lx
a
~any
elec ical
powe
eye lme
hee
b.en
used
o
e
s
he
p oposed
alco l hm.
The
esul e
co espondin&
o
he
81Eh
la ge
sys ems
a e
abula ed
n
Tebles
I
and
II
o
he
T-2
al.o l hm
and
he
one
p opo.ed
he e
••
pec l .ly.
The
"
Appendix
d
••
c ib
••
b ie ly
h.
che ac e l. lc.
a
s1ze.
anE.
om
116
o
661
nod
•••
a.
~ell
••
he
ac o iza ion
o
he
~. lx
(ac ually.
he
~e lx
has
one
nod.
le
••
h.n
he
nu~b.
lndlc. ed
In
he
l a
column
•••
he
e . ence
o
.lack
node
Ie
no
included).
Fo
e e y
po
••
ble
.Incl. on
he
pa h
lenc h
end
he
o .l
I
1/
i
II
II
I
I
I
I
,
!
I1UL
-1
ADD
HOD U U
Ae
118
25)
••
0
425
175
399 2761 710
265
5<9
3622
912
29)
68'
5435
1229
)83
1038
8971
2503
,,8
1189
10667
2~50
596 1710
15688
007
~hl
18~1
17638
<4972
uL
-1
ADO
HOD
U U
Ae
118
256 968
436
175
<l3
1961
B8
265
552
30426
.87
293
121
396]
1414
38)
1048
T012 1602
"8
1197
8-41b
;1916
5'6
1714
1]819
055
661
1855
lS
3H
5;:'10
RASC
O" SlNCLEi
OX
S
CLU
S
TE
R
ED
SI~CLETO~S
I
SINGL
2 SHICl
.5
SING 10
SINC
2
SIHCL
5 SIHCl 10
SINC
A?
1.01
loP
AD
AP
AD A?
AD
AP
AD
AJ'
AD
AP
AD
'.5
21
1·4.
1
))
25.1
62
35 . 9
88
10 . 6 2)
1]
. 1
2'
17.5
,0
16.9
50
22.6
66
33.2
9.
0.3
130
19.0
56
19.8
57
H . 2
68
1~.1
..
21.3
:
10
.36.2
111
51.5
148 16 . 1
53
18.1
56
23.8
70
19.6
57
28.5
85
"".1
129
60.~
175
20.1
56
23.6
68
29.8
.,
2~.5
137
31.~
1
81
~9.7
260
66.1
32<4
26.3
H5
28.2
H9
H . 6 173
2~.8
1.041
H.9
189
50.9
268
70
. 7
lH
25.8
14]
28.]
152
34.3172
27.
04
188 37 . 1 257
58.9
392
53.5
512
28.6
194
30.9
203
36.6
220
21.
7 190
40.3
278
61.8406
86.0
523
27.8
187 32 . 1 210 37 . 3 221
I
Table
I.
Rc.ul .
ob ained
wi h
T-2
~. hod
.
RANOOM
SINGlETCNS
ClDSTERED
SINGLETONS
1
SINGL
2
SIHGL
5
SHiel
10
S NG
2 S HC 5
SINGL-
o
SIHC
AP
AD
AP
AD
AP
AD
AP
AD
AP
AD
AP
AD
AP
1.0
..
)
21
13.8
13
21
. 5
59
l5.~
89
10.6
'
14
12.2
27
11.7
<l
12.)
13
18.3
54
32
.0
.8
46 . 1 138
13.8
.
37
16.1
<l
10.8
56
14.0
<S
20.4
65
304
.9 110
50.8
149
1-4
. 6
,.
17.5
53
22
. 7 65
13.6
,2
21.8
72
37.9
130
58.1
196
15.2
,7
16.8
'
51
2l.9
6
19.4
.1
28.1
147 45 . 9 231
66,8
328
20.9
104
23.1
III
29.1
134
19.8
101
18.0
146 H . 9 24b
70.1
J4"
21.9
ill
23.6
11-4
29.6
IH
14
. 2 153 36 . 3 245
58
. 7 392 81 .
9510
25.3
162
18.7119
3~.9
102
24.2
IS6
36.6
20
60.739J
84.8
519
2(.8
155 1
9.
6
lB
35. 4 203
A~:
A e
.,e
pi h
l~n' h.
A
O:
A
e a,e
o
al
~ul -.dd
• .
Tabl
II.
R .ul .
ob alned
wl h
he
p o
pol d
.l,
o
l ha.
01
R2
OJ
..
5<
12
8 H
56
H
12
II
5<
71
• 23
5<
72
8
26
57
11
13
28
56
70
12
25
56
68
11
23
55 68
10
21
~
Rl
.2
0)
..
~
I
54 H 8
25
9 !
S4
13 8
28
. ,
5-4
H 8 27 2
53
72
6
18
,
55 13 9 H 3
54 H 8 23 "
55 H 9 26 J
5~
J4
8
2'
J !
J22
abula ed,
••
well
a.
he
a lo.
Rl.
R2.
A3.
and
R4
de ined
1n
(21
which
el e
&
~e
••
u .
o
he
el. i e
ad en _,e
o
uelo&
lne .ad
o
he
lull
o wa d
p oc
••••
u he
e. .
wa .
~.d.
ln ol !nc
epa a.
ec o a
wi h
wo,
i e
end
len
.ndo~ly
cho
••
n
nonze o
al.men a.
o
.ach
c...
100
lal.
wa e
pe (o ~ed.
and
he
a a ac_
pa h
lenc h
a.
wall..
he
a e a,e
lo al
mul -add.
1n
a e
abul. ed
he e.
F om
h...
••
ul .
lh.
a ios
R5
and
R6
de ined
1n
(21
may
alao
be
compu ed.
Th
•••
a
loa
indica o
he
loa.
o
. iclency
o
epa
••
ec o
me hode
a.
he
numbe
~(
andom
nonze 08
c owe.
In
p ac ice,
he
nonze 08
1n
he
epa ..
eclo
a e
no
andomlY
chosen.
To
ea
he
. ec .
o
opolo11cally
el. ed
nod
••
ano he
.a
o(
100
ial.
was
done.
Thi.
1me
he
wo.
i e
anJ
en
nonze o
l~~onl&
o
ll ~
spa ae
eclo
we e
cho
••
n
1n
a
.1mil.
way
aa
la
done
In
dl~&onal
b~nd
o do ing.
A
nod
a
i.
andomly
chossn
••
he
i s
one.
The
adjacen
nodes
o
hose
al eady
numbe ed
a e
conaecu l ely
choaen.
and
so
on
unlil
he
equi ed
numb.
o
nodas
is
achie od.
The
a e age
eGul s
om
lhe~.
.ale
a .
aleo
el en
.
I
can
be
seen,
••
poin ed
ou
1n
121.
ha
he
g ow h
1n
pa h
he
nonzo o
enl l.~
The
las
column
in
Table
II
ahowe
he
adop ed
numbe
o
clus o s
o
~ach
n~lwo k.
These
esul s
sugg.e
he
ollowin,
conclueionsl
The
(ill-in
o(
lh.
ma ix
U
is
sligh ly
inc .aead
wi h
••
epec
o
T-2
al~o i hm.
as
was
expec ed
(abou
l~).
OpPosilely,
he
spa si y
o
0-
1 La
ai&ni(ican ly
imp o ed
(abou
IS~)
which
means.
a.
wa.
axplained
p e iously,
ha
he
a e .ce
pa h
~~n h
o
ana
51n la on
ia
p opo ionally
dac s
••
ad
(column
5).
Beslde&.
he
a e age
o al
mul -adds
in
FF
(o
1
81n~le on
(column
b)
1s
abou
20~
lecs
han
wi h
T-2
algo ilhm.
,
,.
J23
As
he
numbe
o
sin~le on.
~ ows,
bo h
a
l~o l hma
end
o
behA
e
simila ly,
hou~h
(o
10
clus e
e d
.in~le ong
he
p opoaed
&
l~o i hm
s 111
sa es
abou
1~%
o
mul -adds
1n
co~p
a
.d
o
T-2
(colu~
18)
•
-When
h~
aingle ona
a
s
andomly
chosen,
he
ad an ~ge
o(
he
p oposed
algo i hm
o e
T-2
is
las8
impo an .
CONCLUSIONS.
Recen ly,
mo e
in e es
hes
boon
de o ed
o
op4C'Qe
VQc o
me hode,
·
whlch
enhance
o wa d
and
backwa d
a
ubs 1 u 10n
~o=e==o=
·
b7
exploi ing
he
spa s1 y
o
he
independen
ec o
and/o
ho
need
o
know
only
a
subse
o
hs
unknown
ec o .
The
sa no
ochnlque
a
con
be
usod
In
h
e
pa ial
ma ix
e
a
c o iz
a
ion
p oco
~Q
.
Do h
ypes
o
p oblems
a ise
equen ly
In
many
eng1nee ing
ields,
pa icula ly
in
elec ic
powe
sys ems
analysis.
The
speedup
in
a
pa icula
applica 10n
dep
en
ds
no
only
on
ho
numbo
o
nonze oo
in
he
oc o
bu
on
he
epa el y
o
U
and
U-
1,
which
may
be
enhanced
wi h
a
p ope
node
o de ing.
In
his
pape ,
an
a
lgo i hm
is
p opoaed
which
clea ly
imp o es
he
spa si y
o
U-
1
compa ed
o
he
minimum
de~ ee
alg
o i hm.
The
imp o emen
10
anola ed
in o
a
educ 10n
on
he
ope a iona
coun
bee
ides
15:1:.
REFERENCES.
[1] -
Chan
S.M., B andwajn V.,
Pa ial
ma ix
e ac o iza ion.
IEEE
T ansac.
on
PWRS-1,
pp. 193-200, 1986.
[2] -Tinney W.F., B andwajn V.,
Chan
S.M., Spa se Vec o Me hods.
IEEE
T ans.
on
PAS-104,
pp. 29S-301, 1985.
[3] -Tinney W.F., Walke J.W.,
Di ec
Solu ions
o
Equa ions
by
Op imally O de ed
T iangula
P ocee.
IEEE
ol.
55
pp. 1801-1809, 1967
Spa se Ne wo k
Fac o iza ion.
[4] -
IEEE
Commi ee Repo ,
IEEE
Reliabili y
Tes s
Sys ems.
IEEE
T ansac.
on
PAS-98,
pp. 2047-2054, 1979.