Pub
.
Ma
.
UAB
N°
20
Se
.
1980
Ac es
.
VII
JMHL
ON
BLOCK-QUASI-TRIDIAGONAL
MATRICES
José
Vi ó ia
Dp o
.
de
Ma emá ica
Uni e sidade
de
Coimb a
Abs ac
- We
in end o
in e
(and
o
s udy
he
eigen alues
o )
block-quasi-
- idíagonal
ma ices,
We also
men ion
ways
o
handling
he
p o-
blem
o
eigen alues
o a
block-quasi- idiagonal
ma ixand
we
ob ain
uppe
bounds o
he
spec al
adius o
á
(ce ain)
block-
-quasi- idiagonal
ma ixwhich
a ises
in
he
disc e iza ion
o
pa ial
di e en ial
equa ions
o
ellip ic
ype,
sel -adjoin
case
.
1
.
Abou
he in e inn
o
block-quasi- idiagonalma ices
1
.1
- In
his
sec ion
ws
in e es
us
o
in e ing
block-quasi- i-
diagonal
ma ices,
ha is o
say,
ma ices
o
he
o m
A
n-l,n
A
nn
*
The
wo k
o
his
pape
was
suppo ed,
.a di e en
s ages,
by
Fundagáo
Calous e
Gulbenkian
(Lisboa,
Po ugal),
Uni e sidade
de
Mapu o
(Mozambique),
and Ins i u o
Nacional
de
In es igagáo
Cien í ica
(Lisboa,
Po ugal)
.
A
11
A12
00
. . .
0
0
0A
1
n
T
:
_
/
A
21
A
22
A
23
0
.. .
0
0
0
0
00 0 0
. .
.
0
A
n-l,n-2
A
n-l,n
- 1
Anl 0 0 0 0A
n,n-1
whe e
ell
blocks
a e o
he-same
o de
and
commu e
in
pai s
.
Such
ma ices
a ise
in
he
disc e iza ion
o
ellip ic
pa ial
di e en ial equa ions
.
We need a
esul
in ol ing
a
ma ix
ob ained om
T
by
aking
i s
de e minan
conside ing
he (commu ing) blocks as
elemen s
.
Le
AT
:
=
de T
be
such
a
ma ixi
AT
is
he o mal
de e minan
o
T
.
I is
known ha
de A
T
=de
T
.
This esul
allows
us o
manipula e
only
ma ices
o low o -
de ,
when
sol ing
la ga
sys ems
o linea
equa ions
and, like in ou case,
in e ing
la ge block-ma ices
.
1
.2
-
To
in e
he
ma iz
T we shall use an hyb id
me hod
:
classi=
cal
pa i ioning
in
ou
blocks
plus
a
ecu ing p ocedu e
.
Le
us
pa i ion
and
no e as
ollows
is
228
T=
I
A
ln
10
'

i
A
n1
0 0
. .
.0
An .n-1(Ann
whe e
P ie a
block- idiagonal
ma ix
.
I is
known ha he
in e sa
o
a
ma iz
pa i ioned
in
ha
manne
wi h he same
pa i ioning
and
whe e
K
:
=P
-1
-
P
-1
QM
.
M
:
_
-
NRP
-1
,
L
:
_ -
P
-1
QN

and

N
:
=
(S-RP-1Q)-l
.
he
j
h
block-column
wi h
need
only o
in e
wo
ma ices
:
P

and

(S-RP
-1 Q)
.
So we
a la ga
ma iz
P
,
wi h
commu ing
blocks
.
Fo
achie ing
his,
P
-1
=(X
ij
)
pa i ioned
commu e
in
pai s,
om
P
by
eplacing
Then, i we pu
A
:
=de
P and
T-1
:
_

-
K
Í
-N-
LM

~
In
his
way we
ha e o
in e
one can
use a
ecu ing
p ocedu e
.
We
ob ain
a
ma ix
in he sama
way
as
P
and
whe e
he
blocks
X
ij
also
Le
us
deno e
by P
I(i,
)
he
ma iz
ob ained
e
hé
ma iz
i
)
.
=
P-Í-Q-
R
I
S
.

1
i
h
block-line
.
--------------------1A4_1_--
1
ij
:
=de
PI(i,j)

-

(i,j=1,2
.
. .
. .
n-1),
we
ob ain
Xi
.)

= A-1

nij

,

(i,j=1
.2-

n-1)
1,3
-
As
we
ha e
sean,
o
in e ing
a
block-quasi- idiagonal
ma ix
we
need
only
o
in e
wo
ma ices
:
S-RP
-1
Q and A
:
=de
P
.
Bu ,
in
p a i-
ce, we
in e
wo
ma ices
o
he same o de , ins ead o
in e ing
a low o
de
ma ix
S-RP
-1
Q and a high o de
ma ix
P
.
We
ema k
ha he
ma ices
S-RP
-1
Q and
A
ha e he
o de
o
he
o iginal
blocks
A¡J,
2
.
Abdu
he
eigen alues
o a
(ce ain)
block-quasi- idiagonal
ma ix
In
his
sec ion
we in e es us
o he
eigen alue
p oblem
in block-
-quasi- idiagonal
ma ices
.
He e
we
ge
uppe
baunds
o
he
absolu a
alue
o he
eigen alues
by
using
a
ma icial
no m
:
Uppe
bounds o he spec al adius o
a
block-quasi- idiagonal
má
ix,
which a ise
in
he
disc e iza ion
o
pa ial
di e en ial
equa ions
o
ellip ic
ype,
sel -adjoín
case
.
Gi en
he
ma ix
A
B
0 0
.

0
0
B
BAB0
...
000
08AB,
. .
000
.
...
...
. . . .
.
. .
.
0
000
.
..,
BAB
B000
. .
.0BA
(1)
Fo

B
=
(s
i
.)
e
M
s
(K),
K=R
(o

C), we
le
J
we look
o
uppe
bounds o p(T),
whe e
p(T)
is
he
spec al
adius
o
he
ma ix
T, ha is o
say,
p(T)
:
=Max
IX
(T)
I,
whe e
a
(T)
is
any
eigen a-
lue
o
T
.
We
ake
he
(scala )
no m

II
"
i11
1)
,
(1=1,°°),
o
each
block,
so
ob ai
ning a
ma icial
no m

M
i
(T)
.

I is
known,
ha

p(T)
_<p(M
i
(TM
,
(i=1,°°)
.
And
i is
also
known ha

p
(A)
<
II
A
II
J
,

(j=l
.
.°°),
o any
ma ix

A,
and any
(subo dina e)
ma ix
no m
11
-Il
i

s
J1$111
:
=
Max

{
E
Isij1
,
IIBII
.==
Max

(
E
1Bij
l}
.
j=1,2,
. .
.,s
i=1

i=1,2,
.
.
.,
j=1
Hence
we
ha e
Rema k
p(T)<¡¡Alli*
2
11Bili
,
(i=1,°°)
a
e y simple
uppe bound
o
he
eigen alues
o
T
.
The
comple e
e sion
o
his
pape
is
o
appea
in
"Re is ada Uni-
e sidade
de
Coimb a"
.
PRINCIPAL
REFERENCES
TENEUR,
J
.
-
É ude
c i ique
de
mé hodes
apides
pou
la
solu ion
de
Z'équa-
ionde
Poisson
dansun
ec angle
.
(P ojec
de
in
d'é udes)
.
Ins i u
Poly hécnique
de
G enoble
-
Ma h
.
Appliquées
.
G enoble
.
1970
.
VITóRIA,
J
.
-
Ma ices
pa i ionnées
en
blocscommu a i s
.
Gaz
.
Ma
.
(Lisboa),
117-120
:
49-57
(1970),
MR 49
#
3281
Zb
278,
15008
.
In e sion
o
ma ices
pa i ioned
in
commu ingblocks
.
Re
.Cienc
.Ma
.
4
:
29-32
(1973)
.
MR
52
#
31871
Zb 322,
15009
.
Ma izes
de
blocos
comu a i os
.
Tesa
.
Uni
.
Lou engo-Ma ques
C
now
Mapu o
:
,
Mozambique
.
1974
.
No mas
ec o iais
de
ec o es
e
de
ma ices
.
(Repo )
.
Uni e sidade
de
Lou enQo-Ma ques
í-
now Mapu o
7,
Mozambique
.
1974
.
G
E O
M
ETRIA

A
L
GEBR
A
I C A
Clas
.
A
.M.S
.

13,14
Mesap esidencial
:
Ped o
Abellanas,
M
.
Luisa
No onha,
F ancisco
Pé ez
Mo-
naso ,
Edua doCasas