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A numerical algorithm for computing the restricted singular value decomposition of matrix triplets
es
Submittedby Richard A. Brualdi
An algoriam is developed for computing the restricted singular value decomposipIets. It consists of three stages: we first show that SVD) of general matrix orthogonal transformations can constructed to extract a regular triplet from a given essing the regular tiplet, we reduce general matrix triplet; after p to computing o1~: ordinary singular value decomposition of a product of matrices having the same number of dimensions; then we apply the tliantz technique to this matrix product. Other structural indices of the obtained by computing the ranks of certain submatrices. NumericaI examples are provided to illustrate the accuracy of the algorithm.
1.
INT
N
singular vahe &mnnposition ( the ordinary matrices [s]. The
singular
*Part of the work was done while
the author was a visitor at
1 is
minimization wi
an efficient met
to preprocess the regular triplet, which will e dimension; in Section 5,
rAB = ran
rA,
= r
c = ran
?
owing nomenclature is the ordinary s
the restricted sin n this paper only the case case of complex matrices are s
s is discussed; extensions to
n this section we brie the concept of more detailed factorization of t can also be considered as the two semiinner is described in the following theorem.
A'
B'
ZZ
C’
re j
k
j
p-j-+-s2
l
tl
s1
‘B
r
sg
q - l*.- t 1 r Sl
Sl
,
=
i
i
Ci =
min (11 DE RPXQ
=
.=
?
a
1 1L%
65)
li
I
=
&9
+
i
zz 9
i=s+r+
C)
‘i
)...,S
i=l,...,
+
4
“9
89
+min(s,,s,
s+r+min(sl,s2
c
ve
n submatrices in some in
3.
LCORITHM. umns
of
CF;
we
obtain
II row rank, and c’,“’ = cl*).
nin
Step 5.
,,s _-WJJCret
of
-.
d
ces
e
result follows
is section we m
aracterization of the restricted singul
some preparations for the next section where we uct of three matrithe same dimension.
now on we will
arge numerical error when
ot explicitly forming the p
a*bove, we obtain
(4) Therefore the problem of finding he singular values of that of finding the singular values o and R, are upper triangular and h 2. m < n. Similarly to the above orthogonal sue
“’x”‘, and R, I E
- ‘AC-’
reduces to
“’x”’ are upper triangular.
Before we proceed to the next section, we will give an alternative preprocessing procedure. up with the triplet (A(;;,
form of one of the m
for the
vens 44-4
X
x X
x
X X
x
X X
X
X X
x
X X
X
X
X X X
X
X X
X
X
X
X
X X
X
X X
X X
X
X X
X
X
X
X
X
X X
-1 Q ii+1
i
=
ai+li+l
-1
X
9
ion concise,
resen
1
=a
we
wi
1
1
G?
%2
es.
;
1 .
a12
Q22 en
=
42 . a^22
a
in
Q-
+ 4 +
e
.
ing
-FB+92,-@es
ri j=r',,
ces are
in s
t
ANZ re consists 0
s.
ive
iagon
-1
=
-1
C
ces
s S
.
e
. -00 I
.
I
I
I
23
Ol’Tt’iJT
FOR
w
2.2904
-I.%508
I.4127
-0.7208
1.1723
-2.0553
-0.4267
0.462
0.0183
0.2217
O/l785
0.364
LA8546
-0.2497
0.2244
-0.4270
0.9989
-0.2949
1.2102
0.0000
0.0000
0.0000
0.0000
0
0
0
0
0.0000
0.9946
.OOOO
0.0000
0.0000
.7009
-0.0774
0.0000
0.0000
0
0
0.0000
0.0000
0.0000
0.5746
0.0000
0
0
0.0000
0.0000
0.0000
0.0000
0.2448
0.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
where P and Q are nonsingular, and ?I and V are orthogonal. particular choice we have CondWl=35.2673,
For our
Csnd~Q~=$32.3483.
After the extraction procedure A is transformed to the form shown in Tablle 1. The matrix B is transformed to the form shown in Table 2, and the matrix C is transformed to the form shown in Table 3.
Columns
4 through 1.45'10
-3.1466 0.2594
-0.0582
0.3997
-0.04n
7 0.0562
0.0408
-0.2734
0.1298
0
-0.5657
-0.7552
-0.4533
0.1773
0
0.%495
-0.5582
-O.'lO91
O.S99
0
-1.3672
0.1429
-0.1206
-0.3251
0.6238
-a.'8328
0
0.4941
-0.0185
0.0110
0.0269
0.1%25
0.4309
0
-0.2445
0
6
0
0
8
2.2266
0
0
0
0
0
Columns
8
through
IO
0
e re
sin
en we
are
values usin
icesa!3 follows: # r-=3&? sl=2,
s2=2.
roblena: Formula-
sition
of matrix
t~~~et~,
rization and the general Gauss-
S
Submittedby Richard A. Brualdi
An algoriam is developed for computing the restricted singular value decomposipIets. It consists of three stages: we first show that SVD) of general matrix orthogonal transformations can constructed to extract a regular triplet from a given essing the regular tiplet, we reduce general matrix triplet; after p to computing o1~: ordinary singular value decomposition of a product of matrices having the same number of dimensions; then we apply the tliantz technique to this matrix product. Other structural indices of the obtained by computing the ranks of certain submatrices. NumericaI examples are provided to illustrate the accuracy of the algorithm.
1.
INT
N
singular vahe &mnnposition ( the ordinary matrices [s]. The
singular
*Part of the work was done while
the author was a visitor at
1 is
minimization wi
an efficient met
to preprocess the regular triplet, which will e dimension; in Section 5,
rAB = ran
rA,
= r
c = ran
?
owing nomenclature is the ordinary s
the restricted sin n this paper only the case case of complex matrices are s
s is discussed; extensions to
n this section we brie the concept of more detailed factorization of t can also be considered as the two semiinner is described in the following theorem.
A'
B'
ZZ
C’
re j
k
j
p-j-+-s2
l
tl
s1
‘B
r
sg
q - l*.- t 1 r Sl
Sl
,
=
i
i
Ci =
min (11 DE RPXQ
=
.=
?
a
1 1L%
65)
li
I
=
&9
+
i
zz 9
i=s+r+
C)
‘i
)...,S
i=l,...,
+
4
“9
89
+min(s,,s,
s+r+min(sl,s2
c
ve
n submatrices in some in
3.
LCORITHM. umns
of
CF;
we
obtain
II row rank, and c’,“’ = cl*).
nin
Step 5.
,,s _-WJJCret
of
-.
d
ces
e
result follows
is section we m
aracterization of the restricted singul
some preparations for the next section where we uct of three matrithe same dimension.
now on we will
arge numerical error when
ot explicitly forming the p
a*bove, we obtain
(4) Therefore the problem of finding he singular values of that of finding the singular values o and R, are upper triangular and h 2. m < n. Similarly to the above orthogonal sue
“’x”‘, and R, I E
- ‘AC-’
reduces to
“’x”’ are upper triangular.
Before we proceed to the next section, we will give an alternative preprocessing procedure. up with the triplet (A(;;,
form of one of the m
for the
vens 44-4
X
x X
x
X X
x
X X
X
X X
x
X X
X
X
X X X
X
X X
X
X
X
X
X X
X
X X
X X
X
X X
X
X
X
X
X
X X
-1 Q ii+1
i
=
ai+li+l
-1
X
9
ion concise,
resen
1
=a
we
wi
1
1
G?
%2
es.
;
1 .
a12
Q22 en
=
42 . a^22
a
in
Q-
+ 4 +
e
.
ing
-FB+92,-@es
ri j=r',,
ces are
in s
t
ANZ re consists 0
s.
ive
iagon
-1
=
-1
C
ces
s S
.
e
. -00 I
.
I
I
I
23
Ol’Tt’iJT
FOR
w
2.2904
-I.%508
I.4127
-0.7208
1.1723
-2.0553
-0.4267
0.462
0.0183
0.2217
O/l785
0.364
LA8546
-0.2497
0.2244
-0.4270
0.9989
-0.2949
1.2102
0.0000
0.0000
0.0000
0.0000
0
0
0
0
0.0000
0.9946
.OOOO
0.0000
0.0000
.7009
-0.0774
0.0000
0.0000
0
0
0.0000
0.0000
0.0000
0.5746
0.0000
0
0
0.0000
0.0000
0.0000
0.0000
0.2448
0.0000
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
where P and Q are nonsingular, and ?I and V are orthogonal. particular choice we have CondWl=35.2673,
For our
Csnd~Q~=$32.3483.
After the extraction procedure A is transformed to the form shown in Tablle 1. The matrix B is transformed to the form shown in Table 2, and the matrix C is transformed to the form shown in Table 3.
Columns
4 through 1.45'10
-3.1466 0.2594
-0.0582
0.3997
-0.04n
7 0.0562
0.0408
-0.2734
0.1298
0
-0.5657
-0.7552
-0.4533
0.1773
0
0.%495
-0.5582
-O.'lO91
O.S99
0
-1.3672
0.1429
-0.1206
-0.3251
0.6238
-a.'8328
0
0.4941
-0.0185
0.0110
0.0269
0.1%25
0.4309
0
-0.2445
0
6
0
0
8
2.2266
0
0
0
0
0
Columns
8
through
IO
0
e re
sin
en we
are
values usin
icesa!3 follows: # r-=3&? sl=2,
s2=2.
roblena: Formula-
sition
of matrix
t~~~et~,
rization and the general Gauss-
S