两类循环分块矩阵及其有关算法
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f (x) g (x) In 0 0 In
% * @
25 S
ZPH <` 0
m q 5N ! U
In 0
a
u(x) s(x) v (x) t(x) ,
u(x)f (x) + v (x)g (x) = In
(4)
# a f (ΠR ) = A, g(ΠR ) = 0, N . (4) I A−1 = u(ΠR ). .$ f 7 U f 3 S >W I } 4 . P M B = S CircR (Am−1, · · · , A1 , A0 ) (R = 0) p ! G = D ] {8 P M u(x) ∈ C n×n [x], 5 I B −1 = Ku(ΠR ). . U f 3 U f 4 W I R- s ^P M Z* R- s ^P M L L d ! ` %n x 1 . A = CircR (A0 , A1 , · · · , Am−1 ) (% B = S CircR (Am−1, · · · , A1 , A0 )) IF
u(x), v (x) ∈ C n×n [x],
D *8 ' u x = ΠR , G f (ΠR ) = A, g(ΠR ) = 0, 5 : 0 u(ΠR )A = Inm , J E s ^P M A p !
R-
282
( 1 . P M A = CircR(A0, A1, · · · , Am−1) (R = 0), G A ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 .$ f 7 U f 1 S >W I } 2 . P M B = S CircR (Am−1, · · · , A1 , A0 ) (R = 0), G B p ! L4 W 8 : In : f (x) 7 g (x) L r @ 2 # 8 ( 2 . P M B = SCircR(Am−1, · · · , A1, A0) (R = 0), G B ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 } 3 . P M A = CircR (A0 , A1 , · · · , Am−1 ) (R = 0) p ! G = D ] {8 P M u(x) ∈ C n×n [x], 5 I A−1 = u(ΠR ). Z # a P M A = CircR(A0, A1, · · · , Am−1) (R = 0) p ! N F s ^ ] {8 P M
A11 (x)f (x) + A12 (x)g (x) = h(x), f (x) = A13 (x)h(x).
D f (x) = A13 (x)h(x) u x = ΠR , # a f (ΠR ) = A p ! N h(ΠR ) p ! 5 : .$ f 3 Q = D ] {8 P M h1 (x) 5 I h1 (ΠR )h(ΠR ) = Imn, <` M 0
2
q \N K ) / J b
,
281
A11 (x) A12 (x) A21 (x) A22 (x)
s ^ md Z (2). 5 :
f (x) g (x) = A11 (x) A21 (x) A12 (x) A22 (x) h(x) 0 ,
,
f (x) = A11 (x)h(x), g (x) = A21 (x)h(x). (3)
u(x) v (x) s(x) t(x) f (x) g (x) = h(x) 0 ,
,0
u(x)f (x) + v (x)g (x) = h(x). u(x) s(x) v (x) t(x)
(1)
3# a s ^ ] {8 P M
(2)
1 < ;" Y e mb : A b M [13] , N . P M a
A= f (x) g (x) In 0 0 In ,
'K
5N ! a
C1 =
h(x) u(x) 0 s(x)
v (x) t(x)
,
G h(x) : f (x) 7 g(x) L r @ 2 # 8 z m u(x)f (x) + v(x)g(x) = h(x). Z # a Zs ^ ] {8 P M A H m q 5N ! a C1, N M 0
A0 RA ⎜ m −1 A=⎝ ··· RA1 ⎛ A1 A0 ··· RA2 ··· ··· ··· ··· Am−2 Am−3 ··· RAm−1 ⎞ Am−1 Am−2 ⎟ ⎠ ··· A0
* a Rs ^P M / u A = CircR (A0 , A1 , · · · , Am−1) ∈ CMR . R d 2000 4 A 11 $<E 2001 3 A 21 $<E y{ F:1<9X]RV z F:1 V-&Dicz
*8E h(x) : f (x) 7 g(x) L " 2 # 8 L t 8 h(x) : f (x) 7 g(x) L r @ 2 # 8 [13] . [10] 6 5− ' p ! P M A a mn ? (m, n) a ^ (R, r) - P M P M R p ! G A 1 #: mn ? (m, n) a ^ (R, r)- P M .$ f 3 r- P M L-d W!,W IF 6 6 ' P M A, B T a mn ? (m, n) a ^ (R, r)- P M G AB = BA, AB a mn ? (m, n) a ^ (R, r)PM 6 7 . A = CircR(A0, A1, · · · , Am−1), B = SCircR(Am−1, · · · , A1, A0), G B = AK % BK = A, K = S CircR (0, · · · , 0, In ) (nZ). 2 / : !4 | w " % ~ } 1 . P M A = CircR (A0 , A1 , · · · , Am−1 ) (R = 0), G A p ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 Z APN . h(x) : f (x) 7 g(x) L |r@2 #8 G=D A11(x), A12(x), A13(x) ∈ C n×n [x] 5 I
P
25 2001
S P
4
@
2
& + A
ACTA MATHEMATICAE APPLICATAE SINICA
Vol. 25 No. 2 Apr., 2002
0 VUY WX
(
7
+V2hzMfV ~B l _ 611930)
(
&
∗
SjV-? ',B > +V 610054)
9 2 ci* y6NKq>1 "6 ~6 R- q\NK X( R- q\N Ko )t k.nefS/_ AJ3 V7 G_HDO} ;t k.n/ _ A;t k.nB q\NKYJefS _ AJJb t k.n { R- q\NK X( R- q\NK y6NK o efS
∗
C R = rIn
% * @ 25 S (% C R = In , % C R = −In ) 3 * a rs ^P M ( s ^P M g s ^P M); C n = 1 3 * a r- P M ) } 3 2 ' A = CircR (A0 , A1 , · · · , Am−1), ` Ai = Circr (a(0i) , a(1i) , · · · , a(ni− 1 ) (i = 0, m − 1), R = Circr (r0 , r1 , · · · , rn−1 ), G * A a mn ? (m, n) a ^ (R, r)PM } 3 3 [9] . A0 , A1 , · · · , Am−1 , R T a n ?P M R 7 Ai (i = 0, m − 1) W! ` R 0 n q 8 L mn ?P M
. (3) 8 Q h(x) : f (x) 7 g(x) L 2 # 8 % . h(x) : f (x) 7 g(x) L 5 h(x) L " | 2 # 8 G = D A11 (x), A21 (x) ∈ C n×n [x][13] , 5 I
¯ (x) = h(x). u(x)f (x) + v (x)g (x) = u(x)A11 (x) + v (x)A21 (x) h
1
51 8 v <$
P M + v m0 p L C D v : " j $ X y l # w . .s s ^P MD . L ' 3 s k l # 3 w K 8 h^ u- [1−3] . s ^P Mu p= ; ` / Y L L Z P M &8 q [4] . [5] Z P M s ^P M L R + 0 ` IL d E M $ p ! ~ L ^ [1−7] . I a ! ~ 3 L s ^P M [8] Z* R- s ^P M [9]. Z pnd a jL s ^P M RRs ^P M .5 P 7 e ) w h ! R 3 L m N W . FFT ` IL d Rs ^P M0 L hU [10] . B 4 0 hU L C Q @] a4C N F 9 4C g auC`)x v ` . L @ o L ( C `% 0j $ % [10], P M R p ! J M 5 I (- @@O = F s X e 8L j - ] {8 P Mr @ 2 # 8 W P Lp ! P M L d v m0 p L C N hU M O X t p * ? r t \ I a .5 4 0 hU L v m0 p g hU L L , a . LC |p& $ ? L Z U [11,12] , B W E { 0 ! L v m0 p Lg hU D s ^P Md 0CLW8 +0C3CL 8 e 9 Q x J; hU } 3 1 [8] . A0 , A1 , · · · , Am−1 , R T a n ?P M R 7 Ai (i = 0, m − 1) W! ` R 0 n q 8 L mn ?P M
h1 (x)A11 (x)f (x) + h1 (x)A12 (x)g (x) = h1 (x)h(x) = In ,
N 5I
: f (x) 7 g(x) L r @ 2 # 8 [13] . BCN # a In : f (x) 7 g(x) L r @ 2 # 8 N = D
In u(x)f (x) + v (x)g Байду номын сангаасx) = In .
280 A0 ⎜ A1 A=⎝ ··· Am−1 ⎛ A1 A2 ··· RA0 ··· ··· ··· ··· Am−2 Am−1 ··· RAm−3 ⎞ Am−1 RA0 ⎟ ⎠ ··· RAm−2
* a Z* Rs ^P M / u A = S CircR (A0 , A1 , · · · , Am−1 ) ∈ SCMR . R C R = rIn (% C R = In , % C R = −In ) 3 * a Z* rs ^P M (Z* s ^P M Z*g s ^P M); C n = 1 3 * a Z* r- P M d e L R- s ^P M | p e [8] Z R p ! L s X J P L (- i ^ j W6 e p e [8]; Z e | p e [9] L ; s X A L ^+ R T a rP M G A * a Z* mn ? } 3 4 ' A ∈ SCMR , (m, n) a ^ (R, r)PM } 3 5 [9] . P M A = (Akj ) ∈ C mn×mn , Akj , B ∈ C n×n . B 7 A L ( : B - A L }| k ^ / a BA. 6 1 ' ΠR = CircR(0, In, 0, · · · , 0), G ΠiR = CircR(0, · · · , 0, iI+1 Πm n , 0, · · · , 0), R = RImn , U Π0 = I . mn R 6 2 ΠR L 8R N ] {8 Det (xImn − ΠR) = xmIn − R = g(x) (nZ). 6 3 [8] . m P M A ∈ C mn×mn , G A = CircR (A0 , A1 , · · · , Am−1 ) L4 W 8 : −1 A = f (ΠR ), f (x) = Ai xi (nZ). i=0 6 4 . s ^ ] {8 P M
% * @
25 S
ZPH <` 0
m q 5N ! U
In 0
a
u(x) s(x) v (x) t(x) ,
u(x)f (x) + v (x)g (x) = In
(4)
# a f (ΠR ) = A, g(ΠR ) = 0, N . (4) I A−1 = u(ΠR ). .$ f 7 U f 3 S >W I } 4 . P M B = S CircR (Am−1, · · · , A1 , A0 ) (R = 0) p ! G = D ] {8 P M u(x) ∈ C n×n [x], 5 I B −1 = Ku(ΠR ). . U f 3 U f 4 W I R- s ^P M Z* R- s ^P M L L d ! ` %n x 1 . A = CircR (A0 , A1 , · · · , Am−1 ) (% B = S CircR (Am−1, · · · , A1 , A0 )) IF
u(x), v (x) ∈ C n×n [x],
D *8 ' u x = ΠR , G f (ΠR ) = A, g(ΠR ) = 0, 5 : 0 u(ΠR )A = Inm , J E s ^P M A p !
R-
282
( 1 . P M A = CircR(A0, A1, · · · , Am−1) (R = 0), G A ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 .$ f 7 U f 1 S >W I } 2 . P M B = S CircR (Am−1, · · · , A1 , A0 ) (R = 0), G B p ! L4 W 8 : In : f (x) 7 g (x) L r @ 2 # 8 ( 2 . P M B = SCircR(Am−1, · · · , A1, A0) (R = 0), G B ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 } 3 . P M A = CircR (A0 , A1 , · · · , Am−1 ) (R = 0) p ! G = D ] {8 P M u(x) ∈ C n×n [x], 5 I A−1 = u(ΠR ). Z # a P M A = CircR(A0, A1, · · · , Am−1) (R = 0) p ! N F s ^ ] {8 P M
A11 (x)f (x) + A12 (x)g (x) = h(x), f (x) = A13 (x)h(x).
D f (x) = A13 (x)h(x) u x = ΠR , # a f (ΠR ) = A p ! N h(ΠR ) p ! 5 : .$ f 3 Q = D ] {8 P M h1 (x) 5 I h1 (ΠR )h(ΠR ) = Imn, <` M 0
2
q \N K ) / J b
,
281
A11 (x) A12 (x) A21 (x) A22 (x)
s ^ md Z (2). 5 :
f (x) g (x) = A11 (x) A21 (x) A12 (x) A22 (x) h(x) 0 ,
,
f (x) = A11 (x)h(x), g (x) = A21 (x)h(x). (3)
u(x) v (x) s(x) t(x) f (x) g (x) = h(x) 0 ,
,0
u(x)f (x) + v (x)g (x) = h(x). u(x) s(x) v (x) t(x)
(1)
3# a s ^ ] {8 P M
(2)
1 < ;" Y e mb : A b M [13] , N . P M a
A= f (x) g (x) In 0 0 In ,
'K
5N ! a
C1 =
h(x) u(x) 0 s(x)
v (x) t(x)
,
G h(x) : f (x) 7 g(x) L r @ 2 # 8 z m u(x)f (x) + v(x)g(x) = h(x). Z # a Zs ^ ] {8 P M A H m q 5N ! a C1, N M 0
A0 RA ⎜ m −1 A=⎝ ··· RA1 ⎛ A1 A0 ··· RA2 ··· ··· ··· ··· Am−2 Am−3 ··· RAm−1 ⎞ Am−1 Am−2 ⎟ ⎠ ··· A0
* a Rs ^P M / u A = CircR (A0 , A1 , · · · , Am−1) ∈ CMR . R d 2000 4 A 11 $<E 2001 3 A 21 $<E y{ F:1<9X]RV z F:1 V-&Dicz
*8E h(x) : f (x) 7 g(x) L " 2 # 8 L t 8 h(x) : f (x) 7 g(x) L r @ 2 # 8 [13] . [10] 6 5− ' p ! P M A a mn ? (m, n) a ^ (R, r) - P M P M R p ! G A 1 #: mn ? (m, n) a ^ (R, r)- P M .$ f 3 r- P M L-d W!,W IF 6 6 ' P M A, B T a mn ? (m, n) a ^ (R, r)- P M G AB = BA, AB a mn ? (m, n) a ^ (R, r)PM 6 7 . A = CircR(A0, A1, · · · , Am−1), B = SCircR(Am−1, · · · , A1, A0), G B = AK % BK = A, K = S CircR (0, · · · , 0, In ) (nZ). 2 / : !4 | w " % ~ } 1 . P M A = CircR (A0 , A1 , · · · , Am−1 ) (R = 0), G A p ! L4 W 8 : In : f (x) 7 g(x) L r @ 2 # 8 Z APN . h(x) : f (x) 7 g(x) L |r@2 #8 G=D A11(x), A12(x), A13(x) ∈ C n×n [x] 5 I
P
25 2001
S P
4
@
2
& + A
ACTA MATHEMATICAE APPLICATAE SINICA
Vol. 25 No. 2 Apr., 2002
0 VUY WX
(
7
+V2hzMfV ~B l _ 611930)
(
&
∗
SjV-? ',B > +V 610054)
9 2 ci* y6NKq>1 "6 ~6 R- q\NK X( R- q\N Ko )t k.nefS/_ AJ3 V7 G_HDO} ;t k.n/ _ A;t k.nB q\NKYJefS _ AJJb t k.n { R- q\NK X( R- q\NK y6NK o efS
∗
C R = rIn
% * @ 25 S (% C R = In , % C R = −In ) 3 * a rs ^P M ( s ^P M g s ^P M); C n = 1 3 * a r- P M ) } 3 2 ' A = CircR (A0 , A1 , · · · , Am−1), ` Ai = Circr (a(0i) , a(1i) , · · · , a(ni− 1 ) (i = 0, m − 1), R = Circr (r0 , r1 , · · · , rn−1 ), G * A a mn ? (m, n) a ^ (R, r)PM } 3 3 [9] . A0 , A1 , · · · , Am−1 , R T a n ?P M R 7 Ai (i = 0, m − 1) W! ` R 0 n q 8 L mn ?P M
. (3) 8 Q h(x) : f (x) 7 g(x) L 2 # 8 % . h(x) : f (x) 7 g(x) L 5 h(x) L " | 2 # 8 G = D A11 (x), A21 (x) ∈ C n×n [x][13] , 5 I
¯ (x) = h(x). u(x)f (x) + v (x)g (x) = u(x)A11 (x) + v (x)A21 (x) h
1
51 8 v <$
P M + v m0 p L C D v : " j $ X y l # w . .s s ^P MD . L ' 3 s k l # 3 w K 8 h^ u- [1−3] . s ^P Mu p= ; ` / Y L L Z P M &8 q [4] . [5] Z P M s ^P M L R + 0 ` IL d E M $ p ! ~ L ^ [1−7] . I a ! ~ 3 L s ^P M [8] Z* R- s ^P M [9]. Z pnd a jL s ^P M RRs ^P M .5 P 7 e ) w h ! R 3 L m N W . FFT ` IL d Rs ^P M0 L hU [10] . B 4 0 hU L C Q @] a4C N F 9 4C g auC`)x v ` . L @ o L ( C `% 0j $ % [10], P M R p ! J M 5 I (- @@O = F s X e 8L j - ] {8 P Mr @ 2 # 8 W P Lp ! P M L d v m0 p L C N hU M O X t p * ? r t \ I a .5 4 0 hU L v m0 p g hU L L , a . LC |p& $ ? L Z U [11,12] , B W E { 0 ! L v m0 p Lg hU D s ^P Md 0CLW8 +0C3CL 8 e 9 Q x J; hU } 3 1 [8] . A0 , A1 , · · · , Am−1 , R T a n ?P M R 7 Ai (i = 0, m − 1) W! ` R 0 n q 8 L mn ?P M
h1 (x)A11 (x)f (x) + h1 (x)A12 (x)g (x) = h1 (x)h(x) = In ,
N 5I
: f (x) 7 g(x) L r @ 2 # 8 [13] . BCN # a In : f (x) 7 g(x) L r @ 2 # 8 N = D
In u(x)f (x) + v (x)g Байду номын сангаасx) = In .
280 A0 ⎜ A1 A=⎝ ··· Am−1 ⎛ A1 A2 ··· RA0 ··· ··· ··· ··· Am−2 Am−1 ··· RAm−3 ⎞ Am−1 RA0 ⎟ ⎠ ··· RAm−2
* a Z* Rs ^P M / u A = S CircR (A0 , A1 , · · · , Am−1 ) ∈ SCMR . R C R = rIn (% C R = In , % C R = −In ) 3 * a Z* rs ^P M (Z* s ^P M Z*g s ^P M); C n = 1 3 * a Z* r- P M d e L R- s ^P M | p e [8] Z R p ! L s X J P L (- i ^ j W6 e p e [8]; Z e | p e [9] L ; s X A L ^+ R T a rP M G A * a Z* mn ? } 3 4 ' A ∈ SCMR , (m, n) a ^ (R, r)PM } 3 5 [9] . P M A = (Akj ) ∈ C mn×mn , Akj , B ∈ C n×n . B 7 A L ( : B - A L }| k ^ / a BA. 6 1 ' ΠR = CircR(0, In, 0, · · · , 0), G ΠiR = CircR(0, · · · , 0, iI+1 Πm n , 0, · · · , 0), R = RImn , U Π0 = I . mn R 6 2 ΠR L 8R N ] {8 Det (xImn − ΠR) = xmIn − R = g(x) (nZ). 6 3 [8] . m P M A ∈ C mn×mn , G A = CircR (A0 , A1 , · · · , Am−1 ) L4 W 8 : −1 A = f (ΠR ), f (x) = Ai xi (nZ). i=0 6 4 . s ^ ] {8 P M