AN ITERATIVE METHOD FOR SYMMETRIC SOLUTIONS OF THE MATRIX EQUATION AXB + CXD = F

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1 010 D 11 i ± Ψ E 7 3 "7 4 I Nov. 010 MATHEMATICA NUMERICA SINICA Vol.3 No.4 BD@: AXB + CXD = F >9A< =;C? * 867 Bο*c'Dfi $ΛDf ffl±fi 5300) 1 - meffi BΞ14OL) ψ_#1<*φnt»h~χstffi AXB + CXD = F 4E ρ» KffΨΨν. X<*ΦN χstffi AXB + CXD = F 4/_>&Om<*lffi Φ?EC. -χstffi AXB + CXD = F _Eρ»i m_4& RffC E R!nEρχs X 1 k<*φn ρl_ &0χsTffi4Eρ»; BVps4!n<*χs v&o<* #}5R Eρ». IR E R_>4χs X 0 χstffi AXB + CXD = F 4ffΨΨνEρ»&Oνl<*T»:4χsTffi A XB + C XD = F 4}5R Eρ»0. ψ 4 - xk1φn4_7>. νο : χstffi; <*ΦN; Eρ»; }5R»; ffψψν MR 000) 4%μff: 65F10 AN ITERATIVE METHOD FOR SYMMETRIC SOLUTIONS OF THE MATRIX EQUATION AXB + CXD = F Zhou Hailin Taizhou Inst of Sci&Tech NJUST Taizhou 5300 Jiangsu China) Abstract Motivated by the conjugate gradient method an iterative algorithm is presented to solve the linear matrix equation AXB + CXD = F over symmetric matrix X and its optimal approximation. By this method the solvability of the equation AXB + CXD = F over symmetric X can be determined automatically. When the equation AXB + CXD = F is consistent over symmetric X its solution can be obtained within finite iteration steps in the absence of round off errors for any initial symmetric matrix X 1 and its least-norm symmetric solution can be derived by choosing a suitable initial iterative matrix. Furthermore its optimal approximation to the given matrix X 0 can be obtained by choosing the least-norm symmetric solution of a new matrix equation A XB + C XD = F. Some numerical examples verify the efficiency of the algorithm. Keywords: matrix equation; iterative algorithm; symmetric solution; least-norm solution; optimal approximation 000 Mathematics Subject Classification: 65F10 * 009 D 11 i 30 v0.

2 414 Π Φ D 010 D 1. 0 R m n ffip m n jffir~o I n ffip n +ffiffir SR n n n ASR n n Wffl ffip R n n ~DßffirnODßffir~o. =S <AB>=trB T A) o ff`.3q fl Frobenius Q A = <AA>=trA T A) 1. Dffir A =a 1 a a n ) R m n J~ a i R m i =1 n veca) =a T 1 a T a T n ) T A B ffipffir A d B 3 Kronecker. ±χλj#6</!. )$ I. ^=ffir A C R m n BD R n p F R m p S X SR n n l1 AXB + CXD = F. 1) )$ II.! I.^h J ~ofl S E D^=3 X 0 R n n S X S E l1 X X 0 = min X S E X X 0. ) ffirsfl3s!^ J3)7QSnr 3WZffz. 8;3-=ffirSfl H` a) AX = B b) AX + XB = C c) AXB = C d) A 1 XB 1 + A XB + + A k XB k = C e) AX + YB = C 5 fl 1) Mßw/0@Gffir7n zw'cp3gξn t KU 10rGffj. lz3smλj^ltzw SVD) 3 hsltzw GSVD) 4 5 n flπ.gw CCD) 6 5. ldflffla conjugate gradient) Λ03NK FF9 7 fn;)mhff0ffirsfl AXB = C ffirsflψ A 1 XB 1 = C 1 A XB = C 3g} ; Peng 8 n Wang 5 9 Wffl S 0ffirSfl AXB + CY D = F flxffir n AXB + CX T D = E 3L Ω4 Jfi. ±χ^";) M S 0ffirSfl AXB + CXD = F 3Dß JΩΦΦμ. χ ~^"3 z±kwj0z M3^6=..!ß*& I # 1. 1) yaffir A R m n B R n p C R m n D R n p n F R m p U X 1 SR n n ; ) Ξ R 1 = F AX 1 B + CX 1 D) P 1 = A T R 1 B T + C T R 1 D T Q 1 = 1 P 1 + P1 T ) k := 1. 3) `j R k =0y R k 0 Q k =0 ; Xo Y k := k +1. 4) Ξ X k = X k 1 + R k 1 Q k 1 Q k 1 R k = F AX k B + CX k D) P k = A T R k B T + C T R k D T

3 4 I m: χstffi AXB + CXD = F Eρ»4<*ΦN 415 Q k = 1 P k + P T k ) trp kq k 1 ) Q k 1 Q k 1. 5) w/ 3). *X Q i SR n n X i SR n n i =1.». 1. Dffir MN R s t b trm T N)=0oßffir M N.tv. /fi 1. Dd ;) M 1 Ωμg3 R i Q i P j i j =1 ^ ψ. b Q T i = Q i o trr T i+1 R j) = trr T i R j) R i Q i trq ip j ). 3) trri+1 T R j)=trf AX i+1 B + CX i+1 D)) T R j = tr F A X i + R i ) Q i Q i B C X i + R i ) ) T Q i Q i D R j = tr R Ti R i ) Q i AQ ib + CQ i D) T R j = trri T R j) R i Q i trbt Q T i AT R j ) R i Q i trdt Q T i CT R j ) = trri T R j) R i Q i trqt i AT R j B T ) R i Q i trqt i CT R j D T ) = trr T i R j) R i Q i trq ip j ). /fi. D M 1 ;)kfl~μg3 R i Q j k ^ trr T i R j)=0 trq T i Q j)=0 ij =1 k i j. 4) ψ. Z CiAMw>. b trr T i R j)=trr T j R i) #;}?w> i>jh 7ff. %. D k =ΞQ/ Q T 1 = Q 1 V) 1 n M 1 ^ trr T R 1)=trR1 T R 1) R 1 Q 1 trq 1P 1 )= R 1 R 1 Q 1 tr P 1 + P1 T Q 1 = R 1 R 1 Q 1 trq 1Q 1 )=0 trr T R 1)=0. 5) trq T P + P Q 1 )=tr T Q 1 trp Q 1 ) Q 1 QT 1 Q 1 = trp Q 1 ) trp Q 1 )=0 P Q = tr 1 + P Q 1 ) T ) trp Q 1 ) trq T Q 1 )=0. 6)

4 416 Π Φ D 010 D Ωf 4) D k = s ff. ΞQ/ Q T s = Q s trq T s Q s 1 )=0 V) 1 n M 1 d 5). Π3w> #;^ trrs+1r T s )= R s R s Q s tr Q T s Q s + trp ) sq s 1 ) Q s 1 Q s 1 = R s R s Q s tr Q T s Q s )+ trp sq s 1 ) Q s 1 trqt s Q s 1 ) = R s R s Q s trqt s Q s )=0 d 6).Π3w> #;%1/ trq T s+1 Q s)=0. D j =1 V) 1 ^ trr T s+1 R j)=0. D j = 3 s 1 ΞQ/ trr T s R j )=0 trq T s Q j)=0trq T s Q j 1) =0 V) 1 n M 1 1/ trr T s+1 R j)=trr T s R j) R s = R s Q s tr Qs P j +Q s P j ) T = R s Q s tr = R s Q s Q T s D j =1 s 1 ^ Q s trq sp j ) ) = R s Q s tr Q j + trp jq j 1 ) Q j 1 ) Q j 1 trq T s Q j )+ trp jq j 1 ) Q j 1 trqt s Q j 1 ) Ps+1 trq T + Ps+1 s+1q j )=tr T Ps+1 + Ps+1 = tr T Q T s =0. trp )T s+1q s ) Q s Q s Q j P j + P T j ) Q j trp s+1q s ) Q s trq T s Q j ) Ps+1 Q = tr j +P s+1 Q j ) T ) = trp s+1 Q j )=trq j P s+1 ). ΞQ/ trr T s+1 R j)=0trr T s+1 R j+1) =0 V) 1 ^ trq T s+1 Q j)=trq j P s+1 )= Q j R j trr T j R s+1 ) trr T j+1 R s+1) = Q j R j trr T s+1 R j ) trr T s+1 R j+1) =0. U$ D k = s +14)off.. CiAMy V) 1/0w>. /fi 3. f X fl! I 3L o trx X k )Q k = R k k =1. 7) ) ψ. Z CiAMw>. k =1h b X X 1 ) T = X X 1 o trx X 1 )Q 1 =tr X X 1 ) P 1 + P1 T = trx X 1 )P 1 = trx X 1 )A T R 1 B T +trx X 1 )C T R 1 D T = trbr1 T AX X 1) + trdr1 T CX X 1)

5 4 I m: χstffi AXB + CXD = F Eρ»4<*ΦN 417 = trr1 T AX X 1 )B+trR1 T CX X 1 )D = trr1 T F AX 1B CX 1 D) = trr1 T R 1)= R 1 Kflr trx X 1 )Q 1 = R 1. 8) Ωf k = s h 7) ff.. Ufl trx X s+1 )Q s =tr X X s R s Q s Q s)q s = R s R s Q s trqt s Q s M 1 ^ trx X s+1 )Q s+1 =tr X Xs+1 )P = tr s+1 +X X s+1 )P s+1 ) T = trx X s+1 )P s+1. a 8) 3w>ρWΠ 1/ Ps+1 + Ps+1 T X X s+1 ) trp ) s+1q s ) Q s Q s trx X s+1 )Q s+1 =trx X s+1 )P s+1 = R s+1. trp s+1q s ) Q s tr X X s+1 )Q s U$ CiAMyV) 3 1w.»fi 1. Ω=! DQ3ψmffir X 1 SR n n! I 3 %Nμk^ + ;)1/. ψ. b R i 0i =1 mp o V) 3 ^ Q i 0i =1 M 1 %NΞ " X mp+1 R mp+1. V) ^ trrmp+1 T R i) = 0 i =1 mp N trri T R j)=0i =1 mp i j. U$ R 1 R R mp rffir'fi R m p 3LΨv z %H R mp+1 =0 X mp+1 r! I 3L.! I.^h %Nw>! I3 %NμkΩG οk t 0 t 0 = minmp n )) 1/. qjd b n mp n R i 0i =1 ;) M Q i 0i =1 n Q %NΞ X n +1 R n +1 Q n +1. ap<3w>3.π 1/ Q n +1 =0N R n +1 =0% H X n +1 flr! I 3L. %=) 1 r^p1/<3ß7. 'χ 1.! I.^3χJ»ffirl M 1 ~&l?vu k l1 R k 0HQ k =0. ψ. χw= b&l?vu k l1 R k 0H Q k =0 V) 3 y! I.^. ΩJ= b! I.^ odω^3vu i ^ R i 0. bdω^3vu Q i 0 =) 1 3w>kfly! I ^ d.^=9f. U$ ΩX&l?vu k l1 R k 0H Q k =0. /fi 4. ffirsfl 1).^Q ffirsflψ { AXB + CXD = F B T XA T + D T XC T = F T 9).^. ψ. bffirsfl 1) ^L X 0 SR n X T 0 = X 0 AX 0 B + CX 0 D = F B T X 0 A T + D T X 0 C T = B T X T 0 AT + D T X T 0 CT = F T. U$ X 0 KrSflΨ 9) 3L.

6 418 Π Φ D 010 D.O6 bffirsflψ 9) ^L X R n n l1 A XB + C XD = F B T XA T + D T XC T = F T Y ˆX = X + X ˆX SR n n Q A ˆXB + C ˆXD = A X + X T B + C X + X T D = A XB + C XD + A X T B + C X T D = 1 F + 1 F T ) T = F. Kflr ffirsfl 1) K^L ˆX. U$ ffirsfl 1) 3% =5ffbffirSflΨ 9) 3% =. %V) 4 3w>%y ffirsfl AXB + CXD = F 3Dß KrffirSflΨ 9) 3. ffirsflψ 9) Ω^3 3~ofl ŚE ffirsfl 1) 3Ω^ 3~ofl S S E ŚE. /fi 5 7. f.^-=sflψ My = b 3L y 0 RM T y 0 fl$.^-=sfl ΨfiL3 4Q.»fi. f! I r.^3 buψm;)ffir X 1 = A T H T B T + BHA + C T H T D T + DHC J~ H r R p m ~3Qffir flffl6 Y X 1 =0 R n n o M 1 μk^+ ;)%N1/! I fil3 4Q. ψ. M 1 n=) 1 y by X 1 = A T H T B T + BHA + C T H T D T + DHC J~ H r R p m ~Qffir oßk^+ ;)%1/! I 3 X Q X %NΠffipff X = A T Y T B T + BY A + C T Y T D T + DY C. V) 4 3w>y ffirsfl AXB + CXD = F 3Dß KrffirSflΨ 9) 3. U $ Jw> X r! I 3 4Q }?w> X rffirsflψ 9) 3 4Q %. vecx) =x vecx )=x vecy T )=y 1 vecy )=y vecf )=c 1 vecf T )=c o ffirsflψ 9) 5ffb`-=SflΨ B T A + D T C) ΞQ/ A B T + C D T ) y x = x =vecx )=veca T Y T B T + C T Y T D T + BY A + DY C) =B A T )y 1 +D C T )y 1 +A T B)y +C T D)y = B A T + D C T ) A T B + C T y 1 D) y T T B T A + D T C y 1 = R B T A + D T C A B T + C D T A B T + C D T c 1 c. 10) V) 5 y x r-=sflψ 10) 3 4Q. a vec ±rοf3 U$ X rffir SflΨ 9) fil3 4Q %H X Kr! I 3fiL 4Q. 3. *& II ß DQ^=3 X S E SR n n bdßffirdodßffir.tv o X X 0 = X X 0 + X0 T ) X 0 X0 T = X X 0 + X0 T + X 0 X0 T U$ flsfimfl %Ωf! II ~^=3 X 0 SR n n..

7 4 I m: χstffi AXB + CXD = F Eρ»4<*ΦN 419! I.^h! I 3 ~ S E V' D X S E *X AXB + CXD = F AX X 0 )B + CX X 0 )D = F AX 0 B CX 0 D. Y X = X X 0 F = F AX 0 B CX 0 D o! II 5ffbS.^ffirSfl A XB + C XD = F 3 4Q Dß X. `ψ M 1 Uψm;)ffir X 1 = A T H T B T + BHA + C T H T D T + DHC J~ H r R p m ~Qffir ypbflffl6y X 1 =0 R n n %1/ffirSfl A XB + C XD = F 3 fil 4Q Dß X %HKfl1/0! II 3 X Q X %ffipff X = X + X "3ffi5 N±@rμkff8 M χffil Matlab7.1 dj< Ω1 jwj0z M3^6=. fl 1. f A = C = B = D = F = I) S.^ffirSfl AXB + CXD = F 3Dß J 4Q Dß. II) ffirsfl AXB + CXD = F 3Ω^Dß ~ofl S E UU X 0 = S X S E l1 X X 0 = min X S E X X 0.

8 40 Π Φ D 010 D I) WZ M 1 %1/ffirSfl AXB + CXD = F Dß n 4Q Dß. b Ξ %»3Y l! 3;)kfl~ c3 R i i =1 ) L x5b3. U$ UχW43 ε H` ε =1.0e 010Matlab fla:u j~*p QΛr ) R k <ε=1.0e 010 h fl ;) t X k flffirsfl AXB + CXD = F 3L. ULψm;)ffir X 1 SR n n q+u X 1 = ;) M X 30 = R 30 =.1319e 01 <ε X 30 = U$ %1ffirSfl AXB + CXD = F 3LDß X 30. buψmffir X 1 = A T H T B T + BHA + C T H T D T + DHC J~ H= X 1 = %;)"ffirsfl AXB + CXD = F 3L 4Q Dß X = X 39 = Q R 39 =.8805e 011 <ε X 39 = bu X 1 =0K1/ffirSfl AXB + CXD = F 3 4Q Dß X = X 9 = Q R 9 =6.0085e 011 <ε X 9 = II) S X 0 l S E ~3ΩΦΦμh 4 X = X X 0 F = F AX0 B CX 0 D Uψm; )ffir X =01/ffirSfl A XB + C XD = F 3 4Q Dß X Q

9 4 I m: χstffi AXB + CXD = F Eρ»4<*ΦN X = X 9 = R 9 =6.399e 011 <ε X 9 = %H1/l S E ~D^=ffir X 0 3ΩΦΦμ ˆX Q ˆX = X + X 0 = fl. f A = C = B = D = F = =) 1 nß7 1 y `j;)kfl~^ R k =0offirSfl AXB + CXD = F ^Dß X k ; `j&l R k 0H Q k =0oSfl$Dß. bξ %»3Y l! 3;)k fl~ R i Q i i =1 ) L x5b3. L^=r43v ε H` ε =1.0e 007 R k <εyp R k >εh Q k <εh o;). Uψm;)ffir X 1 =0WZ±χ3; ) M 1/ R 11 = Q 11 =.407e 008 <ε. U$ ffirsfl AXB + CXD = F $Dß. Y5: MatlabeflA)8 function Solution=SFABCDFX0X1H\varepsilon) F=F-A*X0*B+C*X0*D); X1=A *H *B +B*H*A+C *H *D +D*H*C;

10 4 Π Φ D 010 D R1=F-A*X1*B+C*X1*D); P1=A *R1*B +C *R1*D ; Q1=P1+P1 )/; k=1; while tracer1 *R1)>\varepsilon trr1=tracer1 *R1); trq1=traceq1 *Q1); if traceq1 *Q1)<\varepsilon break end k=k+1 X=X1+traceR1 * R1)/traceQ1 *Q1))*Q1; R=F-A*X*B+C*X*D); P=A *R*B +C *R*D ; Q=P+P )/-tracep*q1)/traceq1 *Q1))*Q1; X1=X; R1=R; Q1=Q; trr1=tracer1 *R1); sqrttrr1) trq1=traceq1 *Q1); sqrt trq1) trx1=tracex1 *X1); sqrttrx1) end X1 out=x1+x0 times=k ffl ρ + 1 Roger A Horn Charles R. Johnson. Topics in Matrix AnalysisM. Ξο: Z=9#Λe Gene H Golub Charles F Van Loan. Matrix ComputationsM. Baltimore: The Johns Hpkins University Press fi u. * ffit P" M. qb: qb$dλz#λe Charles F Van Loan. Generalizing the singular value decompositionj. SIAM J. Numer Anal ): Moody T Chu Robert E Funderlic Gene H Golub. On a variational formulation of the generalized singular value decompositionj. SIAM J. Matrix Anal. Appl ): Gene H Golub Zha Hongyuan. Perturbation analysis of the canonical correlations of marix pairsj. Linear Algebra Appl ): GG:. T»h~χsTffi KffΨΨν4<*N4Ifi D. νc: sb'd Peng Zhenyun Peng Yaxin. An efficient iterative method for solving the matrix equation A B + CY D = EJ. Numer Linear Algebra Appl ): Wang Minghui Cheng Xuehan Wei Musheng. Iterative algorithms for solving the matrix equation A B + CX T D = EJ. Appl. Math. Comput. 187): 6-69.