关于一类特殊 3 × 3 块鞍点系统解的结构向后误差分析 Structured Backward Error Analysis ona Special Class of Block Three-by-ThreeSaddle Point Systems

doi:10.12677/PM.2023.135155

设为首页加入收藏期刊导航网站地图

期刊菜单

文章导航

PureMathematicsnØêÆ,2023,13(5),1528-1547

PublishedOnlineMay2023inHans.https://www.hanspub.org/journal/pm

https://doi.org/10.12677/pm.2023.135155

'u˜aAÏ3×3¬Q:XÚ)(•

Ø©Û

000WWWååå

Ü“‰ŒÆ§êÆ†ÚOÆ§[‹=²

ÂvFÏµ2023c423F¶¹^FÏµ2023c524F¶uÙFÏµ2023c531F

Á‡

Cc5§˜aAÏ3×3 ¬Q:XÚ2•A^u˜Ôn¯K¥"•BuÿÁ¢SêŠŽ{

-½5§©éù«a.3×3 ¬Q:XÚ ?1(•Ø©Û§ ¿‰Ñ(•Ø

ŒOŽäNúª"Äu(•Ø§·‚?1êŠ¢"êŠ¢(JL²TLˆªŒ

^uu¢SŽ{-½5"

'…c

3×3 ¬Q:¯K§•Ø§(•Ø

StructuredBackwardErrorAnalysison

aSpecialClassofBlockThree-by-Three

SaddlePointSystems

JialuXing

CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu

Received:Apr.23

,2023;accepted:May24

,2023;published:May31

,2023

©ÙÚ^:0Wå.'u˜aAÏ3×3¬Q:XÚ)(•Ø©Û[J].nØêÆ,2023,13(5):1528-1547.

DOI:10.12677/pm.2023.135155

0Wå

Abstract

Inrecentyears,aspecialclassofblockthree-by-threesaddlepointsystemsiswidely

appliedtoanumberofphysicalproblems.Inordertoevaluatethestabilityofactual

numericalalgorithms,thispaperperformsthestructuredbackwarderroranalysis

forthistypeofblockthree-by-threesaddlepointsystemandpresentsanexplicit

andcomputableformulaforthestructuredbackwarderror.Basedonthestructured

backwarderror,weperformnumericalexp eriment.Numericalexampleshowsthat

theexpressionsareusefulfortestingthestabilityofpracticalalgorithms.

Keywords

Block3×3SaddlePointProblem,BackwardError,StructuredBackwardError

2023byauthor(s)andHansPublishersInc.

This work is licensed undertheCreative Commons Attribution InternationalLicense (CCBY4.0).

http://creativecommons.org/licenses/by/4.0/

1.Úó

©ïÄ±e3×3 ¬Q:¯K(•Ø







B−EC

0CD































,(1.1)

Ù¥A∈R

n×n

ÚD∈R

p×p

´é¡½Ý,E∈R

m×m

´é¡Œ½Ý,B∈R

m×n

C∈R

p×m

´Ý/Ý.B

“LB=˜,C

“LC=˜.éA,B,C,D,Eåy‚5

XÚ(1.1) )•3•˜5.‰½,‡(¯KCq),(•Ø©Û•)3êâ¥Ïé˜‡

•(6Ä,¦Cq)´(6Ä¯K°(),•(6ÄŒ¡•(•

Ø.(•Ø©ÛŒ ±£‰¢S)û¯K†·‚Ž‡)û¯KkõC,¿«êŠŽ

{-½5[1,2].‚5XÚ(1.1))u¯õ‰ÆOŽ+•¥,'XÍÜ/¢-ñóÄåÆ•Zý

N!ìmu[3],\„ÍÜš YåÆ•§n|S“¦)¬ýN!ìmu[4],-½ÍÜšYåÆ·Ü

¬¬n/ýN!ìmu[5] .

DOI:10.12677/pm.2023.1351551529nØêÆ

0Wå

3×3¬Q:XÚ(1.1) Œ±T/©•2ÂQ:¯K,=:







B−EC

0CD





































B−EC

0CD































Cc5,˜Æö[6–10] ®²é2ÂQ:XÚ?1(•Ø©Û,y²ƒAêŠŽ{ä

kr-½5.¦+3×3 ¬Q:XÚ(1.1) Œ±@•´2Â(2×2 ¬) Q:¯K,Ï•§

äkAÏ3×3¬(,þã(•Ø©Û¿ØUO(/L«XÚ(1.1).3©¥,·‚òé

3×3¬Q:XÚ(1.1) ?1(•Ø©Û.

•{B,r(1.1) PŠ

At= d.

é?¿OŽ)



˜x

,˜y

,˜z



,½ÂÙÃ(•Øη(

t)•

η(

t) =min

∆A,∆d







k∆Ak

kAk

k∆dk

kdk





: (A+∆A)

t= d+∆d



þ¡ªfŒ±?˜ÚL«•[11]

η(

t) =

kd−A

kAk

+kdk

,(1.2)

Ù¥k·k

Úk·k

©OL«ÝFrobenius ‰êÚ2-‰ê.eη(

t)´Åì°ÝÓþ?,KOŽ

)

tOŽL§´•-½§=

t´˜‡•-½).I‡5¿´,XJXêÝAäk,«

AÏ(,g,‡¦A+∆A•äk†AƒÓ(.3ù«œ¹e,g,‡•Ä(•Ø.

©(SüXeµ

312 !¥,Ì‡0Kronecker È§Ú¡y²¥I‡^A‡'…Ún.

313 !¥,í3 ×3 ¬Q:XÚ(1.1) (ÝCØ6Ä) (•ØäNŒOŽ

Lˆª.

314 !¥,í3×3 ¬Q:XÚ(1.1) ((i)˜y= 0,(ii)˜x= 0,(iii)˜z= 0) (•Ø

äNŒOŽLˆª.

315 !¥,ÏLêŠ¢5y²íÑLˆªéuÿÁ¦)3×3 ¬Q:XÚ¢SêŠ

Ž{-½5´k^.

316 !¥,é©óŠ?1o(.

2.ý•£

©^R

m×n

ÚSR

n×n

©OL«m×n¢Ý8ÜÚn×n¢é¡Ý8Ü,^A

†

L«ÝAMoore-Penrose_,X= (x

) ∈R

m×n

,Z∈R

p×q

.XÚZƒmKronecker È½

DOI:10.12677/pm.2023.1351551530nØêÆ

0Wå

Â•(„©z[12],14Ù])

X⊗Z= (x

Z) ∈R

mp×nq

d©z[12] •

vec(XYZ) =



⊗X



vec(Y),

(2.1)

(X⊗Z)

= X

⊗Z

(2.2)

(X⊗Z)(C⊗G) = (XC⊗ZG),(2.3)

Ù¥Y∈R

n×p

,CÚG•Ü·êÝ,vec(Y) ½Â•òÝY¤k•gæU¤˜‡•þ

•þ.

3!¥,·‚ò0˜Ún,ùÚnò3e5Ù!¥¦^.

Ún2.1([8,13])u∈R

Úp∈R

®•.½Â



X∈R

n×m

: Xu= p



K,X6= ∅…=pu

†

u= p,…3dœ¹e,?¿X∈XŒL«•

X= pu

†



−uu

†



,Z∈R

n×m

Ún2.2([8,13])b,c∈R

®•.½Â



H∈SR

n×n

: Hb= c



K,H6= ∅…=bÚc÷vcb

†

b= c,…3d^‡e,?¿H∈HŒL«•

H= cb

†



†





−bb

†





−bb

†





−bb

†



Ù¥T∈SR

n×n

3.¦)XÚ(1.1)(ÝCØ6Ä)(•Ø¯K



˜x

,˜y

,˜z



´XÚ(1.1) OŽ),½Â(•Øη

(˜x,˜y,˜z)•

(˜x,˜y,˜z) =min







∆A,∆B,∆E,

∆D,∆f,∆g,

∆h







∈F









k∆Ak

kAk

k∆Bk

kBk

k∆Ek

kEk

k∆Dk

kDk

k∆fk

kfk

k∆gk

kgk

k∆hk

khk









DOI:10.12677/pm.2023.1351551531nØêÆ

0Wå

Ù¥

















∆A,∆B,∆E,

∆D,∆f,∆g,

∆h













A+∆A(B+∆B)

B+∆B−(E+∆E)C

0CD+∆D













˜x

˜y

˜z













f+∆f

g+∆g

h+∆h







∆A= ∆A

∆D= ∆D

∆E= ∆E











(3.1)

eOŽ)

t(•Ø´Åì°ÝÓþ?,KOŽ)

t´˜‡(•-½),ƒAêŠŽ

{´(•-½(½r-½[14] ).Ïd,‰Ñ(•Øη

(˜x,˜y,˜z) ŒOŽäNLˆª

òkÏuÿÁ¢SêŠŽ{-½5.•d,?˜Ú½Âη

(θ

,θ

,λ

)

(˜x,˜y,˜z)•

(θ

,θ

,λ

)

(˜x,˜y,˜z)

=min







∆A,∆B,∆E,

∆D,∆f,∆g,

∆h







∈F









k∆Ak

k∆Bk

k∆Ek

k∆Dk

k∆fk

k∆gk

k∆hk









(3.2)

Ù¥θ

,θ

,λ

Úλ

•ëê.eA6=0,B6= 0,E6= 0,D6= 0,f6= 0,g6= 0Úh6= 0,K

kAk

kBk

kEk

kDk

kfk

kgk

,and

khk

lk

(˜x,˜y,˜z) = η

(

)

(˜x,˜y,˜z).(3.3)

•‰Ñη

(θ

,θ

,λ

)

(˜x,˜y,˜z) (•Ø²(Lˆª.·‚ÄkïÄÜ©(

•Øη

(θ

,θ

)

(˜x,˜y,˜z),Ù½Â•

(θ

,θ

)

(˜x,˜y,˜z) =min

(∆A,∆B,∆E,∆D)∈F



k∆Ak

k∆Bk

k∆Ek

k∆Dk



,(3.4)

Ù¥











∆A,∆B,

∆E,∆D







A+∆A(B+∆B)

B+∆B−(E+∆E)C

0CD+∆D













˜x

˜y

˜z



















∆A= ∆A

∆D= ∆D

∆E= ∆E











(3.5)

e¡‰Ñη

(θ

,θ

)

(˜x,˜y,˜z)²(Lˆª.

½n3.1b(˜x

,˜y

,˜z

)

÷v˜x6= 0,˜y6= 0 Ú˜z6= 0•XÚ(1.1) ˜‡OŽ).K



(θ

,θ

)

(˜x,˜y,˜z)



2θ

k˜zk



˜x





˜y



−

k˜zk



˜z



2θ



˜x



˜y



(3.6)

DOI:10.12677/pm.2023.1351551532nØêÆ

0Wå

Ù¥

= f−A˜x−B

˜y, r

= −g+B˜x−E˜y+C

˜z,r

= h−C˜y−D˜z,

= θ

k˜xk

+2θ

k˜yk

, γ

= 2θ

k˜xk

+θ

k˜yk

, γ

= θ

k˜xk

+θ

k˜xk

k˜yk

+θ

k˜yk

= 2θ

k˜yk

−θ

k˜xk

−θ

k˜yk

, γ

= 2θ

k˜xk

−θ

k˜yk

−θ

k˜xk

y²d(3.5)•,(∆A,∆B,∆E,∆D) ∈F

…=∆A,∆B,∆EÚ∆D÷v

∆A˜x= r

−∆B

˜y,∆E˜y= r

+∆B˜x,∆D˜z= r

,∆A= ∆A

,∆D= ∆D

,∆E= ∆E

.(3.7)

òÚn2.2 A^u(3.7),Œ

∆A=



−∆B

˜y



˜x

†



˜x

†





−∆B

˜y





−˜x˜x

†





−˜x˜x

†





−˜x˜x

†



,(3.8)

Ù¥T

∈SR

n×n

∆E= (r

+∆B˜x)˜y

†



˜y

†



+∆B˜x)



−˜y˜y

†





−˜y˜y

†





−˜y˜y

†



,(3.9)

Ù¥T

∈SR

m×m

∆D= r

˜z

†



˜z

†





−˜z˜z

†





−˜z˜z

†





−˜z˜z

†



,(3.10)

Ù¥T

∈SR

p×p

é(3.8),(3.9) Ú(3.10) Òü>ÓžFrobenius‰ê,Œ

k∆Ak



−∆B

˜y



k˜xk





−˜x˜x

†





−˜x˜x

†







−˜x˜x

†



−∆B

˜y





k˜xk



−∆B

˜y



k˜xk

−



˜x−˜y

∆B˜x



k˜xk





−˜x˜x

†





−˜x˜x

†





(3.11)

k∆Ek

+∆B˜xk

k˜yk





−˜y˜y

†





−˜y˜y

†







−˜y˜y

†



+∆B˜x)



k˜yk

2kr

+∆B˜xk

k˜yk

−



˜y+˜y

∆B˜x



k˜yk





−˜y˜y

†





−˜y˜y

†





(3.12)

k∆Dk

k˜zk





−˜z˜z

†





−˜z˜z

†







−˜z˜z

†





k˜zk

2kr

k˜zk

−



˜z



k˜zk





−˜z˜z

†





−˜z˜z

†





(3.13)

Šâη

(θ

,θ

)

(˜x,˜y,˜z)½Â(3.4),±9Lˆª(3.11),(3.12)Ú(3.13)Œ±Ñ

DOI:10.12677/pm.2023.1351551533nØêÆ

0Wå



(θ

,θ

)

(˜x,˜y,˜z)



=min

∆B∈R

m×n

∈SR

n×n

∈SR

m×m

∈SR

p×p

k∆Ak

+θ

k∆Bk

+θ

k∆Ek

+θ

k∆Dk

2θ

k˜zk

−



˜z



k˜zk

+min

∆B∈R

m×n

p(∆B),

(3.14)

Ù¥

p(∆B) =

2θ



−∆B

˜y



k˜xk

−



˜x−˜y

∆B˜x



k˜xk

2θ

+∆B˜xk

k˜yk

−



˜y+˜y

∆B˜x



k˜yk

+θ

k∆Bk

2θ

k˜xk

2θ

k˜yk

−





k˜xk

−



˜y



k˜yk

2θ



˜x



˜y

∆B˜x



k˜xk

−

2θ



˜y



˜y

∆B˜x



k˜yk

−

(θ

k˜yk

+θ

k˜xk

)



˜y

∆B˜x



k˜xk

k˜yk

−

4θ



˜y

∆Br



k˜xk

4θ



∆B˜x



k˜yk

2θ

k˜y

∆Bk

k˜xk

2θ

k∆B˜xk

k˜yk

+θ

k∆Bk

Pt= vec(∆B) ∈R

,|^KroneckerÈ5Ÿ(2.1)Ú(2.2),þ¡ªfŒ±?˜Úz•

p(∆B)

2θ

k˜xk

2θ

k˜yk

−





k˜xk

−



˜y



k˜yk

+θ

2θ



˜x



˜x

⊗˜y



k˜xk

−

2θ



˜y



˜x

⊗˜y



k˜yk

−

(θ

k˜xk

+θ

k˜yk

(˜x⊗˜y)



˜x

⊗˜y



k˜xk

k˜yk

−

4θ



⊗˜y



k˜xk

2θ

⊗˜y)



⊗˜y



k˜xk

4θ



˜x

⊗r



k˜yk

2θ

(˜x⊗I

)



˜x

⊗I



k˜yk

òþ¡ª‘\(3.14) ¥,¿-E¦^(2.2),Œ±



(θ

,θ

)

(˜x,˜y,˜z)



2θ

k˜xk

2θ

k˜yk

2θ

k˜zk

−





k˜xk

−



˜y



k˜yk

−



˜z



k˜zk

+min

t∈R

H(t),

ùpH(t) = t

Kt−2k

t,Ù¥

K=θ

2θ

⊗˜y)



⊗˜y



k˜xk

2θ

(˜x⊗I

)



˜x

⊗I



k˜yk

−

(θ

k˜yk

+θ

k˜xk

)(˜x⊗˜y)



˜x

⊗˜y



k˜xk

k˜yk

DOI:10.12677/pm.2023.1351551534nØêÆ

0Wå

2θ

⊗˜y)

k˜xk

−

2θ

(˜x⊗r

)

k˜yk

−



˜x



k˜xk

−



˜y



k˜yk

(˜x⊗˜y),

|^(2.3),¿5¿



−˜x˜x

†





−˜x˜x

†





−˜y˜y

†





−˜y˜y

†



Ú˜x

†

=˜x

/k˜xk

,˜y

†

˜y

/k˜yk

,Œ±Ñ

K=θ

⊗˜y)



⊗˜y



k˜xk

(˜x⊗I

)



˜x

⊗I



k˜yk



−˜x˜x

†



⊗˜y



−˜x˜x

†



⊗˜y



k˜xk



˜x⊗



−˜y˜y

†



˜x

⊗



−˜y˜y

†



k˜yk

²w/,K´˜‡é¡½Ý.lt= K

−1

kž,H(t)U•Š.ƒA,



(θ

,θ

)

(˜x,˜y,˜z)



2θ

k˜xk

2θ

k˜yk

2θ

k˜zk

−





k˜xk

−



˜y



k˜yk

−



˜z



k˜zk

−k

−1

(3.15)

õg|^Sherman-Morrison-Woodbury úª(„©z[15]),²L˜ÐOŽŒ

−1

−

2θ



⊗˜y˜y



(θ

k˜xk

+2θ

k˜yk

)

−

2θ



˜x˜x

⊗I



(2θ

k˜xk

+θ

k˜yk

)



˜x˜x

⊗˜y˜y



(2θ

k˜xk

+θ

k˜yk

)

Ù¥

ω=

(θ

+4θ

)k˜xk

+8θ

k˜xk

k˜yk

+(4θ

+θ

)k˜yk

(θ

k˜xk

+2θ

k˜yk

)(θ

k˜xk

+θ

k˜xk

k˜yk

+θ

k˜yk

)

²L˜„¡OŽ,Œ±Ñ

−1

4θ

k˜yk

k˜xk

4θ

k˜xk

k˜yk

−

k˜yk

k˜xk



˜x



−

k˜xk

k˜yk



˜y



−

2θ



˜x



˜y



Ù¥

= 4θ

k˜xk

+3θ

k˜xk

k˜yk

+2θ

k˜yk

, γ

= 2θ

k˜xk

+3θ

k˜xk

k˜yk

+4θ

k˜yk

Ù¥γ

,γ

Úγ

3½n3.1 c¡Ü©½ÂL,òþª‘\(3.15) ¥,íÑ(3.6).y²..

e¡,|^½n3.1 ¥Ü©(•Øη

(θ

,θ

)

(˜x,˜y,˜z)LˆªíÑ(•Ø

(θ

,θ

,λ

)

(˜x,˜y,˜z)äNLˆª.

½n3.2b(˜x

,˜y

,˜z

)

÷v˜x6= 0,˜y6= 0 Ú˜z6= 0•XÚ(1.1) ˜‡OŽ).…

(θ

,θ

,λ

)

(˜x,˜y,˜z)½Âd(3.1)Ú(3.2) ‰Ñ.K

DOI:10.12677/pm.2023.1351551535nØêÆ

0Wå



(θ

,θ

,λ

)

(˜x,˜y,˜z)



2θ

(µ

−2θ

)

2θ

(µ

−2θ

)

2θ

(µ

−2θ

)

k˜zk



˜x





˜y





˜z





˜x



˜y



(3.16)

Ù¥

= γ

+2θ

, µ

= γ

+2θ

, µ

= k˜zk

+2θ

, µ

= θ

k˜xk

+γ

+2θ

= θ

k˜xk

+θ

+2θ

−θ

k˜xk

, µ

= k˜zk

+θ

Ω

= γ

+k˜yk

, Ω

= θ

k˜yk

−γ

−k˜yk

(γ

Ω

−4θ

k˜xk

Ω

−θ

k˜xk

Ω

−θ

k˜xk

Ω

)

Ω

−

k˜yk

(Ω

−k˜yk

−4θ

)

Ω

−

2θ

(2θ

Ω

+2θ

Ω

−θ

k˜xk

k˜yk

)

Ω

(γ

Ω

−θ

k˜yk

Ω

+4θ

k˜yk

+θ

k˜yk

)

Ω

−

k˜xk

(µ

Ω

+θ

k˜xk

Ω

−4θ

)

Ω

−

2θ

(2θ

Ω

−θ

k˜xk

k˜yk

−2θ

)

Ω

(3θ

−µ

−θ

)

k˜zk

2θ

(Ω

+θ

k˜xk

k˜yk

+θ

k˜xk

k˜yk

)

Ω

−

2θ

k˜xk

(γ

Ω

+θ

k˜xk

Ω

+2θ

Ω

−2θ

)

Ω

−

2θ

k˜yk

(γ

Ω

−2θ

−k˜yk

)

Ω

−

4θ

(θ

Ω

−2θ

+θ

Ω

)

Ω

y²Šâη

(θ

,θ

,λ

)

(˜x,˜y,˜z) ½Â(3.2) ÚÜ©(•Øη

(θ

,θ

)

(˜x,˜y,˜z) 

Lˆª(3.6),Œ±íÑ

DOI:10.12677/pm.2023.1351551536nØêÆ

0Wå



(θ

,θ

,λ

)

(˜x,˜y,˜z)



=min

∆f∈R

,∆g∈R

,∆h∈R

X(∆f,∆g,∆h),

Ù¥

X(∆f,∆g,∆h)

=λ

k∆fk

+λ

k∆gk

+λ

k∆hk

2θ

+∆fk

2θ

+∆gk

2θ

k˜zk

+∆hk

+∆f)

˜x

+∆g)

˜y

−

k˜zk

+∆h)

˜z

2θ

+∆f)

˜x

+∆g)

˜y

²L˜ÄOŽ,Œ±Ñ

X(∆f,∆g,∆h) =



(θ

,θ

)

(˜x,˜y,˜z)









∆f

∆g

∆h













∆f

∆g

∆h













∆f

∆g

∆h







…

Φ =







+2θ

˜x˜x

˜x˜y

˜y˜x

+2θ

˜y˜y

k˜zk

+2θ

k˜zk

−

˜z˜z

k˜zk













2θ



˜x



˜x+



˜y



˜x

2θ



˜y



˜y+



˜x



˜y

2θ

k˜zk

−



˜z



k˜zk

˜z







ÏL„¡OŽ,éu?¿š"•þs=





,Ù¥a∈R

,b∈R

,Úc∈R

,Œ

Φs=

+2θ

˜x˜x

+2θ

˜y˜y

k˜zk

+2θ

k˜zk

c−

k˜zk

˜z˜z

2θ

˜x˜y

=λ

a+λ

k˜zk

+θ

k˜zk

k˜yk

k˜xk

˜x˜x

k˜xk

k˜yk

˜y˜y



˜x+b

˜y



2θ

k˜xk



k˜xk

−˜x˜x



2θ

k˜yk



k˜yk

−˜y˜y



k˜zk



k˜zk

−˜z˜z



>0,

ùL²Φ ´˜‡é¡½Ý.Ïd,X(∆f,∆g,∆h) •Š:•

DOI:10.12677/pm.2023.1351551537nØêÆ

0Wå



∆f

,∆g

,∆h



= −Φ

−1

ƒA,



(θ

,θ

,λ

)

(˜x,˜y,˜z)





(θ

,θ

)

(˜x,˜y,˜z)



−q

−1

q.(3.17)

−1







00Ψ







,Ψ

∈R

n×n

,Ψ

∈R

m×m

,Ψ

∈R

p×p

²L˜XÐ1C†§k

Ω

˜x˜x

, Ψ

= −

Ω

˜x˜y

−

Ω

˜y˜y

, Ψ

k˜zk

˜z˜z

²L˜X„¡Ð?“êOŽ,Œ±

−1

4θ

k˜zk



˜x





˜y





˜z





˜x



˜y



(3.18)

ùp

k˜xk

(4θ

Ω

+γ

Ω

+θ

k˜xk

Ω

)

Ω

k˜yk

(Ω

−k˜yk

−4θ

)

Ω

2θ

(2θ

Ω

+2θ

Ω

−θ

k˜xk

k˜yk

)

Ω

k˜xk

(µ

Ω

+θ

k˜xk

Ω

−4θ

)

Ω

k˜yk

(γ

Ω

−4θ

−k˜yk

)

Ω

2θ

(2θ

Ω

−θ

k˜xk

k˜yk

−2θ

)

Ω

(θ

−3µ

)

k˜zk

2θ

k˜xk

(γ

Ω

+θ

k˜xk

Ω

+2θ

Ω

−2θ

)

Ω

2θ

k˜yk

(γ

Ω

−2θ

−k˜yk

)

Ω

−

2θ

k˜xk

k˜yk

(γ

+θ

)

Ω

4θ

(θ

Ω

−2θ

+θ

Ω

)

Ω

DOI:10.12677/pm.2023.1351551538nØêÆ

0Wå

Ù¥γ

,γ

Úγ

3½n3.1 c¡Ü©½ÂL,µ

,µ

,Ω

ÚΩ

3½n3.2 

c¡Ü©½ÂL,•ò(3.18) Ú(3.6) ‘\(3.17) ¥Ï"Lˆª(3.16).

(•Øη

(θ

,θ

,λ

)

(˜x,˜y,˜z) Lˆª(3.16) •,wå5éE,,´lOŽ

Ý5w,Lˆª(3.16) ´N´OŽ,Ï•§•9\!~!¦!ØùÄ$Ž.

4.¦)XÚ(1.1)(˜y=0,˜x6=0Ú˜z6=0)(•Ø¯K



˜x

,0,˜z



´XÚ(1.1) OŽ),½Â(•Øη

(˜x,0,˜z)•

(˜x,0,˜z) =min







∆A,∆B,∆C,

∆D,∆f,∆g,

∆h







∈F









k∆Ak

kAk

k∆Bk

kBk

k∆Ck

kCk

k∆Dk

kDk

k∆fk

kfk

k∆gk

kgk

k∆hk

khk









Ù¥

















∆A,∆B,∆C,

∆D,∆f,∆g,

∆h













A+∆A(B+∆B)

B+∆B−(E+∆E)(C+∆C)

0C+∆CD+∆D













˜x

˜z













f+∆f

g+∆g

h+∆h







∆A= ∆A

∆D= ∆D











(4.1)

eOŽ)

(•Ø´Åì°ÝÓþ?,KOŽ)

´˜‡(•-½),ƒAê

ŠŽ{´(•-½(½r -½[14] ).Ïd,‰ Ñ(•Øη

(˜x,0,˜z)ŒOŽäNL

ˆªòkÏuÿÁ¢SêŠŽ{-½5.•d,?˜Ú½Âη

(θ

,θ

,λ

)

(˜x,0,˜z)•

(θ

,θ

,λ

)

(˜x,0,˜z)

=min







∆A,∆B,∆C,

∆D,∆f,∆g,

∆h







∈F









k∆Ak

k∆Bk

k∆Ck

k∆Dk

k∆fk

k∆gk

k∆hk









(4.2)

Ù¥θ

,θ

,λ

Úλ

•ëê.lk

(˜x,0,˜z) = η

(

)

(˜x,0,˜z).(4.3)

•‰ Ñη

(θ

,θ

,λ

)

(˜x,0,˜z) (•Ø²(Lˆ ª.·‚ÄkïÄÜ©(

•Øη

(θ

,θ

)

(˜x,0,˜z),Ù½Â•

(θ

,θ

)

(˜x,0,˜z) =min

(∆A,∆B,∆C,∆D)∈F



k∆Ak

k∆Bk

k∆Ck

k∆Dk



,(4.4)

DOI:10.12677/pm.2023.1351551539nØêÆ

0Wå

Ù¥











∆A,∆B,

∆C,∆D







A+∆A(B+∆B)

B+∆B−(E+∆E)(C+∆C)

0C+∆CD+∆D













˜x

˜z



















∆A= ∆A

∆D= ∆D











(4.5)

e¡‰Ñη

(θ

,θ

)

(˜x,0,˜z)²(Lˆª.

½n4.1b(˜x

,0,˜z

)

÷v˜x6= 0 Ú˜z6= 0•XÚ(1.1) ˜‡OŽ).K

(θ

,θ

)

(˜x,0,˜z)

2θ

k˜xk

2θ

k˜zk

−

k˜xk



˜x



−

k˜zk



˜z



(4.6)

Ù¥

= f−A˜x, r

= g−B˜x−C

˜z,r

= h−D˜z,γ

= θ

k˜xk

+θ

k˜zk

y²d(4.5)•,(∆A,∆B,∆C,∆D) ∈F

…=∆A,∆B,∆CÚ∆D÷v

∆A˜x= r

, ∆B˜x= r

−∆C

˜z,∆D˜z= r

, ∆A= ∆A

, ∆D= ∆D

.(4.7)

òÚn2.1 A^u(4.7)1‡ª,Œ

∆B=



−∆C

˜z



˜x

†



−˜x˜x

†



, Z∈R

m×n

(4.8)

òÚn2.2 A^u(4.7)1˜!n‡ª,Œ

∆A= r

˜x

†



˜x

†





−˜x˜x

†





−˜x˜x

†





−˜x˜x

†



, T

∈SR

n×n

(4.9)

∆D= r

˜z

†



˜z

†





−˜z˜z

†





−˜z˜z

†





−˜z˜z

†



, T

∈SR

p×p

.(4.10)

é(4.8),(4.9) Ú(4.10) Òü>ÓžFrobenius‰ê,Œ

k∆Ak

k˜xk





−˜x˜x

†





−˜x˜x

†







−˜x˜x

†





k˜xk

2kr

k˜xk

−



˜x



k˜xk





−˜x˜x

†





−˜x˜x

†





(4.11)

k∆Bk



−∆C

˜z



k˜xk





−˜x˜x

†





(4.12)

k∆Dk

k˜zk





−˜z˜z

†





−˜z˜z

†







−˜z˜z

†





k˜zk

2kr

k˜zk

−



˜z



k˜zk





−˜z˜z

†





−˜z˜z

†





(4.13)

DOI:10.12677/pm.2023.1351551540nØêÆ

0Wå

Šâη

(θ

,θ

)

(˜x,0,˜z)½Â(4.4),±9Lˆª(4.11),(4.12) Ú(4.13)Œ±Ñ

(θ

,θ

)

(˜x,0,˜z)

=min

∆C∈R

p×m

,Z∈R

m×n

∈SR

n×n

∈SR

p×p

k∆Ak

+θ

k∆Bk

+θ

k∆Ck

+θ

k∆Dk

2θ

k˜xk

2θ

k˜zk

−



˜x



k˜xk

−



˜z



k˜zk

+min

∆C∈R

p×m

p(∆C),

(4.14)

Ù¥

p(∆C) =



−∆C

˜z



k˜xk

+θ

k∆Ck

k˜xk

−

2θ



˜z

∆Cr



k˜xk

k˜z

∆Ck

k˜xk

+θ

k∆Ck

= vec(∆C) ∈R

,|^KroneckerÈ5Ÿ(2.1)Ú(2.2),þ¡ªfŒ±?˜Úz•

p(∆C) =

k˜xk

+θ

−

2θ



⊗˜z



k˜xk

⊗˜z)



⊗˜z



k˜xk

òþ¡ª‘\(4.14) ¥,¿-E¦^(2.2),Œ±

(θ

,θ

)

(˜x,˜y,˜z)

2θ

k˜xk

2θ

k˜zk

−



˜x



k˜xk

−



˜z



k˜zk

+min

∈R

H(t

ùpH(t

) = t

−2k

,Ù¥

= θ

⊗˜z)



⊗˜z



k˜xk

⊗˜z)

k˜xk

w,K

´˜‡é¡½Ý.lt

= K

−1

ž,H(t

)U•Š.ƒA,

(θ

,θ

)

(˜x,˜y,˜z)

2θ

k˜xk

2θ

k˜zk

−



˜x



k˜xk

−



˜z



k˜zk

−k

−1

(4.15)

−1

−



⊗˜z˜z



(θ

k˜xk

+θ

k˜zk

)

²L˜ÐOŽ,Œ±Ñ

−1

k˜zk

k˜xk

Ù¥γ

= θ

k˜xk

+θ

k˜zk

.òþª‘\(4.15) ¥,íÑ(4.6).y²..

DOI:10.12677/pm.2023.1351551541nØêÆ

0Wå

(θ

,θ

)

(˜x,0,˜z)LˆªíÑ(•Ø

(θ

,θ

,λ

)

(˜x,0,˜z)äNLˆª.

½n4.2b(˜x

,0,˜z

)

÷v˜x6= 0 Ú˜z6= 0•XÚ(1.1) ˜‡OŽ).…

(θ

,θ

,λ

)

(˜x,0,˜z)½Âd(4.1) Ú(4.2)‰Ñ.K

(θ

,θ

,λ

)

(˜x,0,˜z)

2θ

(µ

−2θ

)

k˜xk

(µ

−θ

)

2θ

(µ

−2θ

)

k˜zk

(3θ

−θ

−µ

)

k˜xk



˜x



(3θ

−θ

−µ

)

k˜zk



˜z



(4.16)

Ù¥

= k˜xk

+2θ

, µ

= γ

+θ

, µ

= k˜zk

+2θ

= k˜xk

+θ

, µ

= k˜zk

+θ

y²Šâη

(θ

,θ

,λ

)

(θ

,θ

)

(˜x,0,˜z)Lˆ

ª(4.6),Œ±íÑ

(θ

,θ

,λ

)

(˜x,0,˜z)

=min

∆f∈R

,∆g∈R

,∆h∈R

(∆f,∆g,∆h),

Ù¥

(∆f,∆g,∆h)

=λ

k∆fk

+λ

k∆gk

+λ

k∆hk

2θ

k˜xk

+∆fk

+∆gk

2θ

k˜zk

+∆hk

−

k˜xk

+∆f)

˜x

−

k˜zk

+∆h)

˜z

²L˜ÄOŽ,Œ±Ñ

(∆f,∆g,∆h) =

(θ

,θ

)

(˜x,0,˜z)







∆f

∆g

∆h













∆f

∆g

∆h













∆f

∆g

∆h







…







k˜xk

+2θ

k˜xk

−

˜x˜x

k˜xk

+θ

k˜zk

+2θ

k˜zk

−

˜z˜z

k˜zk







DOI:10.12677/pm.2023.1351551542nØêÆ

0Wå







2θ

k˜xk

−



˜x



k˜xk

˜x

2θ

k˜zk

−



˜z



k˜zk

˜z







N´y²Φ

´˜‡é¡½Ý.Ïd,X

(∆f,∆g,∆h) •Š:•



∆f

,∆g

,∆h



= −Φ

−1

ƒA,

(θ

,θ

,λ

)

(˜x,0,˜z)

(θ

,θ

)

(˜x,0,˜z)

−q

−1

.(4.17)

Ù¥

−1







k˜xk

˜x˜x

k˜zk

˜z˜z







²L˜XÐ?“êOŽ,Œ±

−1

4θ

k˜xk

4θ

k˜zk

(θ

−3µ

)

k˜xk



˜x



(θ

−3µ

)

k˜zk



˜z



(4.18)

Ù¥γ

,µ

Úµ

(4.18)Ú(4.6)‘\(4.17) ¥Ï"Lˆª(4.16).

½n4.3XJ˜x= 0,˜y6= 0 Ú˜z6= 0,Œ±aqþ¡y²‰Ñ

(θ

,θ

,λ

)

(0,˜y,˜z)

(µ

−θ

)

k˜yk

2θ

(µ

−2θ

)

2θ

(µ

−2θ

)



˜y





˜z





˜y



˜z



(4.19)

Ù¥θ

,θ

,λ

Úλ

•ëê.Ù¥

= f−B

˜y, r

= −g−E˜y+C

˜z,r

= h−C˜y−D˜z,

= θ

k˜yk

+2θ

k˜zk

, γ

= 2θ

k˜yk

+θ

k˜zk

, γ

= θ

k˜yk

+θ

k˜yk

k˜zk

+θ

k˜zk

= 2θ

k˜zk

−θ

k˜yk

−θ

k˜zk

, γ

= 2θ

k˜yk

−θ

k˜zk

−θ

k˜yk

DOI:10.12677/pm.2023.1351551543nØêÆ

0Wå

= k˜yk

+θ

, µ

= γ

+2θ

, µ

= γ

+2θ

, µ

= γ

+2θ

+θ

k˜yk

= θ

+2θ

+θ

k˜yk

−θ

k˜yk

Ω

= γ

+k˜zk

, Ω

= θ

k˜zk

−γ

−k˜zk

(µ

Ω

−2θ

Ω

+2θ

k˜yk

k˜zk

)

Ω

−

(2θ

+k˜yk

)(2θ

Ω

+γ

Ω

+θ

k˜yk

Ω

)

Ω

−

k˜zk

(Ω

−k˜zk

−4θ

)

Ω

(µ

Ω

−2θ

Ω

+2θ

k˜yk

k˜zk

)

Ω

−

(2θ

+k˜zk

)(γ

Ω

−2θ

−k˜zk

)

Ω

−

k˜yk

(µ

Ω

+θ

k˜yk

Ω

−4θ

)

Ω

(2γ

Ω

−k˜zk

Ω

+2θ

k˜zk

+k˜zk

)

Ω

−

(2θ

+k˜yk

)(µ

Ω

+θ

k˜yk

Ω

)

Ω

−

(2θ

+k˜zk

)(Ω

−k˜zk

−4θ

−2θ

k˜yk

)

Ω

−

2θ

(θ

Ω

−θ

k˜yk

k˜zk

+θ

Ω

)

Ω

−

k˜yk

(2θ

Ω

+γ

Ω

+θ

k˜yk

Ω

)

Ω

½n4.4XJ˜z= 0,˜x6= 0 Ú˜y6= 0,Œ±aqþ¡y²‰Ñ

(θ

,θ

,λ

)

(˜x,˜y,0)

2θ

(µ

−2θ

)

2θ

(µ

−2θ

)

(µ

−θ

)

k˜yk



˜x





˜y





˜x



˜y



(4.20)

DOI:10.12677/pm.2023.1351551544nØêÆ

0Wå

Ù¥θ

,θ

,λ

Úλ

•ëê.Ù¥

= f−A˜x−B

˜y, r

= −g+B˜x−E˜y, r

= h−C˜y,µ

= k˜yk

+θ

¿…γ

,γ

,µ

,Ω

5.êŠ¢

!ò‰Ñ˜‡êŠ~f5 '13 !¥íÑ(•Øη

(˜x,˜y,˜z) ÚƒAÃ(•

Øη(

t).êŠ¢3MATLABR2015b¥?1,Åì°Ý•2.2204×10

−16

•Ä‚5XÚ(1.1) ÷v

A= M

,D= M







000100

001000

010000

−3

00000







,C=







1−210

−2−100

1000







E= I

,f=



,10,0,0,0,0



,g=



,1,0,0



,h=



−8

,0,0



Ù¥

= diag(1,5,10,50,100,10000),M

= diag(1,5,10),

P= (p

),p

(i+j−2)!

(i−1)!(j−1)!

ž{Œ±˜‡OŽ)



˜x

,˜y

,˜z



,Ù¥

˜x=







4.0418×10

−1.6445×10

1.0030×10

−1.4400×10

2.8125×10

−4.6236×10







,˜y=







7.3927×10

−3.4302×10

−8.8214×10

4.0418×10







Ú˜z=







7.6232×10

−1.7857×10

3.1926×10







d(1.2) ,Ã(•Ø

η(

t) = 8.8414×10

−22

d(3.3) Ú(3.16),(•Ø

(˜x,˜y,˜z) = 1.4127×10

−8

DOI:10.12677/pm.2023.1351551545nØêÆ

0Wå

6.o(

Øù˜Ý`z¯K=z•½g.•Š¯K§,| ^Sherman-Morrison-Woodbury

úª,•ª¼(•ØŒOŽäNLˆª.•·‚‰Ñ˜‡êŠ¢, ±y²·‚

(JŒ±éN´/^5ÿÁ¢SêŠŽ{-½5.

Ä7‘8

[‹Ž#Ñ“cÄ7(20JR5RA540)"

ë•©z

[1]Bunch, J.R.(1987) TheWeakand StrongStabilityof Algorithmsin NumericalLinear Algebra.

LinearAlgebraandItsApplications,88-89,49-66.

https://doi.org/10.1016/0024-3795(87)90102-9

[2]Higham,N.J.(2002)AccuracyandStabilityofNumericalAlgorithms.2ndEdition,SIAM,

Philadelphia.https://doi.org/10.1137/1.9780898718027

[3]Rhebergen,S.,Wells,G.N.,Wathen,A.J.andKatz,R.F.(2015)Three-FieldBlockPre-

conditionersforModelsofCoupledMagma/MantleDynamics.SIAMJournalonScientiﬁc

Computing,37,A2270-A2294.https://doi.org/10.1137/14099718X

[4]Castelletto,N.,White,J.A.andFerronato,M.(2016)ScalableAlgorithmsforThree-Field

MixedFiniteElementCoupledPoromechanics.JournalofComputationalPhysics,327,894-

918.https://doi.org/10.1016/j.jcp.2016.09.063

[5]Frigo,M.,Castelletto, N.,Ferronato, M.andWhite,J.A.(2021)EﬃcientSolversforHybridized

Three-FieldMixedFinite Element CoupledPoromechanics.ComputersandMathematicswith

Applications,91,36-52.https://doi.org/10.1016/j.camwa.2020.07.010

[6]Chen,X.S.,Li,W.,Chen,X.J. and Liu, J.(2012) StructuredBackward Errorsfor Generalized

SaddlePointSystems.LinearAlgebraandItsApplications,436,3109-3119.

https://doi.org/10.1016/j.laa.2011.10.012

[7]Meng,L.S.,He,Y.W.andMiao,S.X.(2020)StructuredBackwardErrorsforTwoKindsof

GeneralizedSaddlePointSystems.LinearandMultilinearAlgebra,70,1345-1355.

https://doi.org/10.1080/03081087.2020.1760193

[8]Sun,J.G.(1999)StructuredBackwardErrorsforKKTSystems.LinearAlgebraandItsAp-

plications,288,75-88.https://doi.org/10.1016/S0024-3795(98)10184-2

DOI:10.12677/pm.2023.1351551546nØêÆ

0Wå

[9]Xiang,H.andWei,Y.M.(2007)OnNormwiseStructuredBackwardErrorsforSaddlePoint

Systems.SIAMJournalonMatrixAnalysisandApplications,29,838-849.

https://doi.org/10.1137/060663684

[10]Zheng, B.and Lv,P. (2020) Structured Backward Error Analysis forGeneralized Saddle Point

Problems.AdvancesinComputationalMathematics,46,ArticleNo.34.

https://doi.org/10.1007/s10444-020-09787-x

[11]Sun, J.G. (1996) Optimal Backward PerturbationBounds for Linear Systemsand Linear Least

SquaresProblems.Tech.Rep.,UMINF96.15,DepartmentofComputingScience,UmeaUni-

versity,Umea.

[12]Horn,R.A. andJohnson,C.R. (1991)Topics inMatrixAnalysis. Cambridge UniversityPress,

Cambridge.https://doi.org/10.1017/CBO9780511840371

[13]Sun,J.G.(2001)MatrixPerturbationAnalysis.2ndEdition,SciencePress,Beijing.

[14]Boﬃ,D.,Brezzi,F.andFortin,M.(2013)MixedFiniteElementMethodsandApplications.

In:SpringerSeriesinComputationalMathematics,Springer,NewYork.

https://doi.org/10.1007/978-3-642-36519-5

[15]Golub, G.H.andVanLoan,C.F.(2013)MatrixComputations.4thEdition,TheJohnsHopkins

UniversityPress,Baltimore.

[16]Lv,P.andZheng,B.(2022)StructuredBackwardErrorAnalysisforaClassofBlockThree-

by-ThreeSaddlePointProblems.NumericalAlgorithms,90,59-78.

https://doi.org/10.1007/s11075-021-01179-6

DOI:10.12677/pm.2023.1351551547nØêÆ