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PureMathematicsnØêÆ,2023,13(6),1677-1688
PublishedOnlineJune2023inHans.https://www.hanspub.org/journal/pm
https://doi.org/10.12677/pm.2023.136171
E‚5XÚ•Ø
444ŒŒŒ
Ü“‰ŒÆêƆÚOÆ§[‹=²
ÂvFϵ2023c519F¶¹^Fϵ2023c620F¶uÙFϵ2023c628F
Á‡
©ïÄEê•þ˜a2ÂQ:XÚ(z•Ø§¿òÙ¥ü«AÏœ¹§í
Ñ•ØOŽúª"ÏLêŠ~fL²§·‚(JŒ±•B/u¢SŽ{-½5§
íÑ#(z•Ø'~^••Ü·Úk"
'…c
•Ø§Eꕧr-½
BackwardErrorofComplex
LinearSystem
YulingLiu
CollegeofMathematicsandStatistics,NorthwestNormalUniversity,LanzhouGansu
Received:May19
th
,2023;accepted:Jun.20
th
,2023;published:Jun.28
th
,2023
Abstract
Inthispaper,thestructuredbackwarderrorofaclassofgeneralizedsaddle-point
systemsincomplexnumberfieldisstudied,andtwospecialcasesareextended,and
thecalculationformulaofbackwarderrorisderived.Numericalexamplesshowthat
our results caneasily test the stability of the actual algorithm, and thenew structured
backwarderrorismoresuitableandeffectivethanthecommonone.
©ÙÚ^:4Œ .E‚5XÚ•Ø[J].nØêÆ,2023,13(6):1677-1688.
DOI:10.12677/pm.2023.136171
4Œ
Keywords
BackwardError,ComplexNumberField,StrongStability
Copyright
c
2023byauthor(s)andHansPublishersInc.
This work is licensed undertheCreative Commons Attribution International License (CCBY 4.0).
http://creativecommons.org/licenses/by/4.0/
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DOI:10.12677/pm.2023.1361711678nØêÆ
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DOI:10.12677/pm.2023.1361711679nØêÆ
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1
θ
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θ
1
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DOI:10.12677/pm.2023.1361711680nØêÆ
4Œ
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1
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s
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θ
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2
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θ
2
2
kex−eyk
2
2
kr
u
+r
v
k
2
2
.
=•½n3.1Lˆª.
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2
,λ,µ)
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1
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2
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1
−α
2
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k
2
2
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v
k
2
2
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1
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2
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>
u
r
v
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2
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−
1
λ
4
µ
2
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(θ
1
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2
,λ,µ)
s
1
(ex,ey)]
2
=min
∆f∈R
m
,∆g∈R
n
b
X(∆u,∆v),(3.8)
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b
X(∆u,∆v)=
1
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1
k(r
u
+∆u)+(r
v
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2
2
+λ
2
k∆uk
2
2
+µ
2
k∆vk
2
2
=[η
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1
,θ
2
)
s
1
(ex,ey)]
2
+
1
α
1
h
∆u
>
∆v
>
i
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λ
2
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n
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n
I
n
µ
2
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n
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∆v
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+
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>
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>
i
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r
u
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v
r
u
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v
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I
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r
u
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v
r
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DOI:10.12677/pm.2023.1361711681nØêÆ
4Œ
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2
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1
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2
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2
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u
k
2
2
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1
α
1
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v
k
2
2
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2
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1
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u
r
v
,
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(θ
1
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2
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s
1
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1
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2
)(ex,ey)
s
1
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1
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2
)
s
1
(ex,ey) =min
(∆A,∆B)∈F
1
1

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(θ
1
k∆Ak
F
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2
k∆Bk
F
)
>


2
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F
1
1
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∆A,∆B
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B+∆BA+∆A
#"
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ey
#
=
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u
v
#)
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>
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[η
(θ
1
,θ
2
)
s
1
(ex,ey)]
2
=
θ
2
1
kr
u
1
k
2
2
kexk
2
2
+
θ
2
2
kr
v
1
k
2
2
kexk
2
2
,
Ù¥
r
u
1
= u−Aex,r
v
1
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u
1
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v
1
,
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u
1
ex
†
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1
(I
n
−exex
†
),∆B= r
v
1
ex
†
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2
(I
n
−exex
†
).
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1
,Z
2
∈R
n×n
.·‚Œ±Ñ
k∆Ak
2
F
=
kr
u
1
k
2
2
kexk
2
2
+kZ
1
(I−exex
†
)k
2
F
,(4.4)
k∆Bk
2
F
=
kr
v
1
k
2
2
kexk
2
2
+kZ
2
(I−exex
†
)k
2
F
.(4.5)
DOI:10.12677/pm.2023.1361711682nØêÆ
4Œ
•Äη
(θ
1
,θ
2
)
s
1
(ex,ey)½Â(4.1),(Ü(4.4),(4.5),Œ
[η
(θ,θ
1
)
s
1
(ex,ey)]
2
=min


Z
1
∈R
n×n
,Z
2
∈R
n×n
∆A,∆B∈R
n×n



θ
2
1
k∆Ak
2
F
+θ
2
2
k∆Bk
2
F

=
θ
2
1
kr
u
1
k
2
2
kexk
2
2
+
θ
2
2
kr
v
1
k
2
2
kexk
2
2
,(4.6)
=½ny²¤.
½n4.2b½n4.1¥^‡¤á,d(1.2),(1.4),
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
λ
2
θ
2
1
β
1
kr
u
1
k
2
2
+
µ
2
θ
2
2
β
2
kr
v
1
k
2
2
,(4.7)
Ù¥
β
1
= λ
2
kexk
2
2
+θ
2
1
,β
2
= µ
2
kexk
2
2
+θ
2
2
.
y²:d½Â(1.2)Ú½n4.1Lˆª,Œ•
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=min
∆u∈R
n
,∆v∈R
n
b
X(∆u
1
,∆v
1
),(4.8)
Ù¥
b
X(∆u
1
,∆v
1
)=
θ
2
1
kexk
2
2
kr
u
1
+∆u
1
k
2
2
+
θ
2
2
kexk
2
2
kr
v
1
+∆v
1
k
2
2
=[η
(θ
1
,θ
2
)
s
1
(ex,ey)]
2
+∆u
>
1
(
θ
2
1
kexk
2
2
I
n
+λ
2
I
n
)∆u
1
+2∆u
>
1
θ
2
1
r
u
1
kexk
2
2
+∆v
>
1
(
θ
2
2
kexk
2
2
I
n
+µ
2
I
n
)∆v
1
+2∆v
>
1
θ
2
2
r
v
1
kexk
2
2
.
N´y²
θ
2
1
kexk
2
2
I
n
+λ
2
I
n
,
θ
2
2
kexk
2
2
I
n
+µ
2
I
n
´¢é¡½Ý,K
∆u= −(
θ
2
1
kexk
2
2
I
n
+λ
2
I
n
)
−1
θ
2
1
r
u
1
kexk
2
2
,
∆v= −(
θ
2
2
kexk
2
2
I
n
+µ
2
I
n
)
−1
θ
2
2
r
v
1
kexk
2
2
ž,þª•Š.ƒA/,²L˜XE,OŽ,·‚íÑ
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
λ
2
θ
2
1
β
1
kr
u
1
k
2
2
+
µ
2
θ
2
2
β
2
kr
v
1
k
2
2
,
K½ny²¤.
DOI:10.12677/pm.2023.1361711683nØêÆ
4Œ
5.ex6=0,ey=01«œ¹
e5Ü©·‚ò?Øex6= 0,ey= 0ž1«œ¹e(z•Øη
(θ
1
,θ
2
,λ,µ)
s
1
.Ó§
·‚I‡kOŽÜ©(•Øη
(θ
1
,θ
2
)(ex,ey)
s
1
,=
η
(θ
1
,θ
2
)
s
1
(ex,ey) =min
(∆A,∆B)∈F
2
1


(θ
1
k∆Ak
F
,θ
2
k∆Bk
F
)
>


2
,(5.1)
Ù¥
F
2
1
=
(
(
∆A,∆B
) :
"
A+∆A−(B+∆B)
B+∆BA+∆A
#"
ex
ey
#
=
"
u
v
#
: ∆A= ∆A
>
,∆B= ∆B
>
)
. (5.2)
½n5.1(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex6= 0,ey= 0,A= A
>
,B= B
>
.·‚k
[η
(θ
1
,θ
2
)
s
1
(ex,ey)]
2
=
2θ
2
1
kr
u
1
k
2
2
kexk
2
2
+
2θ
2
2
kr
v
1
k
2
2
kexk
2
2
−
θ
2
1
(r
>
u
1
ex)
2
kexk
4
2
−
θ
2
2
(r
>
v
1
ex)
2
kexk
4
2
.
y²:d(5.2)
∆Aex= r
u
1
,∆Bex= r
v
1
.(5.3)
òÚn2.2^u(5.3)Œ
∆A= r
u
1
ex
†
+(ex
†
)
>
r
>
u
1
(I
n
−exex
†
)+(I
n
−exex
†
)Z
3
(I
n
−exex
†
),
∆B= r
v
1
ex
†
+(ex
†
)
>
r
>
v
1
(I
n
−exex
†
)+(I
n
−exex
†
)Z
4
(I
n
−exex
†
).
Ù¥Z
3
,Z
4
∈SR
n×n
,·‚Œ±
k∆Ak
2
F
=
kr
u
k
2
2
kexk
2
2
+k(ex
†
)
>
r
>
u
1
(I
n
−exex
†
)k
2
F
+k(I
n
−exex
†
)Z
3
(I
n
−exex
†
)k
2
F
,(5.4)
k∆Bk
2
F
=
kr
v
k
2
2
kexk
2
2
+k(ex
†
)
>
r
>
v
1
(I
n
−exex
†
)k
2
F
+k(I
n
−exex
†
)Z
4
(I
n
−exex
†
)k
2
F
.(5.5)
•Äη
(θ
1
,θ
2
)
s
1
(ex,ey)½Â(5.1),(Ü(5.4),(5.5),Œ
[η
(θ
1
,θ
2
)
s
1
(ex,ey)]
2
=min

Z
3
∈SR
n×n
,Z
4
∈R
n×n
∆A,∆B∈R
n×n


θ
2
1
k∆Ak
2
F
+θ
2
2
k∆Bk
2
F

=min
∆A,∆B∈R
n×n
p(∆A,∆B),
Ù¥
P(∆A,∆B) =
2θ
2
1
kr
u
1
k
2
2
kexk
2
2
−
θ
2
1
(r
>
u
1
ex)
2
kexk
4
2
+
2θ
2
2
kr
v
1
k
2
2
kexk
2
2
−
θ
2
2
(r
>
v
1
ex)
2
kexk
4
2
,
DOI:10.12677/pm.2023.1361711684nØêÆ
4Œ
=•½n4.3Lˆª.
½n5.2b½n5.1¥^‡¤á.d(1.2),(1.4),Œ
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
2θ
2
1
β
11
+4θ
2
1
β
11
kexk
2
2
kr
u
1
k
2
2
−
θ
2
1
β
1
β
11
+3θ
4
1
β
1
−θ
6
1
kexk
4
2
β
1
β
11
(r
>
u
1
ex)
2
+
2θ
2
2
β
22
+4θ
2
2
β
22
kexk
2
2
kr
v
1
k
2
2
−
θ
2
2
β
2
β
22
+3θ
4
2
β
2
−θ
6
2
kexk
4
2
β
2
β
22
(r
>
v
1
ex)
2
,
Ù¥
β
11
= 2θ
2
1
+λ
2
kexk
2
2
,β
22
= 2θ
2
2
+µ
2
kexk
2
2
.
y²:d½Â(1.2)Ú½n5.1Lˆª,Œ•
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=min
∆u∈R
n
,∆v∈R
n
b
X(∆u
1
,∆v
1
),
Ù¥
b
X(∆u
1
,∆v
1
)=
2θ
2
1
kr
u
1
+∆u
1
k
2
2
kexk
2
2
−
θ
2
1
((r
u
1
+∆u
1
)
>
ex)
2
kexk
4
2
+λ
2
k∆u
1
k
2
2
+
2θ
2
2
kr
v
1
+∆v
1
k
2
2
kexk
2
2
−
θ
2
2
((r
v
1
+∆v
1
)
>
ex)
2
kexk
4
2
+µ
2
k∆v
1
k
2
2
=[η
(θ
1
,θ
2
)
s
1
(ex,ey)]
2
+∆u
>
(
2θ
2
1
kexk
2
2
I
n
+λ
2
I
n
−
θ
2
1
exex
>
kexk
4
2
)∆u+2∆u
>
(
2θ
2
1
r
u
kexk
2
2
−
θ
2
1
(ex
>
r
u
)ex
kexk
4
2
)
+∆v
>
(
2θ
2
2
kexk
2
2
I
n
+µ
2
I
n
−
θ
2
2
exex
>
kexk
4
2
)∆v+2∆v
>
(
2θ
2
2
r
u
kexk
2
2
−
θ
2
2
(ex
>
r
v
)ex
kexk
4
2
).(5.6)
N´y²
2θ
2
1
kexk
2
2
I
n
+λ
2
I
n
−
θ
2
1
exex
>
kexk
4
2
,
2θ
2
2
kexk
2
2
I
n
+µ
2
I
n
−
θ
2
2
exex
>
kexk
4
2
´¢é¡½Ý,K
∆u= −(
2θ
2
1
kexk
2
2
I
n
+λ
2
I
n
−
θ
2
1
exex
>
kexk
4
2
)
−1
(
2θ
2
1
r
u
kexk
2
2
−
θ
2
1
(ex
>
r
u
)ex
kexk
4
2
),
∆v= −(
2θ
2
2
kexk
2
2
I
n
+µ
2
I
n
−
θ
2
2
exex
>
kexk
4
2
)
−1
(
2θ
2
2
r
u
kexk
2
2
−
θ
2
2
(ex
>
r
v
)ex
kexk
4
2
)
ž,(5.6)•Š,²L˜XE,OŽ,·‚íÑ
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
2θ
2
1
β
11
+4θ
2
1
β
11
kexk
2
2
kr
u
1
k
2
2
−
θ
2
1
β
1
β
11
+3θ
4
1
β
1
−θ
6
1
kexk
4
2
β
1
β
11
(r
>
u
1
ex)
2
+
2θ
2
2
β
22
+4θ
2
2
β
22
kexk
2
2
kr
v
1
k
2
2
−
θ
2
2
β
2
β
22
+3θ
4
2
β
2
−θ
6
2
kexk
4
2
β
2
β
22
(r
>
v
1
ex)
2
.
=½ny.
e5ü«œ¹y²†þ¡ƒq,ŽÑ.
DOI:10.12677/pm.2023.1361711685nØêÆ
4Œ
íØ1(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex6= 0,ey= 0,B= B
>
,·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
λ
2
θ
2
1
β
1
kr
u
1
k
2
2
+
2θ
2
2
β
22
+4θ
2
2
β
22
kexk
2
2
kr
v
1
k
2
2
−
θ
2
2
β
2
β
22
+3θ
4
2
β
2
−θ
6
2
kexk
4
2
β
2
β
22
(r
>
v
1
ex)
2
.
íØ2(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex6= 0,ey= 0,A= A
>
,·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
2θ
2
1
β
11
+4θ
2
1
β
11
kexk
2
2
kr
u
1
k
2
2
−
θ
2
1
β
1
β
11
+3θ
4
1
β
1
−θ
6
1
kexk
4
2
β
1
β
11
(r
>
u
1
ex)
2
+
µ
2
θ
2
2
β
2
kr
v
1
k
2
2
.
6.ex=0,ey6=0œ¹
e5?Øex=0,ey6=0œ¹,y²†þ¡ƒq,ŽÑ.½n6.1(ex,ey)
>
´‚5X
Ú(1.1)˜‡OŽ),…ex= 0,ey6= 0.·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
λ
2
θ
2
1
ω
1
kr
v
2
k
2
2
+
µ
2
θ
2
2
ω
2
kr
u
2
k
2
2
,
Ù¥
r
u
2
= u+Bey,r
v
2
= v−Aey,
ω
1
= θ
2
1
+λ
2
keyk
2
2
,ω
2
= θ
2
2
+µ
2
keyk
2
2
.
½n6.2(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex= 0,ey6= 0,A= A
>
,B= B
>
.·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
2θ
2
1
ω
11
+4θ
2
1
ω
11
keyk
2
2
kr
v
2
k
2
2
−
θ
2
1
ω
1
ω
11
+3θ
4
1
ω
1
−θ
6
1
keyk
4
2
ω
1
ω
11
(r
>
v
2
ey)
2
(6.1)
+
2θ
2
2
ω
22
+4θ
2
2
ω
22
keyk
2
2
kr
u
2
k
2
2
−
θ
2
2
ω
2
ω
22
+3θ
4
2
ω
2
−θ
6
2
keyk
4
2
ω
2
ω
22
(r
>
u
2
ey)
2
,(6.2)
Ù¥
ω
11
= 2θ
2
1
+λ
2
keyk
2
2
,ω
22
= 2θ
2
2
+µ
2
keyk
2
2
.
½n6.3(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex= 0,ey6= 0,B= B
>
.·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
λ
2
θ
2
1
ω
1
kr
v
2
k
2
2
+
2θ
2
2
ω
22
+4θ
2
2
ω
22
keyk
2
2
kr
u
2
k
2
2
−
θ
2
2
ω
2
ω
22
+3θ
4
2
ω
2
−θ
6
2
keyk
4
2
ω
2
ω
22
(r
>
u
2
ey)
2
.
½n6.4(ex,ey)
>
´‚5XÚ(1.1)˜‡OŽ),…ex= 0,ey6= 0,A= A
>
.·‚k
[η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey)]
2
=
2θ
2
1
ω
11
+4θ
2
1
ω
11
keyk
2
2
kr
v
2
k
2
2
−
θ
2
1
ω
1
ω
11
+3θ
4
1
ω
1
−θ
6
1
keyk
4
2
ω
1
ω
11
(r
>
v
2
ey)
2
+
µ
2
θ
2
2
ω
2
kr
u
2
k
2
2
.
DOI:10.12677/pm.2023.1361711686nØêÆ
4Œ
7.A^
3ùÜ©,·‚ò ^•[êŠ~fy²½n3.1,3.2,¿…OŽÑ•[êâ.Ù{½n
yaq,ŽÑ.¤kêŠOŽÑ3MATLABR2016a¥±Åì°Ý2.2204×10
−16
?1.
·‚•Ä‚5XÚ(1.1)
"
A−B
BA
#"
x
y
#
=
"
u
v
#
.
Ù¥
A=





2−801
−50−45
4−3−1−2
36−71





, B=





2103
1401
0751
10
−3
021





,
u=





10
6
5
0
0





, v=





10
−6
0
0
1





.
¦^Ü©ÀÌGaussž{,·‚˜‡OŽ)ez= (ex
>
,ey
>
)
>
,Ù¥
ex=





−2.550948811691706×10
4
−8.103528926046949×10
4
−5.127357916480393×10
4
7.085379024155514





,ey=





1.111396285339100×10
5
1.416787831660810×10
4
2.818042030869885×10
4
−1.894433357643171×10
5





.
òþã(J“\½n3.1,3.2,·‚Œ±
η(ez) = 2.767658843372967×10
−17
,
η
(θ
1
,θ
2
)
s
1
(ex,ey) = 1.207459273255117×10
−33
,
η
(θ
1
,θ
2
,λ,µ)
s
1
(ex,ey) = 1.185846126215313×10
4
.
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[1]Benzi,M.,Golub,G.H.andLiesen,J.(2005)NumericalSolutionofSaddlePointProblems.
ActaNumerica,14,1-137.https://doi.org/10.1017/S0962492904000212
[2]Xiang, H., Wei, Y.M.and Diao, H.A.(2006) Perturbation Analysisof GeneralizedSaddle Point
Systems.LinearAlgebraanditsApplications,419,8-23.
DOI:10.12677/pm.2023.1361711687nØêÆ
4Œ
https://doi.org/10.1016/j.laa.2006.03.041
[3]Xu,W.(2009)New PerturbationAnalysisfor GeneralizedSaddlePointSystems.Calcolo,46,
25-36.https://doi.org/10.1007/s10092-009-0157-8
[4]Xu,W.W.,Liu,M.M.,Zhu,L.andZuo,H.F.(2017)NewPerturbationBoundsAnalysisof
aKindofGeneralizedSaddlePointSystems.EastAsianJournalonAppliedMathematics,7,
116-124.https://doi.org/10.4208/eajam.100616.031216a
[5]Sun,J.G.(1999)StructuredBackwardErrorsforKKTSystems.LinearAlgebraanditsAp-
plications,288,75-88.https://doi.org/10.1016/S0024-3795(98)10184-2
[6]Yang,X.D.,Dai,H.andHe,Q.Q.(2011)ConditionNumbersandBackwardPerturbation
Bound for LinearMatrixEquations. NumericalLinearAlgebrawithApplications,18,155-165.
https://doi.org/10.1002/nla.725
[7]Rigal,J.L.andGaches,J.(1967)OntheCompatibilityofaGivenSolutionwiththeDataof
aLinearSystem.JournaloftheACM,14,543-548.https://doi.org/10.1145/321406.321416
[8]Wilkinson,J.(1965)TheAlgebraicEigenvalueProblem.OxfordUniversityPress,Oxford.
[9]Xiang,H.andWei,Y.M.(2007)OnNormwiseStructuredBackwardErrorsforSaddlePoint
Systems.SIAMJournalonMatrixAnalysisandApplications,29,838-849.
https://doi.org/10.1137/060663684
[10]Chen,X.S.,Li,W.,Chen, X.J. and Liu,J.(2012)Structured Backward Errors for Generalized
SaddlePointSystems.LinearAlgebraanditsApplications,436,3109-3119.
https://doi.org/10.1016/j.laa.2011.10.012
[11]Eisenstat,S.C., Gratton,S. and Titley-peloquin, D. (2017) On the Symmetric Componentwise
RelativeBackwardErrorforLinearSystemsofEquations.SIAMJournalonMatrixAnalysis
andApplications,38,1100-1115.https://doi.org/10.1137/140986566
[12]Higham,D.J.andHigham,N.J.(1992)BackwardErrorandConditionofStructuredLinear
Systems.SIAMJournalonMatrixAnalysisandApplications,13,162-175.
https://doi.org/10.1137/0613014
[13]Higham,N.J.(2002)AccuracyandStabilityofNumericalAlgorithms,2ndEdition,SIAM,
Philadelphia.
[14]Rump,S.M.(2015)TheComponentwiseStructuredandUnstructuredBackwardErrorsCan
BeArbitrarilyFarApart.SIAMJournalonMatrixAnalysisandApplications,36,385-392.
https://doi.org/10.1137/140985500
[15]Stewart,G.W.andSun,J.G.(1990)MatrixPerturbationTheory.AcademicPress,Boston.
[16]Rump,S.M.(2015)TheComponentwiseStructuredandUnstructuredBackwardErrorsCan
BeArbitrarilyFarApart.SIAMJournalonMatrixAnalysisandApplications,36,385-392.
https://doi.org/10.1137/140985500
DOI:10.12677/pm.2023.1361711688nØêÆ

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