An algorithm for solving shifted skew-symmetric systems

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13 Οκτ 2013 (πριν από 3 χρόνια και 11 μήνες)

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An algorithm for solving shifted skew-symmetric systems
Jiang Erxiong
Department of Mathematics,Shanghai University,No.99 Shang Da Road,200444,
China [ejiang@fudan.edu.cn]
2000 Mathematics Subject Classification.65F10,65N22
Consider linear algebra equations system
Ax=g
where x ∈ R
n
,is a unknown vector.b ∈ R
n
is a given vector.A is a given n×n,real
matrix and A = (αI+S),where α is a real number,I is the identity matrix and S is
a skew-symmetric matrix namely S
T
= −S.Such system is produced many place
see [1],[2].I put forward an algorithm for solving such linear equations system as
follows:
Skew-symmetric MRES algorithm
Pick x
0
,r
0
= g −Ax
0
s = ￿r
0
￿,q
1
= r
0
/s,p
−1
= p
0
= 0
˜
θ
1
= α,δ
−1
= δ
0
= β
0
= s
0
= 0,c
0
= 1
1 →k
I.q = Sq
k

k−1
q
k−1

k
= ￿q￿
if β
k
= 0 then goto II else
q
k+1
= q/β
k
II.θ
k
=
￿
(
˜
θ
k
2

2
k
),c
k
=
˜
θ
k

k
,s
k
= β
k

k

k−1
= −s
k−1
β
k
,
˜
θ
k+1
= c
k−1
θ
k
p
k
= (q
k
−δ
k−2
p
k−2
)/θ
k
x
k
= x
k−1
+sc
k
p
k
,
s = −ss
k
,￿r
k
∗ ￿ = |s|
if ￿r
k
∗ ￿ < ￿ then goto III,else
k +1 →k goto I.
III.x
k
is a approximate solution.
Theorem:For any eigenvalue of S,iβ
j
,if it holds β
j
∈ [−β,β] then
￿g −Ax
k
￿ = ￿r
k
∗ ￿ ≤ 2(
1
α/β +
￿
(α/β)
2
+1
)
k
￿r
0
￿
[1] Zhong-Zgi Bai,Gene H.Golub and Michael K.Ng,Hermitian and Skew-Hermitian
Splitting Methods for Nin-Hermitian Positive Definite Linear Systems,SIAMJ.Matrix
Anal.Appl.,Vol.24,No.3,(2003),603-626.
[2] Zhong-zhi Bai,Gene H.Golub,Lin-Zhang Lu and Jun-Fen Yin,Block Triangular
and Skew-Hermitian Splitting Methods for Positive-Definite Linear Systems,SIAM
J.Sci.Comput.,Vol.26,No.3,(2006),844-863.