The Orthogonal Rayleigh Quotient Iteration (ORQI) method
This paper presents a new method for computing all the eigenvectors of a real n× n symmetric band matrix T. The algorithm computes an orthogonal matrix Q=[ q 1,…, q n] and a diagonal matrix Λ=diag{ λ 1,…, λ n } such that TQ= QΛ. The basic ideas are rather simple. Assume that q 1,…, q k−1 and λ 1,…,...
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| Published in: | Linear algebra and its applications Vol. 358; no. 1; pp. 23 - 43 |
|---|---|
| Main Author: | |
| Format: | Journal Article |
| Language: | English |
| Published: |
Elsevier Inc
2003
|
| Subjects: | |
| ISSN: | 0024-3795, 1873-1856 |
| Online Access: | Get full text |
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| Summary: | This paper presents a new method for computing all the eigenvectors of a real
n×
n symmetric band matrix
T. The algorithm computes an orthogonal matrix
Q=[
q
1,…,
q
n]
and a diagonal matrix
Λ=diag{
λ
1,…,
λ
n
} such that
TQ=
QΛ. The basic ideas are rather simple. Assume that
q
1,…,
q
k−1
and
λ
1,…,
λ
k−1
have already been computed. Then
q
k
is obtained via the Rayleigh Quotient Iteration (RQI) method. Starting from an arbitrary vector
u
0
the RQI method generates a sequence of vectors
u
ℓ
, ℓ=1,2,… , and a sequence of scalars
ρ
ℓ, ℓ=0,1,2,… The theory tells us that these two sequences converge (almost always) to an eigenpair
(ρ
*,
u
*)
. The appeal of the RQI method comes from the observation that the final rate of convergence is cubic. Furthermore, if the starting point is forced to satisfy
q
T
i
u
0=0
for
i=1,…,
k−1, as our method does, then all the coming vectors,
u
ℓ,ℓ=1,2,…
, and their limit point,
u
*
, should stay orthogonal to
q
1,…,
q
k−1
. In practice orthogonality is lost because of rounding errors. This difficulty is resolved by successive orthogonalization of
u
ℓ
against
q
1,…,
q
k−1
. The key for effective implementation of the algorithm is to use a selective orthogonalization scheme in which
u
ℓ
is orthogonalized only against “close” eigenvectors. That is,
u
ℓ
is orthogonalized against
q
i
only if |
ρ
ℓ−
λ
i
|⩽
γ where
γ is a small threshold value, e.g.,
γ=∥
T∥
∞/1000. An essential feature of the proposed orthogonalization scheme is the use of reorthogonalization.
The ORQI method is supported by forward and backward error analysis. Preliminary experiments on medium-size problems (
n⩽1000) are quite encouraging. The average number of iterations per eigenvector was less than 13, while the overall number of flops required for orthogonalizations is often below
n
3/2. |
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| ISSN: | 0024-3795 1873-1856 |
| DOI: | 10.1016/S0024-3795(01)00330-5 |