Digital Signal Processing Reference
In-Depth Information
Lanczos coupling constants are exceptionally important for spectral analysis,
it is necessary to look for more stable algorithms than the Lanczos recur
sion for state vectors{ψ
n
}
or autocorrelation functions as the only input data. Given that the ma
jor task of the Lanczos algorithm consists of obtaining the couplings, it is
recommended to try to bypass the construction of state vectors{ψ
n
}, but that still rely upon the signal points{c
n
}whose
orthogonality could be destroyed in the course of the recursion. To this end,
one can resort to at least two recursive algorithms that fulfill the mentioned
requirements by using only the signal points and by simultaneously allevi
ating altogether generation of the Lanczos state vectors{ψ
n
}. These are the
Rutishauser QD algorithm [156] and the Gordon [158] productdifference (PD)
algorithm. Both algorithms can compute the entire set of the Lanczos coupling
constants{α
(s
n
,β
(s)
}for arbitrarily large values of n. This fact is important
to emphasize particularly because of a statement from Numerical Recipes [77]
claiming that computing the Lanczos coupling parameters generated by, e.g.,
the power moments{
n
n
}) must be viewed as useless due
to their mathematical illconditioning. This claim does not hold true for the
power moments generated by the QD [156] and PD [158] algorithms.
The most general prescription for the PD algorithm is facilitated by intro
ducing an auxiliary matrix
}(equivalent to{c
n
={λ
(s)
λ
(s)
}with zerovalued elements below
n,m
the main counterdiagonal as
0
@
1
A
.
λ
(s)
1,1
λ
(s)
1,2
λ
(s)
1,3
λ
(s)
1,n−2
λ
(s)
1,n−1
λ
(s)
1,n
λ
(s)
2,1
λ
(s)
2,2
λ
(s)
λ
(s)
2,n−2
λ
(s)
0
2,3
2,n−1
λ
(s)
3,1
λ
(s)
3,2
λ
(s)
3,3
λ
(s)
3,n−2
0
0
.
.
.
.
.
.
.
.
.
λ
(s)
=
(4.29)
λ
(s)
n−2,1
λ
(s)
n−2,2
λ
(s)
0
0
0
n−2,3
λ
(s)
n−1,1
λ
(s)
0 0
0
0
n−1,2
λ
(s)
n,1
0
0 0
0
0
Here, the first column of this matrix is filled with zeros, except the element
λ
(s)
1,1
which is set to unity. The second column contains the signal points with
the alternating sign via
0
1
λ
(s)
1,n−2
λ
(s)
1,n−1
λ
(s)
1
c
s
c
s+1
1,n
@
A
. (4.30)
0 −c
s+1
−c
s+2
λ
(s)
2,n−2
λ
(s)
0
2,n−1
λ
(s)
3,n−2
0
c
s+2
c
s+3
0
0
λ
(s)
.
.
.
.
.
.
=
.
.
.
0 (−1)
n−1
c
n+s−3
(−1)
n−1
c
n+s−2
0
0
0
(−1)
n
c
n+s−2
0
0
0
0
0
0
0
0
0
0
0
Search WWH ::
Custom Search