Digital Signal Processing Reference
In-Depth Information
4.9 The Pade-Lanczos approximant
Here, we define the delayed PadeLanczos approximant (PLA) [5, 17] by
P
(s)
L
c
s
β
(s)
1
(u)
G
PLA(s)
L,K
(u) =
.
(4.66)
Q
(s)
K
(u)
As usual, the special case L = K of (4.66) yields the paradiagonal variant
denoted as
1
P
(s
n
(u)
Q
(s
n
(u)
c
s
β
(s)
1
G
PLA(s)
n,n
(u)≡G
PLA(s)
n
G
PLA(s)
n
(u)
(u) =
.
(4.67)
Functions Q
(s
n
(u) and P
(s
n
(u) are the delayed Lanczos polynomials of the
first and second kind, respectively.
They can be introduced through their
recursion relations
β
(s)
n+1
P
(s)
n+1
(u) = [u−α
(s
n
]P
(s)
(u)−β
(s
n
P
(s)
n−1
(u)
n
(4.68)
P
(s)
0
P
(s)
1
(u) = 0
(u) = 1
β
(s)
n+1
Q
(s)
n+1
(u) = [u−α
(s
n
]Q
(s
n
(u)−β
(s
n
Q
(s)
n−1
(u)
.
(4.69)
Q
(s)
Q
(s
0
(u) = 1
−1
(u) = 0
Alternatively, the polynomials P
(s
n
(u) and Q
(s
n
(u) could be defined via the
corresponding power series representations
n−1
n
p
(s)
q
(s)
P
(s)
n
n,n−r
u
r
Q
(s
n
(u) =
n,n−r
u
r
.
(u) =
(4.70)
r=0
r=0
The polynomial expansion coe
cients p
(s)
n,n−r
and q
(s)
n,n−r
can be generated by
means of the following recursions
β
(s)
n+1
p
(s)
= p
(s)
n,n+1−r
−α
(s
n
p
(s)
−β
(s
n
p
(s)
n+1,n+1−r
n,n−r
n−1,n−1−r
(4.71)
p
(s)
p
(s)
0,0
= 0
1,1
= 1
β
(s)
n+1
q
(s)
n+1,n+1−r
= q
(s)
−α
(s
n
q
(s)
n,n−r
−β
(s
n
q
(s)
n−1,n−1−r
n,n+1−r
(4.72)
q
(s)
0,0
q
(s)
1,1
=−α
(s
0
/β
(s)
= 1
1
p
(s)
q
(s)
p
(s)
q
(s)
n,−1
= 0
n,−1
= 0
n,m
= 0
n,m
= 0
(m > n).
(4.73)
1
Exceptionally, in the Lanczos algorithm, the numerator P
(s)
L
and denominator Q
(s)
K
poly-
nomials in the quotient P
(s)
L
/Q
(s)
K
are of degree L−1, and K, respectively. For this reason,
the case L = K, i.e., P
(s)
K
/Q
(s)
K
, does not actually represent the diagonal, but rather the
para-diagonal Pade approximant, [(K−1)/K] [5].
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