Digital Signal Processing Reference
In-Depth Information
4.8 The Lanczos algorithm for continued fractions
Employing (4.22), the infinite and the m th order delayed continued fractions
G
CF(s)
(u) and G
CF(s)
(u) from (4.17) and (4.18) become, respectively
m
q
(s)
1
1
e
(s)
1
u
−· · ·−
q
(s)
e
(s)
G
CF(s)
(u) =
c
s
r
1
r
u
−· · ·
(4.51)
u
−
−
−
q
(s)
1
1
e
(s)
1
u
−· · ·−
q
(s)
e
(s
m
u
.
(u) =
c
s
m
1
G
CF(s)
m
(4.52)
u
−
−
−
Similarly, the infinite and the m th order of the even part of the associated
Lanczos continued fractions (LCF) can be introduced as
q
(s
1
e
(s)
q
(s
r
e
(s)
c
s
u−q
(s)
1
r
G
LCF(s)
e
1
(u) =
−e
(s
0
−
−e
(s
1
−· · ·−
−e
(s
r
−· · ·
u−q
(s)
2
u−q
(s)
r+1
[β
(s
1
]
2
u−α
(s
1
−· · ·−
[β
(s
r
]
2
u−α
(s)
c
s
u−α
(s
0
−
=
(4.53)
−· · ·
r
q
(s
1
e
(s)
q
(s
m
e
(s
m
u−q
(s)
m+1
c
s
u−q
(s)
1
G
LCF(s)
e,m
1
(u) =
−e
(s
0
−
−e
(s
1
−· · ·−
u−q
(s)
2
−e
(s
m
[β
(s
1
]
2
u−α
(s
1
−· · ·−
[β
(s
m
]
2
u−α
(s
m
c
s
u−α
(s
0
−
=
.
(4.54)
A detailed derivation [5] shows that there is a general relationship between
G
LCF(s)
(u) and G
CF(s)
(u) which by reference to (4.19) can be written as
e,m
m
(u) = G
CF(s)
2n
G
LCF(s)
e,n
(u) = G
CCF(s)
e,n
(u).
(4.55)
There exists also the infiniteorder and the m th order odd part of (4.51) that
are labeled by G
LCF(s)
(u) and G
LCF(s)
(u), respectively
o
o,m
(u) =
c
s
u
G
LCF(s)
o
1 +
q
(s
r
e
(s)
q
(s)
1
u−q
(s)
1
q
(s
2
e
(s)
r−1
1
×
−e
(s
1
−
−e
(s
2
−· · ·−
−e
(s
r
−· · ·
u−q
(s)
2
u−q
(s)
r
(4.56)
1
u
G
LCF(s)
o,m
(u) =
c
s
+
q
(s
2
e
(s)
q
(s
m
e
(s
m−1
u−q
(s)
c
s+1
u−q
(s)
1
1
×
−
−· · ·−
−e
(s)
1
u−q
(s)
2
−e
(s)
2
−e
(s
m
m
[β
(s+1
1
]
2
u−α
(s+1)
1
[β
(s+1)
m−1
]
2
u−α
(s+1)
m−1
1
u
c
s+1
u−α
(s+1)
0
=
c
s
+
(4.57)
−
−· · ·−
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