Digital Signal Processing Reference
In-Depth Information
Here, β
n
becomes proportional to the squared norm of Q
n
(x) and, hence, all
the parameters{β
n
}are strictly positive. Furthermore, since the diagonal case
of the matrix element (2.35) is the normQ
n
b
a
|Q
n
(x)|
2
dσ(x) = w
n
, we
can derive a very simple formula for β
n
from (2.52) via
2
=
w
n
w
n−1
.
β
n
=
(2.53)
Thus, the entire procedure of a numerical construction of the orthonormal
ized polynomials{Q
r
(x)}with the selfgenerating coe
cients{α
r
,β
r
}is rem
iniscent of the wellknown Lanczos algorithm when applied to the quantum
mechanical position operator x [5]. Moreover, the Lanczos algorithm for phys
ical state vectors can be devised by replacing operator x by the Hamiltonian
H in the scalar recursion (2.49). The polynomials{Q
n
(x)}are called the
polynomials of the first kind. Every such polynomial Q
n
(x) of the degree n
has n zeros{x
k
}(1≤k≤n). Due to the mentioned assumptions on W(x),
all the quantities{x
k
}are realvalued simple zeros (x
k
= x
k
′
for k
′
= k) and
belong to the interval (a,b).
We also need the polynomials of the second kind denoted by{P
r
(x)}that
are tightly linked to the basis functions{Q
r
(x)}. Formally, the polynomials
P
r
(x) satisfy the same recursive relation from (2.49), but the initial conditions
are different
β
n+1
P
n+1
(x) = (x−α
n
)P
n
(x)−β
n
P
n−1
(x)
P
0
(x) = 0
.
(2.54)
P
1
(x) = 1
Otherwise, both recursions (2.49) and (2.54) are numerically stable. Alter
natively, the polynomials Q
n
(x) and P
n
(x) can be introduced by their power
series representations
n
n−1
q
r,n−r
x
r
p
r,n−r
x
r
.
Q
n
(x) =
P
n
(x) =
(2.55)
r=0
r=0
Here, the expansion coe
cients p
r,n−r
and q
r,n−r
can also be computed re
cursively by the same kind of relations from (2.49) and (2.54)
β
n+1
p
n+1,n+1−r
= p
n,n+1−r
−α
n
p
n,n−r
−β
n
p
n−1,n−1−r
p
n,−1
= 0
(2.56)
p
n,m
= 0
(m > n)
p
0,0
= 1
p
1,1
= 1
β
n+1
q
n+1,n+1−r
= q
n,n+1−r
−α
n
q
n,n−r
−β
n
q
n−1,n−1−r
q
n,−1
= 0
.
(2.57)
q
n,m
= 0
(m > n)
q
0,0
= 1.
It can be shown that the zeros{x
k
}of P
n
(x) interlace with the zeros x
k
of
Q
n
(x)
x
1
< x
1
< x
2
< x
2
<< x
n
< x
n
.
(2.58)
In the mathematical literature, this is known as the Cauchy-Poincare interlac
ing theorem, which was rediscovered in quantum chemistry as the Hylleraas-
Undheim theorem [124].
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