Computer Generation of Network
Functions:
We want to obtain a network function in the semisymbolic form
Fs  
Ns 
Ds 
This will be useful for direct evaluation of F(s) at
different frequencies (instead of using n3/3 operations
for LU factorization each time) or inverse Laplace
transform to obtain time domain solution.
Assume we know LU factorization at frequency si
and assume that T = LU has all entries with s in the
numerator. Determinant D(si) can be found as
n
Ds i   det Ts i   det Ls i    L ij
j1
Solving LUX =  we can get X and then F(si). The
numerator N(si) can be found as
Ns i   Ds i Fs i 
Changing si we get sets of pairs (si, N(si)) and
(si, D(si)), and we get the coefficients of N and D.
This is a well-known problem of polynomial
interpolation.
Unit circle polynomial interpolation:
Assume we have (n + 1) distinct points (xi, f(xi) = yi).
We want to find coefficients of the polynomial
n
f (x)   a j x i
j0
We have
a 0  a 1 x i  a 2 x i2  ...  a n x in  y i
or in the matrix form
1 x 0

1 x 1
 

1 x n
x 02  x 0n   a 0 

x 12  x 1n   a 1 
 

  
x 2n  x nn  a n 
or
Xa = y

i  0,1,2,..., n
a = X-1y
=
y0 
y 
 1
 
 
y n 
 
j
where X  x i
The most numerically stable selection of xi is on the
unit circle.
Let us define
  exp
2  1
n 1
and xk = k
so
   
X  x ij  ij
1
1
X

X*
and
n 1
Therefore the solution is
a  X 1 y 
 
1
1
X* y 
ij y
n 1
n 1
1 n
 jk
or simply a j  n  1  y k 
k 0
Using this equation all coefficients of the
approximating polynomial can be explicitly
calculated. Finding each coefficient requires an
addition of n + 1 complex numbers - each one easily
obtained from the approximated function values yk.
In addition the polynomial interpolation on the unit
circle is the most numerically stable algorithm.
Condition numbers for interpolation:
Condition number
K X  for
~
the matrix
X
~
is

K x   max /  min
~
where max, min are the largest and smallest
eigenvalues of XX*.
K(X) is a measure of perturbations in y.
In our case
XX *  XX 1 n  1  n  1I
So all the eigenvalues of XX* are equal to (n + 1) and
K(x) = 1.
Note that K(x)  1 always. Therefore, the selection
of all points on the unit circle yields the best possible
accuracy of the polynomial approximation problem.
Example
Numerical value of the numerator and denominator
are
N s0  t1  3
Ds0  1  11
N s p  1  3
Ds p  1  7
Find transfer function (first order polynomials N(s)
and D(s)).
n=1
since
n+1=2
T
a 0  a 1s
b 0  b1s
  exp
2  1
 1
2
so
1 1
 10 k Ns k   1 3  3  0

2 k 0
2
1 1
1
1k
a 1    1 Ns k   3  3  3
2 k 0
2
1 1
1
0k
b 0    1 Ds k   11  7   2
2 k 0
2
1 1
1
1k
b1    1 Ds k   11  7   9
2 k 0
2
a0 
And, we obtain
T
3s
2  9s
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Computer Generation of Network Functions: