1. 1 EE 616 Computer Aided Analysis of Electronic Networks Lecture 7 Instructor: Dr. J. A. Starzyk, Professor School of EECS Ohio University Athens, OH, 45701

2. 2 Outline Computer Generation of Network Functions Unit circle polynomial interpolation Condition numbers for interpolation

3. 3 Computer Generation of Network Functions We want to obtain a network function in the semi-symbolic form This will be useful for direct evaluation of F(s) at different frequencies (instead of using 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

4. 4 Computer Generation of Network Functions (cont’d) Solving LUX =  we can get X and then F(si). The numerator N(si) can be found as 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.

5. 5 Unit circle polynomial interpolation Assume we have (n + 1) distinct points (xi, f( x i) = y i). We want to find coefficients of the polynomial We have or in the matrix form = or Xa = y  a = X -1 y where The most numerically stable selection of is on the unit circle.

6. 6 Unit circle polynomial interpolation (cont’d) Let us define andx k =  k so and Therefore the solution is or simply 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. In addition the polynomial interpolation on the unit circle is the most numerically stable algorithm.

7. 7 Condition numbers for interpolation Condition numberfor the matrix is where  max,  min are the largest and smallest eigenvalues of XX*. K(X) is a measure of perturbations in y. In our case 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.

8. 8 Condition numbers for interpolation - Example Example: Numerical value of the numerator and denominator are Find transfer function (first order polynomials N(s) and D(s)). n = 1n + 1 = 2 since so

9. 9 Condition numbers for interpolation - Example And, we obtain

10. 10 Condition numbers for various interpolations

11. 11 Growth of error versus degree of the polynomial n