On fast evaluation of bivariate polynomials at equispaced arguments

1813 order to use the FFT and IFFT. As b normally has structure, we may wish to model it using an affine model. Following Starer and Nehorai [8], constrain b to be of the form b = Tf + c. Here T is a (p + 1) x /-dimensional constraint matrix with full column rank, c is a real (p + l)-dimensional vector, and/is a real /-dimensional vector. Both T and c are chosen to enforce the structure desired of b. The vector/contains the / degrees of freedom of b and is the vector for which we solve-. d= FFKtA/l, 0 • • • 0]) r = 0 for; = 0 • • • (N-1) d, = \d,\ 2 if dj < t then g(r) = i r = r + 1 d, = 4+1 endif d, = dj x endfor q = IFFT(rf) Form a, 0, and y from q Ml = 7-/3* X aT 1 X 0 if r > 0 Form E 2 using g Ml = Ml X £ 2 Ml = Ml + M2 X (/-£ 2 * x M2)~' X M2* endif M3 = ((r* x ?*) x Ml) x Y Solve the system-(real(M3 x T)) x b = real(M3 X c) for b b, = T x / + c. We are now at a point where we can compare the complexity of the new algorithm with those of other methods. It is easily shown that the new method requires O((p + I + r)N 2) multiplications and 0(/> 3 + / 3 + r 3) computations for the two matrix inversions and the solution of the system of linear equations. This represents an improvement over the method of Kumaresan et al. [1] which requires Q((2pf + / 3) computations for the matrix inversions. Further , the new method is always stable. The Steiglitz-McBride algorithm requires ©((/> + /)iV 2) multiplications and 0(p 3 + / 3) computations for matrix inversions. Thus it is cheaper than the new method only when r =£ 0. On machines with sufficient precision this is a low probability event. Therefore, with our improvement, the computation required by IQML is almost always the same as that required by the Steiglitz-McBride algorithm. Thus, neither Steiglitz-McBride nor IQML should be preferred. This is intuitively pleasing since the iterations have been shown to produce equivalent estimates [4].An algorithm for pole-zero modeling …