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Chapter 14 Orthogonal rational functions In this chapter we adapt the technique laid down in the previous chapters for polynomial (vector) sequences to a specific set of proper rational functions. The goal is the computation of an orthonormal basis of the linear space Rn of proper rational functions φ(z) = n(z)/d(z) with regard to the discrete inner product φ, ψ = n $ i=0|wi|2 φ(zi)ψ(zi) with given points zi and weights wi. The rational function φ(z) is proper, i.e., the degree of the numerator n(z) is less than or equal to the degree of the denominator d(z). Both degrees are not greater than n and the denominator polynomial d(z) has a prescribed set {y1, . . . , yn}, yi ∈ C, of possible zeros, called the poles of the rational function φ(z). The orthonormal basis of rational functions can then be used when solving least squares approximation problems with rational functions with prescribed poles. Moreover, it is also closely related with the computation of an orthogonal factorization of Cauchy-like matrices whose nodes are the points zi and yi [74, 76]. We prove that an orthonormal basis of (Rn,  · , · ) can be generated by means of a suitable recurrence relation. When the points zi as well as the points yi are all real, fast O(n2 ) Stieltjes-like procedures for computing the coefficients of such relation were first devised in [74, 76]. However, like the polynomial (Vandermonde) case [132], these fast algorithms turn out to be quite sensitive to roundoff errors so that the computed functions are far from orthogonal. Therefore, in this chapter we propose a different approach based on the reduction of the considered problem to the following inverse eigenvalue problem (HL-IEP): Find a generator representable Hessenberg-like matrix Z of order n + 1, i.e., a matrix whose lower triangular part is the lower triangular part of a rank one matrix, and a unitary matrix Q of order n + 1 such that QH w = w e1 and QH DzQ = Z + Dy. Here and below w = [w0, . . . , wn]T , Dz = diag([z0, . . . , zn]) and Dy = diag([y0, . . . , yn]), where y0 can be chosen arbitrarily. Moreover, we denote by Hk the class of k × k Hessenberg-like 447 448 Chapter 14. Orthogonal rational functions matrices and by H (g) k the class of k × k generator representable Hessenberg-like matrices. If both Z and ZH belong to Hk, then Z is a semiseparable matrix. In Chapter 12, a quite similar reduction to an inverse eigenvalue problem for a tridiagonal symmetric matrix (T-IEP) or for a unitary Hessenberg matrix (H-IEP) was also exploited in the theory on the construction of orthonormal polynomials with regard to a discrete inner product. We also generalized this theory to orthonormal vector polynomials. Since invertible (generator representable) semiseparable matrices are the inverses of (irreducible) tridiagonal ones as we saw in Chapter 1. We find that HL-IEP gives a generalization of T-IEP and, in particular, it reduces to T-IEP in the case where yi, zi ∈ R and all prescribed poles yi are equal. We devise a method for solving HL-IEP which fully exploits its recursive properties. This method proceeds by applying a sequence of carefully chosen Givens transformations to update the solution at the k-th step by adding a new data (wk+1, zk+1, yk+1). The unitary matrix Q can thus be determined in its factored form as a product of O(n2 ) Givens transformations at the cost of O(n2 ) arithmetic operations (ops). The complexity of forming the matrix Z depends on the structural properties of its upper triangular part and, in general, it requires O(n3 ) ops. In the case where all the points zi lie on the real axis, we show that Z is a generator representable semiseparable matrix so that the computation of Z can be carried out using O(n2 ) ops only. In addition to that, the class H (g) n+1 results to be close under linear fractional (Möbius) transformations of the form z → (αz +β)/(γz +δ). Hence, by combining these two facts together, we are also able to prove that the process of forming Z can be performed at the cost of O(n2 ) ops whenever all points zi belong to a generalized circle (ordinary circles and straight lines) in the complex plane. This chapter is organized in the following way. In Section 14.1 we reduce the computation of a sequence of orthonormal rational...

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