Matrix methods for quadrature formulas on the unit circle. A survey I
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paper_title: Spectral methods for orthogonal rational functions

paper_content:

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and five-diagonal matrices. As an application of this operator approach, we obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation, generalizing to the rational case known results for orthogonal polynomials on the unit circle. Finally, we extend these results to orthogonal polynomials on the real line with poles in the lower half plane.
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paper_title: A matricial computation of rational quadrature formulas on the unit circle

paper_content:

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Mobius transformations of Hessenberg matrices and also as the eigenvalues of the operator Mobius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.
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paper_title: Orthogonal Laurent Polynomials and Gaussian Quadrature

paper_content:

The purpose of this paper is to introduce a class of orthogonal functions similar in many respects to the classical orthogonal polynomials [6]. These functions are linear combinations of integral (positive, negative and zero) powers of a single complex variable z. Hence they are called Laurent polynomials.
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paper_title: A strong Stieltjes moment problem

paper_content:

A data encoding keyboard for use in electronic processing systems such as electronic calculators includes a pushbutton encoded thumbwheel switch. A plurality of cams secured to the switch define different codes in accordance with the angular orientation of the thumbwheel. The cams engage contacts on a printed circuit matrix adjacent the thumbwheel when the thumbwheel is depressed.
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paper_title: Some results about numerical quadrature on the unit circle

paper_content:

In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.
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paper_title: On the construction of Gaussian quadrature rules from modified moments.

paper_content:

Abstract : Given a weight function omega(x) on (alpha, beta), and a system of polynomials (p sub k)(x), k = 0 to infinity, with degree p sub k (x) = k, we consider the problem of constructing Gaussian quadrature rules from 'modified moments'. Classical procedures take p sub k (x) = x, but suffer from progressive ill-conditioning as n increases. A more recent procedure, due to Sack and Donovan, takes for p sub k (x) a system of (classical) orthogonal polynomials. The problem is then remarkably well-conditioned, at least for finite intervals (alpha, beta). In support of this observation, we obtain upper bounds for the respective asymptotic condition number. In special cases, these bounds grow like a fixed power of n. We also derive an algorithm for solving the problem considered which generalizes one due to Golub and Welsch. Finally, some numerical examples are presented.
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paper_title: Laurent–Jacobi Matrices and the Strong Hamburger Moment Problem

paper_content:

AbstractLet ℒ be the linear space of the Laurent polynomials and suppose that <⋅, ⋅<ℒ is a positive-definite Hermitian inner product in ℒ with the additional property that ::: $$ \left\langle {f\left( z \right),g\left( z \right)} \right\rangle {/Mathcal L} = \left\langle {f\left( z \right)\overline {g\left( z \right)} ,1} \right\rangle {/Mathcal L} $$ ::: . Starting from the five-term recurrence relation for orthogonal Laurent polynomials with respect to <⋅, ⋅<ℒ, we derive Laurent–Jacobi matrices ::: $${/Mathcal J} $$ ::: and ::: $${/Mathcal K} $$ ::: for the multiplication operator and its inverse in ℒ. These matrices are real and symmetric, and ::: $${/Mathcal J} $$ ::: generates a symmetric operator in the Hilbert space ℓ2 with natural basis { en }n = 0∞. We show that this operator has deficiency indices (0, 0) or (1, 1) and that every self-adjoint extension A in ℓ2 has simple spectrum with generating vector e0. Let E be the spectral measure of A. Then the measure μe0 given by μe0(Ω) =<E(Ω) e0, e0< for all Borel sets Ω in ::: $${/Mathcal R} $$ ::: , satisfies ::: $$\int_{/Mathcal R} {f\overline g {d\mu }_{e_0 } } = \left\langle {f,g} \right\rangle {/Mathcal L} $$ ::: forf,gℒ. In this way, we obtain a solution μe0 of the Strong Hamburger Moment Problem (SHMP) for which ℒ is dense in L2(μe0). Some results concerning the relation between the deficiency indices andthe set of all solutions of the SHMP are established. Finally, we give an analogue of a theorem by M. H. Stone which tells us which self-adjoint operators are generatedby a Laurent–Jacobi matrix with deficiency indices (0, 0).
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paper_title: Quadrature formulas on the unit circle based on rational functions

paper_content:

Abstract Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szego polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.
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paper_title: Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation

paper_content:

Starting from a strong Stieltjes distribution Φ, general sequences of orthogonal Laurent polynomials are introduced and sonic of their most relevant algebraic properties are studied. From this perspective, the connection between certain quadrature formulas associated with the distribution Φ and two-point Pade approximants to the Stieltjes transform of Φ is revisited. Finally, illustrative numerical examples are discussed.
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paper_title: Orthogonal Laurent polynomials and the strong Hamburger moment problem

paper_content:

Abstract This paper is concerned with the strong Hamburger moment problem (SHMP): For a given double sequence of real numbers C = { c n } ∞ −∞ , does there exist a real-valued, bounded, non-decreasing function ψ on (−∞, ∞) with infinitely many points of increase such that for every integer n , c n = ∝ ∞ −∞ (− t ) n dψ ( t )? Necessary and sufficient conditions for the existence of such a function ψ are given in terms of the positivity of certain Hankel determinants associated with C. Our approach is made through the study of orthogonal (and quasi-orthogonal) Laurent polynomials (referred to here as L-polynomials) and closely related Gaussian-type quadrature formulas. In the proof of sufficiency an inner product for L-polynomials is defined in terms of the given double sequence C . Since orthogonal L-polynomials are believed to be of interest in themselves, some examples of specific systems are considered.
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paper_title: Minimal representations of unitary operators and orthogonal polynomials on the unit circle

paper_content:

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.
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paper_title: L-Polynomials Orthogonal on the Unit Circle

paper_content:

Two sequences {σn} and {τn} of L-polynomials are derived which are orthogonal with respect to a linear functional M on Λ and span Λ. The sequences occur as denominators of approximants of a modified PC-fraction. Three term recursion relations for the sequence {σn} as well as for {τn} are derived.
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paper_title: Two-point Padé expansions for a family of analytic functions

paper_content:

Abstract Each member G(z) of a family of analytic functions defined by Stieltjes transforms is shown to be represented by a positive T-fraction, the approximants of which form the main diagonal in the two-point Pade table of G(z). The positive T-fraction is shown to converge to G(z) throughout a domain D(a, b) = [z: z∋[−b, −a]], uniformly on compact subsets. In addition, truncation error bounds are given for the approximants of the continued function; these bounds supplement previously known bounds and apply in part of the domain of G(z) not covered by other bounds. The proofs of our results employ properties of orthogonal L -polynomials (Laurent polynomials) and L -Gaussian quadrature which are of some interest in themselves. A number of examples are considered.
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paper_title: Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

paper_content:

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.
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paper_title: On Generating Orthogonal Polynomials

paper_content:

We consider the problem of numerically generating the recursion coefficients of orthogonal polynomials, given an arbitrary weight distribution of either discrete, continuous, or mixed type. We discuss two classical methods, respectively due to Stieltjes and Chebyshev, and modern implementations of them, placing particular emphasis on their numerical stability properties. The latter are being studied by analyzing the numerical condition of appropriate finite-dimensional maps. A number of examples are given to illustrate the strengths and weaknesses of the various methods and to test the theory developed for them.
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paper_title: A Christoffel-Darboux formula and a Favard's theorem for orthogonal Laurent polynomials on the unit circle

paper_content:

Let {@f"k(z)}"k"="0^~ be the family of orthonormal Laurent polynomials on the unit circle which spans @D in the ''ordering'' induced by p(n)=E[(n+1)/2]. From the three-term recurrence relation satisfied by {@f"k(z)}"k"="0^~ we deduce a Christoffel-Darboux formula. Particular examples are considered and a Favard-type theorem is proved. A connection with the ordering induced by p(n)=E[n/2] is also established.
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paper_title: Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle

paper_content:

Abstract We show that the well-known Levinson algorithm for computing the inverse Cholesky factorization of positive definite Toeplitz matrices can be viewed as a special case of a more general process. The latter process provides a very efficient implementation of the Arnoldi process when the underlying operator is isometric. This is analogous with the case of Hermitian operators where the Hessenberg matrix becomes tridiagonal and results in the Hermitian Lanczos process. We investigate the structure of the Hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle. These eigenvalues are then interpreted as abscissas for analogs of Gaussian quadrature, now on the unit circle instead of the real line. The trapezoidal rule appears as the analog of the Gauss-Legendre formula.
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paper_title: Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

paper_content:

Let there be given a probability measure @m on the unit circle T of the complex plane and consider the inner product induced by @m. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r^"^k}"k, for a certain order of the r"[email protected]?Z, by means of the Gram-Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {@j"k}"k. We show that the matrix representation with respect to {@j"k}"k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r^"^k}"k are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for @m. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.
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paper_title: Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

paper_content:

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.
---
paper_title: A matricial computation of rational quadrature formulas on the unit circle

paper_content:

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Mobius transformations of Hessenberg matrices and also as the eigenvalues of the operator Mobius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.
---
paper_title: Minimal representations of unitary operators and orthogonal polynomials on the unit circle

paper_content:

In this paper we prove that the simplest band representations of unitary operators on a Hilbert space are five-diagonal. Orthogonal polynomials on the unit circle play an essential role in the development of this result, and also provide a parametrization of such five-diagonal representations which shows specially simple and interesting decomposition and factorization properties. As an application we get the reduction of the spectral problem of any unitary Hessenberg matrix to the spectral problem of a five-diagonal one. Two applications of these results to the study of orthogonal polynomials on the unit circle are presented: the first one concerns Krein's Theorem; the second one deals with the movement of mass points of the orthogonality measure under monoparametric perturbations of the Schur parameters.
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paper_title: Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

paper_content:

Let there be given a probability measure @m on the unit circle T of the complex plane and consider the inner product induced by @m. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r^"^k}"k, for a certain order of the r"[email protected]?Z, by means of the Gram-Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {@j"k}"k. We show that the matrix representation with respect to {@j"k}"k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r^"^k}"k are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for @m. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.
---
paper_title: Some results about numerical quadrature on the unit circle

paper_content:

In this paper, quadrature formulas on the unit circle are considered. Algebraic properties are given and results concerning error and convergence established.
---
paper_title: L-Polynomials Orthogonal on the Unit Circle

paper_content:

Two sequences {σn} and {τn} of L-polynomials are derived which are orthogonal with respect to a linear functional M on Λ and span Λ. The sequences occur as denominators of approximants of a modified PC-fraction. Three term recursion relations for the sequence {σn} as well as for {τn} are derived.
---
paper_title: Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

paper_content:

In this paper, the construction of orthogonal bases in the space of Laurent polynomials on the unit circle is considered. As an application, a connection with the so-called bi-orthogonal systems of trigonometric polynomials is established and quadrature formulas on the unit circle based on Laurent polynomials are studied.
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paper_title: Domain of validity of Szegö quadrature formulas

paper_content:

As is well known, the n-point Szego quadrature formula integrates correctly any Laurent polynomial in the subspace span{1/z^n^-^1,...,1/z,1,z,...,z^n^-^1}. In this paper we enlarge this subspace. We prove that a set of 2n linearly independent Laurent polynomials are integrated correctly. The obtained result is used for the construction of Szego quadrature formulas. Illustrative examples are given.
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paper_title: An application of Szegö quadratures to the computation of the Fourier transform

paper_content:

Abstract Szego quadrature formulas represent the analogue on the unit circle of the well-known Gauss–Christoffel quadratures for intervals of the real line. In the present work, Szego formulas are revised and an application to the computation of the Fourier transform given.
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paper_title: A Stable Divide and Conquer Algorithm for the Unitary Eigenproblem

paper_content:

We present a divide and conquer algorithm for computing the eigendecomposition of a unitary upper Hessenberg matrix H. Previous divide and conquer approaches suffer a potential loss of orthogonality among the computed eigenvectors of H. Using a backward stable method based on previous work by Gu and Eisenstat in the rank-one modification of the symmetric eigenproblem, our algorithm provides a backward stable method for computing the eigenvectors. The method also compares well against the efficiency of other available methods.
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paper_title: Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

paper_content:

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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paper_title: Measures on the unit circle and unitary truncations of unitary operators

paper_content:

In this paper, we obtain new results about the orthogonality measure of orthogonal polynomials on the unit circle, through the study of unitary truncations of the corresponding unitary multiplication operator, and the use of the five-diagonal representation of this operator.Unitary truncations on subspaces with finite co-dimension give information about the derived set of the support of the measure under very general assumptions for the related Schur parameters (an). Among other cases, we study the derived set of the support of the measure when limn|an+1/an| = 1, obtaining a natural generalization of the known result for the Lopez class limnan+1/an ∈ T, limn|an| ∈ (0,1).On the other hand, unitary truncations on subspaces with finite dimension provide sequences of unitary five-diagonal matrices whose spectra asymptotically approach the support of the measure. This answers a conjecture of L. Golinskii concerning the relation between the support of the measure and the strong limit points of the zeros of the para-orthogonal polynomials.Finally, we use the previous results to discuss the domain of convergence of rational approximants of Caratheodory functions, including the convergence on the unit circle.
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paper_title: Positive definite Toeplitz matrices, the Arnoldi process for isometric operators, and Gaussian quadrature on the unit circle

paper_content:

Abstract We show that the well-known Levinson algorithm for computing the inverse Cholesky factorization of positive definite Toeplitz matrices can be viewed as a special case of a more general process. The latter process provides a very efficient implementation of the Arnoldi process when the underlying operator is isometric. This is analogous with the case of Hermitian operators where the Hessenberg matrix becomes tridiagonal and results in the Hermitian Lanczos process. We investigate the structure of the Hessenberg matrices in the isometric case and show that simple modifications of them move all their eigenvalues to the unit circle. These eigenvalues are then interpreted as abscissas for analogs of Gaussian quadrature, now on the unit circle instead of the real line. The trapezoidal rule appears as the analog of the Gauss-Legendre formula.
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paper_title: Some perspectives on the eigenvalue problem

paper_content:

This expository paper explores the relationships among a number of algorithms for solving eigenvalue problems, including the power method, subspace iteration, the $QR$ algorithm, and the Arnoldi and symmetric Lanczos algorithms. The symmetric Lanczos algorithm is shown to be identical to the three-term recursion (Stieltjes procedure) for computing orthogonal polynomials with respect to a measure on the real line. The connection between measures on the line and symmetric tridiagonal (Jacobi) matrices is investigated. If such a matrix is transformed by a step of the $QR$ algorithm, there is a corresponding transformation in the measure. The tridiagonal matrices are also exploited for the construction of Gaussian quadrature formulas for measures on the line. The developments on the real line are replicated with suitable modifications on the unit circle via Lanczos-like procedures for unitary operators. The best-known procedure of this type is the recursion of Szego for computing orthogonal polynomials on the...
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paper_title: Orthogonal Laurent polynomials on the unit circle and snake-shaped matrix factorizations

paper_content:

Let there be given a probability measure @m on the unit circle T of the complex plane and consider the inner product induced by @m. In this paper we consider the problem of orthogonalizing a sequence of monomials {z^r^"^k}"k, for a certain order of the r"[email protected]?Z, by means of the Gram-Schmidt orthogonalization process. This leads to a sequence of orthonormal Laurent polynomials {@j"k}"k. We show that the matrix representation with respect to {@j"k}"k of the operator of multiplication by z is an infinite unitary or isometric matrix allowing a 'snake-shaped' matrix factorization. Here the 'snake shape' of the factorization is to be understood in terms of its graphical representation via sequences of little line segments, following an earlier work of S. Delvaux and M. Van Barel. We show that the shape of the snake is determined by the order in which the monomials {z^r^"^k}"k are orthogonalized, while the 'segments' of the snake are canonically determined in terms of the Schur parameters for @m. Isometric Hessenberg matrices and unitary five-diagonal matrices (CMV matrices) follow as a special case of the presented formalism.
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paper_title: Orthogonal rational matrix‐valued functions on the unit circle: Recurrence relations and a Favard‐type theorem

paper_content:

This paper contains further steps towards a Szego theory for orthogonal rational matrix-valued functions on the unit circle . It continues the investigations started in [18]–[20]. Hereby we are guided by former work of Bultheel, Gonzalez–Vera, Hendriksen, and Njastad on scalar orthogonal rational functions on the one side and by research of Delsarte, Genin, and Kamp on matrix polynomials on the other side. In this paper we derive recursion formulas for Christoffel–Darboux pairs of rational matrix functions which lead us to jqq-recursively connected pairs of rational matrix functions. Moreover, we prove a Favard-type theorem for rational matrix functions. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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paper_title: On Canonical Solutions of a Moment Problem for Rational Matrix-valued Functions

paper_content:

We discuss extremal solutions of a certain finite moment problem for rational matrix functions which satisfy an additional rank condition.W e will see, among other things, that these solutions are molecular nonnegative Hermitian matrix-valued Borel measures on the unit circle and that these measures have a particular structure.W e study the above-mentioned solutions in all generality, but later focus on the nondegenerate case.In this case, the family of these special solutions can be parametrized by the set of unitary matrices.T his realization allows us to further examine the structure of these solutions.H ere, the analysis of the structural properties relies, to a great extent, on the theory of orthogonal rational matrix functions on the unit circle.
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paper_title: Spectral methods for orthogonal rational functions

paper_content:

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and five-diagonal matrices. As an application of this operator approach, we obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation, generalizing to the rational case known results for orthogonal polynomials on the unit circle. Finally, we extend these results to orthogonal polynomials on the real line with poles in the lower half plane.
---
paper_title: Spectral methods for orthogonal rational functions

paper_content:

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and five-diagonal matrices. As an application of this operator approach, we obtain new relations between the support of the orthogonality measure and the location of the poles and parameters of the recurrence relation, generalizing to the rational case known results for orthogonal polynomials on the unit circle. Finally, we extend these results to orthogonal polynomials on the real line with poles in the lower half plane.
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paper_title: Rational quadrature formulas on the unit circle with prescribed nodes and maximal domain of validity

paper_content:

This paper is concerned with rational Szegő quadrature formulas to approximate integrals of the form I μ (f) = ∫ π ―π f(e iθ )dμ(θ) by a formula such as I n (f) = Σ n k=1 λ k f(z k ), where the weights λ k are positive and the nodes z k are carefully chosen on the complex unit circle. It will be shown that, for a given set of poles, the quadrature formulas can be chosen to be exact in certain subspaces of rational functions of dimension 2n. Also, the problem where one node (Radau) or two nodes (Lobatto) are prefixed will be analysed and the corresponding rational Szegő―Radau and rational Szegő―Lobatto quadrature formulas will be characterized.
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paper_title: MATRICES, MOMENTS, AND RATIONAL QUADRATURE

paper_content:

Abstract Many problems in science and engineering require the evaluation of functionals of the form F u ( A ) = u T f ( A ) u , where A is a large symmetric matrix, u a vector, and f a nonlinear function. A popular and fairly inexpensive approach to determining upper and lower bounds for such functionals is based on first carrying out a few steps of the Lanczos procedure applied to A with initial vector u , and then evaluating pairs of Gauss and Gauss–Radau quadrature rules associated with the tridiagonal matrix determined by the Lanczos procedure. The present paper extends this approach to allow the use of rational Gauss quadrature rules.
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paper_title: Orthogonal rational functions and rational modifications of a measure on the unit circle

paper_content:

In this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure.
---
paper_title: Quadrature formulas on the unit circle based on rational functions

paper_content:

Abstract Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szego polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.
---
paper_title: Rational Krylov approximation of matrix functions: Numerical methods and optimal pole selection

paper_content:

Matrix functions are a central topic of linear algebra, and problems of their numerical approximation appear increasingly often in scientific computing. We review various rational Krylov methods for the computation of large-scale matrix functions. Emphasis is put on the rational Arnoldi method and variants thereof, namely, the extended Krylov subspace method and the shift-and-invert Arnoldi method, but we also discuss the nonorthogonal generalized Leja point (or PAIN) method. The issue of optimal pole selection for rational Krylov methods applied for approximating the resolvent and exponential function, and functions of Markov type, is treated in some detail. (© 2013 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)
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paper_title: Orthogonal rational functions and quadrature on the unit circle

paper_content:

In this paper we generalize the notion of orthogonal Laurent polynomials to orthogonal rational functions. Orthogonality is considered with respect to a measure on the positive real line. From this, Gauss-type quadrature formulas are derived and multipoint Pade approximants for the Stieltjes transform of the measure. Convergence of both the quadrature formula and the multipoint Pade approximants is discussed.
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paper_title: A matricial computation of rational quadrature formulas on the unit circle

paper_content:

A matricial computation of quadrature formulas for orthogonal rational functions on the unit circle, is presented in this paper. The nodes of these quadrature formulas are the zeros of the para-orthogonal rational functions with poles in the exterior of the unit circle and the weights are given by the corresponding Christoffel numbers. We show how these nodes can be obtained as the eigenvalues of the operator Mobius transformations of Hessenberg matrices and also as the eigenvalues of the operator Mobius transformations of five-diagonal matrices, recently obtained. We illustrate the preceding results with some numerical examples.
---
paper_title: Orthogonal rational functions and rational modifications of a measure on the unit circle

paper_content:

In this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure.
---
paper_title: Quadrature formulas on the unit circle based on rational functions

paper_content:

Abstract Quadrature formulas on the unit circle were introduced by Jones in 1989. On the other hand, Bultheel also considered such quadratures by giving results concerning error and convergence. In other recent papers, a more general situation was studied by the authors involving orthogonal rational functions on the unit circle which generalize the well-known Szego polynomials. In this paper, these quadratures are again analyzed and results about convergence given. Furthermore, an application to the Poisson integral is also made.
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Title: Matrix methods for quadrature formulas on the unit circle. A survey I

Section 1: Introduction
Description 1: This section introduces the topic, providing background on Gauss-Christoffel quadrature rules and their extension to the unit circle. 

Section 2: Orthogonal Laurent polynomials on the unit circle
Description 2: This section covers the notation, preliminary results, and the construction of orthogonal Laurent polynomial sequences, including various important properties and recurrence relations.

Section 3: Matricial representations
Description 3: This section discusses the multiplication operator in the space of Laurent polynomials on the unit circle and analyzes different types of matrix representations, including five-diagonal and snake-shaped matrices.

Section 4: Matricial computation of Szegő quadrature formulas
Description 4: This section derives formulas for the nodes and weights of the Szegő quadrature formulas and explores methods to practically compute them using different matricial representations.

Section 5: A numerical example
Description 5: This section illustrates the computation of nodes and weights of Szegő quadrature formulas with a specific numerical example.

Section 6: Orthogonal rational functions
Description 6: This section introduces orthogonal rational functions, generalizing orthogonal polynomials to include non-infinite poles and detailing the properties and construction of these functions.

Section 7: The multiplication operator in rational function spaces
Description 7: This section describes the matrix representation for the multiplication operator in rational function spaces, introducing an alternative basis for simplification.

Section 8: Para-orthogonal rational functions
Description 8: This section defines and proves the properties of para-orthogonal rational functions, explaining their use in quadrature formulas and connection to eigenvalue problems.

Section 9: Matricial computation of rational Szegő quadrature formulas
Description 9: This section describes methods for approximating integrals using rational Szegő quadrature formulas, detailing computational techniques based on matrices and eigenvalues.

Section 10: Numerical examples
Description 10: This section presents numerical examples to illustrate the practical application of rational Szegő quadrature formulas and compare results with polynomial cases.