The extended Krylov subspace method and orthogonal Laurent polynomials

Recursion relations for the extended Krylov subspace method

Modern power systems have experienced a significant increase in its complexity, and extremely large-scale system models have to be addressed in the study of stability analysis and control. Model reduction is a technique for developing an approximate system model with lower dimensions that shares similar properties to the original system. This work proposes a computationally efficient approach for linear system model reduction in large-scale power systems that is based on the balanced truncation method using an extended Krylov subspace technique. Key algorithm improvements, including sparsity handling of the linearized power system models and efficient computational techniques for solving dual Lyapunov equations, are discussed in detail. In addition, application of the proposed model reduction is extended to unstable systems, and a power system stabilizer (PSS) parameter optimization method based on reduced system models is used to validate the effectiveness of the proposed method. Case studies show that the resulting reduced models preserve the unique characteristics of the original high-dimensional models, including time and frequency domain responses as well as eigenvalues. Numerical results based on test systems with up to 12 $\thinspace$ 685 buses demonstrate the computational efficiency and validity of the proposed approach.

The Strong Chebyshev Distribution and Orthogonal Laurent Polynomials

Orthogonal Laurent polynomials and strong moment theory: a survey

Orthogonal polynomials and some q-beta integrals of Ramanujan

A three-complex-parameter class of orthogonal Laurent polynomials on the unit circle associated with basic hypergeometric or q q -hypergeometric functions is considered. To be precise, we consider the orthogonality properties of the sequence of polynomials { 2 Φ 1 ( q − n , q b + 1 ; q − c + b − n ; q , q − c + d − 1 z ) } n = 0 ∞ \{\,_2\Phi _1(q^{-n},q^{b+1};q^{-c+b-n}; q, q^{-c+d-1}z)\}_{n=0}^{\infty } , where 0 > q > 1 0 > q > 1 and the complex parameters b b , c c and d d are such that b ≠ − 1 , − 2 , … b \neq -1, -2, \ldots , c − b + 1 ≠ − 1 , − 2 , … c-b+1 \neq -1, -2, \ldots , R e ( d ) > 0 \mathcal {R}e(d) > 0 and R e ( c − d + 2 ) > 0 \mathcal {R}e(c-d+2) > 0 . Explicit expressions for the recurrence coefficients, moments, orthogonality and also asymptotic properties are given. By a special choice of the parameters, results regarding a class of Szegő polynomials are also derived.

PC-fractions and orthogonal Laurent polynomials for log-normal distributions

<abstract> By utilizing an inner-outer iteration strategy, a shift-splitting (SS) iteration method to solve a class of large sparse linear matrix equation $ AXB = C $ is proposed in this work. Two convergence theorems for differential forms are studied in depth. Moreover, the quasi-optimal parameters which minimize the upper bound for the spectral radius of SS iteration matrix are given. Two numerical examples illustrate the high-efficiency of SS iteration method, especially when coefficient matrices are ill-conditioned. </abstract>

Nonsymmetric Askey–Wilson polynomials and Q-polynomial distance-regular graphs

Sequences of orthogonal Laurent polynomials, bi-orthogonality and quadrature formulas on the unit circle

This paper is concerned with double sequencesC={Cn}n=−∞/∞ of Hermitian matrices with complex entriesCn∈Ms×s) and formal Laurent seriesL0(z)=−Σk=1∞C−kzk andL∞(z)=Σk=0∞Ckz−k. Making use of a Favard-type theorem for certain sequences of matrix Laurent polynomials which was obtained previously in [1] we can establish the relation between the matrix counterpart of the so-calledT-fractions and matrix orthogonal Laurent polynomials. The connection with two-point Pade approximants to the pair (L0,L∞) is also exhibited proving that such approximants are Hermitian too. Finally, error formulas are also given.

A projected preconditioned conjugate gradient algorithm for computing many extreme eigenpairs of a Hermitian matrix

The representation of an integral operator in wavelet bases is a large numerical sparse matrix. Thus, a suitable approach is needed that converts the matrix from full storage mode into sparse storage mode, which not only saves the storage, but also saves the execution time. In this paper, the storage of this kind of matrix is discussed in the symmetric case. First, we present a standard storage approach. Then, a modified one is proposed, which is based on a special decomposition of the symmetric matrix. When iterative solvers are used to solve the corresponding linear system, the main cost is the multiplication of the symmetric sparse matrix and a given vector. The modified sparse storage mode is optimized for this purpose and reduces nearly half of the number of multiplications.

Darboux transformations for CMV matrices

We introduce an economical Gram--Schmidt orthogonalization on the extended Krylov subspace originated by actions of a symmetric matrix and its inverse. An error bound for a family of problems arising from the elliptic method of lines is derived. The bound shows that, for the same approximation quality, the diagonal variant of the extended subspaces requires about the square root of the dimension of the standard Krylov subspaces using only positive or negative matrix powers. An example of an application to the solution of a 2.5-D elliptic problem attests to the computational efficiency of the method for large-scale problems.