On the calculation of Jacobi Matrices
TL;DR: This paper presents two methods not requiring the explicit knowledge of the roots of r and obtains various properties of the similarity transformations between Jacobi matrices, which are proved by simple matrix calculus without using the generalized Christoffel theorem.
About: This article is published in Linear Algebra and its Applications.The article was published on 1983-07-01 and is currently open access. It has received 64 citations till now. The article focuses on the topics: Jacobi operator & Jacobi method.
Citations
More filters
TL;DR: A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions.
Abstract: A collection of subroutines and examples of their uses, as well as the underlying numerical methods, are described for generating orthogonal polynomials relative to arbitrary weight functions. The object of these routines is to produce the coefficients in the three-term recurrence relation satisfied by the orthogonal polynomials. Once these are known, additional data can be generated, such as zeros of orthogonal polynomials and Gauss-type quadrature rules, for which routines are also provided.
291 citations
TL;DR: A tribute is paid to those who have made an understanding of the Lanczos and conjugate gradient algorithms possible through their pioneering work, and to review recent solutions of several open problems that have also contributed to knowledge of the subject.
Abstract: The Lanczos and conjugate gradient algorithms were introduced more than five decades ago as tools for numerical computation of dominant eigenvalues of symmetric matrices and for solving linear algebraic systems with symmetric positive definite matrices, respectively. Because of their fundamental relationship with the theory of orthogonal polynomials and Gauss quadrature of the Riemann-Stieltjes integral, the Lanczos and conjugate gradient algorithms represent very interesting general mathematical objects, with highly nonlinear properties which can be conveniently translated from algebraic language into the language of mathematical analysis, and vice versa. The algorithms are also very interesting numerically, since their numerical behaviour can be explained by an elegant mathematical theory, and the interplay between analysis and algebra is useful there too.Motivated by this view, the present contribution wishes to pay a tribute to those who have made an understanding of the Lanczos and conjugate gradient algorithms possible through their pioneering work, and to review recent solutions of several open problems that have also contributed to knowledge of the subject.
171 citations
Cites background from "On the calculation of Jacobi Matric..."
...…somewhat related problem concerning sensitivity of the Lanczos coefficients to perturbations of the distribution function in the Riemann– Stieltjes integral is investigated in Kautsky and Golub (1983); see also Gragg and Harrod (1984), Laurie (1999, 2001) and Druskin, Borcea and Knizhnerman (2005)....
[...]
TL;DR: 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.
Abstract: 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...
155 citations
Cites background or result from "On the calculation of Jacobi Matric..."
...Related results were given in [31] and [50] and were implicit in the earlier works [63], [64], and [71]....
[...]
...This was the motivation for the papers [31] and [50]....
[...]
TL;DR: Numerical methods applicable to computing Gauss-type quadrature rules and Sobolev orthogonal polynomials, although their recurrence relations are more complicated are discussed.
Abstract: We give examples of problem areas in interpolation, approximation, and quadrature, that call for orthogonal polynomials not of the classical kind. We then discuss numerical methods of computing the respective Gauss-type quadrature rules and orthogonal polynomials. The basic task is to compute the coefficients in the three-term recurrence relation for the orthogonal polynomials. This can be done by methods relying either on moment information or on discretization procedures. The effect on the recurrence coefficients of multiplying the weight function by a rational function is also discussed. Similar methods are applicable to computing Sobolev orthogonal polynomials, although their recurrence relations are more complicated. The paper concludes with a brief account of available software.
153 citations
Cites methods from "On the calculation of Jacobi Matric..."
...The procedure (7.14), (7.15) based on QR methodology is due to Kautsky and Golub (1983)....
[...]
TL;DR: In this paper, the authors considered the problem of finding the monic Jacobi matrix associated with the three canonical perturbations in terms of the so-called Jacobi matrices associated with a quasi-definite linear functional.
131 citations
References
More filters
Book•
01 Jan 1960
TL;DR: In this article, the Second Edition Preface is presented, where Maximization, Minimization, and Motivation are discussed, as well as a method of Hermite and Quadratic Form Index.
Abstract: Foreword Preface to the Second Edition Preface 1. Maximization, Minimization, and Motivation 2. Vectors and Matrices 3. Diagonalization and Canonical Forms for Symmetric Matrices 4. Reduction of General Symmetric Matrices to Diagonal Form 5. Constrained Maxima 6. Functions of Matrices 7. Variational Description of Characteristic Roots 8. Inequalities 9. Dynamic Programming 10. Matrices and Differential Equations 11. Explicit Solutions and Canonical Forms 12. Symmetric Function, Kronecker Products and Circulants 13. Stability Theory 14. Markoff Matrices and Probability Theory 15. Stochastic Matrices 16. Positive Matrices, Perron's Theorem, and Mathematical Economics 17. Control Processes 18. Invariant Imbedding 19. Numerical Inversion of the Laplace Transform and Tychonov Regularization Appendix A. Linear Equations and Rank Appendix B. The Quadratic Form of Selberg Appendix C. A Method of Hermite Appendix D. Moments and Quadratic Forms Index.
3,500 citations
TL;DR: In this paper, two algorithms for generating the Gaussian quadrature rule defined by the weight function are presented, assuming that the three term recurrence relation is known for the orthogonal polynomials generated by the weighted function.
Abstract: Most numerical integration techniques consist of approximating the integrand by a polynomial in a region or regions and then integrating the polynomial exactly. Often a complicated integrand can be factored into a non-negative ''weight'' function and another function better approximated by a polynomial, thus $\int_{a}^{b} g(t)dt = \int_{a}^{b} \omega (t)f(t)dt \approx \sum_{i=1}^{N} w_i f(t_i)$. Hopefully, the quadrature rule ${\{w_j, t_j\}}_{j=1}^{N}$ corresponding to the weight function $\omega$(t) is available in tabulated form, but more likely it is not. We present here two algorithms for generating the Gaussian quadrature rule defined by the weight function when: a) the three term recurrence relation is known for the orthogonal polynomials generated by $\omega$(t), and b) the moments of the weight function are known or can be calculated.
1,386 citations
TL;DR: In this paper, the problem of finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one is considered.
Abstract: We consider the numerical calculation of several matrix eigenvalue problems which require some manipulation before the standard algorithms may be used. This includes finding the stationary values of a quadratic form subject to linear constraints and determining the eigenvalues of a matrix which is modified by a matrix of rank one. We also consider several inverse eigenvalue problems. This includes the problem of determining the coefficients for the Gauss–Radau and Gauss–Lobatto quadrature rules. In addition, we study several eigenvalue problems which arise in least squares.
615 citations
TL;DR: The Gauss-Christoffel quadrature formula (GCE) as discussed by the authors is defined as a Gauss Christoffel formula with maximum degree of exactness, i.e. if (1.1) is an exact equality whenever f is a polynomial of degree 2n 1.
Abstract: Each of these rules will be called a Gauss-Christoffel quadrature formula if it has maximum degree of exactness, i.e. if (1.1) is an exact equality whenever f is a polynomial of degree 2n 1. It is a well-known fact, due to Christoffel [3], that such quadrature formulas exist uniquely, provided the weight function w(x) is nonnegative, integrable with Jb w(x)dx > 0, and such that all its moments
198 citations
TL;DR: An algorithm for solving the problem of constructing Gaussian quadrature rules from 'modified moments' is derived, which generalizes one due to Golub and Welsch.
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.
148 citations