The statement in the opposite direction follows directly by Corollaries 7. For nondefinite linear functionals all three properties cannot hold.
Using Theorem 7. Thus property G2 cannot be satisfied. Indeed, using 7. Hence by Theorem 7. However, by Theorem 7. In the quasi-definite case the degree of exactness is uniquely determined; see Theorems 7. The following example demonstrates this fact. In Section 1 we asked how far we can go with the generalization of Gauss quadrature as an approximant for an arbitrary linear functional.
We suggest that any generalization of Gauss quadrature should have properties G1—G3. In this sense, the quasi-definiteness of the linear functional represents a necessary and sufficient condition for the existence of Gauss quadrature. Generalization of Gauss quadrature to a complex plane can be linked with many topics and developed in different ways. As pointed out by the referees, there is a link with Prony's method for exponential sums, Gauss—Konrod quadrature, etc.
Such links cannot be covered within a single paper; we hope to return to them elsewhere. We are grateful to the referees for the insightful reports and for many comments and suggestions that substantially helped in preparing the revised version of the text. Oxford University Press is a department of the University of Oxford.
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Matrices, moments and quadrature with applications
Close mobile search navigation Article Navigation. Volume Article Contents. Gauss quadrature for positive-definite linear functionals. Orthogonality and quasi-definite linear functionals. Complex generalization of Jacobi matrices. Moment matching property for Jacobi matrices.
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Quasi-definite linear functionals and Gauss quadrature under restrictive assumptions. Gauss quadrature for general quasi-definite linear functionals. Gauss quadrature for quasi-definite linear functionals Stefano Pozza. Oxford Academic. Google Scholar. Miroslav S.
Cite Citation. Permissions Icon Permissions. Abstract Gauss quadrature can be formulated as a method for approximating positive-definite linear functionals. Considering the Cholesky factorization see, e. Since the nodes of the quadrature, i. The orthonormal polynomials associated with 2. From 2. Moreover, it can be proved see Wilf, , Sections 2. We recall that Jacobi matrices are linked with Krylov subspace methods via the Hermitian Lanczos algorithm.
This method was introduced by Lanczos ; for a description of the method and its properties we refer, e.
Using the coefficients of the recurrence 2. Continued fractions are directly related to Gauss quadrature and moments.
If we change the variable in 2. The recurrences 3. On the other hand, as shown in Chihara , Chapter I, Theorem 4. In the rest of this section, we recall the relationship between Jacobi matrices and the non-Hermitian Lanczos algorithm for details we refer to Brezinski, , Section 2. Then recall 2. The proof uses an analogy of the property of matrix of moments for the positive-definite case recalled in Section 2.
Matrices, Moments and Quadrature with Applications by Gene H. Golub, Gerard Meurant - dinachstithecoun.ml
Finally, by 3. As recalled in Sections 1 and 2, positive-definite linear functionals lead to Gauss quadrature with properties G1—G3. The quadrature 6. Assume that 6.
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In order to avoid restrictions to quasi-definite linear functionals that produce diagonalizable Jacobi matrices and allow full generality, we have to modify the quadrature concept presented in relation 6. Quadrature 7. Theorem 7. Thus we conclude that the quadrature 7. Using 4. Then, by 7.
By Theorem 7. As shown in the proof of Theorem 5. The presented construction 7. In order to avoid confusion, it should be stressed that the Gauss quadrature proposed in this article quadrature 7. Comparing 7.
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Example 8. Google Preview. Search ADS. Reprinted in Gesammelte mathematische Abhandlungen, vol. Teubner, Leipzig, , pp. Matrix interpretations and applications of the continued fraction algorithm. A completed theory of the unsymmetric Lanczos process and related algorithms, part I. A completed theory of the unsymmetric Lanczos process and related algorithms, part II. Contributions to the analytic theory of continued fractions and infinite matrices. An iteration method for the solution of the eigenvalue problem of linear differential and integral operators. The Lanczos and conjugate gradient algorithms in finite precision arithmetic.
On the construction of cubature formulas with the smallest number of nodes. A theorem on isotropic null vectors and its application to thermoelasticity. Reprinted in Oeuvres, vol. Noordhoff, Groningen, , pp. In order to avoid ambiguity, we take always the principal value of the complex square root, i. All rights reserved. Issue Section:. Download all figures. View Metrics. Email alerts New issue alert. Advance article alerts. Article activity alert. Receive exclusive offers and updates from Oxford Academic. Related articles in Web of Science Google Scholar.
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