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By Gene H. Golub

This computationally orientated publication describes and explains the mathematical relationships between matrices, moments, orthogonal polynomials, quadrature principles, and the Lanczos and conjugate gradient algorithms. The booklet bridges assorted mathematical parts to procure algorithms to estimate bilinear types related to vectors and a functionality of the matrix. the 1st a part of the e-book offers the required mathematical heritage and explains the idea. the second one half describes the purposes and offers numerical examples of the algorithms and methods built within the first part.

Applications addressed within the publication contain computing parts of services of matrices; acquiring estimates of the mistake norm in iterative tools for fixing linear structures and computing parameters in least squares and overall least squares; and fixing ill-posed difficulties utilizing Tikhonov regularization.

This ebook will curiosity researchers in numerical linear algebra and matrix computations, in addition to scientists and engineers engaged on difficulties related to computation of bilinear forms.

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Extra resources for Matrices, moments and quadrature with applications

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We consider a Cholesky-like factorization of Jk . Let ∆k be a diagonal matrix with diagonal elements δj , j = 1, . . , k and let Lk be a lower triangular (bidiagonal) matrix to be determined,   1   l1 1   . .  .. Lk =     1 lk−2 lk−1 1 Let the factorization of Jk be Jk = Lk ∆k LTk . By identification it is easy to see that we have δ1 = α1 , δj = αj − 2 βj−1 , j = 2, . . , k, δj−1 l1 = β1 /δ1 , lj = βj /δj , j = 2, . . , k − 1 The factorization can be completed if no δj is zero for j = 1, .

2 Chebyshev polynomials (second kind) Uk , k = 1, . . 3 Legendre Polynomials The choice δ = β = 0 (that is, a weight function equal to 1) gives the Legendre polynomials Pk . The three-term recurrence is (k + 1)Pk+1 (λ) = (2k + 1)λPk (λ) − kPk−1 (λ), P0 (λ) ≡ 1, P1 (λ) ≡ λ. The Legendre polynomial Pk is bounded by 1 on [−1, 1]. 3. 4 Laguerre and Hermite Polynomials Other classical examples on different intervals are the Laguerre and Hermite polynomials. For Laguerre polynomials Lk , the interval is [0, ∞) and the weight function is e−λ .

7 on [−10, 10] Other interesting polynomials are the Gegenbauer and Meixner–Pollaczek polynomials; see Szeg¨o [323] or Gautschi [131]. Less classical examples, in particular polynomials of a discrete variable, are described in Gautschi [131]. 6 Variable-Signed Weight Functions So far we have assumed that the measure is positive. What happens if the measure is defined by a weight function which changes sign in the interval [a, b]? This problem was considered by G. W. Struble [321]. Regarding existence of the orthogonal polynomials related to such a measure, the result is the following.

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