Matrices, Moments and Quadrature with Applications by Gene H. Golub
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 e-book bridges various mathematical components to acquire algorithms to estimate bilinear kinds concerning vectors and a functionality of the matrix. the 1st a part of the booklet offers the mandatory mathematical history and explains the speculation. the second one half describes the purposes and provides numerical examples of the algorithms and methods constructed within the first part.Applications addressed within the ebook contain computing components of services of matrices; acquiring estimates of the mistake norm in iterative equipment 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 regarding computation of bilinear kinds.
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Extra resources for Matrices, Moments and Quadrature with Applications (Princeton Series in Applied Mathematics)
Using the orthogonality of the Xi ’s and the spectral decomposition of A, we can write i δij I2 = XiT Xj = j (i) (Ck )T X0T Ak (j) Al X0 Cl k=0 l=0 (i) T (j) T k+l T (Ck ) X0 QΛ Q X0 Cl = k,l (i) ˆ k+l X ˆ T C (j) (Ck )T XΛ l = k,l n (i) ˆ ˆT λk+l m Xm X m (Ck )T = m=1 k,l n (i) λkm (Ck )T = m=1 (j) Cl (j) T ˆmX ˆm X k λlm Cl , l ˆ m are the columns of X ˆ = X T Q, which is a 2 × n matrix. Therefore, where X 0 n T ˆmX ˆm pi (λm )T X pj (λm ). XiT Xj = m=1 The sum in the right-hand side can be written as an integral for a 2 × 2 matrix measure, if λ < λ1 , 0, i ˆ ˆT α(λ) = j=1 Xj Xj , if λi ≤ λ < λi+1 , n ˆ ˆT Xj X , if λn ≤ λ.
The last sum can be written as an integral for a measure α which is piecewise constant (here we suppose for the sake of simplicity that the eigenvalues of A are distinct): if λ < λ1 , 0, i 2 [ˆ v ] , if λi ≤ λ < λi+1 , j α(λ) = j=1 n 2 [ˆ v ] , if λn ≤ λ. j=1 j THE LANCZOS AND CONJUGATE GRADIENT ALGORITHMS 43 The measure α has a finite number of points of increase at the (unknown) eigenvalues of A. ✷ The Lanczos algorithm provides an example of orthonormal polynomials for an unknown measure.
PN −1 (λ)]JN + [0, . . , 0, pN (λ)ΓN ], and as P (λ) = [p0 (λ), . . , pN −1 (λ)]T , JN P (λ) = λP (λ) − [0, . . 16). As in the scalar case, the inner product of the matrices Xi can be related to an integral. 6 Considering the matrices Xk , there exists a matrix measure α (defined in the proof) such that b XiT Xj = pi (λ)T dα(λ)pj (λ) = δij I2 , a where a ≤ λ1 = λmin and b ≥ λn = λmax . Proof. Using the orthogonality of the Xi ’s and the spectral decomposition of A, we can write i δij I2 = XiT Xj = j (i) (Ck )T X0T Ak (j) Al X0 Cl k=0 l=0 (i) T (j) T k+l T (Ck ) X0 QΛ Q X0 Cl = k,l (i) ˆ k+l X ˆ T C (j) (Ck )T XΛ l = k,l n (i) ˆ ˆT λk+l m Xm X m (Ck )T = m=1 k,l n (i) λkm (Ck )T = m=1 (j) Cl (j) T ˆmX ˆm X k λlm Cl , l ˆ m are the columns of X ˆ = X T Q, which is a 2 × n matrix.