By Gene H. Golub
This computationally orientated e-book describes and explains the mathematical relationships between matrices, moments, orthogonal polynomials, quadrature ideas, and the Lanczos and conjugate gradient algorithms. The publication bridges varied mathematical components to procure algorithms to estimate bilinear types related to vectors and a functionality of the matrix. the 1st a part of the e-book presents the required mathematical historical past and explains the speculation. the second one half describes the functions and offers numerical examples of the algorithms and strategies constructed within the first part.Applications addressed within the booklet comprise computing parts of features of matrices; acquiring estimates of the mistake norm in iterative tools for fixing linear platforms and computing parameters in least squares and overall least squares; and fixing ill-posed difficulties utilizing Tikhonov regularization.This booklet will curiosity researchers in numerical linear algebra and matrix computations, in addition to scientists and engineers engaged on difficulties related to computation of bilinear types.
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Example text
Conversely, Gauss quadrature rules can be used to estimate norms of the error between the CG approximate solutions and the exact solution during the iterations. This also gives a reliable and cheap way to compute stopping criteria for the CG iterations. 1 The Lanczos Algorithm Let A be a real symmetric matrix of order n. We introduce the Lanczos algorithm as a means of computing an orthogonal basis of a Krylov subspace. 1) be the Krylov matrix of dimension n × k. The subspace that is spanned by the columns of the matrix Kk is called a Krylov subspace and denoted by Kk (A, v) or K(A, v) when no confusion is possible.
2) by VkT and using orthogonality, we have Hk = VkT AVk . Clearly, a symmetric Hessenberg matrix is tridiagonal. Therefore, we denote Hk by Jk (since this is a Jacobi matrix, the elements in the sub- and superdiagonals being strictly positive) and we have hi,j = 0, j = i + 2, . . , k. This implies that v¯k and hence the new vector v k+1 can be computed by using only the two previous vectors v k and v k−1 . This describes the Lanczos algorithm. In fact, Vk is the orthonormal matrix (that is, such that VkT Vk = I) involved in a QR factorization of the Krylov matrix Kk , and the matrix KkT AKk is similar to Jk = VkT AVk .
PROPERTIES OF TRIDIAGONAL MATRICES 29 The last components of the eigenvectors z i of Jk satisfy (k) χ1,k−1 (θi ) (zki )2 = (k) , χ1,k (θi ) that is, (zki )2 = (k) θi (k) θi (k) (k−1) − θ1 (k) − θ1 ··· θi (k) θi (k−1) − θi−1 (k) (k−1) θi − θi−1 (k) (k−1) (k) − θi (k) θi+1 − θi ··· (k) θk−1 − θi (k) (k) θk − θi . The components of the eigenvectors are also related to (the derivatives of) the (k) functions δj (λ) and dj (λ). 5 The first components of the eigenvectors of Jk are given by (z1i )2 = 1 (k) [d1 ] .