Matrices, moments and quadrature with applications - download pdf or read online
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 e-book bridges varied mathematical components to acquire algorithms to estimate bilinear varieties regarding vectors and a functionality of the matrix. the 1st a part of the ebook offers the mandatory mathematical history and explains the speculation. the second one half describes the functions and provides numerical examples of the algorithms and methods built within the first part.
Applications addressed within the publication comprise computing components of capabilities 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 concerning computation of bilinear forms.
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Extra resources for Matrices, moments and quadrature with applications
K−2 αk−1 βk−1 βk−1 αk where the values βj , j = 1, . . , k − 1 are assumed to be nonzero. We remark that the determinant of Jk verifies a three-term recurrence. 2 det(Jk+1 ) = αk+1 det(Jk ) − βk2 det(Jk−1 ) with initial conditions det(J1 ) = α1 , det(J2 ) = α1 α2 − β12 . Proof. This is obtained by expanding the determinant of Jk+1 along the last row ✷ or column of Jk+1 . The eigenvalues of Jk are the zeros of det(Jk − λI). 2, we see that the zeros do not depend on the signs of the coefficients βj , j = 1, .
1 · · · βi−1 δi · · · δk Since, for j ≥ i, we have (Jk−1 )i,j = ui vj , we obtain the result. ✷ The diagonal elements of the inverse of Jk can also be obtained using twisted factorizations. 7 Let l be a fixed index and ωj the diagonal elements of the corresponding twisted factorization of Jk . Then, (Jk−1 )l,l = 1 . ωl Proof. This is obtained by solving Jk y = el and looking at the lth element of the solution. Since all the components of el are zero except the lth one, starting from the top and the bottom, all the components of the solution of the first phase are zero except for the lth one which is 1/ωl .
Therefore, we have the following equation for θ, ξj2 k αk+1 − βk2 (k) j=1 θj −θ = θ, where ξj is the last component of the jth eigenvector of Jk . An equation like this one is called a “secular” equation. The function is monotone in each interval (k) defined by the poles θj . There is only one root in each interval and this proves the result. We will study secular equations in more details in chapter 9. Note that the quadratic form (ek )T (Jk − θI)−1 ek is an essential part of this equation. For bounds on the eigenvalues of tridiagonal matrices, see Golub .
Matrices, moments and quadrature with applications by Gene H. Golub