Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods

نویسنده

  • G W Stewart
چکیده

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon 4] shows that the Rayleigh quotient | i.e., the tridiagonal matrix produced by the Lanczos recursion | contains fully accurate approximations to the Ritz values in spite of the lack of orthogonality. Unfortunately, the same lack of orthogonality can cause the Ritz vectors to fail to converge. It also makes the classical estimate for the residual norm misleadingly small. In this note we show how to adjust the Rayleigh quotient to overcome this problem. Abstract In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon 4] shows that the Rayleigh quotient | i.e., the tridiagonal matrix produced by the Lanczos recursion | contains fully accurate approximations to the Ritz values in spite of the lack of orthogonality. Unfortunately, the same lack of orthogonality can cause the Ritz vectors to fail to converge. It also makes the classical estimate for the residual norm misleadingly small. In this note we show how to adjust the Rayleigh quotient to overcome this problem.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

Adjusting the Rayleigh Quotient in Semiorthogonal Lanczos Methods

In a semiorthogonal Lanczos algorithm, the orthogonality of the Lanczos vectors is allowed to deteriorate to roughly the square root of the rounding unit, after which the current vectors are reorthogonalized. A theorem of Simon 4] shows that the Rayleigh quotient | i.e., the tridiagonal matrix produced by the Lanczos recursion | contains fully accurate approximations to the Ritz values in spite...

متن کامل

On Convergence of the Inexact Rayleigh Quotient Iteration with the Lanczos Method Used for Solving Linear Systems∗

For the Hermitian inexact Rayleigh quotient iteration (RQI), we consider the local convergence of the inexact RQI with the Lanczos method for the linear systems involved. Some attractive properties are derived for the residual, whose norm is ξk, of the linear system obtained by the Lanczos method at outer iteration k + 1. Based on them, we make a refined analysis and establish new local converg...

متن کامل

Preconditioning Eigenvalues and Some Comparison of Solvers

Preconditioning techniques are discussed for symmetric eigenvalue problems. The methods Davidson, Jacobi-Davidson, Rayleigh quotient iteration, and preconditioned Lanczos are considered. Some relationships are given between these different approaches, and some experimental comparisons are done. Jacobi-Davidson appears to be efficient in both expense and storage. A hybrid method may be helpful f...

متن کامل

A harmonic Lanczos bidiagonalization method for computing interior singular triplets of large matrices

This paper proposes a harmonic Lanczos bidiagonalization method for computing some interior singular triplets of large matrices. It is shown that the approximate singular triplets are convergent if a certain Rayleigh quotient matrix is uniformly bounded and the approximate singular values are well separated. Combining with the implicit restarting technique, we develop an implicitly restarted ha...

متن کامل

Faster Fiedler Vector Computation for Laplacians of Well - Shaped Meshes

We present a faster method for finding the Fiedler vector of the Laplacian of a mesh. This problem is the main computational cost of spectral partitioning, a technique for partitioning a graph into pieces of roughly equal size with few crossing edges, which has applications to load balancing on parallel machines. Our experiments show an improvement of 15 to 40 percent in the speed of the standa...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2001