Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm

Kuczyński, Jacek; Woźniakowski, Henryk

doi:10.1137/s0895479892230456

Cited by 22 publications

(18 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The results are displayed in Table 7.1. We see that the upper bound m from (6.2) is much larger than k 1 , the actual number of steps needed to obtain a relative error smaller than tol; this has already been observed in other examples for the upper bound obtained with (6.3) [4,5]. We also observe that m > k 2 , the number of steps needed to obtain (λ up − θ k )/λ up while executing the Lanczos method and check whether this quantity is smaller than tol or not.…”

Section: Numerical Experimentssupporting

confidence: 71%

“…Spectral bounds using Chebyshev polynomials. Chebyshev polynomials are often used to obtain error bounds for the Lanczos method; cf., e.g., [2,5,7]. In this section we explain how these polynomials can be used to obtain probabilistic upper and lower bounds for the spectrum of A, based on computations with the Lanczos method.…”

Section: Re V1mentioning

confidence: 99%

“…In this section we explain how these polynomials can be used to obtain probabilistic upper and lower bounds for the spectrum of A, based on computations with the Lanczos method. One type of bounds follows easily from a result by Kuczyński and Woźniakowski [5,Theorem 3].…”

Section: Re V1mentioning

confidence: 99%

“…In [5,Theorem 3], the following result has been derived for symmetric positive definite matrices. Let t > 1 and v 1 be chosen randomly from the uniform distribution over S n−1 .…”

Section: Re V1mentioning

confidence: 99%

“…Polynomials related to the Lanczos process, namely the Lanczos polynomials and Ritz polynomials, are used to derive two types of such bounds. For symmetric positive definite matrices Kuczyński and Woźniakowski [5,Theorem 3] give, for arbitrary t > 1, an a priori upper bound for the probability that the largest eigenvalue is greater than t times the largest Ritz value; Chebyshev polynomials of the second kind are used to obtain these bounds. This result can be used to compute a posteriori probabilistic bounds for the spectrum while executing the Lanczos process, and bounds based on [5,Theorem 3] can be used for symmetric indefinite matrices as well.…”

Section: Introduction Knowledge About the Extreme Eigenvalues Of Symmentioning

confidence: 99%

See 4 more Smart Citations

Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

Dorsselaer¹,

Hochstenbach²,

Vorst³

2001

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

Abstract. We study the Lanczos method for computing extreme eigenvalues of a symmetric or Hermitian matrix. It is not guaranteed that the extreme Ritz values are close to the extreme eigenvalues-even when the norms of the corresponding residual vectors are small. Assuming that the starting vector has been chosen randomly, we compute probabilistic bounds for the extreme eigenvalues from data available during the execution of the Lanczos process. Four different types of bounds are obtained using Lanczos, Ritz, and Chebyshev polynomials. These bounds are compared theoretically and numerically. Furthermore we show how one can determine, after each Lanczos step, a probabilistic upper bound for the number of steps still needed (without performing these steps) to obtain an approximation to the largest or smallest eigenvalue within a prescribed tolerance.

show abstract

Section: Numerical Experimentssupporting

confidence: 71%

Section: Re V1mentioning

confidence: 99%

Section: Re V1mentioning

confidence: 99%

“…In [5,Theorem 3], the following result has been derived for symmetric positive definite matrices. Let t > 1 and v 1 be chosen randomly from the uniform distribution over S n−1 .…”

Section: Re V1mentioning

confidence: 99%

Section: Introduction Knowledge About the Extreme Eigenvalues Of Symmentioning

confidence: 99%

See 3 more Smart Citations

Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

Dorsselaer¹,

Hochstenbach²,

Vorst³

2001

SIAM J. Matrix Anal. & Appl.

View full text Add to dashboard Cite

show abstract

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

Leyk

Woźniakowski

1998

Numer. Linear Algebra Appl.

View full text Add to dashboard Cite

Assessing the numerical accuracy of the impedance method

Olree

Horch

2007

Bioelectromagnetics

View full text Add to dashboard Cite

The impedance method has been used extensively to calculate induced electric fields and currents in tissue as a result of applied electromagnetic fields. However, there has previously been no known method for an a priori assessment of the numerical accuracy of the results found by this method. Here, we present a method which permits an a priori assessment of the numerical accuracy of the impedance method applied to physiologically meaningful problems in bioengineering. The assessment method relies on estimating the condition number associated with the impedance matrix for problems with varying shapes, sizes, conductivities, anisotropies, and implementation strategies. Equations have been provided which predict the number of significant figures lost due to poor matrix conditioning as a function of these variables. The results show that, for problems of moderate size and uncomplicated geometry, applied fields should be measured or calculated accurately to at least five or six significant figures. As resolutions are increased and material properties are more widely divergent even more significant figures are needed. The equations provided here should ensure that solutions found from the impedance method are calculated accurately.

show abstract

Probabilistic Bounds on the Extremal Eigenvalues and Condition Number by the Lanczos Algorithm

Cited by 22 publications

References 8 publications

Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

Computing Probabilistic Bounds for Extreme Eigenvalues of Symmetric Matrices with the Lanczos Method

Estimating a largest eigenvector by Lanczos and polynomial algorithms with a random start

Assessing the numerical accuracy of the impedance method

Contact Info

Product

Resources

About