Estimation of the bilinear form y⁎f(A)x for Hermitian matrices

Fika, Paraskevi; Mitrouli, Marilena

doi:10.1016/j.laa.2015.08.033

Cited by 15 publications

(50 citation statements)

References 17 publications

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“…Extensions of this approach in least squares and regularization were presented in and in estimating the trace of powers of linear operators in Hilbert spaces in . In , the bilinear form x T A − 1 y was estimated for any real matrix A , and in , estimates for the bilinear form y ∗ f ( A ) x , for a Hermitian matrix

A \in {double-struckC}^{p \times p},

and

x, y \in {double-struckC}^{p}

were developed. Next, we specify how the procedure developed in can lead to a direct estimation of the whole diagonal of a matrix f ( A ).…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

“…In , the following family of estimates for the quadratic form ( x , f ( A ) x ) was obtained.

(x, f (A) x) ≃ e_{f, ν} = f ((), ρ^{ν} \frac{c_{1}}{c_{0}}) c_{0}, where ρ = c_{0} c_{2} / c_{1}^{2} ⩾ 1, ν \in double-struckC and c_{n} = (x, A^{n} x), n \in double-struckR .

For an increasing function f and for

ν \in double-struckR, e_{f, ν}

is also an increasing function of ν for c 1 > 0 and decreasing for c 1 < 0, whereas for a decreasing function f , e f , ν is an increasing function of ν for c 1 < 0 and decreasing for c 1 > 0.…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

“…In practice, this optimal value ν o is not computed, because it requires the a priori knowledge of the exact value of ( x , f ( A ) x ). However, bounds and an approximation of ν o , for symmetric positive definite matrices are given in . Moreover, if ρ = 1, then the estimate becomes exact, that is,

e_{f, ν} = (x, f (A) x), \forall ν \in double-struckC

.…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

“…Throughout the examples, we present the execution time in seconds and the mean relative error of the diagonal entries that is defined as

(\sum_{i = 1}^{p} | {(f (A))}_{i i} - e s t (i) | / | {(f (A))}_{i i} |) / p

, where e s t ( i ) is an estimation of the diagonal entry ( f ( A )) i i given either by Proposition for a value of ν that is selected according to the procedure developed in , or as an iterate of the Lanczos algorithm using the single or the block Gauss quadrature rules. The characteristics of the algorithms that are employed, that is, the value of ν , the number of iterations k , and the block size q are also reported in the tables.…”

Section: Numerical Examplesmentioning

confidence: 99%

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Estimating the diagonal of matrix functions

Fika

Mitrouli

Roupa

2016

Math Methods in App Sciences

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The evaluation of the diagonal of matrix functions arises in many applications and an efficient approximation of it, without estimating the whole matrix f(A), would be useful. In the present paper, we compare and analyze the performance of three numerical methods adjusted to attain the estimation of the diagonal of matrix functions f(A), where A∈double-struckRp×p is a symmetric matrix and f a suitable function. The applied numerical methods are based on extrapolation and Gaussian quadrature rules. Various numerical results illustrating the effectiveness of these methods and insightful remarks about their complexity and accuracy are demonstrated. Copyright © 2016 John Wiley & Sons, Ltd.

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A \in {double-struckC}^{p \times p},

and

x, y \in {double-struckC}^{p}

were developed. Next, we specify how the procedure developed in can lead to a direct estimation of the whole diagonal of a matrix f ( A ).…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

“…In , the following family of estimates for the quadratic form ( x , f ( A ) x ) was obtained.

(x, f (A) x) ≃ e_{f, ν} = f ((), ρ^{ν} \frac{c_{1}}{c_{0}}) c_{0}, where ρ = c_{0} c_{2} / c_{1}^{2} ⩾ 1, ν \in double-struckC and c_{n} = (x, A^{n} x), n \in double-struckR .

For an increasing function f and for

ν \in double-struckR, e_{f, ν}

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

e_{f, ν} = (x, f (A) x), \forall ν \in double-struckC

.…”

Section: Vector Estimates Through An Extrapolation Proceduresmentioning

confidence: 99%

“…Throughout the examples, we present the execution time in seconds and the mean relative error of the diagonal entries that is defined as

(\sum_{i = 1}^{p} | {(f (A))}_{i i} - e s t (i) | / | {(f (A))}_{i i} |) / p

Section: Numerical Examplesmentioning

confidence: 99%

See 2 more Smart Citations

Estimating the diagonal of matrix functions

Fika

Mitrouli

Roupa

2016

Math Methods in App Sciences

Self Cite

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show abstract

“…Computing the trace in this manner, however, requires the computation of all the eigenvalues, which is also often prohibitively expensive. Hence, various methods proposed for approximately computing tr( f ( A )) consist of the following two ingredients: Approximate the trace of f ( A ) by using the average of unbiased samples

u_{i}^{T} f false(A false) u_{i}

, i =1,…, N , where the u i are independent random vectors of some nature. Approximately compute the bilinear form

u_{i}^{T} f false(A false) u_{i}

by using some numerical technique. …”

Section: Introductionmentioning

confidence: 99%

A posteriori error estimate for computing tr(f(A)) by using the Lanczos method

Chen

Saad

2018

Numerical Linear Algebra App

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Summary An outstanding problem when computing a function of a matrix, f(A), by using a Krylov method is to accurately estimate errors when convergence is slow. Apart from the case of the exponential function that has been extensively studied in the past, there are no well‐established solutions to the problem. Often, the quantity of interest in applications is not the matrix f(A) itself but rather the matrix–vector products or bilinear forms. When the computation related to f(A) is a building block of a larger problem (e.g., approximately computing its trace), a consequence of the lack of reliable error estimates is that the accuracy of the computed result is unknown. In this paper, we consider the problem of computing tr(f(A)) for a symmetric positive‐definite matrix A by using the Lanczos method and make two contributions: (a) an error estimate for the bilinear form associated with f(A) and (b) an error estimate for the trace of f(A). We demonstrate the practical usefulness of these estimates for large matrices and, in particular, show that the trace error estimate is indicative of the number of accurate digits. As an application, we compute the log determinant of a covariance matrix in Gaussian process analysis and underline the importance of error tolerance as a stopping criterion as a means of bounding the number of Lanczos steps to achieve a desired accuracy.

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