Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

Gaaf, SW Sarah; Hochstenbach, Michiel E.

doi:10.1137/140975218

Cited by 3 publications

(7 citation statements)

References 19 publications

(32 reference statements)

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“…Although (4.8) was satisfied in all our experiments, it may fail to hold in unlucky cases, for example when poor estimates of α, 0 are used so that α A 2 or 0 σ min (X 0 ) (as discussed in [48, § 5.8], 0 can be estimated using a condition number estimator as 1/condest(A); or by using the estimator of [19,35]. When this happens we suggest running Zolo-pd again on X 2 (not A).…”

Section: Stopping Criterionmentioning

confidence: 93%

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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Abstract. The symmetric eigenvalue decomposition and the singular value decomposition (SVD) are fundamental matrix decompositions with many applications. Conventional algorithms for computing these decompositions are suboptimal in view of recent trends in computer architectures, which require minimizing communication together with arithmetic costs. Spectral divideand-conquer algorithms, which recursively decouple the problem into two smaller subproblems, can achieve both requirements. Such algorithms can be constructed with the polar decomposition playing two key roles: it forms a bridge between the symmetric eigendecomposition and the SVD, and its connection to the matrix sign function naturally leads to spectral-decoupling. For computing the polar decomposition, the scaled Newton and QDWH iterations are two of the most popular algorithms, as they are backward stable and converge in at most nine and six iterations, respectively. Following this framework, we develop a higher-order variant of the QDWH iteration for the polar decomposition. The key idea of this algorithm comes from approximation theory: we use the best rational approximant for the scalar sign function due to Zolotarev in 1877. The algorithm exploits the extraordinary property enjoyed by the sign function that a high-degree Zolotarev function (best rational approximant) can be obtained by appropriately composing low-degree Zolotarev functions. This lets the algorithm converge in just two iterations in double-precision arithmetic, with the whopping rate of convergence seventeen. The resulting algorithms for the symmetric eigendecompositions and the SVD have higher arithmetic costs than the QDWH-based algorithms, but are better-suited for parallel computing and exhibit excellent numerical backward stability.

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Section: Stopping Criterionmentioning

confidence: 93%

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Nakatsukasa¹,

Freund²

2016

SIAM Rev.

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“…Vecharynski proposes a Rayleigh quotient–type iteration that uses a single iteration of a preconditioned iterative solver to compute an approximate singular vector given an approximate singular value (which is estimated from the previous singular‐vector estimate) . Gaaf and Hochstenbach propose to estimate the condition number of square matrices using the extended Krylov subspace . Computing a basis for the extended Krylov subspace requires applying the inverse of the matrix; hence, the method uses an LU factorization of the matrix.…”

Section: Related Workmentioning

confidence: 99%

“…LSQR is a method for solving least squares problems min ||Ax − b|| 2 . At its core, LSQR uses the bidiagonalization procedure of Golub and Kahan to form iterates u (0) , u (1) (1) , · · · ∈ ℝ, and (0) , (1) , · · · ∈ ℝ such that…”

Section: The Algorithmmentioning

confidence: 99%

“…More importantly, dense SVD algorithms require space that is proportional to mn when the matrix is m × n , which is impractical for large sparse matrices. Likewise, if A is square and an LU factorization of A can be computed, then

σ_{\min}

can be estimated efficiently from the factorization; however, for some large sparse matrices, space requirements due to fill‐in make this method impractical.…”

Section: Introductionmentioning

confidence: 99%

“…Suppose b is a vector in the column space of A, and let Ax ⋆ = b. Let x (0) , x (1) , … be the iterates of a Krylov-subspace method applied to the problem arg min ||Ax − b|| (all norms in this paper denote the 2-norm unless stated otherwise). If we knew the forward error d (k) = x ⋆ − x (k) , we could use the Rayleigh quotients ||Ad (k) ||∕||d (k) || as an upper estimate on min .…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Spectral condition‐number estimation of large sparse matrices

Avron

Druinsky

Toledo

2019

Numerical Linear Algebra App

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Summary We describe a randomized Krylov‐subspace method for estimating the spectral condition number of a real matrix A or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value σmin of A. Our method estimates this value by solving a consistent linear least squares problem with a known solution using a specific Krylov‐subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to σmin. Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running dense singular value decomposition would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR), and it works equally well on square and rectangular matrices.

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A note on the condition number of the scaled total least squares problem

Wang

Yang

2018

Calcolo

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In this paper, we consider the explicit expressions of the normwise condition number for the scaled total least squares problem. Some techniques are introduced to simplify the expression of the condition number, and some new results are derived. Based on these new results, new expressions of the condition number for the total least squares problem can be deduced as a special case. New forms of the condition number enjoy some storage and computational advantages. We also proposed three different methods to estimate the condition number. Some numerical experiments are carried out to illustrate the effectiveness of our results.Keywords: condition number, Fréchet derivative, the scaled total least squares problem, power method, probabilistic condition estimation methodwhere λ is a positive real number, · F denotes the Frobenius norm and R(·) is the range space. Let [E S r S ] be the solution to (1.1), then the solution to the linear system (A + E S )λx = λb − r S is called the STLS solution and denoted by x S . As shown in [2], when λ = 1, λ → 0 and λ → ∞, x S becomes the TLS solution x ⊺ , OLS solution x O and DLS solution x D , respectively. The condition number gives a quantitative measurement of the maximum amplification of the resulting change in solution with respect to a perturbation in the data and has been extensively studied for too many topics to list here. For the STLS problem, Zhou et al. [3] considered its perturbation analysis and presented the normwise, mixed and componentwise condition numbers. Based on the ✩

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Probabilistic Bounds for the Matrix Condition Number with Extended Lanczos Bidiagonalization

Cited by 3 publications

References 19 publications

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Computing Fundamental Matrix Decompositions Accurately via the Matrix Sign Function in Two Iterations: The Power of Zolotarev's Functions

Spectral condition‐number estimation of large sparse matrices

A note on the condition number of the scaled total least squares problem

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