arXiv subjects

Victor Y. Pan

Publications and source records attributed to Victor Y. Pan.

At least 19 recordsLinked to original sources

arXiv2026

New Combinations of Polynomial Root-Finding Iterations

Some near-optimal polynomial root-finders of 2024-25, based on subdivision iterations, approximate all complex roots of a polynomial or all roots lying in a fixed Region of Interest in the complex plane. We combine these iterations with Newton's and/or Schroeder's to yield significant empirical acceleration versus each approach standing alone. Like the cited recent algorithms, our root-finders can be applied not only to a polynomial represented in monomial basis by its coefficients but also to a black box polynomial represented by an oracle (black box subroutine) for its evaluation. Some by-products of our study such as an extension of the Gauss-Lucas theorem and a fast black box estimator for root radius can be of independent interest.

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arXiv2025

Superfast iterative refinement of low rank approximation of a matrix based on ALS method and random sampling

A matrix algorithm runs superfast (aka at sublinear cost) if it involves much fewer flops and memory cells than an input matrix has entries. Big Data are frequently represented by matrices of immense sizes that cannot be handled directly but can be approximated with low rank matrices, with which one can operate superfast. Superfast computation of Low Rank Approximations (LRA) of a matrix, however, can be a challenge. Any superfast LRA algorithm fails miserably on worst case matrices. Fortunately, they rarely appear in computational practice. For an important example, superfast Adaptive Cross--Approximation iterations have consistently output accurate LRA of a large and important class of matrices during the decades of their worldwide application. Furthermore, they output CUR LRA, which is a special attractive form of LRA, widely applied in data analysis. %VP Adequate formal support for that empirical behavior, however, is still a challenge. Encouraged by success of these iterations we present a superfast randomized iterative refinement of a crude CUR LRA by means of combining random sampling with the Alternating Least Squares Method, which reduces this refinement to recursive solution of generalized Linear Least Squares Problems (LLSPs). We prove monotone convergence of our iterations with a high probability to a near-optimal LRA of an input matrix under some specified assumptions on it and on a crude approximate solution to an initial generalized LLSP. In our numerical tests two or three iterations of our algorithm have consistently and significantly improved crude initial LRAs of real world inputs.

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arXiv2025

Superfast Low Rank Approximation

Low rank approximation of a matrix (LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA superfast -- by using much fewer memory cells and flops than an input matrix has entries. Can we, however, compute an LRA of a matrix superfast? YES and NO. For worst case inputs, any LRA algorithm fails miserably unless it involves all input entries, but in computational practice worst case inputs seem to appear rarely, and accurate LRA are routinely computed superfast for large and important classes of matrices, in particular in the memory efficient form of CUR, widely used in data analysis. We advance formal study of this YES and NO coexistence by proving novel universal upper bounds on the spectral output error norms of all CUR LRA algorithms and, under a fixed probabilistic structure in the space of input matrices, on both spectral and Frobenius error norms of nearly all sketching LRA algorithms. These bounds imply that superfast LRA algorithms of the two kinds fail miserably only for a very narrow input class. Furthermore, in our numerical tests such superfast algorithms were consistently much more accurate than our upper estimates ensure and usually were reasonably close to optimal.

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arXiv2025

Low Rank Approximation at Sublinear Cost

Low Rank Approximation (LRA) of a matrix is a hot research subject, fundamental for Matrix and Tensor Computations and Big Data Mining and Analysis. Computations with low rank matrices can be performed at sublinear cost -- by using much fewer floating-point operations (flops) than an input matrix has entries, but can we compute LRA at sublinear cost? This is routinely done in computational practice for a large class of inputs, even though any sublinear cost LRA algorithm fails most miserably on worst case matrices. To provide insight into this controversy we first accelerate some popular near-optimal random sketching LRA algorithms -- to run them at sublinear cost. Then we define two probabilistic structures in the space of input matrices and estimate that the expected spectral and Frobenius error norms for the output LRA of the accelerated algorithms stay within a reasonable factor from their optima under both models, and so these sublinear cost algorithms only fail for a very narrow input class. Our upper estimates for their output accuracy are still quite high, but under some additional semi-heuristic amendments the algorithms have consistently output accurate LRA of various synthetic and real-world matrices in our numerical tests.

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arXiv2025

CUR Low Rank Approximation of a Matrix at Sublinear Cost

Low rank approximation of a matrix (hereafter LRA) is a highly important area of Numerical Linear and Multilinear Algebra and Data Mining and Analysis. One can operate with an LRA at sublinear cost -- by using much fewer memory cells and flops than an input matrix M has entries. For worst case inputs one cannot compute even a reasonably close LRA at sublinear cost, but in computational practice accurate LRAs, even in their memory efficient form of CUR LRAs, are routinely obtained at sublinear cost for large and important classes of matrices, in particular by means of Cross-Approximation iterations, which specialize Alternating Direction techniques to LRA. We identify some classes of matrices for which CUR LRA are computed at sublinear cost as well as some sublinear cost LRA algorithms that are empirically accurate for large classes of inputs. Some of our techniques and concepts can be of independent interests.

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arXiv2025

Superfast 1-Norm Estimation

A matrix algorithm is said to be superfast (that is, runs at sublinear cost) if it involves much fewer scalars and flops than the input matrix has entries. Such algorithms have been extensively studied and widely applied in modern computations for matrices with low displacement rank and more recently for low-rank approximation of matrices, even though they are known to fail on worst-case inputs in the latter application. We devise novel superfast algorithms that consistently produce accurate 1-norm estimates for real-world matrices and discuss some promising extensions of our surprisingly simple techniques. With further testing and refinement, our algorithms can potentially be adopted in practical computations.

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arXiv2024

Iterative Refinement and Oversampling for Low Rank Approximation

Iterative refinement is particularly popular for numerical solution of linear systems of equations. We extend it to Low Rank Approximation of a matrix (LRA) and observe close link of the resulting algorithm to oversampling techniques, commonly used in randomized LRA algorithms. We elaborate upon this link and revisit oversampling and some efficient randomized LRA algorithms. Applied with sparse sketch matrices they run significantly faster and in particular yield Very Low Rank Approximation (VLRA) at sublinear cost, using much fewer scalars and flops than the input matrix has entries. This is achieved at the price of deterioration of output accuracy, but according to our formal and empirical study subsequent oversampling improves accuracy to near-optimal level under the spectral norm for a large sub-class of matrices with fast decaying spectra of singular values.

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arXiv2024

Polynomial Root-Finding and Algebraic Eigenvalue Problem

Univariate polynomial root-finding has been studied for four millennia and very intensively in the last decades. Our new near-optimal root-finders approximate all zeros of a polynomial p almost as fast as one accesses its coefficients with the precision required for the solution within a prescribed error bound. Furthermore, our root-finders can be applied to a black box polynomial, defined by an oracle (black box subroutine) for its evaluation rather than by its coefficients. Due to this feature our root-finders support approximation of the eigenvalues of a matrix in a record Las Vegas expected bit operation time and are particularly fast for a polynomial that can be evaluated fast such as the sum of a few shifted monomials or a Mandelbrot-like polynomial defined by a recurrence. Our divide and conquer algorithm of ACM STOC 1995 is the only other known near-optimal polynomial root-finder, but it extensively uses the coefficients, is quite involved, and has never been implemented, while according to extensive numerical experiments with standard test polynomials, already a slower initial implementation of our new root-finders competes with user's choice package of root-finding subroutines MPSolve and supersedes it more and more significantly as the degree of a polynomial grows large. We elaborate upon the design and analysis of our algorithms, comment on their potential heuristic acceleration, and briefly cover polynomial root-finding by means of functional iterations. Our techniques can be of independent interest.

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arXiv2023

Fast Approximation of Polynomial Zeros and Matrix Eigenvalues

We approximate the d complex zeros of a univariate polynomial p(x) of a degree d or those zeros that lie in a fixed region of interest on the complex plane such as a disc or a square. Our divide and conquer algorithm of STOC 1995 supports solution of this problem in optimal Boolean time (up to a poly-logarithmic factor), that is, runs nearly as fast as one can access the coefficients of p with the precision necessary to support required accuracy of the output. That record complexity has not been matched by any other algorithm yet, but our root-finder of 1995 is quite involved and has never been implemented. We present alternative nearly optimal root-finders based on our novel variants of the classical subdivision iterations. Unlike our predecessor of 1995, we require randomization of Las Vegas type, allowing us to detect any output error at a dominated computational cost, but our new root-finders are much simpler to implement than their predecessor of 1995. According to the results of extensive test with standard test polynomials for their preliminary version, which incorporates only a part of our novel techniques, the new root-finders compete and for a large class of inputs significantly supersedes the package of root-finding subroutines MPSolve, which for decades has been user's choice package. Unlike our predecessor of 1995 and all known fast algorithms for the cited tasks of polynomial root-finding, our new algorithms can be also applied to a polynomial given by a black box oracle for its evaluation rather than by its coefficients. This makes our root-finders particularly efficient for polynomials p(x) that can be evaluated fast such as the Mandelbrot polynomials or those given by the sum of a small number of shifted monomials. Our algorithm can be readily extended to fast approximation of the eigenvalues of a matrix or a matrix polynomial.

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arXiv2022

New Progress in Classic Area: Polynomial Root-squaring and Root-finding

The DLG root-squaring iterations, due to Dandelin 1826 and rediscovered by Lobachevsky 1834 and Graeffe 1837, have been the main approach to root-finding for a univariate polynomial p(x) in the 19th century and beyond, but not so nowadays because these iterations are prone to severe numerical stability problems. Trying to avoid these problems we have found simple but novel reduction of the iterations applied for Newton's inverse ratio -p'(x)/p(x) to approximation of the power sums of the zeros of p(x) and its reverse polynomial. The resulting polynomial root-finders can be devised and performed independently of DLG iterations, based on Newton's identities or Cauchy integrals. In the former case the computation involve a set of leading or tailing coefficients of an input polynomial. In the latter case we must scale the variable and increase the arithmetic computational cost to ensure numerical stability. Nevertheless the cost is still manageable, at least for fast root-refinement, and the algorithms can be applied to a black box polynomial p(x)--given by a black box for the evaluation of the ratio p'(x)/p(x) rather than by its coefficients. This enables important computational benefits, including efficient recursive as well as concurrent approximation of a set of zeros of p(x) or even all of its zeros, acceleration where an input polynomial can be evaluated fast, and extension to approximation of the eigenvalues of a matrix or a polynomial matrix, being efficient if the matrix can be inverted fast, e.g., is data sparse. We also recall our recent fast algorithms for approximation of the root radii, that is, the distances to the roots from the origin or any complex value to the zeros of p(x), and apply it for fast black box initialization of polynomial root-finding by means of functional iterations, e.g., Newton's, Ehrlich's, and Weierstrass's.

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arXiv2022

Accelerated Subdivision for Clustering Roots of Polynomials given by Evaluation Oracles

In our quest for the design, the analysis and the implementation of a subdivision algorithm for finding the complex roots of univariate polynomials given by oracles for their evaluation, we present sub-algorithms allowing substantial acceleration of subdivision for complex roots clustering for such polynomials. We rely on Cauchy sums which approximate power sums of the roots in a fixed complex disc and can be computed in a small number of evaluations --polylogarithmic in the degree. We describe root exclusion, root counting, root radius approximation and a procedure for contracting a disc towards the cluster of root it contains, called $\varepsilon$-compression. To demonstrate the efficiency of our algorithms, we combine them in a prototype root clustering algorithm. For computing clusters of roots of polynomials that can be evaluated fast, our implementation competes advantageously with user's choice for root finding, MPsolve.

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arXiv2021

Root Radii and Subdivision for Polynomial Root-Finding

We depart from our approximation of 2000 of all root radii of a polynomial, which has readily extended Sch{ö}nhage's efficient algorithm of 1982 for a single root radius. We revisit this extension, advance it, based on our simple but novel idea, and yield significant practical acceleration of the known near optimal subdivision algorithms for complex and real root-finding of user's choice. We achieve this by means of significant saving of exclusion tests and Taylor's shifts, which are the bottleneck of subdivision root-finders. This saving relies on our novel recipes for the initialization of root-finding iterations of independent interest. We demonstrate our practical progress with numerical tests, provide extensive analysis of the resulting algorithms, and show that, like the preceding subdivision root-finders, they support near optimal Boolean complexity bounds.

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arXiv2021

Superfast Approximate Linear Least Squares Solution of a Highly Overdetermined Linear System of Equations

With a high probability the Sarlos randomized algorithm of 2006 outputs a nearly optimal least squares solution of a highly overdeterminedlinear system of equations. We propose its simple deterministic variation which computes such a solution for a random input whp and therefore computes it deterministically for a large input class. Unlike the Sarlos original algorithm our variation performs computations at sublinear cost or, as we say, superfast, that is, by using much fewer memory cells and arithmetic operations than an input matrix has entries. Our extensive tests are in good accordance with this result.

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arXiv2021

CUR Low Rank Approximation at Deterministic Sublinear Cost

A matrix algorithm runs at {\em sublinear cost} if it uses much fewer memory cells and arithmetic operations than the input matrix has entries. Such algorithms are indispensable for Big Data Mining and Analysis. Quite typically in that area the input matrices are so immense that realistically one can only access a small fraction of all their entries but can access and process at sublinear cost their Low Rank Approximation {\em (LRA)}. Can, however, we compute LRA at sublinear cost? Adversary argument shows that the output of any algorithm running at sublinear cost is extremely far from LRA of the worst case input matrices and even of the matrices of small families of our Appendix, but we prove that some deterministic sublinear cost algorithms output reasonably close LRA in a memory efficient form of CUR LRA if an input matrix admits LRA and is Symmetric Positive Semidefinite or is very close to a low rank matrix. The latter result is technically simple but provides some (very limited but long overdue) support for the well-known empirical efficiency of sublinear cost LRA by means of Cross-Approximation. We demonstrate the power of application of such LRA by turning the Fast Multipole celebrated Method into Superfast Multipole Method. The design and analysis of our algorithms rely on extensive prior study of the link of LRA of a matrix to maximization of its volume.

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arXiv2019

QRP Variation of Cross--Approximation Iterations for Low Rank Approximation

We call matrix algorithms superfast if they use much fewer flops and memory cells than the input matrix has entries. Using such algorithms is indispensable for Big Data Mining and Analysis, where the input matrices are so immense that one can only access a small fraction of all their entries. A natural remedy is Low Rank Approximation (LRA) of these matrices, which is routinely computed by means of Cross-Approximation iterations for more than a decade of worldwide application in computational practice. We point out and extensively test an important application of superfast LRA to significant acceleration of the celebrated Fast Multipole Method, which turns it into Superfast Multipole Method.

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arXiv2019

Polynomial root clustering and explicit deflation

We seek complex roots of a univariate polynomial $P$ with real or complex coefficients. We address this problem based on recent algorithms that use subdivision and have a nearly optimal complexity. They are particularly efficient when only roots in a given Region Of Interest (ROI) are sought. We propose two improvements for root finders. The first one is applied to polynomials having only real coefficients; their roots are either real or appear in complex conjugate pairs. We show how to adapt the subdivision scheme to focus the computational effort on the imaginary positive part of the ROI. In our second improvement we deflate $P$ to decrease its degree and the arithmetic cost of the subdivision.

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arXiv2019

New practical advances in polynomial root clustering

We report an ongoing work on clustering algorithms for complex roots of a univariate polynomial $p$ of degree $d$ with real or complex coefficients. As in their previous best subdivision algorithms our root-finders are robust even for multiple roots of a polynomial given by a black box for the approximation of its coefficients, and their complexity decreases at least proportionally to the number of roots in a region of interest (ROI) on the complex plane, such as a disc or a square, but we greatly strengthen the main ingredient of the previous algorithms. Namely our new counting test essentially amounts to the evaluation of a polynomial $p$ and its derivative $p'$, which is a major benefit, e.g., for sparse polynomials $p$. Moreover with evaluation at about $\log(d)$ points (versus the previous record of order $d$) we output correct number of roots in a disc whose contour has no roots of $p$ nearby. Moreover we greatly soften the latter requirement versus the known subdivision algorithms. Our second and less significant contribution concerns subdivision algorithms for polynomials with real coefficients. Our tests demonstrate the power of the proposed algorithms.

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arXiv2019

Root-finding with Implicit Deflation

Functional iterations such as Newton's are a popular tool for polynomial root-finding. We consider realistic situation where some (e.g., better-conditioned) roots have already been approximated and where further computations is directed to approximation of the remaining roots. Such situation is also realistic for root-finding by means of subdivision. A natural approach of applying explicit deflation has been much studied and recently advanced by one of the authors of this paper, but presently we contribute to the alternative approach of applying implicit deflation, which we combine with mapping the variable and reversion of an input polynomial. We also show another unexplored direction for substantial further progress in this long and extensively studied area. Namely we dramatically increase their local efficiency by means of the incorporation of fast algorithms for multipoint polynomial evaluation and Fast Multipole Method.

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