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• W3126495890 abstract "We exhibit a randomized algorithm which given a square matrix A ∈ mathbbC <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>n×n</sup> with ||A|| ≤ 1 and , computes with high probability an invertible V and diagonal D such that ||A-VDV <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>-1</sup> || ≤ δ in O(TMM(n)log <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> (n/δ)) arithmetic operations on a floating point machine with O(log <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>4</sup> (n/δ)logn) bits of precision. The computed similarity V additionally satisfies ||V||||V <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>-1</sup> || ≤ O(n <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2.5</sup> /δ). Here TMM(n) is the number of arithmetic operations required to multiply two n×n complex matrices numerically stably, known to satisfy TMM(n)=O(n <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>ω+η</sup> ) for every where ω is the exponent of matrix multiplication [1]. The algorithm is a variant of the spectral bisection algorithm in numerical linear algebra [2] with a crucial Gaussian perturbation preprocessing step. Our running time is optimal up to polylogarithmic factors, in the sense that verifying that a given similarity diagonalizes a matrix requires at least matrix multiplication time. It significantly improves the previously best known provable running times of O(n <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>10</sup> /δ <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>2</sup> ) arithmetic operations for diagonalization of general matrices [3], and (with regards to the dependence on n) O(n <sup xmlns:mml=http://www.w3.org/1998/Math/MathML xmlns:xlink=http://www.w3.org/1999/xlink>3</sup> ) arithmetic operations for Hermitian matrices [4], and is the first algorithm to achieve nearly matrix multiplication time for diagonalization in any model of computation (real arithmetic, rational arithmetic, or finite arithmetic). The proof rests on two new ingredients. (1) We show that adding a small complex Gaussian perturbation to any matrix splits its pseudospectrum into n small well-separated components. In particular, this implies that the eigenvalues of the perturbed matrix have a large minimum gap, a property of independent interest in random matrix theory. (2) We give a rigorous analysis of Roberts' [5] Newton iteration method for computing the sign function of a matrix in finite arithmetic, itself an open problem in numerical analysis since at least 1986 [6]. This is achieved by controlling the evolution of the pseudospectra of the iterates using a carefully chosen sequence of shrinking contour integrals in the complex plane." @default.
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• W3126495890 date "2020-11-01" @default.
• W3126495890 modified "2023-09-25" @default.
• W3126495890 title "Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time" @default.
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• W3126495890 doi "https://doi.org/10.1109/focs46700.2020.00056" @default.
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