Pseudospectral Shattering, the Sign Function, and Diagonalization in Nearly Matrix Multiplication Time

Abstract We exhibit a randomized algorithm which, given square matrix $$A\in \mathbb {C}^{n\times n}$$ A∈Cn×n with $$\Vert A\Vert \le 1$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">‖A‖≤1 and $$\delta >0$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">δ>0 , computes high probability an invertible V diagonal D such that $$ \Vert A-VDV^{-1}\Vert \delta xmlns:mml="http://www.w3.org/1998/Math/MathML">‖A-VDV-1‖≤δ using $$O(T_\mathsf {MM}(n)\log ^2(n/\delta ))$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(TMM(n)log2(n/δ)) arithmetic operations, in finite $$O(\log ^4(n/\delta )\log n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(log4(n/δ)logn) bits of precision. The computed similarity additionally satisfies V\Vert V^{-1}\Vert O(n^{2.5}/\delta )$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">‖V‖‖V-1‖≤O(n2.5/δ) . Here $$T_\mathsf {MM}(n)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">TMM(n) is the number operations required to multiply two $$n\times n$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">n×n complex matrices numerically stably, known satisfy {MM}(n)=O(n^{\omega +\eta })$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">TMM(n)=O(nω+η) for every $$\eta xmlns:mml="http://www.w3.org/1998/Math/MathML">η>0 where $$\omega xmlns:mml="http://www.w3.org/1998/Math/MathML">ω exponent multiplication (Demmel et al. Numer Math 108(1):59–91, 2007). variant spectral bisection numerical linear algebra (Beavers Jr. Denman 21(1-2):143–169, 1974) crucial Gaussian perturbation preprocessing step. Our result significantly improves previously best-known provable running times $$O(n^{10}/\delta ^2)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(n10/δ2) diagonalization general (Armentano J Eur Soc 20(6):1375–1437, 2018) (with regard dependence on n ) $$O(n^3)$$ xmlns:mml="http://www.w3.org/1998/Math/MathML">O(n3) Hermitian (Dekker Traub Linear Algebra Appl 4:137–154, 1971). It first achieve nearly time any model computation (real arithmetic, rational or arithmetic), thereby matching complexity other dense as inversion QR factorization up polylogarithmic factors. proof rests new ingredients. (1) show adding small splits its pseudospectrum into well-separated components. In particular, this implies eigenvalues perturbed have large minimum gap, property independent interest random theory. (2) give rigorous analysis Roberts’ Newton iteration method (Roberts Int Control 32(4):677–687, 1980) computing sign function itself open problem since at least 1986.