Solving algebraic problems with high accuracy
Proc. of the symposium on A new approach to scientific computation
Topics in matrix analysis
Newton's method for the matrix square root
Mathematics of Computation
Expansion and estimation of the range of nonlinear functions
Mathematics of Computation
Solution of the matrix equation AX + XB = C [F4]
Communications of the ACM
C-XSC: A C++ Class Library for Extended Scientific Computing
C-XSC: A C++ Class Library for Extended Scientific Computing
SIAM Journal on Matrix Analysis and Applications
SIAM Journal on Scientific Computing
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Functions of Matrices: Theory and Computation (Other Titles in Applied Mathematics)
Introduction to Interval Analysis
Introduction to Interval Analysis
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We present methods to compute verified square roots of a square matrix $A$. Given an approximation $X$ to the square root, obtained by a classical floating point algorithm, we use interval arithmetic to find an interval matrix which is guaranteed to contain the error of $X$. Our approach is based on the Krawczyk method, which we modify in two different ways in such a manner that the computational complexity for an $n\times n$ matrix is reduced to $n^3$. The methods are based on the spectral decomposition or, in the case that the eigenvector matrix is ill conditioned, on a similarity transformation to block diagonal form. Numerical experiments prove that our methods are computationally efficient and that they yield narrow enclosures provided $X$ is a good approximation. This is particularly true for symmetric matrices, since their eigenvector matrix is perfectly conditioned.