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Accurate Sum and Dot Product
Algorithms for summation and dot product of floating-point numbers are presented which are fast in terms of measured computing time. We show that the computed results are as accurate as if computed…
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Accurate Floating-Point Summation Part I: Faithful Rounding
TLDR
This paper presents an algorithm for calculating a faithful rounding of a vector of floating-point numbers, which adapts to the condition number of the sum, and proves certain constants used in the algorithm to be optimal.
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Verified Eigenvalue Evaluation for the Laplacian over Polygonal Domains of Arbitrary Shape
TLDR
The proposed algorithm can provide concrete eigenvalue bounds for domain of arbitrary shape, even in the case that eigenfunction has singularity, when the finite element method is applied to bound leading eigenvalues of Laplace operator over polygonal domain.
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Self-synchronization of coupled oscillators with hysteretic responses
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Accurate Floating-Point Summation Part II: Sign, K-Fold Faithful and Rounding to Nearest
TLDR
An algorithm for calculating the rounded-to-nearest result of $s:=\sum p_i$ for a given vector of floating-point numbers $p_i$, as well as algorithms for directed rounding, working for huge dimensions.
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A First Order Phase Transition Resulting from Finite Inertia in Coupled Oscillator Systems
We analyze the collective behavior of a set of coupled damped driven pendula with finite (large) inertia, and show that the synchronization of the oscillators exhibits a first order phase transition…
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Fast high precision summation
TLDR
Given a vector pi of ∞oating-point numbers with exact sum s, a new algorithm is presented that is fast in terms of measured computing time because it allows good instruction-level parallelism and more accurate and faster than competitors such as XBLAS.
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Fast verification of solutions of matrix equations
  • S. Oishi, S. Rump
  • Mathematics, Computer Science
    Numerische Mathematik
  • 1 February 2002
TLDR
Different algorithms are proposed, the fastest with $\frac{2}{3} n^3$ flops computational cost for the verification step, the same as for the LU decomposition.
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A parallel algorithm for accurate dot product
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ACCURATE FLOATING-POINT SUMMATION
TLDR
These algorithms are fast in terms of measured computing time because they neither require special operations such as access to mantissa or exponent, they contain no branch in the inner loop, nor do they require extra precision: the only operations used are standard floating-point addition, subtraction and multiplication in one working precision.
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