Ultimately Fast Accurate Summation
نویسنده
چکیده
We present two new algorithms FastAccSum and FastPrecSum, one to compute a faithful rounding of the sum of floating-point numbers and the other for a result “as if” computed in K-fold precision. Faithful rounding means the computed result either is one of the immediate floating-point neighbors of the exact result or is equal to the exact sum if this is a floating-point number. The algorithms are based on our previous algorithms AccSum and PrecSum and improve them by up to 25%. The first algorithm adapts to the condition number of the sum; i.e., the computing time is proportional to the difficulty of the problem. The second algorithm does not need extra memory, and the computing time depends only on the number of summands and K. Both algorithms are the fastest known in terms of flops. They allow good instruction-level parallelism so that they are also fast in terms of measured computing time. The algorithms require only standard floating-point addition, subtraction, and multiplication in one working precision, for example, double precision.
منابع مشابه
Fast and accurate integral equation methods with applications in microfluidics
This thesis is concerned with computational methods for fluid flows on the microscale, also known as microfluidics. This is motivated by current research in biological physics and miniaturization technology, where there is a need to understand complex flows involving microscale structures. Numerical simulations are an important tool for doing this. The first, and smaller, part of the thesis pre...
متن کاملTwofold fast summation
Debugging accumulation of floating-point errors is hard; ideally, computer should track it automatically. Here we consider twofold approximation of exact real with value + error pair of floating-point numbers. Normally, value + error sum is more accurate than value alone, so error can estimate deviation between value and its exact target. Fast summation algorithm, that provides twofold sum of ∑...
متن کاملSeismic modeling using the frozen Gaussian approximation
We adopt the frozen Gaussian approximation (FGA) for modeling seismic waves. The method belongs to the category of ray-based beam methods. It decomposes seismic wavefield into a set of Gaussian functions and propagates these Gaussian functions along appropriate ray paths. As opposed to the classic Gaussian-beam method, FGA keeps the Gaussians frozen (at a fixed width) during the propagation pro...
متن کاملA fast algorithm for multilinear operators
This paper introduces a fast algorithm for computing multilinear integrals, which are defined through Fourier multipliers. The algorithm is based on generating a hierarchical decomposition of summation domain into squares, constructing a low-rank approximation for the multiplier function within each square, and applying FFT based fast convolution algorithm for the computation associated with ea...
متن کاملFast Ewald summation for free-space Stokes potentials
We present a spectrally accurate method for the rapid evaluation of free-space Stokes potentials, i.e., sums involving a large number of free space Green’s functions. We consider sums involving stokeslets, stresslets and rotlets that appear in boundary integral methods and potential methods for solving Stokes equations. The method combines the framework of the Spectral Ewald method for periodic...
متن کاملImplementation of Fast Multipole Algorithm on Special-Purpose Computer MDGRAPE-2
N -body simulation is a time consuming task in which force calculation part is most dominant part. The simplest and most accurate algorithm for force calculation is direct summation which has time complexity O(N). It is not practically suitable for large-scale simulations on most general-purpose computers. To cut down cost of force calculation one applies fast algorithms or performs force calcu...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید
ثبت ناماگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید
ورودعنوان ژورنال:
- SIAM J. Scientific Computing
دوره 31 شماره
صفحات -
تاریخ انتشار 2009