Bernhard Langer
The majority of computer applications employ numerical data types with a fixed amount of precision for their computations. Their limited numerical range and precision are sufficient for most use cases. However, for some purposes, such as cryptography or geometrical computations, the required range and precision can become arbitrarily large. Numerical types that can handle such […]
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Jan Verschelde, Xiangcheng Yu
Numerical continuation methods apply predictor-corrector algorithms to track a solution path defined by a family of systems, the so-called homotopy. The systems we consider are defined by polynomials in several variables with complex coefficients. For larger dimensions and degrees, the numerical conditioning worsens and hardware double precision becomes often insufficient to reach the end of […]
Valentina Popescu
Many numerical problems require higher precision than the conventional floating-point (single, double) formats. One solution is to use multiple precision libraries such as GNU MPFR, which allow the manipulation of very high precision numbers. But their generality (they are able to handle numbers with millions of digits), is a quite heavy alternative when medium precision […]
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Umar Ibrahim Minhas, Samuel Bayliss, George A. Constantinides
FPGAs and GPUs are increasingly used in a range of high performance computing applications. When implementing numerical algorithms on either platform, we can choose to represent operands with different levels of accuracy. A trade-off exists between the numerical accuracy of arithmetic operators and the resources needed to implement them. Where algorithmic requirements for numerical stability […]
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Mioara Joldes, Valentina Popescu, Warwick Tucker
Today, GPUs represent an important hardware development platform for many problems in dynamical systems, where massive parallel computations are needed. Beside that, many numerical studies of chaotic dynamical systems require a computing precision higher than common floating point (FP) formats. One such application is locating invariant sets for chaotic dynamical systems. In particular, we focus […]
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Sylvain Collange, David Defour, Stef Graillat, Roman Iakymchuk
On modern multi-core, many-core, and heterogeneous architectures, floating-point computations, especially reductions, may become non-deterministic and thus non-reproducible mainly due to non-associativity of floating-point operations. We introduce a solution to compute deterministic sums of floating-point numbers efficiently and with the best possible accuracy. Our multi-level algorithm consists of two main stages: a filtering stage that uses […]
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Vadim Demchik, Alexey Gulov
A general method to produce uniformly distributed pseudorandom numbers with extended precision by combining two pseudorandom numbers with lower precision is proposed. In particular, this method can be used for pseudorandom number generation with extended precision on graphics processing units (GPU), where the performance of single and double precision operations can vary significantly.
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Jan Verschelde, Xiangcheng Yu
In order to compensate for the higher cost of double double and quad double arithmetic when solving large polynomial systems, we investigate the application of NVIDIA Tesla C2050, K20C, and K40 general purpose graphics processing units. As the dimension equals several thousands, the cost to compute one QR decomposition is sufficiently large so that the […]
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Jan Verschelde, Genady Yoffe
Our problem is to accurately solve linear systems on a general purpose graphics processing unit with double double and quad double arithmetic. The linear systems originate from the application of Newton’s method on polynomial systems. Newton’s method is applied as a corrector in a path following method, so the linear systems are solved in sequence […]
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Jon Wilkening, Jia Yu
A high-performance shooting algorithm is developed to compute time-periodic solutions of the free-surface Euler equations with spectral accuracy in double and quadruple precision. The method is used to study resonance and its effect on standing water waves. We identify new nucleation mechanisms in which isolated large-amplitude solutions, and closed loops of such solutions, suddenly exist […]
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Jan Verschelde, Genady Yoffe
Our problem is to accurately solve linear systems of modest dimensions (typically, the number of variables equals 32) on a general purpose graphics processing unit. The linear systems originate from the application of Newton’s method on polynomial systems of (moderately) large degrees. Newton’s method is applied as a corrector in a path following method, so […]
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Scott A. Sarra, Clyde Meador
Multiple results in the literature exist that indicate that all computed solutions to chaotic dynamical systems are time-step dependent. That is, solutions with small but different time steps will decouple from each other after a certain (small) finite amount of simulation time. When using double precision floating point arithmetic time step independent solutions have been […]
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