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ZKCM: A C plus plus library for multiprecision matrix computation with applications in quantum information

COMPUTER PHYSICS COMMUNICATIONS 2013年第8期184卷 2005-2020页

作者： SaiToh, Akira Natl Inst Informat Quantum Informat Sci Theory Grp Chiyoda Ku Tokyo 1018430 Japan

ZKCM is a C++ library developed for the purpose of multiprecision matrix computation, on the basis of the GNU MP and MPFR libraries. It provides an easy-to-use syntax and convenient functions for matrix manipulations including those often used in numerical simulations in quantum physics. Its extension library, ZKCM_QC, is developed for simulating quantum computing using the time-dependent matrix-product-state simulation method. This paper gives an introduction about the libraries with practical sample programs. Program Summary Program title: ZKCM Catalogue identifier: AEPI_v1_0 Program summary URL: http://***/summaries/AEPI_v1_*** Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU Lesser General Public License, version 3 No. of lines in distributed program, including test data, etc.: 95600 No. of bytes in distributed program, including test data, etc.: 1133481 Distribution format: *** Programming language: C++. Computer: General computers. Operating system: Unix-like systems, such as Linux, Free BSD, Cygwin on Windows OS, etc. RAM: Several mega bytes several giga bytes, dependent on the problem instance Classification: 4.8, 4.15. External routines: GNU MP (GMP) [1], MPFR [2] Ver. 3.0.0 or later Nature of problem: multiprecision computation is helpful to guarantee and/or evaluate the accuracy of simulation results in numerical physics. There is a potential demand for a programming library focusing on matrix computation usable for this purpose with a user-friendly syntax. Solution method: A C++ library ZKCM has been developed for multiprecision matrix computation. It provides matrix operations useful for numerical studies of physics, e.g., the tensor product (Kronecker product), the tracing-out operation, the inner product, the LU decomposition, the Hermitian-matrix diagonalization, the singular-value decomposition, and the discrete Fourier transform. For basic floating-point operations, G

关键词： multiprecision computing Linear algebra Time-dependent matrix product state Quantum information

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Numerical calculations accuracy comparison of the Inverse Laplace Transform algorithms for solutions of fractional order differential equations

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NONLINEAR DYNAMICS 2016年第1期84卷 65-77页

作者： Brzezinski, Dariusz W. Ostalczyk, Piotr Lodz Univ Technol Inst Appl Comp Sci 18-22 Stefanowskiego St PL-90924 Lodz Poland

In the paper we present results of a numerical experiment in which we evaluate and compare some numerical algorithms of the Inverse Laplace Transform for inversion accuracy of some fractional order differential equations solutions. The algorithms represent diverse lines of approach to the subject of the numerical inversion and include methods by Stehfest, Abate and Whitt, Vlach and Singhai, De Hoog, Talbot, Zakian and a one in which the FFT is applied for Fourier series convergence acceleration. We used C++ and Python languages and applied arbitrary precision mathematical libraries to address some crucial issues of an numerical implementation. Introductory test set includes Laplace transforms which are considered as difficult to compute as well as some others commonly applied in fractional calculus. In the main part of the evaluation, there is assessed accuracy of the numerical Inverse Laplace Transform of some popular fractional differential equations solutions, e.g., the initial value problem in case of the inhomogeneous Bagley-Torvik equation and composite fractional oscillation equation. Evaluation results enable to conclude that the Talbot method which involves deformed Bromwich contour integration, the De Hoog and the Abate and Whitt methods which use Fourier series expansion with accelerated convergence provide the most accurate inversions. They can be applied to a wide variety of inversion problems.

关键词： Numerical accuracy of the Inverse Laplace Transform Fractional order differential equations multiprecision computing

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A Unified Root Algorithm in Bignum Arithmetic 3

A Unified Root Algorithm in Bignum Arithmetic

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3rd IEEE Advanced Information Technology, Electronic and Automation Control Conference (IAEAC)

作者： Cheng, Yiping Cheng, Guizhi Beijing Jiaotong Univ Sch Elect & Informat Engn Beijing 100044 Peoples R China Beijing Informat Sci & Technol Univ Sch Econ & Management Beijing 100192 Peoples R China

ISBN: (纸本)9781538645093

In bignum arithmetic, the cube root and higher order root operations are conventionally treated as elementary functions, meaning that for them, correct rounding is not required, usually even without an error bound. In this paper we provide a unified algorithm RootRem to find the (arbitrary ordered) root of bignums. This algorithm first finds the reciprocal root by using Newton iteration, and then obtains the root through a reciprocal operation. It has guaranteed correct rounding in all cases. RootRem(2), which is to find the square root, is currently found to be less efficient than the existing algorithm SqrtRem proposed by Brent and Zimmermann. However, RootRem(p) with p >= 3 fill an important theoretical gap and are major improvements over the existing algorithms.

关键词： arbitrary precision Newton iteration computer arithmetic multiprecision computing correct rounding

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Refinement of a Newton Reciprocal Algorithm for Arbitrary Precision Numbers 4

Refinement of a Newton Reciprocal Algorithm for Arbitrary Pr...

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4th IEEE International Conference on Progress in Informatics and computing (IEEE PIC)

作者： Cheng, Yiping Liu, Ze Beijing Jiaotong Univ Sch Elect & Informat Engn Beijing 100044 Peoples R China

ISBN: (纸本)9781509034840

For arbitrary precision numbers, reciprocal computing algorithms based on Newton iteration is asymptotically the fastest. In this work we provide a refined algorithm based on the Newton reciprocal algorithm by Brent and Zimmermann in their MCA book. The key techniques used in the refinement are D1 balancing, clear specification, remainder operation, and economical multiplication. The refined algorithm is more general, and gives exact and unique result. Numerical results show that these improvements are made without the cost of time efficiency. There is still the potential to further improve the efficiency by utilizing a short multiplication algorithm.

关键词： Newton iteration numerical algorithms computer algebra multiprecision computing

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