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Hidehiko Hasegawa - One of the best experts on this subject based on the ideXlab platform.
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Comparison of Tridiagonalization Methods Using High-Precision Arithmetic with MuPAT
Lecture Notes in Computational Science and Engineering, 2017Co-Authors: Ryoya Ino, Emiko Ishiwata, Kohei Asami, Hidehiko HasegawaAbstract:In general, when computing the eigenvalues of symmetric matrices, a matrix is tridiagonalized using some orthogonal transformation. The Householder transformation, which is a tridiagonalization method, is accurate and stable for dense matrices, but is not applicable to sparse matrices because of the required memory space. The Lanczos and Arnoldi methods are also used for tridiagonalization and are applicable to sparse matrices, but these methods are sensitive to computational errors. In order to obtain a stable algorithm, it is necessary to apply numerous techniques to the original algorithm, or to simply use accurate Arithmetic in the original algorithm. In floating-point Arithmetic, computation errors are unavoidable, but can be reduced by using high-Precision Arithmetic, such as double-double (DD) Arithmetic or quad-double (QD) Arithmetic. In the present study, we compare double, double-double, and quad-double Arithmetic for three tridiagonalization methods; the Householder method, the Lanczos method, and the Arnoldi method. To evaluate the robustness of these methods, we applied them to dense matrices that are appropriate for the Householder method. It was found that using high-Precision Arithmetic, the Arnoldi method can produce good tridiagonal matrices for some problems whereas the Lanczos method cannot.
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Analysis of the GCR method with mixed Precision Arithmetic using QuPAT
Journal of Computational Science, 2012Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:Abstract To verify computation results of double Precision Arithmetic, a high Precision Arithmetic environment is needed. However, it is difficult to use high Precision Arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple Precision Arithmetic environment QuPAT on Scilab to satisfy the following requirements: (i) to enable programs to be written simply using quadruple Precision Arithmetic; (ii) to enable the use of both double and quadruple Precision Arithmetic at the same time; (iii) to be independent of any hardware and operating systems. To confirm the effectiveness of QuPAT, we applied the GCR method for ill-conditioned matrices and focused on the scalar parameters α and β in GCR, partially using DD Arithmetic. We found that the use of DD Arithmetic only for β leads to almost the same results as when DD Arithmetic is used for all computations. We conclude that QuPAT is an excellent interactive tool for using double Precision and DD Arithmetic at the same time.
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development of quadruple Precision Arithmetic toolbox qupat on scilab
International Conference on Computational Science and Its Applications, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
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ICCSA (2) - Development of quadruple Precision Arithmetic toolbox QuPAT on scilab
Computational Science and Its Applications – ICCSA 2010, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
Tsubasa Saito - One of the best experts on this subject based on the ideXlab platform.
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Analysis of the GCR method with mixed Precision Arithmetic using QuPAT
Journal of Computational Science, 2012Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:Abstract To verify computation results of double Precision Arithmetic, a high Precision Arithmetic environment is needed. However, it is difficult to use high Precision Arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple Precision Arithmetic environment QuPAT on Scilab to satisfy the following requirements: (i) to enable programs to be written simply using quadruple Precision Arithmetic; (ii) to enable the use of both double and quadruple Precision Arithmetic at the same time; (iii) to be independent of any hardware and operating systems. To confirm the effectiveness of QuPAT, we applied the GCR method for ill-conditioned matrices and focused on the scalar parameters α and β in GCR, partially using DD Arithmetic. We found that the use of DD Arithmetic only for β leads to almost the same results as when DD Arithmetic is used for all computations. We conclude that QuPAT is an excellent interactive tool for using double Precision and DD Arithmetic at the same time.
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development of quadruple Precision Arithmetic toolbox qupat on scilab
International Conference on Computational Science and Its Applications, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
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ICCSA (2) - Development of quadruple Precision Arithmetic toolbox QuPAT on scilab
Computational Science and Its Applications – ICCSA 2010, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
Emiko Ishiwata - One of the best experts on this subject based on the ideXlab platform.
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Comparison of Tridiagonalization Methods Using High-Precision Arithmetic with MuPAT
Lecture Notes in Computational Science and Engineering, 2017Co-Authors: Ryoya Ino, Emiko Ishiwata, Kohei Asami, Hidehiko HasegawaAbstract:In general, when computing the eigenvalues of symmetric matrices, a matrix is tridiagonalized using some orthogonal transformation. The Householder transformation, which is a tridiagonalization method, is accurate and stable for dense matrices, but is not applicable to sparse matrices because of the required memory space. The Lanczos and Arnoldi methods are also used for tridiagonalization and are applicable to sparse matrices, but these methods are sensitive to computational errors. In order to obtain a stable algorithm, it is necessary to apply numerous techniques to the original algorithm, or to simply use accurate Arithmetic in the original algorithm. In floating-point Arithmetic, computation errors are unavoidable, but can be reduced by using high-Precision Arithmetic, such as double-double (DD) Arithmetic or quad-double (QD) Arithmetic. In the present study, we compare double, double-double, and quad-double Arithmetic for three tridiagonalization methods; the Householder method, the Lanczos method, and the Arnoldi method. To evaluate the robustness of these methods, we applied them to dense matrices that are appropriate for the Householder method. It was found that using high-Precision Arithmetic, the Arnoldi method can produce good tridiagonal matrices for some problems whereas the Lanczos method cannot.
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Analysis of the GCR method with mixed Precision Arithmetic using QuPAT
Journal of Computational Science, 2012Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:Abstract To verify computation results of double Precision Arithmetic, a high Precision Arithmetic environment is needed. However, it is difficult to use high Precision Arithmetic in ordinary computing environments without any special hardware or libraries. Hence, we designed the quadruple Precision Arithmetic environment QuPAT on Scilab to satisfy the following requirements: (i) to enable programs to be written simply using quadruple Precision Arithmetic; (ii) to enable the use of both double and quadruple Precision Arithmetic at the same time; (iii) to be independent of any hardware and operating systems. To confirm the effectiveness of QuPAT, we applied the GCR method for ill-conditioned matrices and focused on the scalar parameters α and β in GCR, partially using DD Arithmetic. We found that the use of DD Arithmetic only for β leads to almost the same results as when DD Arithmetic is used for all computations. We conclude that QuPAT is an excellent interactive tool for using double Precision and DD Arithmetic at the same time.
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development of quadruple Precision Arithmetic toolbox qupat on scilab
International Conference on Computational Science and Its Applications, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
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ICCSA (2) - Development of quadruple Precision Arithmetic toolbox QuPAT on scilab
Computational Science and Its Applications – ICCSA 2010, 2010Co-Authors: Tsubasa Saito, Emiko Ishiwata, Hidehiko HasegawaAbstract:When floating point Arithmetic is used in numerical computation, cancellation of significant digits, round-off errors and information loss cannot be avoided. In some cases it becomes necessary to use multiple Precision Arithmetic; however some operations of this Arithmetic are difficult to implement within conventional computing environments. In this paper we consider implementation of a quadruple Precision Arithmetic environment QuPAT (Quadruple Precision Arithmetic Toolbox) using the interactive numerical software package Scilab as a toolbox. Based on Double-Double (DD) Arithmetic, QuPAT uses only a combination of double Precision Arithmetic operations. QuPAT has three main characteristics: (1) the same operator is used for both double and quadruple Precision Arithmetic; (2) both double and quadruple Precision Arithmetic can be used at the same time, and also mixed Precision Arithmetic is available; (3) QuPAT is independent of which hardware and operating systems are used. Finally we show the effectiveness of QuPAT in the case of analyzing a convergence property of the GCR(m) method for a system of linear equations.
Miodrag Potkonjak - One of the best experts on this subject based on the ideXlab platform.
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low power behavioral synthesis optimization using multiple Precision Arithmetic
Design Automation Conference, 1999Co-Authors: Milos D. Ercegovac, Darko Kirovski, Miodrag PotkonjakAbstract:Many modern multimedia applications such as image and video processing are characterized by a unique combination of Arithmetic and computational features: fixed-point Arithmetic, a variety of short data types, high degree of instruction-level parallelism, strict timing constraints, and high computational requirements. Computationally intensive algorithms usually boost device's power dissipation which is often key to the efficiency of many communications and multimedia applications. Although recently virtually all general-purpose processors have been equipped with multiPrecision operations, the current generation of behavioral synthesis tools for application-specific systems does not utilize this power/performance optimization paradigm. In this paper, we explore the potential of using multiple Precision Arithmetic units to effectively support synthesis of low-power application-specific integrated circuits. We propose a new architectural scheme for collaborate addition of sets of variable Precision data. We have developed a novel resource allocation and computation assignment methodology for a set of multiple Precision Arithmetic units. The optimization algorithms explore the trade-off of allocating low-width bus structures and executing multiple-cycle operations. Experimental results indicate strong advantages of the proposed approach.
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DAC - Low-power behavioral synthesis optimization using multiple Precision Arithmetic
Proceedings of the 36th ACM IEEE conference on Design automation conference - DAC '99, 1999Co-Authors: Milos D. Ercegovac, Darko Kirovski, Miodrag PotkonjakAbstract:Many modern multimedia applications such as image and video processing are characterized by a unique combination of Arithmetic and computational features: fixed-point Arithmetic, a variety of short data types, high degree of instruction-level parallelism, strict timing constraints, and high computational requirements. Computationally intensive algorithms usually boost device's power dissipation which is often key to the efficiency of many communications and multimedia applications. Although recently virtually all general-purpose processors have been equipped with multiPrecision operations, the current generation of behavioral synthesis tools for application-specific systems does not utilize this power/performance optimization paradigm. In this paper, we explore the potential of using multiple Precision Arithmetic units to effectively support synthesis of low-power application-specific integrated circuits. We propose a new architectural scheme for collaborate addition of sets of variable Precision data. We have developed a novel resource allocation and computation assignment methodology for a set of multiple Precision Arithmetic units. The optimization algorithms explore the trade-off of allocating low-width bus structures and executing multiple-cycle operations. Experimental results indicate strong advantages of the proposed approach.
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ICASSP - Behavioral synthesis optimization using multiple Precision Arithmetic
Proceedings of the 1998 IEEE International Conference on Acoustics Speech and Signal Processing ICASSP '98 (Cat. No.98CH36181), 1Co-Authors: Milos D. Ercegovac, Darko Kirovski, G. Mustafa, Miodrag PotkonjakAbstract:Modern image and video processing applications are characterized by a unique combination of Arithmetic and computational features: fixed point Arithmetic, a variety of short data types, high degree of instruction-level parallelism, strict timing constraints, high computational requirements, and high cost sensitivity. The current generation of behavioral synthesis tools does not address well this type of application. In this paper we explore the potential of using multiple Precision Arithmetic units to effectively support implementation of image and video processing applications as application specific integrated circuits. A new architectural scheme for collaborate addition of sets of variable Precision data is proposed as well as an allocation and assignment methodology for multiple Precision Arithmetic units. Experimental results indicate the strong advantages of the proposed approach.
Pascal Giorgi - One of the best experts on this subject based on the ideXlab platform.
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Recursive double-size fixed Precision Arithmetic
2016Co-Authors: Alexis Breust, Christophe Chabot, Jean-guillaume Dumas, Laurent Fousse, Pascal GiorgiAbstract:This work is a part of the SHIVA (Secured Hardware Immune Versatile Architecture) project whose purpose is to provide a programmable and reconfigurable hardware module with high level of security. We propose a recursive double-size fixed Precision Arithmetic called RecInt. Our work can be split in two parts. First we developped a C++ software library with performances comparable to GMP ones. Secondly our simple representation of the integers allows an implementation on FPGA. Our idea is to consider sizes that are a power of 2 and to apply doubling techniques to implement them efficiently: we design a recursive data structure where integers of size 2^k, for k>k0 can be stored as two integers of size 2^{k-1}. Obviously for k
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ICMS - Recursive double-size fixed Precision Arithmetic
Mathematical Software – ICMS 2016, 2016Co-Authors: Alexis Breust, Christophe Chabot, Jean-guillaume Dumas, Laurent Fousse, Pascal GiorgiAbstract:We propose a new fixed Precision Arithmetic package called RecInt. It uses a recursive double-size data-structure. Contrary to arbitrary Precision packages like GMP, that create vectors of words on the heap, RecInt large integers are created on the stack. The space allocated for these integers is a power of two and Arithmetic is performed modulo that power. Operations are thus easily implemented recursively by a divide and conquer strategy. Among those, we show that this packages is particularly well adapted to Newton-Raphson like iterations or Montgomery reduction. Recursivity is implemented via doubling algorithms on templated data-types. The idea is to extend machine word functionality to any power of two and to use template partial specialization to adapt the implemented routines to some specific sizes and thresholds. The main target Precision is for cryptographic sizes, that is up to several tens of machine words. Preliminary experiments show that good performance can be attained when comparing to the state of art GMP library: it can be several order of magnitude faster when used with very few machine words. This package is now integrated within the Givaro C++ library and has been used for efficient exact linear algebra computations.
