• Title, Summary, Keyword: rounding error analysis

### Blockwise analysis for solving linear systems of equations

• Smoktunowicz, Alicja
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.3 no.1
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• pp.31-41
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• 1999
• We investigate some techniques of iterative refinement of solutions of a nonsingular system Ax = b with A partitioned into blocks using only single precision arithmetic. We prove that iterative refinement improves a blockwise measure of backward stability. Some applications of the results for the least squares problem (LS) will be also considered.

### A Fixed-Point Error Analysis of fast DCT Algorithms (고정 소수점 연산에 의한 고속 DCT 알고리듬의 오차해석)

• 연일동;이상욱
• The Transactions of the Korean Institute of Electrical Engineers
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• v.40 no.4
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• pp.331-341
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• 1991
• The discrete cosine transform (DCT) is widely used in many signal processing areas, including image and speech data compression. In this paper, we investigate a fixed-point error analysis for fast DCT algorithms, namely, Lee [6], Hou [7] and Vetterli [8]. A statistical model for fixed-point error is analyzed to predict the output noise due to the fixed-point implementation. This paper deals with two's complement fixed-point data representation with truncation and rounding. For a comparison purpose, we also investigate the direct form DCT algorithm. We also propose a suitable scaling model for the fixed-point implementation to avoid an overflow occurring in the addition operation. Computer simulation results reveal that there is a close agreement between the theoretical and the experimental results. The result shows that Vetterli's algorithm is better than the other algorithms in terms of SNR.

### Maximum Error Reduction for Fixed-width Modified Booth Multipliers Based on Error Bound Analysis (오차범위 분석을 통한 고정길이 modified Booth 곱셈기의 최대오차 감소)

• Cho, Kyung-Ju;Chung, Jin-Gyun
• Journal of the Institute of Electronics Engineers of Korea SD
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• v.42 no.10
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• pp.29-34
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• 2005
• The maximum quantization error has serious effect on the performance of fixed-width multipliers that receive W-bit inputs and produce W-bit products. In this paper, we analyze the error bound of fixed-width modified Booth multipliers. Then, the estimation method for the number of additional columns for fixed-width multipliers is proposed to limit the maximum quantization error within a desired bound. In addition, it is shown that our methodology can be extended to reduced-width multipliers. By simulations, it is shown that the proposed error analysis method is useful to the practical design of fixed-width modified Booth multipliers.

### ON APPROXIMATIONS BY IRRATIONAL SPLINES

• LEVIN, MIKHAIL P.
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.5 no.1
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• pp.47-53
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• 2001
• A problem of approximation by irrational splines is considered. These splines have a constant curvature between interpolation nodes and need only one additional boundary condition for derivatives, which should be set only at one of two boundary nodes, that is impossible for usual polynomial splines required boundary conditions at both boundary nodal points. Some estimations for numerical differentiation and rounding error analysis are presented.

### EFFICIENT LATTICE REDUCTION UPDATING AND DOWNDATING METHODS AND ANALYSIS

• PARK, JAEHYUN;PARK, YUNJU
• Journal of the Korean Society for Industrial and Applied Mathematics
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• v.19 no.2
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• pp.171-188
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• 2015
• In this paper, the efficient column-wise/row-wise lattice reduction (LR) updating and downdating methods are developed and their complexities are analyzed. The well-known LLL algorithm, developed by Lenstra, Lenstra, and Lov${\acute{a}}$sz, is considered as a LR method. When the column or the row is appended/deleted in the given lattice basis matrix H, the proposed updating and downdating methods modify the preconditioning matrix that is primarily computed for the LR with H and provide the initial parameters to reduce the updated lattice basis matrix efficiently. Since the modified preconditioning matrix keeps the information of the original reduced lattice bases, the redundant computational complexities can be eliminated when reducing the lattice by using the proposed methods. In addition, the rounding error analysis of the proposed methods is studied. The numerical results demonstrate that the proposed methods drastically reduce the computational load without any performance loss in terms of the condition number of the reduced lattice basis matrix.

### Analysis of Some Strange Behaviors of Floating Point Arithmetic using MATLAB Programs (MATLAB을 이용한 부동소수점 연산의 특이사항 분석)

• Chung, Tae-Sang
• The Transactions of The Korean Institute of Electrical Engineers
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• v.56 no.2
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• pp.428-431
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• 2007
• A floating-point number system is used to represent a wide range of real numbers using finite number of bits. The standard the IEEE adopted in 1987 divides the range of real numbers into intervals of [$2^E,2^{E+1}$), where E is an Integer represented with finite bits, and defines equally spaced equal counts of discrete numbers in each interval. Since the numbers are defined discretely, not only the number representation itself includes errors but in floating-point arithmetic some strange behaviors are observed which cannot be agreed with the real world arithmetic. In this paper errors with floating-point number representation, those with arithmetic operations, and those due to order of arithmetic operations are analyzed theoretically with help of and verification with the results of some MATLAB program executions.