「数値線形代数における高精度計算」

日時:
2007年10月9日 15：00－16：30、10月11日15：00－18：00

場所:
10/9:総合校舎406セミナー室 10/11:総合校舎102講義室（京都大学・吉田キャンパス)

講師:
大石 進一氏/荻田 武史氏/Siegfried M. Rump氏/中村 佳正氏

講師所属:


講師略歴:


講演概要:
■第1回
10月9日（火）15：00－16：30
講師：大石 進一（早稲田大学理工学部）
題目：「精度保証線形数値計算の基礎から現状まで」（日本語による解説講演）

■第2回
日時：10月11日（木）15：00－18：00
講師：
中村 佳正（京都大学情報学研究科／科学技術振興機構）
荻田 武史（科学技術振興機構／早稲田大学理工学術院）
Siegfried M. Rump (Hamburg University of Technology)

概要：
15:00+
Yoshimasa Nakamura (Kyoto University/JST)
Recent Developments of the mdLVs Algorithm for Computing Matrix Singular Values

The mdLVs algorithm (Iwasaki-Nakamura 2006) is designed by using a discrete time integrable dynamical system. The mdLVs is a new stable algorithm with shift for computing bidiagonal matrix singular values. A new shift strategy for the mdLVs is now invented which makes the original mdLVs faster and more accurate. An implementation of the mdLVs is compared with LAPACK codes. On accuracy the mdLVs is the second after the bisection method. On speed the mdLVs isthe second after the dqds algorithm. The mdLVs is a highly credible algorithm. Namely, it is exponentially stable and can converge to multiple singular values. This is a joint work with M. Iwasaki, K. Kimura and M. Takata.

16:00+
Takeshi Ogita (JST/Waseda University)
Fast and Accurate Summation of Floating-point Numbers

Abstract: In this some years, we develop an accurate summation algorithm. For a floating-point vector p, the algorithm returns a result faithfully rounded from the exact summation of p. Moreover, it is very fast in terms of not only flop counts but also measured computing time. First in the talk, we briefly review our accurate summation algorithm. Next, we explain how to accelerate the algorithm. Finally, numerical results are presented showing performance of the proposed algorithms.

17:00+
Siegfried M. Rump (Hamburg University of Technology)
Inversion of Extremely Ill-Conditioned Matrices Using a Faithfully Rounded Dot Product

Abstract: A well known rule of thumb in numerical analysis tells that matrices up to a condition number 1/eps can numerically be inverted, where eps denotes the relative rounding error unit. In IEEE 754 double precision this limits the condition number to about 1016. In this talk we present an algorithm going way beyond this limit of condition number. The only additional ingredient is a new summation algorithm. All computations are performed solely in double precision. Numerical examples with extremely ill-conditioned matrices will be shown.