Research of Algorithms in terms of Information Geometry Structure and Discrete Time Integrable Systems

Nakamura提出了算术调和平均(AHM)算法，该算法通过在正矩阵空间上连续使用算术平均和调和平均运算来收敛到给定正矩阵的平方。从信息几何的角度出发，AHM算法绘制出连接空间上两点的相互对偶测地线的中点。Ohara将正矩阵空间推广到对称锥空间，明确了广义空间的信息几何结构，例如，这些平均运算确定了其上测地线的中点，当失去正性时，AHM算法一般不收敛。Kondo和Nakamura发现递归关系的第n项采用行列式形式。可解Logistic映射也具有类似的性质。基于行列式表达式和可解性，证明了递推关系的相应Lyapunov指数是正的，而不使用不变度量和积分计算。Nakamura和Tsujimoto开始研究离散时间Toda方程(QD算法)的并行计算，QD算法是离散时间可积系统的原型，用作数值算法。他们构建了一个具有分散存储器和两个CPU的并行计算机系统。将QD表从一边分解到另一边，他们计算出每个CPU的两个部分。结果表明，三对角矩阵特征值问题的计算时间减少到单CPU情况的近60%。此外，通过对QD表的倾斜分解，结果表明并行计算速度变得更快。这些结果将有助于设计新的离散时间可积系统的数值算法。

Nakamura formulated the arithmetic-harmonic mean (AHM) algorithm which converges to the square of a given positive matrix by using a successive use of arithmetic mean and harmonic mean operations on the space of positive matrices. From the viewpoint of information geometry the AHM algorithm plots the midpoints of mutually dual geodesics which connect 2-points on the space. Ohara generalized the space of positive matrices to that of symmetric cones and made clear the information geometry structure of the generalized space, for example, these mean operations determine midpoints of geodesics on it.When the positivity is lost, the AHM algorithm does not converge in general. Kondo and Nakamura found that the n-th terms of the recurrence relation takes a determinantal form. The solvable logistic map has a similar property. Based on the determinantal expression and solvability it is shown that the corresponding Lyapunov exponent of the recurrence relation is positive without using invariant measure and computation of integrations.Nakamura and Tsujimoto started to investigate parallel computing by the discrete-time Toda equation (the qd algorithm), a prototype of discrete-time integrable systems which work as numerical algorithms. They construct a parallel computer system with a dispersive memory and two CPUs. Decomposing the qd table from side to side they computed two pieces by each CPU. Then it is shown that the computation time of a tri-diagonal matrix eigenvalue problem decreases to almost 60 percent of that of one CPU case. Moreover by decomposing the qd table aslant they showed that the parallel computation rate becomes better.These results will be useful for designing new numerical algorithms in terms of discrete-time integrable systems.