DFTによる悪条件行列の改善

概要

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A system specified by ill-conditioned equations has the property that small changes in the coefficients give rise to large changes in the solution. In fact, if the coefficients of ill-conditioned equations are given by some experiments, then no information at all may be, in general, available about the solution. In this case, the coefficient matrix is near singular. As the system is unstable, it is advisable to change the system into the stable one which is equivalent to the original system. This subject is important not only in numerical calculations but also in engineering problems. Let us consider the following system represented by linear equations. [numerical formula] If the each column of coefficients in above equations is transformed by DFT (Discrete Fourier Transform) method respectively, the Fourier coefficients obtained can be arranged so as to make the following simultaneous equations. [numerical formula] where a_<ij>,a_i are Fourier cosine coefficients, b_<ij>,b_i are Fourier sine coefficients and n=2N. In this paper, we will prove that the set of equation (1) is entirely equivalent to equation (2). Next, it will be described that even if the equation (1) ill-conditioned, the transformed equation (2) may become to be well-conditioned.
福山大学の論文