時計と原子の比較に戻ります。 Let r' be the rate of a clock in K' and r be its rate in K. Then r = r'(1+phi/c**2). In the previous discussion, phi was related to angular velocity and the radius of rotation. Apparently, the vibrations of lig…

最近アインシュタインの相対性理論を再読しています。 Relativity basically involves the differences in our perception of temporal and spatial position depending on our frame of reference. In an appendix to his relativity treatise, Einstein di…

六件前に解説しようとした記事は先のアルゴリズムと関係がありそうです。 The recursion given by Kung was: a(ij)(1) = a(ij) a(ij)**(k+1) = a(ij)**(k)+l(ik)*(-u(kj)) l and u are elements in the lower triangular matrix L and upper triangular matr…

1. Note that, in the Doolittle algorithm, it is assumed that a(n,n) is non-zero for all n. Otherwise, the l(i,j)=-a(i,j)/a(n,n) term is undefined. 2. Proof that the inverse of a lower triangular. The proof is by contradiction. Let A be an …

アルゴリズムの詳細と証明を書きたいです。 In each iteration of the Doolittle algorithm, elements below the main diagonal in one of A's columns are eliminated. The result is an upper triangular matrix U, and the complexity is (2/3)n**3 (yet …

I have three objectives in examining LU factorization: 1. Consider the concepts and applications of this technique. 2. Examine at least one specific algorithm and possibly prove it works. 3. Compare the algorithm with the recurrence used b…

While I have worked on programs to infer numerical solutions to ordinary differential equations of a single variable, I have yet to complete any work involving partial differential equations. These are somewhat more complex, as the derivat…

In a recent programming competition, I wrote a program to execute a breadth first search through a state space of 9-character strings, in which each character was a distinct digit 1-9. The goal was, given a 3 by 3 matrix of digits 1-9, per…

I mentioned positive definite symmetric matrices in conjunction with LU factorization via systolic arrays in the previous post. A symmetric matrix is one which equals its transpose, e.g. 1 2 3 2 9 4 3 4 7 A positive definite matrix is one …