Tridiagonal matrix algorithm

Abstract: Suppose that an indefinite symmetric tridiagonal matrix permits triangular factorization T = LDL t We provide individual condition numbers for the eigenvalues and eigenvectors of T when the parameters in L and D suffer small relative perturbations When there is element growth in the factorization, then some pairs may be robust while others are sensitive A 4 × 4 example shows the limitations of standard multiplicative perturbation theory and the efficacy of our new condition numbers

Abstract: Generalizing Muller and Scheerer's method which is used to parallelize the tridiagonal solvers, this paper presents a parallel method for solving the circulant block-tridiagonal systems. The applications of our result to solve the block-tridiagonal systems, the banded systems, and the circulant tridiagonal systems (for example, solving the closed B-spline curve fitting) are also addressed.

Abstract: A Crank-Nicolson-type difference scheme is presented for the spatial variable coefficient subdiffusion equation with Riemann-Liouville fractional derivative. The truncation errors in temporal and spatial directions are analyzed rigorously. At each time level, it results in a linear system in which the coefficient matrix is tridiagonal and strictly diagonally dominant, so it can be solved by the Thomas algorithm. The unconditional stability and convergence of the scheme are proved in the discrete $L_{2}$ norm by the energy method. The convergence order is $\min \{2-\frac{\alpha}{2}, 1+\alpha \}$ in the temporal direction and two in the spatial one. Finally, numerical examples are presented to verify the efficiency of our method.