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Contact: F.X.Timmes
my one page vitae,
full vitae,
research statement, and
teaching statement.
Tridiagonal matrices arise from using a 3 point finite difference stencil in one-dimension. Pentadiagonal matrices arise from using a 5 point stencil in one-dimension or a 3 point stencil in two-dimensions. The tool pentadiagonal.tbz contains routine to solve pentadiagonal linear system of equations \begin{equation} a_i u_{i-2} + b_i u_{i-1} + c_i u_i + d_i u_{i+1} + e_iu_{i+2} = f_i \label{eq1} \tag{1} \end{equation} and cyclic pentadiagonal systems with nonzero entries in the lower left and upper right corners of the matrix:
\begin{equation} \left[\begin{array}{ccccccccccc} c_{1} & d_{1} & e_1 & 0 & 0 & \ldots & & & & p_1 & p_2\\ b_{2} & c_{2} & d_2 & e_2 & 0 & \ldots & & & & & p_3\\ a_{3} & b_{3} & c_3 & d_3 & e_3 & \ldots & & & & & \\ & & & & & \ldots & & & & & \\ & & & & & \ldots & & & & & \\ & & & & & \ldots & & & & & \\ & & & & & \ldots & a_{n-2} & b_{n-2} & c_{n-2} & d_{n-2} & e_{n-2} \\ p_4 & & & & & \ldots & 0 & a_{n-1} & b_{n-1} & c_{n-1} & e_{n-1} \\ p_5 & p_6 & & & & \ldots & 0 & 0 & a_{n} & b_{n} & c_{n} \end{array}\right] \left[\begin{array}{c} u_1 \\ u_2 \\ u_3 \\ \ldots \\ \ldots \\ \ldots \\ u_{n-2}\\ u_{n-1}\\ u_n \end{array}\right] = \left[\begin{array}{c} r_1 \\ r_2 \\ r_3 \\ \ldots \\ \ldots \\ \ldots \\ r_{n-2}\\ r_{n-1}\\ r_n \end{array}\right] \label{eq2} \tag{2} \end{equation}
Such cyclic forms usually arise from periodic boundary conditions.
Heptadiagonal matrices arise from using a 7 point stencil in one-dimension or a 3 point stencil in three-dimensions. The tool heptadiagonal.tbz contains routines to solve heptadiagonal linear system of equations \begin{equation} D_i u_{i-2} + B_i u_{i-2} + b_i u_{i-1} + d_i u_i + a_i u_{i+1} + A_iu_{i+2} + C_iu_{i+2} = r_i \label{eq3} \tag{3} \end{equation} and cyclic heptadiagonal systems with nonzero entries in the lower left and upper right corners of the matrix: \begin{equation} \left[\begin{array}{ccccccccccc} d_{1} & a_{1} & A_1 & C_1 & 0 & 0 & \ldots & & p_1 & p_2 \\ b_{2} & d_{2} & a_2 & A_2 & C_2 & \ddots & \ddots & \ldots & & p_3 \\ B_{3} & b_{3} & d_3 & a_3 & A_3 & C_3 & \ddots & \ddots & \dots & \\ D_{4} & B_{4} & b_4 & d_4 & a_4 & A_4 & C_4 & \ddots & \ddots & \\ 0 & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \vdots \\ \vdots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \\ 0 & \ddots & \ddots & D_{n-3} & B_{n-3} & b_{n-3} & d_{n-3} & a_{n-3} & A_{n-3} & C_{n-3} \\ 0 & \ldots & \ddots & \ddots & D_{n-2} & B_{n-2} & b_{n-2} & d_{n-2} & a_{n-2} & A_{n-2} \\ p_4 & 0 & \ldots & \ddots & 0 & D_{n-1} & B_{n-1} & b_{n-1} & d_{n-1} & a_{n-1} \\ p_5 & p_6 & 0 & \ldots & 0 & 0 & D_n & B_{n} & b_{n} & d_{n} \\ \end{array}\right] \left[\begin{array}{c} u_1 \\ u_2 \\ u_3 \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ u_{n-2}\\ u_{n-1}\\ u_n \end{array}\right] = \left[\begin{array}{c} r_1 \\ r_2 \\ r_3 \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ \ldots \\ r_{n-2}\\ r_{n-1}\\ r_n \end{array}\right] \label{eq4} \tag{4} \end{equation}