Explicit solution of block tridiagonal systems of linear equations (Q1106614)

The authors consider matrix equations \(Az=k\) with A a block tridiagonal matrix of the form \[ A = \left[\begin{matrix} B_1 & C_1 \\ A_2 & B_2 & C_2 && 0 \\ & \ddots & \ddots & \ddots \\ && \ddots & \ddots & \ddots \\ & 0 && \ddots & \ddots & C_{N-1} \\ & &&& A_ N & B_ N \end{matrix}\right] \] where \(A_ i\), \(B_ i\), \(C_ i\), \(i=1,2,...,N\), with \(A_ 1=C_ N=0\), are \(2\times 2\) submatrices. Such matrix equations arise from finite discretizations to two-point boundary value problems. If the vectors z and k are partitioned relative to the matrix A, then the components \(Z_ i=(z\) \(i_ 1z\) \(i_ 2)\) T of the solution of \(Az=k\) are, as known, given recursively by (1) \(Z_ N=G_ N\), \(Z_ i=G-W_ iZ_{i-1},\) \(1\leq i\leq N-1\) where \(W_ 1=B_ 1^{-1}C_ 1,\) \(W_ i=(B_ i-A_ iW_{i-1})^{-1}C_ i,\) \(G_ 1=B_ 1^{-1}K_ 1\), \(G_ i=(B_ i-A_ iW_{i-1})^{-1}(K_ i-A_ iG_{i-1}),\) \(2\leq i\leq N\). As the contribution of the present paper the authors express the recursion formula (1) direct in terms of the elements of the submatrices \(B_ i=(b\) \(i_{mn})\), \(C_ i=A\) \(T_{i+1}=(c\) \(i_{mn})\) and of the subvectors \(K_ i=(k\) \(i_ m)\), \(i=1,2,...,N\), \(m=1,2\), \(n=1,2\).