Matrix homographic iterations and bounds for the inverses of certain band matrices (Q1113970)

The author considers an mn\(\times mn\) block-tridiagonal matrix \(S_ n=(S_{i,j})_{i,j=1,...,n}\) with the \(m\times m\) block entries \(S_{1,1}=A_ 1\), \(S_{i,i}=A\) \((i=2,...,n-1)\), \(S_{n,n}=A_{\infty}\), \(S_{i,i-1}=B\), \(S_{i,i+1}=C\) and \(S_{i,j}=\emptyset\) for \(| i-j| \geq 2\). Under certain conditions imposed on the block entries, the author proves the estimates \((1)\quad \| S^{-1}_{n;i,j}\| \leq k\quad r^{| i-j|}\) \(\forall\) \(i,j=1,2,...,n\) for some constants \(k>0\), \(0<r<1\), independent of n, where \(S^{-1}_{n;i,j}\) denotes the generic block term of \(S_ n^{-1}.\) The technique used to obtain (1) is connected with the convergence of the matrix iteration scheme \(U_{i+1}=A-BU_ i^{-1}C\), \(U_ 1=A_ 1\). Here \(U_ i\) are exactly the generic diagonal block terms of the block- diagonal matrix U in the factorization \(S_ n=VUW\), where V(W) is a block-bidiagonal, lower (upper) triangular matrix. Estimates of the type (1) are needed in bounding spline interpolation errors in the sup-norm.