John Palmer

The solution of linear systems having real, symmetric, diagonally dominant, tridiagonal coefficient matrices with constant diagonals is considered. It is proved that the diagonals of the LU decomposition of the coefficient matrix rapidly converge to full floating-point precision. It is also proved that the computed LU decomposition converges when floating-point arithmetic is used and that the limits of the LU diagonals using floating point are roughly within machine precision of the limits using real arithmetic. This fact is exploited to reduce the number of floating-point operations required to solve a linear system from 8n - 7 to 5n + 2k - 3, where k is much less than n, the order of the matrix. If the elements of the subdiagnals and superdiagonals are 1, then only 4n + 2k - 3 operations are needed. The entire LU decomposition takes k words of storage, and considerable savings in array subscripting are achieved. Upper and lower bounds on k are obtained in terms of the ratio of the coefficient matrix diagonal constants and parameters of the floating-point number system. Various generalizations of these results are discussed.