We deal with the solution of the almost Toeplitz tridiagonal systems that arise from the problem of curve fitting by Natural Cubic Splines and B-Splines. We propose the TJ decomposition that gives rise to a method which is more accurate and faster than other previously proposed methods as we prove along the work. For the solution of the recurrences that arise from the TJ decomposition, we propose a specialization of the Overlapped Partitions Method (OPM). We show that OPM compares favourably in the context of the problem to the classic Divide and Conquer and R-Cyclic Reduction on the Convex C-3480 supercomputer.
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