Summary
A rigorous error analysis is given of both truncation and rounding errors in Miller's algorithm for three-term scalar recursions and 2×2 matrix-vector recursions. The error bounds are shown to be very realistic and this will be supported by examples. The results are generalized to recursions of higher order.
Similar content being viewed by others
References
Gautschi, W.: Computational aspects of three term recurrence relations. SIAM Review9, 24–82 (1967)
Gautschi, W.: Zur Numerik rekurrenter Relationen. Computing9, 107–126 (1972)
Mattheij, R. M. M.: Accurate estimates of solutions of second order recursions. Lin. Algebra and its Applications12, 29–54 (1975)
Miller, J. C. P.: Bessel functions, Part II in Mathematical Tables, vol. X. British Assoc. Advancement Sci., Cambridge-New York: Cambridge University Press 1952
Oliver, J.: Relative error propagation in the recursive solution of linear recurrence relations. Num. Math.9, 323–340 (1967)
Olver, F. W. J.: Error analysis of Miller's recurrence algorithm. Math. Comp.18, 65–74 (1964)
Olver, F. W. J.: Numerical solution of second order linear difference equations. J. Res. NBS71B, 111–129 (1967)
Olver, F. W. J.: Bounds for the solutions of second order linear difference equations. J. Res. NBS71B, 161–166 (1967)
Scraton, R. E.: A modification of Miller's recurrence algorithm. BIT12, 242–251 (1972)
van der Sluis, A.: Condition, equilibration and pivoting in linear algebraic systems. Num. Math.15, 74–86 (1970)
van der Sluis, A.: Estimating the solutions of slowly varying recursions. To appear in SIAM J. Math. Anal.7 (1976)
Tait, R.: Error analysis of recurrence equations. Math. Comp.21, 629–638 (1967)
Wilkinson, J. H.: Rounding errors in algebraic processes. London: Her Majesty's stationery office (1963)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mattheij, R.M.M., van der Sluis, A. Error estimates for Miller's algorithm. Numer. Math. 26, 61–78 (1976). https://doi.org/10.1007/BF01396566
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01396566