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## Summary

Introduction

The previous chapter dwelled on the fundamental methods of matrix computations. In this chapter, more specialized methods are considered. The first topic is an alternative approach to solving general systems of equations – full elimination (often the method taught in beginning linear algebra courses), which has some advantages whenever the inverse is required. Next, our goal is reducing the effort in solving equations by exploiting the structure of a matrix. One such structure is bandedness, and the Cholesky factorization of a banded positive definite matrix is then applied to time-series computations, cutting the work from O(n3) to O(n). Next is the Toeplitz structure, also arising in time-series analysis, where the work can be reduced to O(n2) in a more general setting. Sparse matrix methods are designed to exploit unstructured patterns of zeros and so avoid unneeded work. Finally, iterative methods are discussed, beginning with iterative improvement.

Full Elimination with Complete Pivoting

Gaussian elimination creates an upper triangular matrix, column by column, by adding multiples of a row to the rows below it and placing zeros below the diagonal of each column. An alternative is to place zeros throughout that column – with the exception of the pivot position, which could be made equal to one.

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Publisher: Cambridge University Press
Print publication year: 2011

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## References

(1979), “An Algorithm for the Exact Likelihood of a Mixed Autoregressive-Moving Average Process,” Biometrika 66: 59–65.CrossRefGoogle Scholar
and (1969), “The Simplex Method of Linear Programming Using LU Decomposition,” Communications of the ACM, 12: 266–68.CrossRefGoogle Scholar
and (1972), Time Series Analysis: Forecasting and Control. San Francisco: Holden-Day.Google Scholar
(1980), “The Numerical Stability of the Levinson–Durbin Algorithm for Toeplitz Systems of Equations,” SIAM Journal of Scientific and Statistical Computing 1: 303–19.CrossRefGoogle Scholar
(1960), “The Fitting of Time Series Models,” Revue Internationale Institut de Statistique 28: 233–43.CrossRefGoogle Scholar
and (1996), Matrix Computations, 3rd ed. Baltimore: Johns Hopkins University Press.Google Scholar
and (1984), “Comparison of Some Direct Methods for Computing Stationary Distributions of Markov Chains,” SIAM Journal of Scientific and Statistical Computing 5: 453–69.CrossRefGoogle Scholar
and (1989), “Space-Time Modelling with Long-Memory Dependence: Assessing Ireland's Wind Power Resource,” Applied Statistics 38: 1–50.CrossRefGoogle Scholar
(1981), “Fractional Differencing,” Biometrika 68: 165–76.CrossRefGoogle Scholar
(1977), Matrix Computation for Engineers and Scientists. New York: Wiley.Google Scholar