An Efficient Algorithm to Compute the Inverse of a Fourth Order Positive Definite Symmetric Matrix

Prakash A. Kulkarni and K. N. Sudharshan

Abstract:	This paper presents an efficient and simple algorithm to compute the inverse of a fourth order positive definite symmetric matrix. In GPS related algorithms (whether iterative or non-iterative), the computations of user position or velocity require determination of the inverse of such 4x4 symmetric matrices. The standard Gaussian elimination technique based on LU decomposition technique does not exploit the advantage of matrix symmetry. The approach described here involves a single step LDLT decomposition [Golub 1996] to obtain a block diagonal matrix with two blocks. The blocks will be of the size 1x1 and 3x3. The inverse of the first block is obvious, where as the inverse of the second block may be computed using the classical formula Adj(G)/ det(G). Pivotation can be performed efficiently, again by exploiting the symmetry. This algorithm requires considerably lesser floating-point operations than the standard algorithms.
Published in:	Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000) September 19 - 22, 2000 Salt Palace Convention Center Salt Lake City, UT
Pages:	1962 - 1967
Cite this article:	Kulkarni, Prakash A., Sudharshan, K. N., "An Efficient Algorithm to Compute the Inverse of a Fourth Order Positive Definite Symmetric Matrix," Proceedings of the 13th International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GPS 2000), Salt Lake City, UT, September 2000, pp. 1962-1967.
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