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Updating the inverse of a matrix leibniz laboratory for radiometric dating

A procedure for implementing this algorithm in existing computer code is presented.This paper presents an approach for updating the inverse of the matrix. Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer.* Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED.IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE.Dear All, I need to solve a matrix equation Ax=b, where the matrix A is a lower triangular matrix and its dimension is very big (could be 10000 by 10000).

Changes in value of elements of a matrix may occur due to variation in operating conditions of the system.

Solving the problem x = A\b is a forward substitution, so fast as hell.

On the order of a matrix*vector multiply in terms of the computational load. Wasting the time to do do an update of an inverse matrix, so that you can then do a matrix multiply is just silly.

This paper suggests certain methodologies to update the given matrix inverse for variation of operating conditions.

The Sherman-Woodbury-Morrison (SWM) formula gives an explicit formula for the inverse perturbation of a matrix in terms of the inverse of the original matrix and the perturbation. We have produced similar results, giving an expression for the inverse of a matrix when the $i$th row and column are removed.


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