One Step Inverse Updating

One Step Inverse Updating - File: One Step Inverse Updating...

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File: One Step Inverse Updating Consider that we have an mm × matrix 1 B that we have computed the inverse (here we are assuming that it exists) 1 1 B . We could have done this inverse by the product form method: 1 11 2 1 B MM = " Now assume that we want to replace column i of 1 B with the column vector a . The reason for this exchange in a simplex type algorithm is to change basic variables. So let’s denote the two matrices by 12 1 2 ,,,,, ,,, ,, im m B aa a a B a a ⎡⎤ = ⎣⎦ = "" Now we also have 1 1 B such that 1 1 B BBa a a a e e e e B a a a e e e α −− == The if we form an M matrix such that mi m Mee e ee e e ⎤⎡ = ⎦⎣ , then 21 B MB = since 1 m m MB B MB a a a a e e e = Thus, we can efficiently compute the new inverse from the old inverse by one step of the product form of the inverse process. The M matrix is created as and identity matrix I with the i th column replaced by 1 2 1 / / ,, , 1/ / i i i M e e αα γγ ⎢⎥ =→ = #
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One Step Inverse Updating - File: One Step Inverse Updating...

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