Multiple systems with the same matrix

It is often the case that we need to solve consecutively several linear systems that share the same coefficient matrix

We can save substantial amounts of CPU time by computing the decomposition of the matrix once, and then re-using it to obtain different solutions with different right hand sides.

In the standard solution, we applied to the right hand vector all the permutations and row operations applied to , then solved the upper tringular system

To re-use the decomposition, we need to apply the permutations and row operations (stored in and respectively) to the new right hand vector - that is, compute , then solve the upper triangular system

Now, all permutations are stored in and all row operations in , hence we can use this information directly. In matrix language,

For example, the system

has the same coefficient matrix as (), but the right hand side vector is now instead of .

Therefore, we use the decomposition () as follows.

First, interchange rows (elements) 1 and in the right hand side vector

Then, multiply first row by and subtract it from the second row, and by and subtract it from the third row, to obtain

Note that and are now stored in and respectively.

Next, we need to interchange rows 2 and - nothing to do. Multiply second row by and subtract it from the third row

So far we have applied the permutations () and row transformations () to the new RHS vector . It remains to solve the system

or

It is easy to see that .