Consider the lower triangular matrix A:

Divide the COLUMNS with the diagonal elements in matrix A:

Which gives us matrix B:

Now replace the all the ones on the main diagonal with zeros:

Then calculate matrix powers as follows: Which is exactly as the binomial series: except that here it is applied to a triangular matrix and the result is a new triangular matrix D. Notice that is the identity matrix.

And then finally divide the ROWS in matrix D with the diagonal elements in A:

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What is the rate of convergence for calculating the matrix D above, i.e. – how many powers of C is it necessary to compute before arriving at a reasonable approximation fo D? Similarly, is this a feasible algorithm for inverting large, sparse, triangular matrices? What is it’s computational complexity?

I believe that the rate of convergence is at minimum one correct row per matrix power in the binomial series. Could be faster or slower but I don’t remember for sure. This is probably not the fastest way to invert a matrix, but is a very nice way since I believe that this called a “formal power series” when you take powers of matrices. Gaussian elimination is probably faster. I have no idea what the computational complexity could be.

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