Here follows a technique for inverting triangular matrices that was discovered together with Gary W. Adamson. His blog can be found at http://qntmpkt.blogspot.com/
Consider the lower triangular matrix A:
Divide the COLUMNS with the diagonal elements in matrix A:
Which gives us:
Then replace the first element with -1 and the rest of the elements on the main diagonal with zeros. Call this matrix B:
Now by matrix multiplication successively square matrix B. That is: Let C=B*B, D=C*C, E=D*D and so on, then the first column in the last matrix (E) will be a convergent. Let those elements be symbolized by the letter r.
Then we only need to divide the ROWS with the elements in the main diagonal of matrix A, to get the first column of the matrix inverse of A.
To calculate the rest of the columns in the inverse of A, repeat for submatrices:
and so on.
Edit 21.2.2011: This way of viewing the inverse of a triangular matrix seems to find its explanation when looking at the binomial series expression for inverting a triangular matrix. The inverse of triangular matrix as a binomial series
Mats Granvik mats.granvik(AT)abo.fi