Inverse matrix representation with one triangular array

AbstractWe describe a technique that permits the representation of the inverse of a matrix A with only one additional triangular array. Let L∗A = U, with L lower and U upper triangular arrays of order N. Algorithms are presented that use A and L to compute the matrix-vector products A-1∗b and bT∗A-1 with N2 multiplications and additions. The array L can be computed, with N3/3 multiplications, with a technique that avoids the computation of U. Standard Gaussian elimination simultaneously computes L and U as follows: start with I∗A = A, where I is the identity matrix; perform identical linear combinations of rows on I and on the right hand side array A; gradually transform I into L and A into U. At an intermediate stage, where A has not yet been fully triangularized, we have L′∗A = U′.L′ and Ú represent one of the pairs of arrays present before each linear combination of rows is performed. The key observation is that we only need two elements of Ú to compute each linear combination of rows of Ĺ. Compute them with a scalar product of the appropriate rows of Ĺ and columns of A. Instead of storing the arrays Ú, recompute their few needed elements whenever necessary