On computing the inverse of a sparse matrix

Abstract

SUMMARYThe Gauss-Jordan inversion algorithm requires O(n3) arithmetic operations. In some practical applications, like state estimation or short-circuit calculation in power systems, the given matrix is sparse and only part of the inverse is needed. Most frequently in the diagonal dominant case this is the diagonal. There are two ways to exploit sparsity to determine elements of the inverse:1. Columnwise inversion via the solution of sparse linear systems with columns of the unit matrix as Application of the algorithm of Takahashi et aL6The latter algorithm arises very naturally from multiplying the left-and right-hand factors of the Zollenkopf bifactorization in reverse order. We indicate how computing time can be gained in the important symmetric diagonal dominant case if only part of the diagonal is needed and compare computing times of the method of Takahasi et a!.' with several variants of columnwise inversion. Whereas most of the theory holds for general matrices, experiments are performed on the symmetric diagonal dominant case. For band matrices the operation count is O(n) both for computing a column of the inverse by columnwise inversion and the diagonal by the algorithm of Takahashi et ale6 right-hand sides. THE SPARISITY OF THE INVERSE AND THE 'SPARSE INVERSE'The simplest algorithm to compute the inverse of a matrix A of order n is Gauss-Jordan elimination (see, for example, Reference 2). It can be interpreted as a sequence of n exchange steps.' Each of them subtracts a multiple of the dyadic product of the respective pivot column and pivot row from the whole matrix, except from that pivot column and pivot row. A step in triangular decomposition performs this subtraction in part of the matrix only. Thus, in case of sparse matrices considered in the following the Gauss-Jordan algorithm destroys more zero elements than does a triangular decomposition. In fact the inverse of a band matrix is generally completely filled, whereas the triangular factors show absolutely no fill-in if the pivots are taken from the diagonal in the sequence from top left to bottom right, as is possible with dominant diagonals. In the case of random sparse matrices the probability of being nonzero increases from step to step with Gauss-Jordan elimination uniformly for all elements except pivots, with triangular decomposition on the other hand this probability increases for a decreasing number of elements only.4Large fill-in has an effect on computing costs. Even for a band matrix the Gauss-Jordan method requires O(n 3, arithmetic operations; inversion via the solution of sparse linear systems requires O(n *) operations, whereas triangular decomposition just needs O(n ).

Investigation of the adaptability of transient stability assessment methods to real-time operation

“…In order to use the algorithm proposed by Takahishi et al [47], the matrix has to be transformed into a banded matrix by applying the reverse Cuthill-McKee algorithm, which complexity was shown in [48] to be proportional to the number of edges and, hence, proportional to ( ). The complexity of calculating the whole inverse of a banded matrix can be approximated by ( 2 ) [49], where is the bandwidth. It can be argued that in the case of power system networks the bandwidth of the ordered admittance matrix is small compared to the number of nodes and in large system smaller than the number of machines.…”

“…By performing a LU-factorization and through forward and backward substitution the process can be considerably accelerated. The LU-factorization is, however, the operation with the highest computational cost and can be approximated by ( 2 ) [49]. After the values of the state variables at fault clearance are determined, the transient energy of the system at that point in time can be determined.…”

On Computing Inverse Entries of a Sparse Matrix in an Out-of-Core Environment

“…At every step, an entry of the inverse is computed using the factors L and U and the already computed entries of the inverse. This approach is later extended [23] for a set of entries of the inverse, rather than the whole set in the pattern of (L + U )…”

Differentiation of matrix functionals using triangular factorization

“…The factors L −1 and U −1 can be written as products of elementary row and column operations, and Niessner and Reichert [18] have used this to propose an algorithm for the efficient calculation of some of the elements of A −1 .…”

“…In fact, (3.9) is essentially the same as equations (1) and (2) in Erisman and Tinney [5]. A more detailed discussion of the various options that can be used to solve the Erisman and Tinney equations was subsequently given by Niessner and Reichert [18].…”