The Exact Solution of Systems of Linear Equations with Polynomial Coefficients
Journal of the ACM (JACM)
On Computing the Exact Determinant of Matrices with Polynomial Entries
Journal of the ACM (JACM)
The Algebraic Solution of Sparse Linear Systems via Minor Expansion
ACM Transactions on Mathematical Software (TOMS)
Analysis of algorithms, a case study: Determinants of polynomials
STOC '73 Proceedings of the fifth annual ACM symposium on Theory of computing
| Hi-index | 0.00 |
An important use of a computer symbolic mathematical system lies in the area of matrix manipulation. This includes determinant [1,5] and linear equations [2,6,8]. Computing the inverse is one of the more complicated matrix operations. A principal difficulty in the exact inversion of a matrix is intermediate expression growth. This is particularly true if the given matrix contains a number of variables. The MACSYMA system [7] has been using an inversion scheme based on Gaussian elimination [6]. When the given matrix is sparse, it is desirable to have a way of taking advantage of the presence of the many zero entries. We used the technique of reduction to block triangular form [3]. The reduction is done by finding the strong components in a directed graph [4] associated with the given sparse matrix. In section two we shall describe how this scheme is employed to compute the exact inverse. A proof will be given in section three to show that, as far as symbolic inversion is concerned, this algorithm gives all the “predictable” zero entries in the inverse. Section four contains timing data comparing this method with MACSYMA's Bareiss-type fraction free elimination scheme.