Correlation between matrix multipilcation and other computational problems. Bit-time, bit-space, stability, and condition
In this part of the book we will extend our asymptotically fast algorithms for MM to the algorithms for fast computation for some related problems. We will also estimate the amount of storage-space, the numbers of bit-operations involved in our algorithms, and the condition of those algorithms that characterizes their stability. In particular in Section 18 (whose results are isolated in this book) we will consider some combinatorial computational problems, that is, Boolean matrix multiplication (hereafter referred to as Boolean MM) and the all pair shortest distance problem for a digraph (hereafter referred to as the APSD-problem). In Sections 19–21 and 26–30 we will consider the solution of a system of linear equations, matrix inversion, evaluation of the determinant of a matrix (hereafter referred to as SLE or SLE(n), MI or MI(n), Det or Det(n), respectively; n indicates the dimension of the nXn input matrix) and some other computational problems of linear algebra. We will reduce the solution of all of those combinatorial and algebraic problems to MM, defining the algorithms for those problems that involve O(nω+ɛ) arithmetical operations where ω is the exponent of MM and ɛ is arbitrary positive number, see Sections 18–21. Also we will estimate the bit-time of those algorithms, see Sections 18, 26–30, and of MM itself, see Sections 23 and 24. (The bit-time is defined as the number of bit-operations required for the approximate evaluation of the solution with a prescribed precision E in a given domain D; bit-time depends on E and D.) Estimating the bit-complexity is sometimes quite tedious but, rather unexpectedly, we finally show that the arithmetical complexity classification of the above problems differs from their bit-complexity classification. Specifically, for the customary choices of E and D the bit-time of our algorithms for Boolean MM, SLE, and MI remains at worst of the same order as in the case of MM. Furthermore, see Section 30, SLE can be solved faster than MM (and, in fact, in a near optimum way) in terms of the bit-time involved in the cases where E sufficiently rapidly converges to 0. On the other hand, our algorithm that reduces the APSD-problem for a digraph with n-vertices to MM(n) involves at least n times more bit-operations than it is required for nXn MM and so does also any algorithm for Det(n). In Section 22 we will show that fast MM is compatible with the minimization of the storage space involved. (That result can be extended to the case of the related computational problems.) In Section 25 we will study the condition of our algebraic algorithms and will see some correlation between the condition and the quoitient: bit-time divided by the number of arithmetical operations.
KeywordsArithmetical Operation Storage Space Triangular Matrix Computational Problem Output Error
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