Automated simplification of large symbolic expressions

Abstract

We present a set of algorithms for automated simplification of constants of the form P i αixi with αi rational and xi complex. The included algorithms, called SimplifySums and implemented in Mathematica, remove redundant terms, attempt to make terms and the full expression real, and remove terms using repeated application of the integer relation detection algorithm PSLQ. Also included are facilities for making substitutions according to user-specified identities. We illustrate this toolset by giving some real-world examples of its usage, including one, for instance, where the tool reduced a symbolic expression of approximately 100,000 characters in size enough to enable manual manipulation to one with just four simple terms.

Calcium

“…Algorithm 2 could be improved in many ways, most notably through preprocessing to avoid redundant work and reduce the dimension or improve numerical conditioning of LLL matrices. Some preprocessing strategies are discussed in [1].…”

“…The PSLQ algorithm is often claimed to be superior to LLL for integer relations (see for example [1]), but this is not obviously true with modern floating-point LLL implementations. One benefit of LLL is that we obtain a matrix of all integer relations at once, whereas PSLQ has to be run repeatedly to eliminate relations one by one.…”

“…This paper presents Calcium, 1 a C library for exact computation in R and C. Numbers are represented as elements of fields Q(a 1 , . .…”

Computation and theory of Mordell–Tornheim–Witten sums II

“…We note that even these remaining relations can, as far as we know, with some effort be manually proved using the rigorously proven results given in this paper. Automating this process has proven harder than we anticipated-but equality verification in a computer algebra system is known to be very hard; see [10].…”

Closed Forms: What They Are and Why We Care

“…Indeed, we had cognate results for In a lovely recent paper [52] Stephan Steinerberger has shown that in the limit, as the dimension goes to infinity, A quite recent result is that all box integrals | r | n for integer n and dimensions 1, 2, 3, 4, 5 are hyperclosed, in the sense of our "Sixth Approach" Automating such reductions will require a sophisticated simplification scheme with a very large and extensible knowledge base. With a current research assistant, Alex Kaiser at Berkeley, we have started to design software to refine and automate this process and to run it before submission of any equation-rich paper (see [9]). This semi-automated integrity checking becomes pressing when, as above, verifiable output from a symbolic manipulation can be the length of a Salinger novella.…”