Are there any large-scale simulation studies of the statistics of symbolic problem-solution pairs?"size" of an integration problem by the number of leafs in its tree-based representation, and likewise for the problem's anti-derivative.
Let's consider just symbolic integration of some class of functions, for instance the class covered by the Risch algorithm (exponentials, logarithms, trigonometric functions, addition, subtraction, multiplication, and division). Suppose we quantify the
Problems of a given size may have solutions spanning a range of sizes, of course. Thus there is some statistical distribution of the sizes, and thus a statistical relation (perhaps correlation) between problem and solution sizes.must use large-scale computer-algebra systems.
So if we double the size of an integration problem will the size of its solution double? or increase faster than linear? or slower than linear? What about the variance in sizes? Or other statistics?
Naturally there is an infinite number of integration problems of a given size, so any simulation study will have inherent uncertainties. Nevertheless, this seems like an interesting, and potentially fruitful, class of simulation problems, for which we
Anyone interested in working on this?From my experience implementing symbolic integrators, there is little to no correlation between the size of integrands and their optimal antiderivatives. Derivatives of relatively small expressions can be huge. Conversely antiderivatives of relatively
--David G. Stork
On Wednesday, March 1, 2023 at 3:57:46 PM UTC-10, David Stork wrote:the "size" of an integration problem by the number of leafs in its tree-based representation, and likewise for the problem's anti-derivative.
Are there any large-scale simulation studies of the statistics of symbolic problem-solution pairs?
Let's consider just symbolic integration of some class of functions, for instance the class covered by the Risch algorithm (exponentials, logarithms, trigonometric functions, addition, subtraction, multiplication, and division). Suppose we quantify
we must use large-scale computer-algebra systems.Problems of a given size may have solutions spanning a range of sizes, of course. Thus there is some statistical distribution of the sizes, and thus a statistical relation (perhaps correlation) between problem and solution sizes.
So if we double the size of an integration problem will the size of its solution double? or increase faster than linear? or slower than linear? What about the variance in sizes? Or other statistics?
Naturally there is an infinite number of integration problems of a given size, so any simulation study will have inherent uncertainties. Nevertheless, this seems like an interesting, and potentially fruitful, class of simulation problems, for which
small expressions can be huge.Anyone interested in working on this?
--David G. StorkFrom my experience implementing symbolic integrators, there is little to no correlation between the size of integrands and their optimal antiderivatives. Derivatives of relatively small expressions can be huge. Conversely antiderivatives of relatively
Albert
On Wednesday, March 1, 2023 at 7:12:47 PM UTC-8, Albert Rich wrote:the "size" of an integration problem by the number of leafs in its tree-based representation, and likewise for the problem's anti-derivative.
On Wednesday, March 1, 2023 at 3:57:46 PM UTC-10, David Stork wrote:
Are there any large-scale simulation studies of the statistics of symbolic problem-solution pairs?
Let's consider just symbolic integration of some class of functions, for instance the class covered by the Risch algorithm (exponentials, logarithms, trigonometric functions, addition, subtraction, multiplication, and division). Suppose we quantify
we must use large-scale computer-algebra systems.Problems of a given size may have solutions spanning a range of sizes, of course. Thus there is some statistical distribution of the sizes, and thus a statistical relation (perhaps correlation) between problem and solution sizes.
So if we double the size of an integration problem will the size of its solution double? or increase faster than linear? or slower than linear? What about the variance in sizes? Or other statistics?
Naturally there is an infinite number of integration problems of a given size, so any simulation study will have inherent uncertainties. Nevertheless, this seems like an interesting, and potentially fruitful, class of simulation problems, for which
relatively small expressions can be huge.Anyone interested in working on this?
--David G. StorkFrom my experience implementing symbolic integrators, there is little to no correlation between the size of integrands and their optimal antiderivatives. Derivatives of relatively small expressions can be huge. Conversely antiderivatives of
Your example validates my point. The antiderivative of this relatively small integrand is huge. But the antiderivative of many other small integrands are small. Thus no correlation.AlbertAlbert,
My experience over decades suggests instead that there IS (or at least MAY BE) interesting structure in the relation between problems and solutions, and a principled exploratory study might give unexpected results.
Here's one:
Integrate[((c + d Tan[e + f x])^{5/2}(a + b Tan[e + f x] + c Tan[e + f x]^2))/(a + b Tan[ e + f x])^{9/2},x]
has a solution that takes 36 Mbytes to write out!
--David Stork
Are there any large-scale simulation studies of the statistics of symbolic problem-solution pairs?"size" of an integration problem by the number of leafs in its tree-based representation, and likewise for the problem's anti-derivative.
Let's consider just symbolic integration of some class of functions, for instance the class covered by the Risch algorithm (exponentials, logarithms, trigonometric functions, addition, subtraction, multiplication, and division). Suppose we quantify the
Problems of a given size may have solutions spanning a range of sizes, of course. Thus there is some statistical distribution of the sizes, and thus a statistical relation (perhaps correlation) between problem and solution sizes.must use large-scale computer-algebra systems.
So if we double the size of an integration problem will the size of its solution double? or increase faster than linear? or slower than linear? What about the variance in sizes? Or other statistics?
Naturally there is an infinite number of integration problems of a given size, so any simulation study will have inherent uncertainties. Nevertheless, this seems like an interesting, and potentially fruitful, class of simulation problems, for which we
Anyone interested in working on this?
--David G. Stork
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