Coding Period Week 2

System of Inequalities

• Right now solve uses reduce_inequalities to solve system of equations. But it is not designed for more than one variable.

• ex: system = [2*x - 3*y <= 12, x+5*y <=20, x>0] symbols = [x,y]) you will get : NotImplementedError: inequality has more than one symbol of interest We know answer is [0 < x <120/13 and 2(x-6)/3 <= y <= (20-x)/5] [x = 120/13 and y = 28/13]

• But SymPy already have project on this Cylindrical algebraic decomposition to Provide an interface for solving systems of polynomial inequalities.

• If we want to feature like old solve then just need to pass system of inequalities having one variable to solve for one variable. After checking this use reduce_inequalitie(system, symbol) it will return solution. It is easy to implement.

General solution for Trigonometric equations

• PR #11188

• Previous work in this problem : #10898, #10713, #10733

Problems in old method :

• _solve_trig changes the trig equation in exp form (it’s fine). But then fraction and solving equation for it’s parts makes more number of exp. If we have more number of exp then we get more number of imageset, Since we _invert for each exp factors.

• It retuns ConditionSet when it can’t solve, but its expression is in ` exp form mostly with I` and in complicated form.

• some times _solve_as_poly can solve Trig equations, but it is not using it.

New implemention :

• I added reduce_imageset in solveset.py to reduce the number of union returned by _solve_trig method. As Harsh said it is no specific to solveset, so I moved the method to sets/sets.py. I added the doctests and test-cases in test_sets.py.

• Now solving the exp form directly using solveset. This makes the less number of ImageSet in many cases. But to solve equation having tan in it, will be complicated, for that I changed the solve_rational method, so that it can handle exp with I. in denominator.

• Now using _solve_as_poly when solveset can’t solve it using exp form. If this also can’t solve then retuns the ConditionSet having simple trig functions, which is understandable.

reduce_imageset :

• reduce_imageset take the ImageSet or Union of ImageSet and returns the minimum number of ImageSet.

• First we extract the Lambda expression from the each ImageSet and take principle values. Separate the positive and negative principle values and sort them.

• +ve and -ve values are passed into interpolate method defined in polys/polyfuncs.py to get the general function in terms of Dummy n.

• If only one Imageset or value in +ve or -ve value list then return that as it is. Here we need to note that Dummy n defined here is different then original lambda expression, so need to return original imageset. So I am storing original ImageSet in dict with its principle vales.

• If more than one +ve/-ve values are there then interpolate expression will be ImageSet expression and return.

• Here I separe the -ve and +ve values because interpolate returns complicated expressions if we both types of values.

• Also I sorted the +ve and -ve list, to get simplified expression from interpolate.

Problems :

• In this PR solveset_real(tan(x), x) returns imageset(Lambda(n, pi*(n - 1)), S.Integers) but I want imageset(Lambda(n, n*pi), S.Integers).

Issues :

• Meanwhile I found a issue in solveset. issues/11194. 2*sin(x) - 2*sqrt(3)*cos(x) - sqrt(3)*tan(x) +3 = 0 can be easily solved if we factorize it correctly. But I haven’t found a good way to get its factor. I tried factor, expand(Trig + True) , expand_trig, rewrite(sin).

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