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The package provides some functions for finding polynomial analytical and numerical approximations of functions and solutions of linear differential equations (LDEs). The main function (ApproxSol) allows one to find polynomial approximation for solutions of an LDE with polynomial coefficients on a given interval. The main algorithm applied is so called a-method (see [1]) which has the following special properties. The method is numerically analytical one. It means that LDE coefficients, boundary or initial conditions and parameters of ApproxSol function (except degree of approximation polynomial) can be either symbolical or numerical expressions (see symbolic examples of the package). The method gives the polynomial approximation which is very close to the uniform one. It means that the result is such a polynomial that minimizes the maximum value of the absolute error between the solution and a polynomial of the given degree over the interval under consideration (see numerical examples of the package). The method is fast saturated. It means that the required precision can be reached very fast by increasing the degree of approximation polynomial [2]. The method can be applied to get approximation of either initial value problems (IVP) or boundary value problems (BVP) (see [3]). To receive normalized results of the solution of a homogeneous BVP one can use the package function Normalizes (see BVP examples of the package). The method can be applied to get approximation of LDE's with regular singular points (see RSP examples of the package). The method can be applied to get approximation of IVP or BVP even if the LDE has non-polynomial coefficients. To do this one can use the package function ToRatCoeffs (which gives rational approximation of the LDE coefficients) and then apply ApproxSol function to the result. The so called modified a-method is also applied in the package (see modified a-method examples). Exact mode of calculation gives the result with no real numbers involved. Moreover any intermediate calculations are performed using exact arithmetic. This mode is available only if the LDE, boundary or initial conditions and boundary points of the interval has no real numbers involved. It should be pointed out that this mode (as a rule) leads to time and resources consuming computation and gives more bulky results. Nevertheless in some cases the Exact mode of calculation is absolutely necessary to get right results. In a case of homogeneous LDE and BVP the appropriate eigenvalue variable must be specified by option Eigenvalue→variable (e.g. Eigenvalue→λ) to get solutions for different eigen values. Nevertheless in some simple cases the eigen value variable can be determined automatically by the algorithm (see example below). The approximation problem can be treated as initial value problem (IVP) or boundary value problem (BVP) with regular singular points (RSP) or without them. The treatment is as follows. If no condition (Cond) presented the problem treated as IVP and the initial point of the approximation is the middle point of the giving interval: x0=(a+b)/2. If condition (Cond) is one point condition (for instance cond is y[x0]=0, y'[x0]=1, ... with one point x0) thenthe problem treated as IVP and x0 must be the middle point or the left point of the approximation interval, thusx0 = (a+b)/2 or x0 = a. If condition (Cond) is two points condition (for instance cond is y[x0]=0, y'[x1]=1, ... with two points x0 and x1) then the problem is treated as BVP and it must be x0 = a and x1 = b or x0 = b and x1 = a. If an LDE has an RSP then it must be the left point of the interval of approximation.
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