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#Mathematica taylor expansion code
#Mathematica taylor expansion series

In short, the two-point expansion increases the accuracy at point at the cost of sacrificing precision at point, and this could increase the accuracy of the expansion in the interval when compared to the standard Taylor expansion as shown in the figure above. On the other hand, the approximation of a polynomial of degree will be pretty good for both the one-point or the two-point Taylor expansions of degree, and, of course, will be exact for any attempt to approximate it with any m-point Taylor expansion (see below) of degree higher than, as it can be demonstrated with the polynomial division. A 9-degree polynomial will never be a good fit for this interval because it can have at most 8 turning points, no matter if this approximation was obtained using the one-point or the two-point Taylor expansion. For example, let’s say that you want to approximate the sine function in the interval, which contains 9 turning points.

Nevertheless, the two-point expansion is not a better or worse alternative than the single point expansion for the curve as a whole. The latter seems to converge faster than the former in the interval shown. Sine function (blue) in the interval being approximated by a one-point (yellow) and a two-point (green) Taylor Expansion (TE) at points and respectively. In the figure below I show a comparison between the one-point and the two-point Taylor expansions of the sine function. The conditions above, written as a system of equations of derivatives, can be used for any analytical function, not only polynomials.

#Mathematica taylor expansion series

From this paper you would notice that it is more convenient to write the coefficient of each term of the two point Taylor series in the form. Ī quick Google search about this topic results in a publication by NASA about the two-point Taylor expansion followed by more recent papers.

It has two degrees of freedom to solve for the two equations given above for.

It already satisfies the equations above for and.

This is a polynomial of order at most, as required by the division algorithm.

For we already saw that is a line that crosses at and and, therefore, the best approximation to 0th order to at these points. Using induction, we can show that has the shape described above. Notice from the division equation above that is accurate to order and fulfills the equations Is the curve that fits the best at points and to th order at each of these points. What happens when we combine these two ideas? Let’s take a look at the remainder of divided by for increasing values of and then generalize this idea to obtain the two-point Taylor expansion of analytical functions!Ĭonsider the remainder of for. We saw two ideas in the previous section: The first one is a way to obtain the best curve of only degree 1 (line) that crosses two points in a polynomial function, and the second is a way to obtain the best curve (polynomial) of degree that approximates our function at only one point.

The result follows from the uniqueness of.

Taking we know that is at most a -degree polynomial by definition of the division algorithm and observe that the conditions for all are satisfied.From we observe that is a 1-degree polynomial (a line) that fulfills, and is thus fully determined by, and.Here I computed the remainder of divided by to get the line tangent at (green). Also, you don’t need calculus to find the line tangent to the curve of a polynomial function. In this figure I divided (blue) by to obtain a remainder equals to the line through points and (yellow).

#Mathematica taylor expansion code

The formula may be obtained with the following code that attempts to do simple replacements sequentially ( var below is a formal wrapper that formally replaces a matrix with a real valued variable) : Fold[ReplaceAll, Then if we define an identity tensor for the dot product: Dot ^:= Dot