- #TAYLOR SERIES MATHEMATICA DRIVER#
- #TAYLOR SERIES MATHEMATICA SOFTWARE#
- #TAYLOR SERIES MATHEMATICA CODE#
- #TAYLOR SERIES MATHEMATICA FREE#
The software has been done to be extremely easy to use. The precompiler has been written in Mathematica and Sage (which includes it by default since version 6.4).
#TAYLOR SERIES MATHEMATICA CODE#
In this paper we present the symbolic methods, implemented in a computer algebra system (CAS), used to write, automatically, the code based on the automatic differentiation processes that integrates a particular differential system by means of the Taylor method.
#TAYLOR SERIES MATHEMATICA DRIVER#
The kernel of this software consists of a C library that permits to compute up to any precision level (by using multiple precision libraries for high precision when needed) the solution of an ordinary differential system from a C driver program containing the equations of the ODE.
#TAYLOR SERIES MATHEMATICA FREE#
TIDES is a free software based on the Taylor series method that uses an optimized variable-stepsize variable-order formulation.
One method that can solve most of these problems is the Taylor series method. , there's no way for you to know if the original function was sin(x), sin(x)+g(x), or something else.In the last few years, the requirements in the numerical solution of ordinary differential equations in physics and in dynamical systems have pointed to new kind of methods capable to maintain geometric properties of the equations, or looking for high-precision, or solving variational equations. Similarly, if I give you the Taylor series x-x 3/6 + x 5/120. if I give you the Taylor series 0+0x+0x 2+., there's no way for you to determine whether the original function was f(x)=0, or f(x)=g(x), or f(x)=5g(x), or one of infinitely many other possibilities. (It's not immediately obvious that such a function exists, but this is proven in undergraduate analysis.) Obviously every derivative of g at 0 is 0, so the Taylor series at 0 for g is 0+0x+0x 2+. For example, let g(x) be any smooth function on R which is not the zero function, but which is identically 0 on a neighborhood of 0. The answer to question 1 is no, because there are infinitely many smooth functions with the same Taylor series. The way you phrased this question sounds to me like the first question, but most people here seem to be answering the second question. Given a Taylor series, can you find a closed-form expression for the function it defines? Given the Taylor series for f(x), can you determine f(x)?
There are two different questions you might want to ask here: But if your series is something familiar enough, you might be able to do it. You need to be clever to come up with some differential equation that will fit the terms, and most of the time no amount of cleverness will get you such an equation. Obviously, this kind of thing is by no means general. That's an easy differential equation to solve: just integrate both sides. This is a geometric series, with sum 1/(1 + x) (obviously only when |x| < 1). Take a derivative, and you get f'(x) = 1 - x + x 2 - x 3 = ∑ (-x) n (from n = 0). That's enough to get f(x) = sin(x) from the differential equation.į(x) = x - x 2/2 + x 3/3. You can get boundary conditions from plugging in values into f(x). Now you can solve that differential equation to get back your original function. Look familiar? What we get is that f''(x) = -f(x). Change the letter from m to n and it's -(-1) nx 2n+1/(2n+1)!. However, if we let m = n - 1, then m starts from 0, and we get -(-1) mx 2m+1/(2m+1)!. So we might as well start counting from n = 1. This is all from n = 0 to ∞, but I had to cancel out 2n in the second time I took a derivative which I can't do for n = 0 the n = 0 term is actually 0 because it's the derivative of a constant.
Not in general, but you might be able to pull off some trickery using differential equations.
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