måndag 25 november 2013

Constructive and Symbolic Calculus

The basic idea of BodyandSoul is to support a mathematical symbol like $\sqrt{2}$ and $e=\exp(1)$ by a constructive numerical algorithm through which the decimal expansion of $\sqrt{2}=1.421356237...$ and $e=2.718281828...$ can be computed to arbitrary precision.

The symbol $\sqrt{2}$ thus is assigned to the positive root of the algebraic equation $x^2=2$ computable by e.g. Newton's method and $\exp(t)$ for $t > 0$ is assigned to the function $u(t)$ which by time stepping solves the differential equation $Du(t) = u(t)$ for $t > 0$ such that $u(0)=1$, where
  • $Du(t)=\lim_{\Delta t\rightarrow 0}\frac{u(t+\Delta t)-u(t)}{\Delta t}$,
is the derivative of $u(t)$ defined as a limit in infinite precision. 

The symbols can sometimes be used in symbolic computation as symbol manipulation according to given rules, without using the actual numerical values, only the defining algebraic/differential equations.

For example, one can compute $\sqrt{2}\times\sqrt{2}\times\sqrt{2}\times\sqrt{2}=2\times 2 = 4$ knowing that $\sqrt{2}\times\sqrt{2}=2$ without knowing that $\sqrt{2}\approx 1.414$. 

More importantly, one can compute derivatives symbolically by using defining differential equations and the chain rule, for example:
  • $D\exp(2t)= \exp(2t)\times 2$ 
using that $Dexp(s) = exp(s)$ and $D2t = 2$ with $s=2t$. Doing so we circumvent the ill-posed unstable aspect of numerical computation of derivatives. This is done in symbolic computation software as Maple.

However, there is no corresponding general symbolic method for solving differential equations as different forms of integration, which thus in general is performed by constructive numerical methods. This works because integration is a well-posed stable process which can be performed in finite precision.  

Inga kommentarer:

Skicka en kommentar