Changeset 306 for trunk/doc/pslib/psLibADD.tex

Timestamp:

Mar 25, 2004, 3:28:34 PM (22 years ago)

Author:

Paul Price

Message:

First go at defining polynomials

File:

• trunk/doc/pslib/psLibADD.tex (modified) (1 diff)

trunk/doc/pslib/psLibADD.tex

@@ -434 +434 @@
 quadratic which returns the 50\% value.
 
+
+\section{Polynomials}
+
+We will employ Chebyshev polynomials (NR \S 5.8) to approximate functions:
+\begin{equation}
+f(x) = \Sum_{i=0}^{n} c_i T_i(x)
+\end{equation}
+These have some desirable features:
+\begin{itemize}
+\item They are bounded on $-1 < x < 1$, with the maxima and minima
+over this range being 1 and -1, respectively;
+\item Truncation of the higher-order terms leaves one with the most accurate
+lower-order polynomial representation of the desired function.
+\end{itemize}
+
+The first few Chebyshev polynomials are:
+\begin{equation}
+T_0(x) = 1
+
+T_1(x) = x
+
+T_2(x) = 2x^2 - 1
+
+T_3(x) = 4x^3 - 3x
+
+T_4(x) = 8x^4 - 8x^2 + 1
+\end{equation}
+Chebyshev polynomials follow the recurrence relation:
+\begin{equation}
+T_{n+1} = 2xT_n - T_{n-1}
+\end{equation}
+
+Practically, Chebyshev polynomials should be evaluated using Clenshaw's recurrence
+formula (NR \S 5.5):
+\begin{equation}
+d_j = 2xd_{j+1} - d_{j+2} + c_j
+
+f(x) = x*d_1 - d_2 + 1/2 c_0
+\end{equation}
+
+It shall be the responsibility of the user to convert the domain into the range
+$-1 < x < 1$.
+
+\subsection{Multi-dimensional polynomials}
+
+Multi-dimensional polynomials shall be composed of multiplications of 1D polynomials.
+
+
 \end{document}