From: Michael Orlitzky Date: Sun, 1 May 2011 06:51:28 +0000 (-0400) Subject: Add the beginning of the final paper. X-Git-Tag: 0.0.1~353 X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=da1663eaf5574c0acb9944e3c028fc143669dd2b;p=spline3.git Add the beginning of the final paper. --- diff --git a/doc/spline3.lyx b/doc/spline3.lyx new file mode 100644 index 0000000..2536cef --- /dev/null +++ b/doc/spline3.lyx @@ -0,0 +1,1014 @@ +#LyX 1.6.8 created this file. For more info see http://www.lyx.org/ +\lyxformat 345 +\begin_document +\begin_header +\textclass amsart +\use_default_options false +\begin_modules +theorems-ams +eqs-within-sections +figs-within-sections +\end_modules +\language english +\inputencoding auto +\font_roman ae +\font_sans default +\font_typewriter default +\font_default_family default +\font_sc false +\font_osf false +\font_sf_scale 100 +\font_tt_scale 100 + +\graphics default +\paperfontsize default +\spacing single +\use_hyperref false +\papersize letterpaper +\use_geometry false +\use_amsmath 1 +\use_esint 1 +\cite_engine basic +\use_bibtopic false +\paperorientation portrait +\secnumdepth 3 +\tocdepth 3 +\paragraph_separation skip +\defskip bigskip +\quotes_language english +\papercolumns 1 +\papersides 1 +\paperpagestyle default +\tracking_changes false +\output_changes false +\author "" +\author "" +\end_header + +\begin_body + +\begin_layout Title +Spline3 +\end_layout + +\begin_layout Author +Michael Orlitzky +\end_layout + +\begin_layout Example* +First, will will define our grid +\begin_inset Formula $G$ +\end_inset + + to be over a +\begin_inset Formula $3\times3\times3$ +\end_inset + + cube having grid size +\begin_inset Formula $h=1$ +\end_inset + +. +\end_layout + +\begin_layout Example* +To test the reproduction of trilinears, we will take the function (per Sorokina + and Zeilfelder, p. + 88), +\end_layout + +\begin_layout Example* +\begin_inset Formula \[ +f\left(x,y,z\right)=1+x+xy+xyz\] + +\end_inset + + +\end_layout + +\begin_layout Example* +and compute its value at the 27 points, +\begin_inset Formula $\left(0,0,0\right),\left(0,0,1\right),\dots\left(2,2,2\right)$ +\end_inset + +. +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: x,y,z = var('x,y,z') +\end_layout + +\begin_layout Plain Layout + +sage: f = 1 + x + x*y + x*y*z +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=0, z=0) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=0, z=0) +\end_layout + +\begin_layout Plain Layout + +2 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=0, z=0) +\end_layout + +\begin_layout Plain Layout + +3 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=1, z=0) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=1, z=0) +\end_layout + +\begin_layout Plain Layout + +3 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=1, z=0) +\end_layout + +\begin_layout Plain Layout + +5 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=2, z=0) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=2, z=0) +\end_layout + +\begin_layout Plain Layout + +4 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=2, z=0) +\end_layout + +\begin_layout Plain Layout + +7 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=0, z=1) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=0, z=1) +\end_layout + +\begin_layout Plain Layout + +2 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=0, z=1) +\end_layout + +\begin_layout Plain Layout + +3 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=1, z=1) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=1, z=1) +\end_layout + +\begin_layout Plain Layout + +4 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=1, z=1) +\end_layout + +\begin_layout Plain Layout + +7 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=2, z=1) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=2, z=1) +\end_layout + +\begin_layout Plain Layout + +6 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=2, z=1) +\end_layout + +\begin_layout Plain Layout + +11 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=0, z=2) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=0, z=2) +\end_layout + +\begin_layout Plain Layout + +2 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=0, z=2) +\end_layout + +\begin_layout Plain Layout + +3 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=1, z=2) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=1, z=2) +\end_layout + +\begin_layout Plain Layout + +5 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=1, z=2) +\end_layout + +\begin_layout Plain Layout + +9 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=0, y=2, z=2) +\end_layout + +\begin_layout Plain Layout + +1 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=1, y=2, z=2) +\end_layout + +\begin_layout Plain Layout + +8 +\end_layout + +\begin_layout Plain Layout + +sage: f(x=2, y=2, z=2) +\end_layout + +\begin_layout Plain Layout + +15 +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +We are most interested in the +\begin_inset Quotes eld +\end_inset + +interior +\begin_inset Quotes erd +\end_inset + + cube centered at +\begin_inset Formula $\left(1,1,1\right)$ +\end_inset + + with datum +\begin_inset Formula $4$ +\end_inset + +. + We can enter the data above into a list, +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: g = [[[ f(x=a, y=b, z=c) for a in range(0,3) ] for b in range(0,3) + ] for c in range(0,3) ] +\end_layout + +\begin_layout Plain Layout + +sage: g +\end_layout + +\begin_layout Plain Layout + +[[[1, 2, 3], [1, 3, 5], [1, 4, 7]], +\end_layout + +\begin_layout Plain Layout + + [[1, 2, 3], [1, 4, 7], [1, 6, 11]], +\end_layout + +\begin_layout Plain Layout + + [[1, 2, 3], [1, 5, 9], [1, 8, 15]]] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +although the list will be indexed by +\begin_inset Formula $\left(z,y,x\right)$ +\end_inset + + so we define a function to access it by +\begin_inset Formula $\left(x,y,z\right)$ +\end_inset + +, +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: def grid(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return g[z][y][x] +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +and define directional functions according to Sorokina and Zeilfelder, p. + 81. +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: def I(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x,y,z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def F(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def B(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def L(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y-1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def R(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y+1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def T(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def D(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FL(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y-1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FR(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y+1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BL(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y-1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BR(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y+1, z) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def LD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y-1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def LT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y-1, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def RD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y+1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def RT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x, y+1, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FLD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y-1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FLT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y-1, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FRD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y+1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def FRT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x-1, y+1, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BLD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y-1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BLT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y-1, z+1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BRD(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y+1, z-1) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: def BRT(x,y,z): +\end_layout + +\begin_layout Plain Layout + +....: return grid(x+1, y+1, z+1) +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +Next, we define the coefficients for the cube centered on +\begin_inset Formula $\left(1,1,1\right)$ +\end_inset + + based on these directional functions. +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: c0030 = (1/8)*( I(1,1,1) + F(1,1,1) + L(1,1,1) + T(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + LT(1,1,1) + FL(1,1,1) + FT(1,1,1) + FLT(1,1,1) ) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0003 = (1/8)*( I(1,1,1) + F(1,1,1) + R(1,1,1) + T(1,1,1) + +\end_layout + +\begin_layout Plain Layout + + RT(1,1,1) + FR(1,1,1) + FT(1,1,1) + FRT(1,1,1) ) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0021 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/24)*(L(1,1,1) + FL(1,1,1) + LT(1,1,1) + FLT(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0012 = (5/24)*(I(1,1,1) + F(1,1,1) + T(1,1,1) + FT(1,1,1)) + +\end_layout + +\begin_layout Plain Layout + + (1/24)*(R(1,1,1) + FR(1,1,1) + RT(1,1,1) + FRT(1,1,1)) +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Example* +We can see what the constant values are now: +\end_layout + +\begin_layout Example* +\begin_inset listings +inline false +status open + +\begin_layout Plain Layout + +sage: c0030 +\end_layout + +\begin_layout Plain Layout + +17/8 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0003 +\end_layout + +\begin_layout Plain Layout + +27/8 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0021 +\end_layout + +\begin_layout Plain Layout + +61/24 +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\begin_layout Plain Layout + +sage: c0012 +\end_layout + +\begin_layout Plain Layout + +71/24 +\end_layout + +\end_inset + + +\end_layout + +\end_body +\end_document