readme, fixup

fdb.lua

@@ -3,7 +3,7 @@
 local ORDER = 100
 
 local binCoefs = {}
-
+--memoized binonmial coefficient
 local function Choose( n, k )
   if k < 1 then return 1 end
   if n < 1 then return 0 end

main.lua

@@ -72,17 +72,24 @@ local function PlotFunction( f, color )
     love.graphics.translate( - plot.x, - plot.y )
     love.graphics.setLineWidth( 0.003 * (plot.X - plot.x) )
     
+    love.graphics.setColor( 0.5, 0.5, 0.5, 0.3 )
+    love.graphics.line( -10, -10, 10, 10 )
+    love.graphics.line( -10, 0, 10, 0 )
+    love.graphics.line( 0, -10, 0, 10 )
+
     if color then love.graphics.setColor( color ) end
     love.graphics.line( plot.inverse and inverse or points )
+    
     love.graphics.circle( "fill", a, a, 0.003 * (plot.X - plot.x) )
     love.graphics.pop()
 
+
+    love.graphics.setFont( love.graphics.getFont( 48 ) )
     love.graphics.print( a )
     love.graphics.print( plot.fdbOrder, 0, 15 )
     love.graphics.print( f(a), 0, 30 )
     love.graphics.print( f(-1.0), 0, 45 )
     love.graphics.setColor( 1,1,1,1 )
-    love.graphics.setFont( love.graphics.getFont( 48 ) )
   end
 
 end
@@ -92,7 +99,7 @@ love.wheelmoved = function( x, y )
 end
 
 love.mousepressed = function(x, y, button)
-  --plot.inverse = not( plot.inverse )
+  plot.inverse = not( plot.inverse )
   plot.fdbOrder = plot.fdbOrder + (( button == 1 ) and 1 or -1 )
   return love.keypressed()
 end 
@@ -116,18 +123,15 @@ love.keypressed = function( key, code )
   love.draw = nil
 
   local f, df, fn = Poly( FaaDiBruno( a, plot.fdbOrder ) )
-  --PlotFunction( f, {1, 0, 0, 0.3} )  --Function in red.
+  PlotFunction( f, {1, 0, 0, 0.7} )  --Function in red.
-  --PlotFunction( df, {0, 1, 0, 0.3} ) --First derivative in green.
+  PlotFunction( df, {0, 1, 0, 0.7} ) --First derivative in green.
-  --local lowF, lowDF = Poly( FaaDiBruno( a, 15 ) )
-  --PlotFunction( lowF, {1, 0, 0, 0.3} )  --Function in red.
-  --PlotFunction( lowDF, {0, 1, 0, 0.3} ) --First derivative in green.
   
   --PlotFunction( function(x) return df( f ( x ) ) end, {0, 0, 1, 0.3})
   --PlotFunction( function(x) return f( df ( x ) ) end, {1, 1, 1, 0.3})
   --PlotFunction( function(x) return x end, {1, 1, 1, 0.3})
   local af, an = Poly( Ansatz.coefs )
-  PlotFunction( af, {1,1,1, 0.3} )
+  --PlotFunction( af, {1,1,1, 0.3} )
-  PlotFunction( an, {1,1,1, 0.3} )
+  --PlotFunction( an, {1,1,1, 0.3} )
   PlotFunction( Ansatz.dpo, {1, 1, 1, 0.3} )
   PlotFunction( Ansatz.pos, {1,1,1,0.3} )
 end

readme.md

@@ -0,0 +1,23 @@
+Bespoke script for calculating and plotting some series solutions
+of the functional equation f'( f( x ) ) = x.
+
+Idea: suppose f has fixed point p,
+apply the chain rule to functional equation
+get values of f's derivatives at p
+get truncated taylor series expansion of f at p,
+plot series to get some idea about values, convergence.
+
+We use Faa Di Bruno's formula for iterated derivatives, in terms of Bell numbers.
+We also plot the "ansatz" solution, i.e. the solution of the form x -> a * x ^ b where a and b are positive.
+
+Uses LOVE as a dependency for its plotting: https://love2d.org
+Solution is plotted in red, first derivative is plotted in green.
+
+Keys:
+Q - Increase p
+E - Decrease p
+WASD - Translate view
+Z - Zoom in
+C - Zoom out
+F - Increase order of series expansion
+R - Decrease order