Large commit, fixed large mistake in calculating Ansatz values, added tests for FDB series plot.

ansatz.lua

@@ -2,20 +2,31 @@
 --assume f(x) = ax ^ b. Then
 --f' o f (x) = ba^b * x ^ ( b ( b-1 ) )
 
-local phi = 1 + math.sqrt( 5 ) / 2
+local phi = 0.5 + math.sqrt( 5 ) / 2
-local ihp = 1 - math.sqrt( 5 ) / 2
+local ihp = 0.5 - math.sqrt( 5 ) / 2
+local a, b = math.pow( phi, ihp ), phi
 local function SeriesCoefficients( )
   
   local a, b = math.pow( phi, ihp ), phi
+  local coefs = {[0] = phi}
   local c, d = a, b
-  for i = 1, 100 do
+  for i = 1, 150 do
     
-    --print( c, d )
     c = c * d
     d = d - 1
+    coefs[i] = c * math.pow( phi, d )
     
   end
-  
+  return coefs
 end
 
-SeriesCoefficients()
+--SeriesCoefficients()
+
+local c, d = math.pow( ihp, phi ), ihp
+return {
+  pos = function( x ) return a * math.pow( x, b ) end,
+  dpo = function( x ) return a * b * math.pow( x, b - 1 ) end,
+  neg = function( x ) return c * math.pow( x, d ) end,
+  dne = function( x ) return c * d * math.pow( x, d - 1 ) end,
+  coefs = SeriesCoefficients()
+}

fdb.lua

@@ -1,6 +1,6 @@
 --Compute derivatives at the fixed point using Faa Di Bruno's formula.
 
-local ORDER = 21
+local ORDER = 100
 
 local binCoefs = {}
 
@@ -16,8 +16,8 @@ local function Choose( n, k )
   return z
 end
 
-for i = 13, 1, -1 do 
+for i = ORDER, 1, -1 do 
-  for j = 13, 1, -1 do Choose(i, j) end
+  for j = ORDER, 1, -1 do Choose(i, j) end
 end
 
 Choose = function( n, k )
@@ -25,8 +25,10 @@ Choose = function( n, k )
   return c
 end
 
-local function Series( p )
+local function Series( p, order )
-  local d = { p, 1.0 / p } -- q[i] := f^(i) (p)
+  order = order or ORDER
+  order = math.min( order, ORDER )
+  local d = { [0] = p, p, 1.0 / p } -- d[i] := f^(i) (p)
   local bellPolynomial = {}
 
   --Generate incomplete Bell polynomials by recursive formula:
@@ -43,7 +45,7 @@ local function Series( p )
     end
 
     --Cached values.
-    local idx = n * ORDER + k
+    local idx = n * order + k
     if bellPolynomial[idx] then return bellPolynomial[idx] end
 
     --Sum.
@@ -69,35 +71,11 @@ local function Series( p )
     return new
   end
 
-  for i = 1, ORDER - 2 do NextDerivative() end
+  for i = 1, order - 2 do NextDerivative() end
 
-  --Function that evaluates Taylor series.
+  --First few derivatives evaluated at fixed point p.
-  return {
-    Interpolant = function( x )
-      x = x - p
-      local pow = 1
-      local fact = 1
-      local y = p
-      for i = 1, ORDER do
-        pow = pow * x
-        fact = fact * i
-        y = y + pow * d[i] / fact
-      end
-      return y
-    end,
   
-    Derivative = function( x )
+  return d
-      x = x - p
-      local pow = 1
-      local fact = 1
-      local y = 0
-      for i = 1, ORDER do
-        pow = pow * x
-        fact = fact * i
-        y = y + pow * d[i] / fact
-      end
-    end
-  }
 end
 
 return Series

interpolant.lua

@@ -2,6 +2,7 @@
 --returns two functions which evaluate the polynomial and its derivative, respectively
 return function( coefs )
   local fixedPoint = coefs[0] or error( "Must have constant coefficient!" )
+  --Interpolant, naive
   return function(x) 
 
     x = x - fixedPoint
@@ -16,6 +17,7 @@ return function( coefs )
     return y
   end, 
 
+  --Derivative, naive
   function(x) 
 
     x = x - fixedPoint
@@ -31,5 +33,4 @@ return function( coefs )
   end
 
 
-
 end

main.lua

@@ -7,36 +7,57 @@
 --get truncated taylor series expansion of f at p
 --plot to get some idea about values, convergence.
 
-local RESOLUTION = 1000
+local plot = {
-local EXTENT = 16
+  x = 0,
+  X = 1, 
+  y = 0,
+  Y = 1,
+  dx = 0.001,
+  dy = 0.001,
+  inverse = false,
+  fdbOrder = 50,}
+
+plot.Zoom = function( zoomFactor )
+
+  plot.dx, plot.dy = plot.dx * zoomFactor, plot.dy * zoomFactor
+
+  local cx, cy = 0.5 * ( plot.x + plot.X ), 0.5 * ( plot.y + plot.Y )
+  local lx, ly = zoomFactor * 0.5 * ( plot.X - plot.x ), zoomFactor * 0.5 * ( plot.Y - plot.y )
+  plot.x, plot.X = cx - lx, cx + lx
+  plot.y, plot.Y = cy - ly, cy + ly
+end
+
+plot.Translate = function( x, y )
+  x = 0.01 * (plot.X - plot.x) * x
+  y = 0.01 * (plot.Y - plot.y) * y
+  plot.x, plot.X = plot.x + x, plot.X + x
+  plot.y, plot.Y = plot.y + y, plot.Y + y
+end
 
 local FaaDiBruno = require( 'fdb' )
 local Ansatz = require( 'ansatz' )
 local Poly = require( 'interpolant' )
 
-local a = 0.2
+local a = 1.619
-local function PlotFunction( f )
+local function PlotFunction( f, color )
-
-
 
   local points = {}
-  for n = 1, RESOLUTION do
+  local n = 1
-    points[ 2 * n - 1 ] = EXTENT * n / RESOLUTION
+  for x = plot.x - 0.1, plot.X + 0.1, plot.dx do
-    points[ 2 * n ] = f( EXTENT * n / RESOLUTION )
+    points[ 2 * n - 1 ] = x
+    points[ 2 * n ] = f( x )
+    n = n + 1
 
-    --inverse
-    --points[ 2 * RESOLUTION + 2 * n ] = points[ 2 * n - 1]
-    --points[ 2 * RESOLUTION + 2 * n - 1] = points[ 2 * n ]
   end
   
-  local x, y = love.graphics.getDimensions()
+  --inverse
-  local tf = love.math.newTransform(
+  local inverse = {}
-    x / EXTENT, y,-- / EXTENT,
+  for i = 1, n, 2 do
-    0, x / EXTENT,
+    inverse[i], inverse[i + 1] = points[ i + 1], points[i]
-    -y / EXTENT)
+  end
-  local dtf = love.math.newTransform( 0, 0, 0, 1, 1 )
 
-  --love.graphics.setColor( 1, 1, 1, 0.5 )
+
+  local x, y = love.graphics.getDimensions()
   love.graphics.setLineWidth( 0.1 )
   love.graphics.setLineJoin( "miter" )
   love.graphics.setLineStyle( "smooth" )
@@ -47,38 +68,74 @@ local function PlotFunction( f )
     love.graphics.push("transform")
     love.graphics.translate( 0, y )
     love.graphics.scale( x, -y )
-    love.graphics.scale( 1 / EXTENT, 1 / EXTENT )
+    love.graphics.scale( 1 / (plot.X - plot.x), 1/ (plot.Y - plot.y) )
+    love.graphics.translate( - plot.x, - plot.y )
+    love.graphics.setLineWidth( 0.003 * (plot.X - plot.x) )
 
-    --love.graphics.scale( 1.0 / EXTENT, 1.0 / EXTENT )
+    if color then love.graphics.setColor( color ) end
-    love.graphics.line( points )
+    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.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
 
+love.wheelmoved = function( x, y )
+  return plot.Zoom( (y > 0) and 0.95 or 1.05 ) or love.keypressed()
+end
+
+love.mousepressed = function(x, y, button)
+  --plot.inverse = not( plot.inverse )
+  plot.fdbOrder = plot.fdbOrder + (( button == 1 ) and 1 or -1 )
+  return love.keypressed()
+end 
+
+--[[local f, df = Poly( FaaDiBruno( a ) )
+PlotFunction( f, {1, 0, 0, 0.5} )  --Function
+PlotFunction( df, {0, 1, 0, 0.5} ) --First derivative
+PlotFunction( function(x) return x end, {0, 0, 1, 0.5})]]
+
+love.keypressed = function( key, code )
+  if code == "q" then a = a + 0.001 * (plot.X - plot.x) end
+  if code == "e" then a = a - 0.001 * (plot.X - plot.x) end
+  if code == "w" then plot.Translate( 0, 1 ) end
+  if code == "a" then plot.Translate( -1, 0 ) end
+  if code == "s" then plot.Translate( 0, -1 ) end
+  if code == "d" then plot.Translate( 1, 0 ) end
+  if code == "z" then plot.Zoom( 0.95 ) end
+  if code == "c" then plot.Zoom( 1.05 ) end
+  if code == "f" then plot.fdbOrder = plot.fdbOrder - 1 end
+  if code == "r" then plot.fdbOrder = plot.fdbOrder + 1 end
+  love.draw = nil
+
+  local f, df, fn = Poly( FaaDiBruno( a, plot.fdbOrder ) )
+  --PlotFunction( f, {1, 0, 0, 0.3} )  --Function in red.
+  --PlotFunction( df, {0, 1, 0, 0.3} ) --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( an, {1,1,1, 0.3} )
+  PlotFunction( Ansatz.dpo, {1, 1, 1, 0.3} )
+  PlotFunction( Ansatz.pos, {1,1,1,0.3} )
+end
+
 love.update = function( dt )
-  if love.keyboard.isScancodeDown("w") then
+  for _, code in ipairs{ 'q', 'e', 'w', 'a', 's', 'd', 'z', 'c', 'f', 'r' } do
-    a = a + 0.05
+    if love.keyboard.isScancodeDown( code ) then love.keypressed( nil, code ) end
-    love.graphics.setColor( a, 0, 0, a )
+  end
-    love.draw = nil
-    --PlotFunction( FaaDiBruno( a ) )
-    --PlotFunction( function(x)return x end)
 end
 
-  if love.keyboard.isScancodeDown("s") then
+love.keypressed( nil, nil )
-    a = a - 0.05
-    love.graphics.setColor( 1, 0, 0, 1 )
-    love.draw = nil
-    --[[local f = FaaDiBruno(a)
-    PlotFunction( f.Interpolant )
-    PlotFunction( f.Derivative )]]
-    PlotFunction( function(x)return x end)
-    local f, df = Poly({[0] = 1.5, 1 + a, 2 })
-    PlotFunction( f )
-    PlotFunction( df )
-  end
-  if love.keyboard.isScancodeDown("q") then EXTENT = EXTENT + 0.5 end
-  if love.keyboard.isScancodeDown("e") then EXTENT = EXTENT * 0.99 end
-end