Large commit, fixed large mistake in calculating Ansatz values, added tests for FDB series plot.
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25
ansatz.lua
25
ansatz.lua
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@ -2,20 +2,31 @@
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--assume f(x) = ax ^ b. Then
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--f' o f (x) = ba^b * x ^ ( b ( b-1 ) )
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local phi = 1 + math.sqrt( 5 ) / 2
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local ihp = 1 - math.sqrt( 5 ) / 2
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local phi = 0.5 + math.sqrt( 5 ) / 2
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local ihp = 0.5 - math.sqrt( 5 ) / 2
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local a, b = math.pow( phi, ihp ), phi
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local function SeriesCoefficients( )
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local a, b = math.pow( phi, ihp ), phi
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local coefs = {[0] = phi}
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local c, d = a, b
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for i = 1, 100 do
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for i = 1, 150 do
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--print( c, d )
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c = c * d
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d = d - 1
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coefs[i] = c * math.pow( phi, d )
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end
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return coefs
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end
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SeriesCoefficients()
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--SeriesCoefficients()
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local c, d = math.pow( ihp, phi ), ihp
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return {
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pos = function( x ) return a * math.pow( x, b ) end,
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dpo = function( x ) return a * b * math.pow( x, b - 1 ) end,
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neg = function( x ) return c * math.pow( x, d ) end,
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dne = function( x ) return c * d * math.pow( x, d - 1 ) end,
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coefs = SeriesCoefficients()
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}
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46
fdb.lua
46
fdb.lua
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@ -1,6 +1,6 @@
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--Compute derivatives at the fixed point using Faa Di Bruno's formula.
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local ORDER = 21
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local ORDER = 100
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local binCoefs = {}
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@ -16,8 +16,8 @@ local function Choose( n, k )
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return z
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end
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for i = 13, 1, -1 do
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for j = 13, 1, -1 do Choose(i, j) end
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for i = ORDER, 1, -1 do
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for j = ORDER, 1, -1 do Choose(i, j) end
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end
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Choose = function( n, k )
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@ -25,8 +25,10 @@ Choose = function( n, k )
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return c
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end
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local function Series( p )
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local d = { p, 1.0 / p } -- q[i] := f^(i) (p)
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local function Series( p, order )
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order = order or ORDER
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order = math.min( order, ORDER )
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local d = { [0] = p, p, 1.0 / p } -- d[i] := f^(i) (p)
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local bellPolynomial = {}
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--Generate incomplete Bell polynomials by recursive formula:
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@ -43,7 +45,7 @@ local function Series( p )
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end
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--Cached values.
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local idx = n * ORDER + k
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local idx = n * order + k
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if bellPolynomial[idx] then return bellPolynomial[idx] end
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--Sum.
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@ -69,35 +71,11 @@ local function Series( p )
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return new
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end
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for i = 1, ORDER - 2 do NextDerivative() end
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for i = 1, order - 2 do NextDerivative() end
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--Function that evaluates Taylor series.
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return {
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Interpolant = function( x )
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x = x - p
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local pow = 1
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local fact = 1
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local y = p
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for i = 1, ORDER do
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pow = pow * x
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fact = fact * i
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y = y + pow * d[i] / fact
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end
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return y
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end,
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Derivative = function( x )
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x = x - p
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local pow = 1
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local fact = 1
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local y = 0
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for i = 1, ORDER do
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pow = pow * x
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fact = fact * i
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y = y + pow * d[i] / fact
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end
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end
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}
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--First few derivatives evaluated at fixed point p.
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return d
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end
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return Series
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@ -2,6 +2,7 @@
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--returns two functions which evaluate the polynomial and its derivative, respectively
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return function( coefs )
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local fixedPoint = coefs[0] or error( "Must have constant coefficient!" )
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--Interpolant, naive
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return function(x)
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x = x - fixedPoint
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@ -16,6 +17,7 @@ return function( coefs )
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return y
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end,
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--Derivative, naive
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function(x)
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x = x - fixedPoint
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@ -31,5 +33,4 @@ return function( coefs )
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end
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end
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145
main.lua
145
main.lua
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@ -7,36 +7,57 @@
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--get truncated taylor series expansion of f at p
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--plot to get some idea about values, convergence.
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local RESOLUTION = 1000
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local EXTENT = 16
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local plot = {
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x = 0,
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X = 1,
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y = 0,
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Y = 1,
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dx = 0.001,
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dy = 0.001,
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inverse = false,
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fdbOrder = 50,}
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plot.Zoom = function( zoomFactor )
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plot.dx, plot.dy = plot.dx * zoomFactor, plot.dy * zoomFactor
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local cx, cy = 0.5 * ( plot.x + plot.X ), 0.5 * ( plot.y + plot.Y )
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local lx, ly = zoomFactor * 0.5 * ( plot.X - plot.x ), zoomFactor * 0.5 * ( plot.Y - plot.y )
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plot.x, plot.X = cx - lx, cx + lx
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plot.y, plot.Y = cy - ly, cy + ly
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end
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plot.Translate = function( x, y )
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x = 0.01 * (plot.X - plot.x) * x
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y = 0.01 * (plot.Y - plot.y) * y
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plot.x, plot.X = plot.x + x, plot.X + x
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plot.y, plot.Y = plot.y + y, plot.Y + y
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end
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local FaaDiBruno = require( 'fdb' )
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local Ansatz = require( 'ansatz' )
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local Poly = require( 'interpolant' )
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local a = 0.2
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local function PlotFunction( f )
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local a = 1.619
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local function PlotFunction( f, color )
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local points = {}
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for n = 1, RESOLUTION do
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points[ 2 * n - 1 ] = EXTENT * n / RESOLUTION
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points[ 2 * n ] = f( EXTENT * n / RESOLUTION )
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local n = 1
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for x = plot.x - 0.1, plot.X + 0.1, plot.dx do
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points[ 2 * n - 1 ] = x
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points[ 2 * n ] = f( x )
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n = n + 1
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--inverse
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--points[ 2 * RESOLUTION + 2 * n ] = points[ 2 * n - 1]
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--points[ 2 * RESOLUTION + 2 * n - 1] = points[ 2 * n ]
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end
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--inverse
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local inverse = {}
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for i = 1, n, 2 do
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inverse[i], inverse[i + 1] = points[ i + 1], points[i]
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end
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local x, y = love.graphics.getDimensions()
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local tf = love.math.newTransform(
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x / EXTENT, y,-- / EXTENT,
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0, x / EXTENT,
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-y / EXTENT)
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local dtf = love.math.newTransform( 0, 0, 0, 1, 1 )
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--love.graphics.setColor( 1, 1, 1, 0.5 )
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local x, y = love.graphics.getDimensions()
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love.graphics.setLineWidth( 0.1 )
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love.graphics.setLineJoin( "miter" )
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love.graphics.setLineStyle( "smooth" )
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@ -47,38 +68,74 @@ local function PlotFunction( f )
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love.graphics.push("transform")
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love.graphics.translate( 0, y )
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love.graphics.scale( x, -y )
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love.graphics.scale( 1 / EXTENT, 1 / EXTENT )
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love.graphics.scale( 1 / (plot.X - plot.x), 1/ (plot.Y - plot.y) )
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love.graphics.translate( - plot.x, - plot.y )
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love.graphics.setLineWidth( 0.003 * (plot.X - plot.x) )
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--love.graphics.scale( 1.0 / EXTENT, 1.0 / EXTENT )
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love.graphics.line( points )
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if color then love.graphics.setColor( color ) end
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love.graphics.line( plot.inverse and inverse or points )
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love.graphics.circle( "fill", a, a, 0.003 * (plot.X - plot.x) )
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love.graphics.pop()
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love.graphics.print( a )
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love.graphics.print( plot.fdbOrder, 0, 15 )
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love.graphics.print( f(a), 0, 30 )
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love.graphics.print( f(-1.0), 0, 45 )
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love.graphics.setColor( 1,1,1,1 )
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love.graphics.setFont( love.graphics.getFont( 48 ) )
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end
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end
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love.update = function( dt )
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if love.keyboard.isScancodeDown("w") then
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a = a + 0.05
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love.graphics.setColor( a, 0, 0, a )
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love.draw = nil
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--PlotFunction( FaaDiBruno( a ) )
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--PlotFunction( function(x)return x end)
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end
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love.wheelmoved = function( x, y )
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return plot.Zoom( (y > 0) and 0.95 or 1.05 ) or love.keypressed()
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end
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love.mousepressed = function(x, y, button)
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--plot.inverse = not( plot.inverse )
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plot.fdbOrder = plot.fdbOrder + (( button == 1 ) and 1 or -1 )
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return love.keypressed()
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end
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--[[local f, df = Poly( FaaDiBruno( a ) )
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PlotFunction( f, {1, 0, 0, 0.5} ) --Function
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PlotFunction( df, {0, 1, 0, 0.5} ) --First derivative
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PlotFunction( function(x) return x end, {0, 0, 1, 0.5})]]
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love.keypressed = function( key, code )
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if code == "q" then a = a + 0.001 * (plot.X - plot.x) end
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if code == "e" then a = a - 0.001 * (plot.X - plot.x) end
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if code == "w" then plot.Translate( 0, 1 ) end
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if code == "a" then plot.Translate( -1, 0 ) end
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if code == "s" then plot.Translate( 0, -1 ) end
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if code == "d" then plot.Translate( 1, 0 ) end
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if code == "z" then plot.Zoom( 0.95 ) end
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if code == "c" then plot.Zoom( 1.05 ) end
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if code == "f" then plot.fdbOrder = plot.fdbOrder - 1 end
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if code == "r" then plot.fdbOrder = plot.fdbOrder + 1 end
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love.draw = nil
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local f, df, fn = Poly( FaaDiBruno( a, plot.fdbOrder ) )
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--PlotFunction( f, {1, 0, 0, 0.3} ) --Function in red.
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--PlotFunction( df, {0, 1, 0, 0.3} ) --First derivative in green.
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--local lowF, lowDF = Poly( FaaDiBruno( a, 15 ) )
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--PlotFunction( lowF, {1, 0, 0, 0.3} ) --Function in red.
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--PlotFunction( lowDF, {0, 1, 0, 0.3} ) --First derivative in green.
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if love.keyboard.isScancodeDown("s") then
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a = a - 0.05
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love.graphics.setColor( 1, 0, 0, 1 )
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love.draw = nil
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--[[local f = FaaDiBruno(a)
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PlotFunction( f.Interpolant )
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PlotFunction( f.Derivative )]]
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PlotFunction( function(x)return x end)
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local f, df = Poly({[0] = 1.5, 1 + a, 2 })
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PlotFunction( f )
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PlotFunction( df )
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--PlotFunction( function(x) return df( f ( x ) ) end, {0, 0, 1, 0.3})
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--PlotFunction( function(x) return f( df ( x ) ) end, {1, 1, 1, 0.3})
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--PlotFunction( function(x) return x end, {1, 1, 1, 0.3})
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local af, an = Poly( Ansatz.coefs )
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PlotFunction( af, {1,1,1, 0.3} )
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PlotFunction( an, {1,1,1, 0.3} )
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PlotFunction( Ansatz.dpo, {1, 1, 1, 0.3} )
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PlotFunction( Ansatz.pos, {1,1,1,0.3} )
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end
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love.update = function( dt )
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for _, code in ipairs{ 'q', 'e', 'w', 'a', 's', 'd', 'z', 'c', 'f', 'r' } do
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if love.keyboard.isScancodeDown( code ) then love.keypressed( nil, code ) end
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end
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if love.keyboard.isScancodeDown("q") then EXTENT = EXTENT + 0.5 end
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if love.keyboard.isScancodeDown("e") then EXTENT = EXTENT * 0.99 end
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end
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end
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love.keypressed( nil, nil )
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