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

This commit is contained in:
yaw-man 2023-02-09 00:20:46 -04:00
parent b7a2e67ace
commit 98c27e0725
4 changed files with 133 additions and 86 deletions

View File

@ -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 ihp = 1 - math.sqrt( 5 ) / 2
local phi = 0.5 + 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()
}

44
fdb.lua
View File

@ -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 j = 13, 1, -1 do Choose(i, j) end
for i = ORDER, 1, -1 do
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 d = { p, 1.0 / p } -- q[i] := f^(i) (p)
local function Series( p, order )
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.
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,
--First few derivatives evaluated at fixed point p.
Derivative = function( x )
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
}
return d
end
return Series

View File

@ -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

143
main.lua
View File

@ -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 EXTENT = 16
local plot = {
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 function PlotFunction( f )
local a = 1.619
local function PlotFunction( f, color )
local points = {}
for n = 1, RESOLUTION do
points[ 2 * n - 1 ] = EXTENT * n / RESOLUTION
points[ 2 * n ] = f( EXTENT * n / RESOLUTION )
local n = 1
for x = plot.x - 0.1, plot.X + 0.1, plot.dx do
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()
local tf = love.math.newTransform(
x / EXTENT, y,-- / EXTENT,
0, x / EXTENT,
-y / EXTENT)
local dtf = love.math.newTransform( 0, 0, 0, 1, 1 )
--inverse
local inverse = {}
for i = 1, n, 2 do
inverse[i], inverse[i + 1] = points[ i + 1], points[i]
end
--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 )
love.graphics.line( points )
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.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
a = a + 0.05
love.graphics.setColor( a, 0, 0, a )
love.draw = nil
--PlotFunction( FaaDiBruno( a ) )
--PlotFunction( function(x)return x end)
for _, code in ipairs{ 'q', 'e', 'w', 'a', 's', 'd', 'z', 'c', 'f', 'r' } do
if love.keyboard.isScancodeDown( code ) then love.keypressed( nil, code ) end
end
if love.keyboard.isScancodeDown("s") then
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
love.keypressed( nil, nil )