Feverish insensate spurt:
- Added basic AST calculator for derivatives. - Added direct derivation via Faa Di Bruno formula. - Added ansatz method. - Factored out polynomial interpolation functionality.
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@ -0,0 +1,21 @@
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--Solve the functional equation f' o f = id by ansatz:
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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 function SeriesCoefficients( )
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local a, b = math.pow( phi, ihp ), phi
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local c, d = a, b
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for i = 1, 100 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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end
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end
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SeriesCoefficients()
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@ -0,0 +1,110 @@
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--Bespoke symbolic differentiation.
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--Little prototypes for building a syntax tree containing only
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--multiplication, addition, and composition,
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--evaluating its value, and expanding it according to the chain rule.
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local op;
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local function d( x ) return x:diff() end
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local function D( x ) return setmetatable( { ["n"] = x }, op.atom ) end
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local computedDerivatives = {}
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local __call = D
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local __add = function( a, b ) return setmetatable( { a, b }, op.add ) end
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local __mul = function( a, b ) return setmetatable( { a, b }, op.mul ) end
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local __pow = function( a, b ) return setmetatable( { a, b }, op.compose ) end
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op = {
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atom = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self.n ) end,
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__index = {
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eval = function( self )
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return computedDerivatives[ self.n ]
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end,
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diff = function( self )
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return D( self.n + 1 )
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end,
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}},
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compose = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return "("..tostring( self[1] ).." o "..tostring( self[2] )..")" end,
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__index = {
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eval = function( self )
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--All compositions are of the form f^(n) o f,
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--whose value at a fixed point p is f^(n)(p)
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return self[1]:eval()
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end,
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diff = function( self )
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--All compositions are of the form f^(n) o f,
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--whose derivatives are always ( f^(n+1) o f )
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return d( self[1] ) ^ D(0) * D(1)
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end,
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}},
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add = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self[1] ).." + "..tostring( self[2] ) end,
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__index = {
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name = "add",
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eval = function( self )
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return self[1]:eval() + self[2]:eval()
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end,
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diff = function( self )
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return d(self[1]) + d(self[2])
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end,
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}},
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mul = {
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__add = __add,
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__mul = __mul,
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__pow = __pow,
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__call = __call,
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__tostring = function( self ) return tostring( self[1] ).." x "..tostring( self[2] ) end,
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__index = {
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eval = function( self )
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return self[1]:eval() * self[2]:eval()
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end,
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diff = function( self )
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return d(self[1]) * self[2] + self[1] * d(self[2])
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end,
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}},
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}
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op.New = function( nodeType, a, b )
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return setmetatable( { a, b }, op[nodeType] or error( "Node Type Not Recognized" ) )
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end
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local a = D(1) ^ D(0)
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for i = 1, 5 do
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print( a )
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a = a:diff()
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end
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return op
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@ -0,0 +1,103 @@
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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 binCoefs = {}
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local function Choose( n, k )
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if k < 1 then return 1 end
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if n < 1 then return 0 end
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if k > n then return 0 end
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local idx = n * ORDER + k
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if binCoefs[idx] then return binCoefs[idx] end
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local z = Choose( n - 1, k - 1) + Choose( n - 1, k )
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binCoefs[ idx ] = z
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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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end
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Choose = function( n, k )
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local c = binCoefs[ n * ORDER + k ] or ( ( k < 1 ) and 1 or 0 )
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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 bellPolynomial = {}
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--Generate incomplete Bell polynomials by recursive formula:
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--d[n+1] = - \sum_{k=1}^{n-1} d[k+1] * B(n,k) / p^n
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--B[n][k] = \sum_{i=1}^{n-k+1} Choose( n-1, i-1 ) * d[i] * B( n-i, k-1 )
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--sum_i=1^(n-k+1)
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local function Bell( n, k )
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--Base cases.
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if n == 0 then
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return ( k == 0 ) and 1 or 0
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elseif k == 0 then
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return 0
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end
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--Cached values.
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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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local b = 0
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for i = 1, n - k + 1 do
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local a = Choose( n - 1, i - 1 )
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a = a * d[i]
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a = a * Bell( n - i, k - 1 )
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b = b + a
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end
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bellPolynomial[idx] = b
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return b
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end
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local NextDerivative = function()
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local n = #d
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local new = 0
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for k = 1, n - 1 do
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new = new + d[k + 1] * Bell(n, k)
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end
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new = -new / math.pow( p, n )
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d[n + 1] = new
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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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--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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end
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return Series
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@ -0,0 +1,35 @@
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--Returns a function which takes a sequence of polynomial coefficients and
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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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return function(x)
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x = x - fixedPoint
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local y = fixedPoint
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local pow = 1
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local fact = 1
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for i = 1, #coefs do
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pow = pow * x
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fact = fact / i
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y = y + pow * fact * coefs[i]
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end
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return y
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end,
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function(x)
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x = x - fixedPoint
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local y = 0
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local pow = 1
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local fact = 1
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for i = 1, #coefs do
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y = y + pow * fact * coefs[i]
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pow = pow * x
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fact = fact / i
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end
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return y
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end
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end
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67
main.lua
67
main.lua
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@ -7,41 +7,78 @@
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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 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 RESOLUTION = 1000
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local points = {}
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for n = 1, RESOLUTION do
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points[ 2 * n - 1 ] = n / RESOLUTION
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points[ 2 * n ] = f( n / RESOLUTION )
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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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--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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local x, y = love.graphics.getDimensions()
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local tf = love.math.newTransform(
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0, love.graphics.getHeight(),
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0, love.graphics.getWidth(),
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-love.graphics.getHeight())
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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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love.graphics.setLineWidth( 0.003 )
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--love.graphics.setColor( 1, 1, 1, 0.5 )
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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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local draw = love.draw or function() end
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love.draw = function()
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draw()
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love.graphics.replaceTransform( tf )
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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( 1.0 / EXTENT, 1.0 / EXTENT )
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love.graphics.line( points )
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love.graphics.pop()
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love.graphics.print( a )
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end
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end
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love.update = function( dt )
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print( love.mouse.getPosition( ) )
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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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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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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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PlotFunction( function( x ) return x * x end )
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PlotFunction( function( x ) return x * x * x end )
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PlotFunction( math.sin )
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PlotFunction( function( x ) return math.exp( x ) - 1 end )
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