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Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
Numerical integration/Gauss-Legendre Quadrature (view source)
Revision as of 21:12, 26 June 2017
, 6 years agoAdded Lua entry
(Added Lua entry) |
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Line 1,009:
compared to actual:
20.03574985</pre>
=={{header|Lua}}==
<lang Lua>local order = 0
local legendreRoots = {}
local legendreWeights = {}
local function legendre(term, z)
if (term == 0) then
return 1
elseif (term == 1) then
return z
else
return ((2 * term - 1) * z * legendre(term - 1, z) - (term - 1) * legendre(term - 2, z)) / term
end
end
local function legendreDerivative(term, z)
if (term == 0) then
return 0
elseif (term == 1) then
return 1
else
return ( term * ((z * legendre(term, z)) - legendre(term - 1, z))) / (z * z - 1)
end
end
local function getLegendreRoots()
local y, y1
for index = 1, order do
y = math.cos(math.pi * (index - 0.25) / (order + 0.5))
repeat
y1 = y
y = y - (legendre(order, y) / legendreDerivative(order, y))
until y == y1
table.insert(legendreRoots, y)
end
end
local function getLegendreWeights()
for index = 1, order do
local weight = 2 / ((1 - (legendreRoots[index]) ^ 2) * (legendreDerivative(order, legendreRoots[index])) ^ 2)
table.insert(legendreWeights, weight)
end
end
function gaussLegendreQuadrature(f, lowerLimit, upperLimit, n)
order = n
do
getLegendreRoots()
getLegendreWeights()
end
local c1 = (upperLimit - lowerLimit) / 2
local c2 = (upperLimit + lowerLimit) / 2
local sum = 0
for i = 1, order do
sum = sum + legendreWeights[i] * f(c1 * legendreRoots[i] + c2)
end
return c1 * sum
end
do
print(gaussLegendreQuadrature(function(x) return math.exp(x) end, -3, 3, 5))
end</lang>
{{out}}<pre>20.035577718386</pre>
=={{header|Mathematica}}==
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