Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
Numerical integration/Gauss-Legendre Quadrature (view source)
Revision as of 08:53, 2 December 2021
, 2 years agoAdded 11l
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<big><big><math>\int_{-3}^{3} \exp(x) \, dx \approx \sum_{i=1}^5 w_i \; \exp(x_i) \approx 20.036</math></big></big>
<br><br>
=={{header|11l}}==
{{trans|Nim}}
<lang 11l>F legendreIn(x, n)
F prev1(idx, pn1)
R (2 * idx - 1) * @x * pn1
F prev2(idx, pn2)
R (idx - 1) * pn2
I n == 0
R 1.0
E I n == 1
R x
E
V result = 0.0
V p1 = x
V p2 = 1.0
L(i) 2 .. n
result = (prev1(i, p1) - prev2(i, p2)) / i
p2 = p1
p1 = result
R result
F deriveLegendreIn(x, n)
F calcresult(curr, prev)
R Float(@n) / (@x ^ 2 - 1) * (@x * curr - prev)
R calcresult(legendreIn(x, n), legendreIn(x, n - 1))
F guess(n, i)
R cos(math:pi * (i - 0.25) / (n + 0.5))
F nodes(n)
V result = [(0.0, 0.0)] * n
F calc(x)
R legendreIn(x, @n) / deriveLegendreIn(x, @n)
L(i) 0 .< n
V x = guess(n, i + 1)
V x0 = x
x -= calc(x)
L abs(x - x0) > 1e-12
x0 = x
x -= calc(x)
result[i] = (x, 2 / ((1.0 - x ^ 2) * (deriveLegendreIn(x, n)) ^ 2))
R result
F integ(f, ns, p1, p2)
F dist()
R (@p2 - @p1) / 2
F avg()
R (@p1 + @p2) / 2
V result = dist()
V sum = 0.0
V thenodes = [0.0] * ns
V weights = [0.0] * ns
L(nw) nodes(ns)
sum += nw[1] * f(dist() * nw[0] + avg())
thenodes[L.index] = nw[0]
weights[L.index] = nw[1]
print(‘ nodes:’, end' ‘’)
L(n) thenodes
print(‘ #.5’.format(n), end' ‘’)
print()
print(‘ weights:’, end' ‘’)
L(w) weights
print(‘ #.5’.format(w), end' ‘’)
print()
R result * sum
print(‘integral: ’integ(x -> exp(x), 5, -3, 3))</lang>
{{out}}
<pre>
nodes: 0.90618 0.53847 0.00000 -0.53847 -0.90618
weights: 0.23693 0.47863 0.56889 0.47863 0.23693
integral: 20.035577718
</pre>
=={{header|Axiom}}==
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