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Numerical integration/Gauss-Legendre Quadrature: Difference between revisions
Numerical integration/Gauss-Legendre Quadrature (view source)
Revision as of 14:36, 22 August 2019
, 4 years agoScala contribution added.
(→{{header|Julia}}: function eig was depreciated and replaced with eigen, the dot operator handling has also changed since this example was put up, added the needed module import, removed the return keyword as it is unnecessary in julia) |
(Scala contribution added.) |
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</pre>
=={{header|Scala}}==
{{Out}}Best seen in running your browser either by [https://scalafiddle.io/sf/rrvzhH1/0 ScalaFiddle (ES aka JavaScript, non JVM)] or [https://scastie.scala-lang.org/yYqRqizfSZip2DhYbdfZ2w Scastie (remote JVM)].
<lang Scala>import scala.math.{Pi, cos, exp}
object GaussLegendreQuadrature extends App {
private val N = 5
private def legeInte(a: Double, b: Double): Double = {
val (c1, c2) = ((b - a) / 2, (b + a) / 2)
val tuples: IndexedSeq[(Double, Double)] = {
val lcoef = {
val lcoef = Array.ofDim[Double](N + 1, N + 1)
lcoef(0)(0) = 1
lcoef(1)(1) = 1
for (i <- 2 to N) {
lcoef(i)(0) = -(i - 1) * lcoef(i - 2)(0) / i
for (j <- 1 to i) lcoef(i)(j) =
((2 * i - 1) * lcoef(i - 1)(j - 1) - (i - 1) * lcoef(i - 2)(j)) / i
}
lcoef
}
def legeEval(n: Int, x: Double): Double =
lcoef(n).take(n).foldRight(lcoef(n)(n))((o, s) => s * x + o)
def legeDiff(n: Int, x: Double): Double =
n * (x * legeEval(n, x) - legeEval(n - 1, x)) / (x * x - 1)
@scala.annotation.tailrec
def convergention(x0: Double, x1: Double): Double = {
if (x0 == x1) x1
else convergention(x1, x1 - legeEval(N, x1) / legeDiff(N, x1))
}
for {i <- 0 until 5
x = convergention(0.0, cos(Pi * (i + 1 - 0.25) / (N + 0.5)))
x1 = legeDiff(N, x)
} yield (x, 2 / ((1 - x * x) * x1 * x1))
}
println(s"Roots: ${tuples.map(el => f" ${el._1}%10.6f").mkString}")
println(s"Weight:${tuples.map(el => f" ${el._2}%10.6f").mkString}")
c1 * tuples.map { case (lroot, weight) => weight * exp(c1 * lroot + c2) }.sum
}
println(f"Integrating exp(x) over [-3, 3]:\n\t${legeInte(-3, 3)}%10.8f,")
println(f"compared to actual%n\t${exp(3) - exp(-3)}%10.8f")
}</lang>
=={{header|Sidef}}==
{{trans|Perl 6}}
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}</lang>
{{out}}
<pre>Gauss-Legendre 5-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035577718385561▼
▲Gauss-Legendre 5-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035577718385561
Gauss-Legendre 6-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035746975092344
Gauss-Legendre 7-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749819726600
Line 2,376 ⟶ 2,426:
Gauss-Legendre 9-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854817432
Gauss-Legendre 10-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854819791
Gauss-Legendre 20-point quadrature ∫₋₃⁺³ exp(x) dx ≈ 20.035749854819805</pre>
=={{header|Tcl}}==
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