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test_integrate.nim
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import unittest, math, sequtils
import arraymancer
import numericalnim
proc f(x: float, ctx: NumContext[float, float]): float = ctx["a"] * cos(x)
proc fVector(x: float, ctx: NumContext[Vector[float], float]): Vector[float] = cos(x) * ctx["a"]
proc fTensor(x: float, ctx: NumContext[Tensor[float], float]): Tensor[float] = @[ctx.getF("a") * cos(x), ctx.getF("a") * cos(x), ctx.getF("a") * cos(x)].toTensor()
let xStart = 0.0
let xEnd = 3.0/2.0*PI
let optional = newNumContext[float, float]()
optional["a"] = 2.0
let optionalVector = newNumContext[Vector[float], float]()
optionalVector["a"] = newVector([2.0, 2.0, 2.0])
let optionalTensor = newNumContext[Tensor[float], float]()
optionalTensor.setF("a", 2.0)
let correct = optional["a"] * sin(xEnd)
let correctVector = newVector([optional["a"] * sin(xEnd), optional["a"] * sin(xEnd), optional["a"] * sin(xEnd)])
let correctTensor = @[optional["a"] * sin(xEnd), optional["a"] * sin(xEnd), optional["a"] * sin(xEnd)].toTensor()
let X = linspace(xStart, xEnd, 17)
let Y = map(X, proc(x: float): float = f(x, optional))
let cumY = map(X, proc(x: float): float = optional["a"] * sin(x))
test "trapz func, N = 10":
let value = trapz(f, xStart, xEnd, N = 10, ctx = optional)
check isClose(value, correct, tol = 1e-1)
test "trapz func, N = 100":
let value = trapz(f, xStart, xEnd, N = 100, ctx = optional)
check isClose(value, correct, tol = 1e-3)
test "simpson func, N = 10":
let value = simpson(f, xStart, xEnd, N = 10, ctx = optional)
check isClose(value, correct, tol = 1e-3)
test "simpson func, N = 100":
let value = simpson(f, xStart, xEnd, N = 100, ctx = optional)
check isClose(value, correct, tol = 1e-6)
test "adaptive simpson, tol=1e-3":
let value = adaptiveSimpson(f, xStart, xEnd, tol=1e-3, ctx = optional)
check isClose(value, correct, tol=1e-3)
test "adaptive simpson, tol=1e-8":
let value = adaptiveSimpson(f, xStart, xEnd, tol=1e-8, ctx = optional)
check isClose(value, correct, tol=1e-8)
test "romberg func, depth = 3":
let value = romberg(f, xStart, xEnd, depth = 3, ctx = optional, tol=1e-8)
check isClose(value, correct, tol = 1e-1)
test "romberg func, default":
let value = romberg(f, xStart, xEnd, ctx = optional)
check isClose(value, correct, tol = 1e-8)
test "trapz discrete points":
let value = trapz(Y, X)
check isClose(correct, value, tol = 1e-1)
test "simpson discrete points":
let value = simpson(Y, X)
check isClose(correct, value, tol = 1e-3)
test "romberg discrete points":
let value = romberg(Y, X)
check isClose(correct, value, tol = 1e-3)
test "cumtrapz discrete points":
let values = cumtrapz(Y, X)
for i, value in values:
check isClose(value, cumY[i], tol=1e-1)
test "cumtrapz func, discrete points":
let values = cumtrapz(f, X, ctx = optional)
for i, value in values:
check isClose(value, cumY[i], tol=1e-1)
test "cumtrapz func, dx = 0.1":
let values = cumtrapz(f, X, dx = 0.1, ctx = optional)
for i, value in values:
check isClose(value, cumY[i], tol=1e-1)
test "cumsimpson discrete points":
let values = cumsimpson(Y, X)
for i, value in values:
check isClose(value, cumY[i], tol=1e-3)
test "cumsimpson func, discrete points":
let values = cumsimpson(f, X, ctx = optional)
for i, value in values:
check isClose(value, cumY[i], tol=1e-3)
test "cumsimpson func, dx = 0.1":
let values = cumsimpson(f, X, dx = 0.1, ctx = optional)
for i, value in values:
check isClose(value, cumY[i], tol=1e-3)
test "gaussQuad func n=2":
let value = gaussQuad(f, xStart, xEnd, nPoints=2, N=1, ctx =optional)
check isClose(value, correct, tol=1)
test "gaussQuad func n=2 N=10":
let value = gaussQuad(f, xStart, xEnd, nPoints=2, N=10, ctx =optional)
check isClose(value, correct, tol=1e-2)
test "gaussQuad func n=3":
let value = gaussQuad(f, xStart, xEnd, nPoints=3, N=1, ctx =optional)
check isClose(value, correct, tol=1e-1)
test "gaussQuad func n=4":
let value = gaussQuad(f, xStart, xEnd, nPoints=4, N=1, ctx =optional)
check isClose(value, correct, tol=1e-3)
test "gaussQuad func n=5":
let value = gaussQuad(f, xStart, xEnd, nPoints=5, N=1, ctx =optional)
check isClose(value, correct, tol=1e-4)
test "gaussQuad func n=5 N=10":
let value = gaussQuad(f, xStart, xEnd, nPoints=5, N=10, ctx =optional)
check isClose(value, correct, tol=1e-6)
test "gaussQuad func n=6":
let value = gaussQuad(f, xStart, xEnd, nPoints=6, N=1, ctx =optional)
check isClose(value, correct, tol=1e-5)
test "gaussQuad func n=7":
let value = gaussQuad(f, xStart, xEnd, nPoints=7, N=1, ctx =optional)
check isClose(value, correct, tol=1e-6)
test "gaussQuad func n=8":
let value = gaussQuad(f, xStart, xEnd, nPoints=8, N=1, ctx =optional)
check isClose(value, correct, tol=1e-6)
test "gaussQuad func n=9":
let value = gaussQuad(f, xStart, xEnd, nPoints=9, N=1, ctx =optional)
check isClose(value, correct, tol=1e-7)
test "gaussQuad func n=10":
let value = gaussQuad(f, xStart, xEnd, nPoints=10, N=1, ctx =optional)
check isClose(value, correct, tol=1e-7)
test "gaussQuad func n=10 N=10":
let value = gaussQuad(f, xStart, xEnd, nPoints=10, N=10, ctx =optional)
check isClose(value, correct, tol=1e-9)
test "gaussQuad func n=11":
let value = gaussQuad(f, xStart, xEnd, nPoints=11, N=1, ctx =optional)
check isClose(value, correct, tol=1e-8)
test "gaussQuad func n=12":
let value = gaussQuad(f, xStart, xEnd, nPoints=12, N=1, ctx =optional)
check isClose(value, correct, tol=1e-8)
test "gaussQuad func n=13":
let value = gaussQuad(f, xStart, xEnd, nPoints=13, N=1, ctx =optional)
check isClose(value, correct, tol=1e-9)
test "gaussQuad func n=14":
let value = gaussQuad(f, xStart, xEnd, nPoints=14, N=1, ctx =optional)
check isClose(value, correct, tol=1e-9)
test "gaussQuad func n=15":
let value = gaussQuad(f, xStart, xEnd, nPoints=15, N=1, ctx =optional)
check isClose(value, correct, tol=1e-10)
test "gaussQuad func n=16":
let value = gaussQuad(f, xStart, xEnd, nPoints=16, N=1, ctx =optional)
check isClose(value, correct, tol=1e-10)
test "gaussQuad func n=17":
let value = gaussQuad(f, xStart, xEnd, nPoints=17, N=1, ctx =optional)
check isClose(value, correct, tol=1e-11)
test "gaussQuad func n=18":
let value = gaussQuad(f, xStart, xEnd, nPoints=18, N=1, ctx =optional)
check isClose(value, correct, tol=1e-12)
test "gaussQuad func n=19":
let value = gaussQuad(f, xStart, xEnd, nPoints=19, N=1, ctx =optional)
check isClose(value, correct, tol=1e-13)
test "gaussQuad func n=20":
let value = gaussQuad(f, xStart, xEnd, nPoints=20, N=1, ctx =optional)
check isClose(value, correct, tol=1e-14)
test "gaussQuad func n=20 N=10":
let value = gaussQuad(f, xStart, xEnd, nPoints=20, N=10, ctx =optional)
check isClose(value, correct, tol=1e-14)
test "gaussAdaptive func default":
let value = adaptiveGauss(f, xStart, xEnd, ctx =optional)
check isClose(value, correct, 1e-8)
test "trapz Vector func, N = 10":
let value = trapz(fVector, xStart, xEnd, N = 10, ctx = optionalVector)
check isClose(value, correctVector, tol = 1e-1)
test "trapz Vector func, N = 100":
let value = trapz(fVector, xStart, xEnd, N = 100, ctx = optionalVector)
check isClose(value, correctVector, tol = 1e-3)
test "simpson Vector func, N = 10":
let value = simpson(fVector, xStart, xEnd, N = 10, ctx = optionalVector)
check isClose(value, correctVector, tol = 1e-3)
test "simpson Vector func, N = 100":
let value = simpson(fVector, xStart, xEnd, N = 100, ctx = optionalVector)
check isClose(value, correctVector, tol = 1e-6)
test "adaptive Vector simpson, tol=1e-3":
let value = adaptiveSimpson(fVector, xStart, xEnd, tol=1e-3, ctx = optionalVector)
check isClose(value, correctVector, tol=1e-3)
test "adaptive Vector simpson, tol=1e-8":
let value = adaptiveSimpson(fVector, xStart, xEnd, tol=1e-8, ctx = optionalVector)
check isClose(value, correctVector, tol=1e-8)
test "romberg Vector func, depth = 3":
let value = romberg(fVector, xStart, xEnd, depth = 3, ctx = optionalVector, tol=1e-8)
check isClose(value, correctVector, tol = 1e-1)
test "romberg Vector func, default":
let value = romberg(fVector, xStart, xEnd, ctx = optionalVector)
check isClose(value, correctVector, tol = 1e-8)
test "gaussQuad Vector n=2":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=2, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1.5)
test "gaussQuad Vector n=3":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=3, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-1)
test "gaussQuad Vector n=4":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=4, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-2)
test "gaussQuad Vector n=5":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=5, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-4)
test "gaussQuad Vector n=6":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=6, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-5)
test "gaussQuad Vector n=7":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=7, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-6)
test "gaussQuad Vector n=8":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=8, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-6)
test "gaussQuad Vector n=9":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=9, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-7)
test "gaussQuad Vector n=10":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=10, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-7)
test "gaussQuad Vector n=10 N=10":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=10, N=10, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-8)
test "gaussQuad Vector n=11":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=11, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-8)
test "gaussQuad Vector n=12":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=12, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-8)
test "gaussQuad Vector n=13":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=13, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-9)
test "gaussQuad Vector n=14":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=14, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-9)
test "gaussQuad Vector n=15":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=15, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-10)
test "gaussQuad Vector n=16":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=16, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-10)
test "gaussQuad Vector n=17":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=17, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-11)
test "gaussQuad Vector n=18":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=18, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-12)
test "gaussQuad Vector n=19":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=19, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-13)
test "gaussQuad Vector n=20":
let value = gaussQuad(fVector, xStart, xEnd, nPoints=20, N=1, ctx =optionalVector)
check isClose(value, correctVector, tol=1e-14)
test "adaptiveGauss Vector default":
let value = adaptiveGauss(fVector, xStart, xEnd, ctx = optionalVector)
check isClose(value, correctVector, tol=1e-8)
test "trapz Tensor func, N = 10":
let value = trapz(fTensor, xStart, xEnd, N = 10, ctx = optionalTensor)
check isClose(value, correctTensor, tol = 1e-1)
test "trapz Tensor func, N = 100":
let value = trapz(fTensor, xStart, xEnd, N = 100, ctx = optionalTensor)
check isClose(value, correctTensor, tol = 1e-3)
test "simpson Tensor func, N = 10":
let value = simpson(fTensor, xStart, xEnd, N = 10, ctx = optionalTensor)
check isClose(value, correctTensor, tol = 1e-3)
test "simpson Tensor func, N = 100":
let value = simpson(fTensor, xStart, xEnd, N = 100, ctx = optionalTensor)
check isClose(value, correctTensor, tol = 1e-6)
test "adaptive Tensor simpson, tol=1e-3":
let value = adaptiveSimpson(fTensor, xStart, xEnd, tol=1e-3, ctx = optionalTensor)
check isClose(value, correctTensor, tol=1e-3)
test "adaptive Tensor simpson, tol=1e-8":
let value = adaptiveSimpson(fTensor, xStart, xEnd, tol=1e-8, ctx = optionalTensor)
check isClose(value, correctTensor, tol=1e-8)
test "romberg Tensor func, depth = 3":
let value = romberg(fTensor, xStart, xEnd, depth = 3, ctx = optionalTensor, tol=1e-8)
check isClose(value, correctTensor, tol = 1e-1)
test "romberg Tensor func, default":
let value = romberg(fTensor, xStart, xEnd, ctx = optionalTensor)
check isClose(value, correctTensor, tol = 1e-8)
test "gaussQuad Tensor n=2":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=2, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1.5)
test "gaussQuad Tensor n=3":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=3, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-1)
test "gaussQuad Tensor n=4":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=4, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-2)
test "gaussQuad Tensor n=5":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=5, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-4)
test "gaussQuad Tensor n=6":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=6, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-5)
test "gaussQuad Tensor n=7":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=7, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-6)
test "gaussQuad Tensor n=8":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=8, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-6)
test "gaussQuad Tensor n=9":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=9, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-7)
test "gaussQuad Tensor n=10":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=10, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-7)
test "gaussQuad Tensor n=10 N=10":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=10, N=10, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-8)
test "gaussQuad Tensor n=11":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=11, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-8)
test "gaussQuad Tensor n=12":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=12, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-8)
test "gaussQuad Tensor n=13":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=13, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-9)
test "gaussQuad Tensor n=14":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=14, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-9)
test "gaussQuad Tensor n=15":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=15, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-10)
test "gaussQuad Tensor n=16":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=16, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-10)
test "gaussQuad Tensor n=17":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=17, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-11)
test "gaussQuad Tensor n=18":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=18, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-12)
test "gaussQuad Tensor n=19":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=19, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-13)
test "gaussQuad Tensor n=20":
let value = gaussQuad(fTensor, xStart, xEnd, nPoints=20, N=1, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-14)
test "adaptiveGauss Tensor default":
let value = adaptiveGauss(fTensor, xStart, xEnd, ctx =optionalTensor)
check isClose(value, correctTensor, tol=1e-8)
test "cumGaussSpline":
let spline = cumGaussSpline(f, xStart, xEnd, ctx=optional)
let x = linspace(xStart, xEnd, 23)
var correct: seq[float]
for i in 0..x.high:
correct.add optional["a"] * sin(x[i])
for i in 0..correct.high:
check abs(spline.eval(x[i]) - correct[i]) < 1e-6
test "cumGauss":
let x = linspace(xStart, xEnd, 23)
let spline = cumGauss(f, x, ctx=optional)
var correct: seq[float]
for i in 0..x.high:
correct.add optional["a"] * sin(x[i])
for i in 0..correct.high:
check abs(spline[i] - correct[i]) < 1e-6
import measuremancer
suite "External integration tests: Measurement (Measuremancer) - integrating a complex type":
## tests integration of a complex type, in this case `Measurement[T]`.
# define two procedure we can integrate over with different algorithms
proc foo[T; U](x: U, ctx: NumContext[T, U]): T =
result = exp(x / ctx["a"])
proc sinFo[T; U](x: U, ctx: NumContext[T, U]): T = sin(x)
# define a `Measurement`
let a = measurement(4.71, 0.01) # we construct it with the proc instead of `±` to avoid having to
# compile with `--experimental:unicodeOperators`, which is only available
# from nim version 1.6
type T = typeof(a) # its type for convenience
var ctx = newNumContext[T, float]()
ctx["a"] = a
var exp = measurement(14.995, 0.031231)
template genTest(fn: untyped): untyped =
test "Measurement: " & $astToStr(fn):
let res = fn[T](foo, 1, 7, ctx = ctx)
# very rough comparison is fine for this test. It's supposed to test functionality, not
# exactness of the integration methods
check abs(res.value - exp.value) < 1e-3
check abs(res.error - exp.error) < 1e-3
genTest(simpson)
genTest(trapz)
genTest(romberg)
genTest(gaussQuad)
genTest(adaptiveSimpson)
genTest(adaptiveGaussLocal)
test "Measurement: adaptiveGauss":
# adaptive requires 2 generics
let res = adaptiveGauss[T, float](foo, 1, 7, ctx = ctx)
check abs(res.value - exp.value) < 1e-3
check abs(res.error - exp.error) < 1e-3
test "Measurement: adaptiveGauss in `Measurement` range":
let b = measurement(6.42, 0.03)
block SameVar:
# we integrate from `b` to `b`, that means (due to the nature of
# sin) our uncertainty should vanish. The value of the integration
# is =~= 0 too of course
let res = adaptiveGauss[T, T](sinFo, -b, b)
let exp = measurement(7.23e-17, 1.0065e-18)
check abs(res.value - exp.value) < 1e-16
check abs(res.error - exp.error) < 1e-16
block DifferentVar:
# for two *different* variables, the errors are correlated though. That
# means while the integration result is still =~= 0, the uncertainty isn't
let b2 = measurement(6.42, 0.03)
let res = adaptiveGauss[T, T](sinFo, -b, b2)
let exp = measurement(7.2390e-17, 0.0057865)
check abs(res.value - exp.value) < 1e-16
check abs(res.error - exp.error) < 1e-4