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Diffstat (limited to 'semestr-3/anm/pracownia2/brut.jl')
| -rw-r--r-- | semestr-3/anm/pracownia2/brut.jl | 177 |
1 files changed, 177 insertions, 0 deletions
diff --git a/semestr-3/anm/pracownia2/brut.jl b/semestr-3/anm/pracownia2/brut.jl new file mode 100644 index 0000000..d8eb0a1 --- /dev/null +++ b/semestr-3/anm/pracownia2/brut.jl @@ -0,0 +1,177 @@ +using LinearAlgebra
+using Polynomials
+using Printf
+using OffsetArrays
+using FFTW
+
+function emptyArray(type, size)
+ return OffsetVector{type}(undef, 0:(size - 1))
+end
+
+function naive_cc(f, N)
+ a = emptyArray(Float64, Int(N) + 1)
+ for j in 0.0:1.0:N
+ for k in 0.0:1.0:N
+ u = cos(k * pi / N)
+ uj = cos(j * k * pi / N)
+ val = f(u) * uj
+ if k == 0 || k == N
+ val /= 2.0
+ end
+ a[Int(j)] += val
+ end
+ end
+ return a * (2.0 / N)
+end
+
+function naive_approx(f, N)
+ b = emptyArray(Float64, N)
+ a = naive_cc(f, N)
+ a[0] /= 2.0
+ a[N] /= 2.0
+
+ print(a)
+ result = 0.0
+
+ for k in 0:2:N
+ result += a[k] / Float64(1 - k * k)
+ end
+ return result * 2.0
+end
+
+function naive_approx2(f, N)
+ b = emptyArray(Float64, N)
+ N = Float64(N)
+ a = naive_cc(f, N)
+ result = 0.0
+
+ for k in 1:1:Int(floor(N / 2.0))
+ b[2 * k - 1] = a[2 * k - 2] - a[2 * k]
+ b[2 * k - 1] /= (Float64(k) * 4.0 - 2.0)
+ result += b[2 * k - 1]
+ end
+ return result * 2.0
+end
+
+function smart_cc(f, N)
+ x = [cos(pi * i / N) for i in 0:N]
+ fx = f.(x) / (2 * N)
+ g = real(fft(vcat(fx, fx[N:-1:2])))
+ a = [g[1] * 2; g[2:N] + g[(2 * N):-1:(N + 2)]; g[N + 1] * 2]
+ return a
+end
+
+function clenshaw_curtis_coeffs(f, N)
+ x = [cos(pi * i / N) for i in 0:N]
+ fx = f.(x) / (2 * N)
+ g = real(fft(vcat(fx, fx[N:-1:2])))
+ return [g[1]; g[2:N] + g[(2 * N):-1:(N + 2)]; g[N + 1]]
+end
+
+function clenshaw_curtis(f, N)
+ a = clenshaw_curtis_coeffs(f, N)
+ w = zeros(length(a))
+ w[1:2:end] = 2 ./ (1 .- (0:2:N).^2 )
+ # w oraz a są malejące, więc lepiej dodawać ich iloczyny od końca
+ LinearAlgebra.dot(reverse(w), reverse(a))
+end
+
+global MAX_ITER = 2^18
+
+function clenshaw_curtis_with_eps(f, eps)
+ N = 4
+ while N <= MAX_ITER
+ res1 = clenshaw_curtis(f, N)
+ res2 = clenshaw_curtis(f, N - 1)
+ if abs(res1 - res2) < eps * res1 || N >= MAX_ITER
+ return res1
+ end
+ N *= 2
+ end
+end
+
+
+# zera n-tego wielomianu Czebyszewa
+function chebyshev_nodes(n)
+ return [cos((2.0 * k - 1.0) * pi / (2.0 * n)) for k in 1:n]
+end
+
+# kwadratura Czebyszewa-Gaussa
+# współczynniki stałe równe π/N
+# węzły to zera n-tego wielomianu Czebyszewa
+function gauss_chebyshev(f, N)
+ x = chebyshev_nodes(N)
+ res = 0.0
+ for i in 1:N
+ res += f(x[i]) * sqrt(1.0 - x[i] * x[i])
+ end
+ return res * pi / N
+end
+
+# kwadratura Gaussa-Legendre'a:
+# współczynniki w_i = 2(q_1,i)^2, gdzie q_1,i to pierwsza współrzędna
+# i-tego wektora własnego macierzy trójprzekątniowej
+# węzły x_i to zera N-tego wielomianu Legendre'a
+function gauss_legendre(f, N)
+ # wyliczanie wartości i wektorów własnych macierzy trójprzekątniowej
+ # (Golub-Welsch algorithm)
+ # wartości własne to zera wielomianu Legendre'a
+ X, Q = eigen(SymTridiagonal(zeros(N), [n / sqrt(4.0n^2 - 1.0) for n = 1:N - 1]))
+
+ res = 0.0
+ for i in 1:N
+ w = 2.0 * (Q[1, i])^2
+ res += f(X[i]) * w
+ end
+
+ return res
+end
+
+# uruchamianie kwadratury dla funkcji f, N węzłów,
+# na przedziale (a, b)
+function quadrature(f, quadrature_fun, N=ITER, a=-1.0, b=1.0)
+ return (b - a) / 2 * quadrature_fun(x -> f(x * (b - a) / 2 + (a + b) / 2), N)
+end
+
+# Testowanie-------------------------------------------------------------
+
+funs = [exp, x -> 1.0 / ((x - 1.01) * (x - 1.01)), x -> 10x^4 + 4x^3 + 2x - 1, x -> cos(1000.0x),
+ x -> (3x^2 + 4) / (x - 1.1), abs, x -> 1.0, x -> cos(100.0x) * cos(100.0x), x -> 1 / (x^4 + x^2 + 0.9),
+ x -> 1.0 / (1.0 + x^4), x -> 2.0 / (2.0 + sin(10pi * x))]
+
+# f - całka nieoznaczona
+# obliczanie całki oznaczonej na przedziale (a, b)
+function definete(f)
+ return (a, b) -> f(b) - f(a)
+end
+
+
+# ręcznie policzone całki oznaczone (lub obliczone wyniki)
+# dla testowanych funkcji
+results = [definete(exp), definete(x -> 1.0 / (1.01 - x)), definete(x -> 2x^5 + x^4 + x^2 - x), definete(x -> sin(1000.0x) / 1000.0),
+ definete(x -> 1.5x^2 + 3.3x + 7.63log(abs(x - 1.1))), definete(x -> (x >= 0) ? ((x^2) / 2.0) : (-(x^2) / 2.0)), definete(x -> x),
+ definete(x -> (200.0x + sin(200.0x)) / 400.0), definete(x -> -0.278185log(x^2 - 0.947294x + 0.948683) +
+ 0.278185log(x^2 + 0.947294x + 0.948683) + 0.309633atan(0.587487 * (2x - 0.947294)) + 0.309633atan(0.587487 * (2.0x + 0.947294))),
+ definete(x -> 1.0 / (4.0 * sqrt(2)) * (-log(abs(x^2 - sqrt(2)x + 1)) + log(abs(x^2 + sqrt(2)x + 1)) - 2.0atan(1 - sqrt(2)x)
+ + 2.0atan(1 + sqrt(2)x))), (a, b) -> 4.0 / sqrt(3)]
+
+
+function rel_error(a, b)
+ return abs((a - b) / a)
+end
+
+function test(quadrature_fun, fun_nr, a=-1.0, b=1.0, N=ITER)
+ my_res = quadrature(funs[fun_nr], quadrature_fun, N, a, b)
+ rel_e = rel_error(results[fun_nr](a, b), my_res)
+ abs_e = abs(results[fun_nr](a, b) - my_res)
+
+ return (rel_e, abs_e)
+end
+
+function print_coefficients_of_cc(f, N, ile_pierwszych)
+ a = clenshaw_curtis_coeffs(f, N)
+ for i in 1:ile_pierwszych
+ # println(string(i - 1, " & ", a[i], " \\\\"))
+ @printf("%d & %.20f \\\\\n", i - 1, a[i])
+ end
+end
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