From c5fcf7179a83ef65c86c6a4a390029149e518649 Mon Sep 17 00:00:00 2001
From: Franciszek Malinka <franciszek.malinka@gmail.com>
Date: Tue, 5 Oct 2021 21:49:54 +0200
Subject: Duzy commit ze smieciami

---
 semestr-3/anm/pracownia2/brut.jl | 177 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 177 insertions(+)
 create mode 100644 semestr-3/anm/pracownia2/brut.jl

(limited to 'semestr-3/anm/pracownia2')

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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