Stare semestry, niepoukladane

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
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+++ 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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