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May 22, 2017 21:03
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function quad42(f,a,b,tol; α=0.9, Θ=3.0) | |
x, val, h = a, 0, (b-a)/10 | |
xx, f₀ = [x], f(x) | |
err, evals = 0.0, 1 | |
while x < b | |
fₘ, f₁ = f(x+h/2), f(x+h) # effektive Stufenzahl 2 (2 f-Auswertungen pro Schritt) | |
evals += 2 | |
Q2 = h*(f₀+f₁)/2 # Trapezregel (Ordnung 2) | |
Q4 = h*(f₀+4*fₘ+f₁)/6 # Simpson-Regel (Ordnung 4) | |
if (ϵ = abs(Q4-Q2)) < tol | |
err += ϵ | |
x += h | |
push!(xx,x) # füge Knoten zum adaptiven Gitter hinzu | |
f₀ = f₁ # FSAL-Trick | |
val += Q4 | |
end | |
h = min(0.9h*(tol/ϵ)^(1/3),3h,b-x) # adaptive Schrittweitenformel | |
end | |
return val, err, evals | |
end | |
function quad42₃(f,a,b,tol; α=0.9, Θ=3.0) | |
x, val, h = a, 0, (b-a)/10 | |
xx, f₀ = [x], f(x) | |
err, evals = 0.0, 1 | |
while x < b | |
fₘ, f₁ = f(x+h/2), f(x+h) # effektive Stufenzahl 2 (2 f-Auswertungen pro Schritt) | |
evals += 2 | |
Q2 = h*(f₀+f₁)/2 # Trapezregel (Ordnung 2) | |
Q4 = h*(f₀+4*fₘ+f₁)/6 # Simpson-Regel (Ordnung 4) | |
ϵₘ = abs(Q4-Q2) | |
if (ϵ = ϵₘ ^ 2 / abs(Q4)) < tol | |
err += ϵ | |
x += h | |
push!(xx,x) # füge Knoten zum adaptiven Gitter hinzu | |
f₀ = f₁ # FSAL-Trick | |
val += Q4 | |
end | |
h = min(0.9h*(tol/ϵ)^(1/3),3h,b-x) # adaptive Schrittweitenformel | |
end | |
return val, err, evals | |
end | |
using FastGaussQuadrature | |
function gauss_legendre(s) | |
# für Interval [0,1]; Methode von Golub/Welsh (1969) | |
k = 1:s-1 | |
β = k./sqrt((2*k-1).*(2*k+1)); | |
T = SymTridiagonal(zeros(s),β) | |
λ, Q = eig(T) | |
c = (1+λ)/2 # Knoten | |
w = Q[1,:].^2 # Gewichte | |
P = Q./(Q[1,:]') # Matrix für interpolatorische Formeln | |
return c,w,P | |
end | |
function embedded_gauss_legendre(m) | |
# eingebette Formeln der Stufenzahlen (4m+3,4m+2,2m) und Ordnungen (8m+3,4m+2,2m) | |
global c, w, dw1, dw2 | |
s, s1, s2 = 4m+3, 4m+2, 2m | |
w1, w2, e1 = zeros(s), zeros(s), zeros(s) | |
e1[1] = 1 | |
ind1 = collect(1:s) | |
ind2 = collect(1:s) | |
deleteat!(ind2,1:2:s) | |
deleteat!(ind2,m+1) | |
deleteat!(ind1,2m+2) | |
c, w, P = gauss_legendre(s) | |
k1 = length(ind1); w1[ind1] = P[1:k1,ind1]\e1[1:k1] | |
k2 = length(ind2); w2[ind2] = P[1:k2,ind2]\e1[1:k2] | |
dw1 = w-w1 | |
dw2 = w-w2 | |
return 2s | |
end | |
function quadrature_step(f,x,h) | |
global nfcn | |
fval = f.(x+h*c) | |
evals = length(x) | |
nfcn += length(c) | |
err1 = h*dot(dw1,fval) | |
err2 = h*dot(dw2,fval) | |
return h*dot(w,fval), abs(err1)*(err1/err2)^2, evals # Wert der Quadraturformel, lokale Fehlerschätzung | |
end | |
function AdaptiveGaussQuadrature(f,a,b,tol,m) | |
global nfcn = 0 | |
p = embedded_gauss_legendre(m) | |
x, val, err, h = a, 0, 0, (b-a)/10 | |
evals = 0 | |
xx = [x] | |
while x < b | |
Qp, ϵ, evs = quadrature_step(f,x,h) | |
evals += evs | |
if ϵ <= tol | |
x += h | |
push!(xx,x) # füge Knoten zum adaptiven Gitter hinzu | |
val += Qp | |
err += ϵ | |
end | |
h = min(0.75h*(tol/ϵ)^(1/(p+1)),2h,b-x) # adaptive Schrittweitenformel | |
end | |
return val, err, evals | |
end | |
fns = [ | |
(-1.0, 1.0, (x) -> 1.0/(1e-4 + x^2)), | |
( 0.0, 1.0, (x) -> sqrt(x)*log(x)), | |
(10.0, 110.0, (x) -> 2 + sin(3*cos(0.002 * (x-40)^2))) | |
] | |
Qs = [ | |
("quad42", quad42), | |
("quad42₂", quad42₃), | |
("gauss₁", (f,a,b,tol) -> AdaptiveGaussQuadrature(f,a,b,tol,1)), | |
("gauss₂", (f,a,b,tol) -> AdaptiveGaussQuadrature(f,a,b,tol,2)), | |
("gauss₃", (f,a,b,tol) -> AdaptiveGaussQuadrature(f,a,b,tol,3)) | |
] | |
function plot_it(fn, Qs) | |
a,b,f = fn | |
pows = 10.^(-1.0 .* (2:15)) | |
tols = reshape([pows 0.5.*pows]', 28) | |
xscale("log") | |
yscale("log") | |
for Q in Qs | |
name, I = Q | |
vals = ((tol) -> | |
let | |
_,err,evals = I(f, a, b, tol) | |
return 1.0*evals, err | |
end).(tols) | |
vals = [p[i] for p in vals for i=1:2] | |
vals = reshape(vals, (2, trunc(Int, length(vals)/2))) | |
loglog(vals[1,:], vals[2,:], ".", label=name) | |
end | |
grid("on") | |
xlabel("# f Auswertungen") | |
ylabel("Fehler") | |
legend() | |
end | |
plot_it(fns[1], Qs) | |
plot_it(fns[2], Qs) | |
plot_it(fns[3], Qs) |
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