MINOTAUR (coMposition INtegral cOmpuTing quAdratURe)

Julia package for computing quadrature for integrals of composed functions

$$\int_{\tau_1}^{\tau_2} f(g(\tau)) \mathrm{d}\tau$$

Examples

Error w.r.t. the numebr of discretization nodes

The computation of the integral

$$\int_{e^{-\frac{11\pi}{4}}}^{e^{\frac{\pi}{4}}} \sin\left(\ln \tau \right) \mathrm{d}\tau = \left[\frac{\tau}{2}\left(\sin\left(\ln x\right) - \cos\left(\ln x\right)\right)\right]_{e^{-\frac{11\pi}{4}}}^{e^{\frac{\pi}{4}}}$$

can be carried out using MINOTAUR and compared to the result of another quadrature with the same number of discretization nodes not adjusted to the $\sin$ function (linearly spaced nodes $\tau_i = \tau_{i-1} + \Delta\tau$) with the code:

# adjusted description of the integral
f     = (r::Cdouble->sin(r))
g     = (τ::Cdouble->log(τ))
g_inv = (r::Cdouble->exp(r))

# non-adjusted description of the integral
f_lin     = (r::Cdouble->sin(log(r)))
g_lin     = (τ::Cdouble->τ)
g_lin_inv = (r::Cdouble->r)

# primitive
FG    = (τ::Cdouble->0.5τ*(sin(log(τ)) - cos(log(τ)))) 

# limits
τ_min = exp(-3π/4-2π) # 0.001 
τ_max = exp(π/4) # 2.0
Δr0 = π/2.0 # big enough so that the optional argument Nr_min is the number of discretization nodes to compute the quadrature

# reference value of the integral
val = FG(τ_max) - FG(τ_min)

# compute both quadratures for several level of resolution
Nr_min_array     = [10; 20; 50; 100; 200; 500; 1000; 2000; 5000; 10000; 20000]
int_fg_array     = zeros(Cdouble,length(Nr_min_array))
int_fg_lin_array = zeros(Cdouble,length(Nr_min_array))
for j in eachindex(Nr_min_array)
  # adjusted quadrature
  int_fg_array[j]     = quadrature_fg(f,g,g_inv,τ_min,τ_max,Δr0;Nr_min=Nr_min_array[j])
  # linear quadrature
  int_fg_lin_array[j] = quadrature_fg(f_lin,g_lin,g_lin_inv,τ_min,τ_max,Δr0;Nr_min=Nr_min_array[j])
end

In this figure: left) the discretization adjusted for $f$ is plotted against the linear discretization , and right) the absolute value of the error between the quadrature and the true value of the integral (in blue, using the adjusted discretization, in orange, using the linear discretization of $\tau$)

Application to different function (using 100 discretization nodes)

Some cases are favorable to the adjusted discretization, but some are better suited for linear discretization.

	$f(r)=\sin r$	$f(r)=\cos r$	$f(r)=r^2$	$f(r)=\frac{1}{r}$	$f(r)=\frac{1}{r}$
	$g(\tau)=\log(\tau)$	$g(\tau)=\log(\tau)$	$g(\tau)=\log(\tau)$	$g(\tau) = \sqrt{\tau}$	$g(\tau)=\tan \tau$
Method	$[e^{-\frac{3\pi}{4}-2\pi},e^{\frac{\pi}{4}}]$	$[e^{-\frac{3\pi}{4}-2\pi},e^{\frac{\pi}{4}}]$	$[\frac{1}{4},2]$	$[\frac{1}{100},1]$	$[\frac{\pi}{100},\frac{\pi}{4}]$
Analytical	0.0	1.551	0.5147	1.8	3.1140
linear quadrature error	0.015	0.014	0.0003	0.0004	0.005
adjusted quadrature error	0.001	0.001	0.0001	0.0040	0.008

Install

In the Julia REPL:

] add https://github.com/matthewozon/MINOTAUR