local vegas_prepare = num.vegas_prepare
local ilist = iter.ilist
local function testdim(n)
 local lo,hi = 0,2
 local exact = n*(n+1)/2 * (hi^3 - lo^3)/3 * (hi-lo)^(n-1)
 local function integrand(x)
 local s = 0
 for k= 1, n do s = s + k*x[k]^2 end
 return s
 end
 local a, b = ilist(|| lo, n), ilist(|| hi, n)
 print("Integrating SUM_(k=1,"..n..") k*x[k]^2")
 local calls = 1e4*n
 local vegas_integ = vegas_prepare({N=n})
 local result,sigma,runs,cont=vegas_integ(integrand,a,b,calls)
 print( string.format([[
result = %.6f
sigma = %.6f
exact = %.6f
error = %.6f = %.2g sigma
calls = %.0f
==========================
]] ,result,sigma,exact, result - exact, math.abs(result - exact)/sigma,runs*calls))
 return result
end
local function demo1()
 local maxdim = 10
 local lo,hi = 0,2
 local results = {}
 local p = graph.plot('Integral of sum (i*x_i^2) (i=1..n)')
 p.clip, p.pad = false, true
 local exact = graph.filine(|n| n*(n+1)/2 * (hi^3 - lo^3)/3 * (hi-lo)^(n-1),maxdim)
 local computed = graph.filine(testdim,1,maxdim)
 p:addline(exact)
 p:add(computed, "blue", {{'marker', size=8}})
 p.xtitle="n"
 p:show()
end
local function getunitsphere(n)
 return function(x)
	 local s = 0
	 for k= 1, n do s = s + x[k]^2 end
	 return s < 1 and 1 or 0 end end local function demo2() local ln = graph.path(1, 2) -- 1-sphere = [-1, 1] (length 2) local max_dim = 14 for d=2, max_dim do print("==========================================") print("Calculating the volume of a unit "..d.."-sphere.") local a, b = ilist(&#124;&#124; 0, d), ilist(&#124;&#124; 1, d) local calls, n = d*1e4,1 local vegas_integ = num.vegas_prepare({N=d}) local res,sig,num,cont = vegas_integ(getunitsphere(d),a,b,calls) local fmt = "Volume = %.3f +/- %.3f " print(string.format(fmt,res*2^d,sig*2^d)) while(sig/res> 0.005) do
 print("Increasing accuracy, doubling number of calls...")
 res,sig,num = cont(calls*(2^n))
 print(string.format(fmt,res*2^d,sig*2^d))
 n=n+1
 end
 ln:line_to(d,res*2^d)
 end
 local p = graph.plot('Volume of a unit n-sphere')
 p.clip, p.pad = false, true
 p:addline(graph.fxline(&#124;n&#124; math.pi^(n/2) / sf.gamma(1+n/2), 1, max_dim))
 p:legend('exact value', 'red', 'line')
 p:addline(ln, "blue", {{'marker', size=6}})
 p:legend('calculated', 'blue', 'circle', {{'stroke'}})
 p.xtitle="n"
 p.ytitle="V"
 p:show()
end
return {'VEGAS Monte Carlo integration', {
 {
 name= 'vegas',
 f = demo1,
 description = 'Integrate 9 n-dimensional functions sum(i*(x_i^2))'
 },
 {
 name= 'sphere',
 f = demo2,
 description = 'Calculate the volume of a unit n-sphere (n=2..14)'
 }
}}
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