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Question: How to speed up Maximize and fsolve on this integral

Question:How to speed up Maximize and fsolve on this integral

C_R 2592 Maple 2023
performance integration fsolve numeric maximize
December 12 2023
1

The attached integral is plotted quite quickly, but the search for the maximum and for the inverses of function values takes time. Can the search be accelerated or is the integrand simply too complicated?

Int(22.89730452*exp(-3.373250126*10^6*x^2)*sqrt(2)*(erf(1298.701299*sqrt(2)*(sqrt(2.500000000*10^(-7)-x^2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*sqrt(2)*(sqrt(2.500000000*10^(-7)-x^2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3, method = _d01ajc)

Int(22.89730452*exp(-3373250.126*x^2)*2^(1/2)*(erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3, method = _d01ajc)

 (1)

plot(Int(22.89730452*exp(-3373250.126*x^2)*2^(1/2)*(erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3, method = _d01ajc), y = 0 .. .25)

  

unapply(int(op(1 .. 2, Int(22.89730452*exp(-3373250.126*x^2)*2^(1/2)*(erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3, method = _d01ajc))), y)

proc (y) options operator, arrow; int(22.89730452*exp(-3373250.126*x^2)*2^(1/2)*(erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*2^(1/2)*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3) end proc

 (2)

CodeTools:-Usage(Optimization:-Maximize(proc (y) options operator, arrow; int(22.89730452*exp(-3373250.126*(x^2))*(2^(1/2))*(erf(1298.701299*(2^(1/2))*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*(2^(1/2))*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3) end proc, 0 .. .15))

[HFloat(0.0356073228322897), Vector[column](%id = 36893490875463313756)]

 (3)

CodeTools:-Usage(fsolve((proc (y) options operator, arrow; int(22.89730452*exp(-3373250.126*(x^2))*(2^(1/2))*(erf(1298.701299*(2^(1/2))*((0.2500000000e-6-x^2)^(1/2)+(1/40)*y-0.1540000000e-2))+erf(1298.701299*(2^(1/2))*((0.2500000000e-6-x^2)^(1/2)-(1/40)*y+0.1540000000e-2))), x = -0.5000000000e-3 .. 0.5000000000e-3) end proc)-0.5e-2, 0 .. .15, maxsols = 2, method = subdivide))

0.2456248637e-1, 0.9863751363e-1

 (4)

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