Zernike Transform - Elliptical Gaussian 10th order.ipynb

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       "toarray (generic function with 2 methods)"
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   "source": [
    "using SpecialFunctions\n",
    "using QuantumOptics\n",
    "\n",
    "function fidelity(ρ, σ)\n",
    "    tr(sqrt(sqrt(ρ) * σ * sqrt(ρ)))^(1/2)\n",
    "end\n",
    "\n",
    "function R(n::Int64, m::Int64, ρ::Float64)\n",
    "    if (n - m) % 2 != 0\n",
    "        0\n",
    "    else\n",
    "        function summand(k)\n",
    "            n = big(n)\n",
    "            k = big(k)\n",
    "            (-1)^k * factorial(n-k)/(factorial(k)*factorial(Int((n+m)/2) - k)*factorial(Int((n-m)/2) - k))*(ρ)^(n-2*k)\n",
    "        end\n",
    "        mapreduce(summand, +, Array(range(0, stop=Int((n-m)/2), step=1)))\n",
    "    end\n",
    "end\n",
    "\n",
    "function sample_across_unit_disk(f, discretization)\n",
    "    x = y = Array(range(-1, 1, length=discretization))\n",
    "    z = Array{typeof(f(0.,0.))}(UndefInitializer(), length(x), length(y))\n",
    "    for (i, xx) in enumerate(x)\n",
    "        for (j, yy) in enumerate(y)\n",
    "            if xx^2 + yy^2 > 1\n",
    "                if typeof(f(0.,0.)) == Tuple{Float64,Float64}\n",
    "                    z[i, j] = (0.,0.)\n",
    "                else\n",
    "                    z[i, j] = 0.\n",
    "                end\n",
    "            else\n",
    "                ρ = sqrt(xx^2 + yy^2)\n",
    "                ϕ = atan(yy, xx)\n",
    "                res = f(ρ, ϕ)\n",
    "                z[i, j] = res\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    x, y, z\n",
    "end\n",
    "\n",
    "function unpack_zernike(zernike_coefficients_even, zernike_coefficients_odd, ρ, ω, t)\n",
    "    \"\"\"Each Zernike array is of arbitrary length, but they are ordered by order. The first is Z00 to Zn0 where\n",
    "    n is the number of 0th order zernike polynomials being considered.\"\"\"\n",
    "    Pevens = []\n",
    "    eventuples = []\n",
    "    for (n, zeven) in enumerate(zernike_coefficients_even)\n",
    "        neven = n - 1\n",
    "        push!(Pevens, [zeven[i] * R(neven, i-1, ρ) for i in Array(range(1, length(zeven), step=1))])\n",
    "    end\n",
    "    for i in range(1, length(Pevens), step=1)\n",
    "        rotationeven = - (i - 1) * ω * t\n",
    "        total = 0\n",
    "        for peven in Pevens\n",
    "            if length(peven) >= i\n",
    "                total += peven[i]\n",
    "            end\n",
    "        end    \n",
    "        push!(eventuples, (i - 1, total, rotationeven))\n",
    "    end\n",
    "\n",
    "    Podds = []\n",
    "    oddtuples = []\n",
    "    for (n, zodd) in enumerate(zernike_coefficients_odd)\n",
    "        nodd = n - 1\n",
    "        push!(Podds, [zodd[i] * R(nodd, i-1, ρ) for i in Array(range(1, length(zodd), step=1))])\n",
    "    end\n",
    "    for i in range(1, length(Podds), step=1)\n",
    "        rotationodd = - (i - 1) * ω * t\n",
    "        total = 0\n",
    "        for podd in Podds\n",
    "            if length(podd) >= i\n",
    "                total += podd[i]\n",
    "            end\n",
    "        end    \n",
    "        push!(oddtuples, (i - 1, total, rotationodd))\n",
    "    end\n",
    "    eventuples, oddtuples\n",
    "end\n",
    "\n",
    "function infidelity_across_disk(F1, F2)\n",
    "    function infidelity_polar(ρ, ϕ)\n",
    "        ψ1 = F1(ρ, ϕ).data\n",
    "        ψ2 = F2(ρ, ϕ).data\n",
    "        1 - real(fidelity(ψ1, ψ2))\n",
    "    end\n",
    "end\n",
    "\n",
    "function H_odf(ρ, ϕ, t, zernike_coefficients_even, zernike_coefficients_odd, U, ψ, μ, ω)\n",
    "    eventuples, oddtuples = unpack_zernike(zernike_coefficients_even, zernike_coefficients_odd, ρ, ω, t)\n",
    "    eventerm = oddterm = 0\n",
    "    if length(eventuples) > 0\n",
    "        eventerm = mapreduce(x->x[2]*cos(x[1]*ϕ + x[3]), +, eventuples)\n",
    "    end\n",
    "    if length(oddtuples) > 0\n",
    "        oddterm = mapreduce(x->x[2]*sin(x[1]*ϕ + x[3]), +, oddtuples)\n",
    "    end\n",
    "    U * cos(-μ*t + ψ + eventerm + oddterm) \n",
    "end\n",
    "\n",
    "function least_squares_solution(mat, y)\n",
    "    inv(transpose(mat) * mat)*transpose(mat) * y\n",
    "end\n",
    "\n",
    "toarray(s::Union{Set, Array}) = [toarray.(s)...]\n",
    "toarray(v::Number) = v"
   ]
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   "source": [
    "max_order = 10\n",
    "discretization = 50\n",
    "zernike_samples = []\n",
    "ordering = []\n",
    "for n in range(0, max_order, step=1)\n",
    "    for m in range(0, n, step=1)\n",
    "        if (n-m)%2 == 0\n",
    "            Zeven(ρ, ϕ) = R(n, m, ρ) * cos(m * ϕ)\n",
    "            push!(zernike_samples, sample_across_unit_disk(Zeven, discretization)[3][:])\n",
    "            push!(ordering, (n, m))\n",
    "            if m !=0\n",
    "                Zodd(ρ, ϕ) = R(n, m, ρ) * sin(m * ϕ)\n",
    "                push!(zernike_samples, sample_across_unit_disk(Zodd, discretization)[3][:])\n",
    "                push!(ordering, (n, -m))\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "end\n",
    "Z = reduce(hcat, zernike_samples);"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "σ1 = .1\n",
    "σ2 = 1\n",
    "amp = 1 # Used to make the small angle approximation work better.\n",
    "function gaussian(σ1, σ2)\n",
    "    function func(ρ, ϕ)\n",
    "        x = ρ*cos(ϕ)\n",
    "        y = ρ*sin(ϕ)\n",
    "        amp * exp(-x^2/σ1^2 + -y^2/σ2^2)\n",
    "    end\n",
    "end\n",
    "xx, yy, zz = sample_across_unit_disk(gaussian(σ1, σ2), discretization)\n",
    "G = zz[:];"
   ]
  },
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    "coefficients = least_squares_solution(Z, G)"
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   "source": [
    "reconstruction = zeros(discretization*discretization)\n",
    "for (i, zernike_sample) in enumerate(zernike_samples)\n",
    "    reconstruction += coefficients[i] * zernike_sample\n",
    "end"
   ]
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      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(heatmap(xx, yy, reshape(reconstruction, (discretization, discretization))), aspectratio = true, xlabel=\"x\", ylabel=\"y\", title=\"Reconstruction of a Gaussian using n <= 10\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
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       "  2232.76,1349.85 2232.76,1349.85 2256.76,1349.85 2232.76,1349.85 2232.76,1123.06 2256.76,1123.06 2232.76,1123.06 2232.76,896.268 2256.76,896.268 2232.76,896.268 \n",
       "  2232.76,669.474 2256.76,669.474 2232.76,669.474 2232.76,442.681 2256.76,442.681 2232.76,442.681 2232.76,215.887 2256.76,215.887 2232.76,215.887 2232.76,196.797 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "diff = sample_across_unit_disk(gaussian(σ1,σ2), discretization)[3] - reshape(reconstruction, (discretization, discretization))\n",
    "diff = diff.*diff\n",
    "plot(heatmap(xx, yy, diff), aspectratio = true, xlabel=\"x\", ylabel=\"y\", title=\"Error in Reconstruction of a Gaussian using n <= 10\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
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    "plot(reshape(reconstruction, (discretization, discretization))[:,Int64(discretization/2)])"
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   "execution_count": 9,
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       "  1908.39,523.821 1950.83,570.72 1993.27,617.095 2035.72,662.158 2078.16,705.596 2120.6,748.203 2163.04,792.86 2205.48,845.971 2247.92,919.486 2290.37,1445.72 \n",
       "  \n",
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      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(reshape(reconstruction, (discretization, discretization))[Int64(discretization/2), :])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "lookup = Dict(zip(ordering, coefficients));\n",
    "zernike_even_coefficients = []\n",
    "zernike_odd_coefficients = []\n",
    "curr_even = []\n",
    "curr_odd = []\n",
    "for n in range(0, max_order, step=1)\n",
    "    for m in range(0, n, step=1)\n",
    "        if (n-m)%2 != 0\n",
    "            push!(curr_even, 0)\n",
    "            push!(curr_odd, 0)\n",
    "        else\n",
    "            if m == 0\n",
    "                push!(curr_even, lookup[(n, m)])\n",
    "                push!(curr_odd, lookup[(n, -m)])\n",
    "            else\n",
    "                push!(curr_even, lookup[(n, m)])\n",
    "                push!(curr_odd, lookup[(n, -m)])\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    push!(zernike_even_coefficients, curr_even)\n",
    "    push!(zernike_odd_coefficients, curr_odd)\n",
    "    curr_even = []\n",
    "    curr_odd = []\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gaussian_spin_profile (generic function with 1 method)"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sequential_exact_evolution_evaluator_factory(ψ0, T, zernikeeven, zernikeodd, U, θ, ω, b)\n",
    "    \"\"\"Apply all the zernike coefficients given, in order, for time T each.\"\"\"\n",
    "    orders = range(0, max(length(zernikeeven), length(zernikeodd)) - 1, step=1)\n",
    "    function evaluator(ρ, ϕ)\n",
    "        ψ = ψ0\n",
    "        for order in orders\n",
    "            μ = order * ω\n",
    "            H(t, _) = H_odf(ρ, ϕ, t, zernikeeven, zernikeodd, U, θ, μ, ω)*sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "            _, ψ = timeevolution.master_dynamic(T, ψ, H; rates=[Γ])\n",
    "            ψ = last(ψ)\n",
    "        end\n",
    "        ψ\n",
    "    end\n",
    "end\n",
    "\n",
    "function gaussian_spin_profile(ρ, ϕ)\n",
    "    ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "    H(t, _) = gaussian(σ1, σ2)(ρ, ϕ) * sigmaz(b), [sigmam(b)], [dagger(sigmam(b))]\n",
    "    evolution_time = π/2\n",
    "    step_size = evolution_time/1\n",
    "    T = [0.0:step_size:evolution_time;];\n",
    "    _, ψ = timeevolution.master_dynamic(T, ψ0, H; rates=[Γ])\n",
    "    last(ψ)\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [],
   "source": [
    "Γ = 1/62\n",
    "ω = 2*π*180E3\n",
    "θ = -π/2;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "(::var\"#evaluator#14\"{Ket{SpinBasis{1//2,Int64},Array{Complex{Float64},1}},Array{Float64,1},Array{Any,1},Array{Any,1},Float64,Float64,Float64,SpinBasis{1//2,Int64},StepRange{Int64,Int64}}) (generic function with 1 method)"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "b = SpinBasis(1//2)\n",
    "ψ0 = 1/sqrt(2) * (spindown(b) + spinup(b))\n",
    "evolution_time =  50E-6\n",
    "U = π/(evolution_time)\n",
    "step_size = evolution_time\n",
    "T = [0.0:step_size:evolution_time;];\n",
    "sequential_exact_evolution = sequential_exact_evolution_evaluator_factory(ψ0, T, zernike_even_coefficients, zernike_odd_coefficients, U, θ, ω, b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "max_ρ = 1\n",
    "min_ρ = .0\n",
    "ρs = range(min_ρ, max_ρ, step=max_ρ/20)\n",
    "angle = 0\n",
    "target2 = [Real.(expect(sigmax(b), gaussian_spin_profile(ρ, angle))) for ρ in ρs]\n",
    "y2s = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, angle))) for ρ in ρs];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ρs, [target2, y2s], labels=[\"Gaussian Hamiltonian\" \"Zernike Reconstruction (Exact Evolution)\"], legend=:bottomright, ylabel=\"<σx>\", xlabel=\"x\", title=\"Comparing Evolution Under a Gaussian Hamiltonian\\n to Evolution Under a Zernike-decomposed Hamiltonian\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
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      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rec = reshape(-2 * reconstruction .+ 1, (discretization, discretization))[Int64(discretization/2):discretization,Int64(discretization/2)]\n",
    "exact = [-2 * gaussian(σ1, σ2)(2/discretization * i , 0) + 1 for i in range(0., discretization/2, step=1)]\n",
    "plot([exact, rec])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [],
   "source": [
    "using Plots\n",
    "max_ρ = 1\n",
    "min_ρ = .0\n",
    "ρs = range(min_ρ, max_ρ, step=max_ρ/20)\n",
    "angle = π/2\n",
    "target = [Real.(expect(sigmax(b), gaussian_spin_profile(ρ, angle))) for ρ in ρs]\n",
    "ys = [Real.(expect(sigmax(b), sequential_exact_evolution(ρ, angle))) for ρ in ρs];"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "plot(ρs, [target, ys], labels=[\"Gaussian Hamiltonian\" \"Zernike Reconstruction (Exact Evolution)\"], legend=:bottomright, ylabel=\"<σx>\", xlabel=\"y\", title=\"Comparing Evolution Under a Gaussian Hamiltonian\\n to Evolution Under a Zernike-decomposed Hamiltonian (y)\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "scrolled": false
   },
   "outputs": [
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      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rec = reshape(-2 * reconstruction .+ 1, (discretization, discretization))[Int64(discretization/2), Int64(discretization/2):discretization]\n",
    "exact = [-2 * gaussian(σ1, σ2)(2/discretization * i , π/2) + 1 for i in range(0., discretization/2, step=1)]\n",
    "plot([exact, rec])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#  Plot \\sigma as a function of Zernike polynomials that we need to converge."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "gaussian (generic function with 1 method)"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "discretization = 50\n",
    "function Zmat(max_order)\n",
    "    zernike_samples = []\n",
    "    ordering = []\n",
    "    for n in range(0, max_order, step=1)\n",
    "        for m in range(0, n, step=1)\n",
    "            if (n-m)%2 == 0\n",
    "                Zeven(ρ, ϕ) = R(n, m, ρ) * cos(m * ϕ)\n",
    "                push!(zernike_samples, sample_across_unit_disk(Zeven, discretization)[3][:])\n",
    "                push!(ordering, (n, m))\n",
    "                if m !=0\n",
    "                    Zodd(ρ, ϕ) = R(n, m, ρ) * sin(m * ϕ)\n",
    "                    push!(zernike_samples, sample_across_unit_disk(Zodd, discretization)[3][:])\n",
    "                    push!(ordering, (n, -m))\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    Z = reduce(hcat, zernike_samples);\n",
    "end\n",
    "\n",
    "function Zsamples(max_order)\n",
    "    zernike_samples = []\n",
    "    ordering = []\n",
    "    for n in range(0, max_order, step=1)\n",
    "        for m in range(0, n, step=1)\n",
    "            if (n-m)%2 == 0\n",
    "                Zeven(ρ, ϕ) = R(n, m, ρ) * cos(m * ϕ)\n",
    "                push!(zernike_samples, sample_across_unit_disk(Zeven, discretization)[3][:])\n",
    "                push!(ordering, (n, m))\n",
    "                if m !=0\n",
    "                    Zodd(ρ, ϕ) = R(n, m, ρ) * sin(m * ϕ)\n",
    "                    push!(zernike_samples, sample_across_unit_disk(Zodd, discretization)[3][:])\n",
    "                    push!(ordering, (n, -m))\n",
    "                end\n",
    "            end\n",
    "        end\n",
    "    end\n",
    "    zernike_samples\n",
    "end\n",
    "\n",
    "function gaussian(σ1, σ2)\n",
    "    function func(ρ, ϕ)\n",
    "        x = ρ*cos(ϕ)\n",
    "        y = ρ*sin(ϕ)\n",
    "        exp(-x^2/σ1^2 + -y^2/σ2^2)\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
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   ],
   "source": [
    "for (i, el) in enumerate(zz)\n",
    "    for (j, ell) in enumerate(el)\n",
    "        print(length(el), \"\\n\")\n",
    "        print(el, \"\\n\")\n",
    "    end\n",
    "end"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 59,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "23456789234567234567823456789234567234232342345678232323234567892342323"
     ]
    }
   ],
   "source": [
    "using Plots\n",
    "all_ns = []\n",
    "all_rs = []\n",
    "σs = range(.1, 1, step=.3)\n",
    "r2_vals = [.01]\n",
    "tolerance = .01\n",
    "for r2_val in r2_vals\n",
    "    r2s = []\n",
    "    ns = []\n",
    "    for σ1 in σs\n",
    "        for σ2 in σs\n",
    "            count = 0\n",
    "            r2 = 1\n",
    "            max_n = 1\n",
    "            while r2 > r2_val\n",
    "                Z = Zmat(max_n)\n",
    "                xx, yy, zz = sample_across_unit_disk(gaussian(σ1, σ2), discretization)\n",
    "                for el in zz\n",
    "                    if !isapprox(el, 0, atol=tolerance)\n",
    "                        count += 1\n",
    "                    end\n",
    "                end\n",
    "                G = zz[:];\n",
    "                coefficients = least_squares_solution(Z, G)\n",
    "                reconstruction = zeros(discretization*discretization)\n",
    "                zernike_samples = Zsamples(max_n)\n",
    "                for (i, zernike_sample) in enumerate(zernike_samples)\n",
    "                    reconstruction += coefficients[i] * zernike_sample\n",
    "                end\n",
    "                max_n += 1\n",
    "                print(max_n)\n",
    "                r2 = sum((reconstruction - G).*(reconstruction - G))/count\n",
    "            end\n",
    "            push!(ns, max_n)\n",
    "            push!(r2s, r2)\n",
    "        end\n",
    "    end\n",
    "    push!(all_ns, ns)\n",
    "    push!(all_rs, r2s)\n",
    "end\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 60,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4×4 Array{Any,2}:\n",
       " 9  7  8  9\n",
       " 7  4  3  4\n",
       " 8  3  3  3\n",
       " 9  4  3  3"
      ]
     },
     "execution_count": 60,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "reshape(all_ns[1], (length(σs), length(σs)))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 62,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 62,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(heatmap(σs, σs, reshape(all_ns[1], (length(σs), length(σs)))), aspectratio = true, xlabel=\"σx\", ylabel=\"σy\", title=\"Number of Zernike polynomials required for R^2=.01\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
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       "  2232.76,1349.42 2232.76,1305.11 2256.76,1305.11 2232.76,1305.11 2232.76,1083.53 2256.76,1083.53 2232.76,1083.53 2232.76,861.957 2256.76,861.957 2232.76,861.957 \n",
       "  2232.76,640.38 2256.76,640.38 2232.76,640.38 2232.76,418.804 2256.76,418.804 2232.76,418.804 2232.76,197.227 2256.76,197.227 2232.76,197.227 \n",
       "  \n",
       "  \"/>\n",
       "</svg>\n"
      ]
     },
     "execution_count": 25,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "using Plots\n",
    "plot(heatmap(σs, σs, reshape(all_ns[1], (length(σs), length(σs)))), aspectratio = true, xlabel=\"σx\", ylabel=\"σy\", title=\"Number of Zernike polynomials required for R^2=.01\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gaussian(0., 0.)(2.,1.)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.0"
      ]
     },
     "execution_count": 27,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "exp(-1/0)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
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   "mimetype": "application/julia",
   "name": "julia",
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