{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This function computes a discrete version of the Hankel transform" } ] }, { "__type": "Math", "__tag": 4058, "value": "A(k) = \\int_{0}^{\\infty} \\! a(r) \\, J_\\mu(kr) \\, k \\, dr \\;," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "J_\\mu" }, { "__type": "Text", "__tag": 4046, "value": " is the Bessel function of order " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu" }, { "__type": "Text", "__tag": 4046, "value": ". The index " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\mu" }, { "__type": "Text", "__tag": 4046, "value": " may be any real number, positive or negative. Note that the numerical Hankel transform uses an integrand of " }, { "__type": "InlineMath", "__tag": 4057, "value": "k \\, dr" }, { "__type": "Text", "__tag": 4046, "value": ", while the mathematical Hankel transform is commonly defined using " }, { "__type": "InlineMath", "__tag": 4057, "value": "r \\, dr" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The input array " }, { "__type": "ParamRef", "__tag": 4071, "name": "a" }, { "__type": "Text", "__tag": 4046, "value": " is a periodic sequence of length " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ", uniformly logarithmically spaced with spacing " }, { "__type": "ParamRef", "__tag": 4071, "name": "dln" }, { "__type": "Text", "__tag": 4046, "value": "," } ] }, { "__type": "Math", "__tag": 4058, "value": "a_j = a(r_j) \\;, \\quad\nr_j = r_c \\exp[(j-j_c) \\, \\mathtt{dln}]" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "centred about the point " }, { "__type": "InlineMath", "__tag": 4057, "value": "r_c" }, { "__type": "Text", "__tag": 4046, "value": ". Note that the central index " }, { "__type": "InlineMath", "__tag": 4057, "value": "j_c = (n-1)/2" }, { "__type": "Text", "__tag": 4046, "value": " is half-integral if " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " is even, so that " }, { "__type": "InlineMath", "__tag": 4057, "value": "r_c" }, { "__type": "Text", "__tag": 4046, "value": " falls between two input elements. Similarly, the output array " }, { "__type": "CrossRef", "__tag": 4002, "value": "A", "reference": { "__type": "RefInfo", "__tag": 4000, "module": null, "version": null, "kind": "local", "path": "A" }, "kind": "local" }, { "__type": "Text", "__tag": 4046, "value": " is a periodic sequence of length " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ", also uniformly logarithmically spaced with spacing " }, { "__type": "ParamRef", "__tag": 4071, "name": "dln" } ] }, { "__type": "Math", "__tag": 4058, "value": "A_j = A(k_j) \\;, \\quad\nk_j = k_c \\exp[(j-j_c) \\, \\mathtt{dln}]" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "centred about the point " }, { "__type": "InlineMath", "__tag": 4057, "value": "k_c" }, { "__type": "Text", "__tag": 4046, "value": "." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The centre points " }, { "__type": "InlineMath", "__tag": 4057, "value": "r_c" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "InlineMath", "__tag": 4057, "value": "k_c" }, { "__type": "Text", "__tag": 4046, "value": " of the periodic intervals may be chosen arbitrarily, but it would be usual to choose the product " }, { "__type": "InlineMath", "__tag": 4057, "value": "k_c r_c = k_j r_{n-1-j} = k_{n-1-j} r_j" }, { "__type": "Text", "__tag": 4046, "value": " to be unity. This can be changed using the " }, { "__type": "ParamRef", "__tag": 4071, "name": "offset" }, { "__type": "Text", "__tag": 4046, "value": " parameter, which controls the logarithmic offset " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\log(k_c) = \\mathtt{offset} - \\log(r_c)" }, { "__type": "Text", "__tag": 4046, "value": " of the output array. Choosing an optimal value for " }, { "__type": "ParamRef", "__tag": 4071, "name": "offset" }, { "__type": "Text", "__tag": 4046, "value": " may reduce ringing of the discrete Hankel transform." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If the " }, { "__type": "ParamRef", "__tag": 4071, "name": "bias" }, { "__type": "Text", "__tag": 4046, "value": " parameter is nonzero, this function computes a discrete version of the biased Hankel transform" } ] }, { "__type": "Math", "__tag": 4058, "value": "A(k) = \\int_{0}^{\\infty} \\! a_q(r) \\, (kr)^q \\, J_\\mu(kr) \\, k \\, dr" }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "q" }, { "__type": "Text", "__tag": 4046, "value": " is the value of " }, { "__type": "ParamRef", "__tag": 4071, "name": "bias" }, { "__type": "Text", "__tag": 4046, "value": ", and a power law bias " }, { "__type": "InlineMath", "__tag": 4057, "value": "a_q(r) = a(r) \\, (kr)^{-q}" }, { "__type": "Text", "__tag": 4046, "value": " is applied to the input sequence. Biasing the transform can help approximate the continuous transform of " }, { "__type": "InlineMath", "__tag": 4057, "value": "a(r)" }, { "__type": "Text", "__tag": 4046, "value": " if there is a value " }, { "__type": "InlineMath", "__tag": 4057, "value": "q" }, { "__type": "Text", "__tag": 4046, "value": " such that " }, { "__type": "InlineMath", "__tag": 4057, "value": "a_q(r)" }, { "__type": "Text", "__tag": 4046, "value": " is close to a periodic sequence, in which case the resulting " }, { "__type": "InlineMath", "__tag": 4057, "value": "A(k)" }, { "__type": "Text", "__tag": 4046, "value": " will be close to the continuous transform." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Strong", "__tag": 4048, "children": [ { "__type": "Text", "__tag": 4046, "value": "Array API Standard Support" } ] } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineRole", "__tag": 4003, "value": "fht", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " has experimental support for Python Array API Standard compatible backends in addition to NumPy. 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It evaluates the integral\n\n.. math::\n\n \\int^\\infty_0 r^{\\mu+1} \\exp(-r^2/2) J_\\mu(kr) k dr\n = k^{\\mu+1} \\exp(-k^2/2) .\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nfrom scipy import fft\nimport matplotlib.pyplot as plt\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nParameters for the transform.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "mu = 0.0 # Order mu of Bessel function\nr = np.logspace(-7, 1, 128) # Input evaluation points\ndln = np.log(r[1]/r[0]) # Step size\noffset = fft.fhtoffset(dln, initial=-6*np.log(10), mu=mu)\nk = np.exp(offset)/r[::-1] # Output evaluation points\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nDefine the analytical function.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "def f(x, mu):\n \"\"\"Analytical function: x^(mu+1) exp(-x^2/2).\"\"\"\n return x**(mu + 1)*np.exp(-x**2/2)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nEvaluate the function at ``r`` and compute the corresponding values at\n``k`` using FFTLog.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a_r = f(r, mu)\nfht = fft.fht(a_r, dln, mu=mu, offset=offset)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nFor this example we can actually compute the analytical response (which in\nthis case is the same as the input function) for comparison and compute the\nrelative error.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "a_k = f(k, mu)\nrel_err = abs((fht-a_k)/a_k)\n", "execution_status": "success" }, { "__type": "Text", "__tag": 4046, "value": "\nPlot the result.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "figargs = {'sharex': True, 'sharey': True, 'constrained_layout': True}\nfig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 4), **figargs)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax1.set_title(r'$r^{\\mu+1}\\ \\exp(-r^2/2)$')\nax1.loglog(r, a_r, 'k', lw=2)\nax1.set_xlabel('r')\nax2.set_title(r'$k^{\\mu+1} \\exp(-k^2/2)$')\nax2.loglog(k, a_k, 'k', lw=2, label='Analytical')\nax2.loglog(k, fht, 'C3--', lw=2, label='FFTLog')\nax2.set_xlabel('k')\nax2.legend(loc=3, framealpha=1)\nax2.set_ylim([1e-10, 1e1])\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "ax2b = ax2.twinx()\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax2b.loglog(k, rel_err, 'C0', label='Rel. 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[1] Talman J. D., 1978, J. Comp. Phys., 29, 35", ".. [2] Hamilton A. J. S., 2000, MNRAS, 312, 257 (astro-ph/9905191)" ], "qa": "scipy.fft._fftlog:fht", "arbitrary": [], "local_refs": [ "A", "a", "bias", "dln", "mu", "offset" ] }