{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This method is a variant of Lagrange polynomial interpolation " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": " based on " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": ". Instead of using Lagrange's or Newton's formula, the polynomial is represented by the barycentric formula" } ] }, { "__type": "Math", "__tag": 4058, "value": "p(x) =\n\\frac{\\sum_{i=1}^m\\ w_i y_i / (x - x_i)}{\\sum_{i=1}^m w_i / (x - x_i)}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "w_i" }, { "__type": "Text", "__tag": 4046, "value": " are the barycentric weights computed with the general formula" } ] }, { "__type": "Math", "__tag": 4058, "value": "w_i = \\left( \\prod_{k \\neq i} x_i - x_k \\right)^{-1}." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "This is the same barycentric form used by " }, { "__type": "CrossRef", "__tag": 4002, "value": "AAA", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.interpolate._bary_rational:AAA" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "CrossRef", "__tag": 4002, "value": "FloaterHormannInterpolator", "reference": { "__type": "LocalRef", "__tag": 4022, "kind": "module", "path": "scipy.interpolate._bary_rational:FloaterHormannInterpolator" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": ". However, in contrast, the weights " }, { "__type": "InlineMath", "__tag": 4057, "value": "w_i" }, { "__type": "Text", "__tag": 4046, "value": " are defined such that " }, { "__type": "InlineMath", "__tag": 4057, "value": "p(x)" }, { "__type": "Text", "__tag": 4046, "value": " is a polynomial rather than a rational function." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The barycentric representation avoids many of the problems associated with polynomial interpolation caused by floating-point arithmetic. However, it does not avoid issues that are intrinsic to polynomial interpolation. Namely, if the x-coordinates are equally spaced, then the weights can be computed explicitly using the formula from " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" } ] }, { "__type": "Math", "__tag": 4058, "value": "w_i = (-1)^i {n \\choose i}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " is the number of x-coordinates. As noted in " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": ", this means that for large " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": " the weights vary by exponentially large factors, leading to the Runge phenomenon." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "To avoid this ill-conditioning, the x-coordinates should be clustered at the endpoints of the interval. An excellent choice of points on the interval " }, { "__type": "InlineMath", "__tag": 4057, "value": "[a,b]" }, { "__type": "Text", "__tag": 4046, "value": " are Chebyshev points of the second kind" } ] }, { "__type": "Math", "__tag": 4058, "value": "x_i = \\frac{a + b}{2} + \\frac{a - b}{2}\\cos(i\\pi/n)." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "in which case the weights can be computed explicitly as" } ] }, { "__type": "Math", "__tag": 4058, "value": "w_i = \\begin{cases}\n (-1)^i/2 & i = 0,n \\\\\n (-1)^i & \\text{otherwise}\n \\end{cases}." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "See " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "2" }, { "__type": "Text", "__tag": 4046, "value": " for more infomation. Note that for large " }, { "__type": "InlineMath", "__tag": 4057, "value": "n" }, { "__type": "Text", "__tag": 4046, "value": ", computing the weights explicitly (see examples) will be faster than the generic formula." } ] } ], "title": [], "level": 0, "target": null }, "Warns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Raises": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Yields": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Methods": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Returns": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Barycentric (Lagrange with improved stability) interpolator (C∞ smooth)." } ] } ], "title": [], "level": 0, "target": null }, "Receives": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Warnings": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Attributes": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "xi", "annotation": "array_like, shape (npoints, )", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "1-D array of x-coordinates of the points the polynomial should pass through" } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "yi", "annotation": "array_like, shape (..., npoints, ...), optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "N-D array of y-coordinates of the points the polynomial should pass through. If None, the y values will be supplied later via the " }, { "__type": "InlineRole", "__tag": 4003, "value": "set_y", "domain": null, "role": null, "inventory": null }, { "__type": "Text", "__tag": 4046, "value": " method. The length of " }, { "__type": "ParamRef", "__tag": 4071, "name": "yi" }, { "__type": "Text", "__tag": 4046, "value": " along the interpolation axis must be equal to the length of " }, { "__type": "ParamRef", "__tag": 4071, "name": "xi" }, { "__type": "Text", "__tag": 4046, "value": ". Use the " }, { "__type": "InlineCode", "__tag": 4051, "value": "axis" }, { "__type": "Text", "__tag": 4046, "value": " parameter to select correct axis." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "axis", "annotation": "int, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Axis in the yi array corresponding to the x-coordinate values. Defaults to " }, { "__type": "InlineCode", "__tag": 4051, "value": "axis=0" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "wi", "annotation": "array_like, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The barycentric weights for the chosen interpolation points " }, { "__type": "ParamRef", "__tag": 4071, "name": "xi" }, { "__type": "Text", "__tag": 4046, "value": ". If absent or None, the weights will be computed from " }, { "__type": "ParamRef", "__tag": 4071, "name": "xi" }, { "__type": "Text", "__tag": 4046, "value": " (default). This allows for the reuse of the weights " }, { "__type": "ParamRef", "__tag": 4071, "name": "wi" }, { "__type": "Text", "__tag": 4046, "value": " if several interpolants are being calculated using the same nodes " }, { "__type": "ParamRef", "__tag": 4071, "name": "xi" }, { "__type": "Text", "__tag": 4046, "value": ", without re-computation. This also allows for computing the weights explicitly for some choices of " }, { "__type": "ParamRef", "__tag": 4071, "name": "xi" }, { "__type": "Text", "__tag": 4046, "value": " (see notes)." } ] } ] }, { "__type": "DocParam", "__tag": 4016, "name": "rng", "annotation": "{None, int, `numpy.random.Generator`}, optional", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "rng" }, { "__type": "Text", "__tag": 4046, "value": " is passed by keyword, types other than " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random.Generator", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random._generator:Generator" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " are passed to " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random.default_rng", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random:default_rng" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " to instantiate a " }, { "__type": "InlineCode", "__tag": 4051, "value": "Generator" }, { "__type": "Text", "__tag": 4046, "value": ". If " }, { "__type": "ParamRef", "__tag": 4071, "name": "rng" }, { "__type": "Text", "__tag": 4046, "value": " is already a " }, { "__type": "InlineCode", "__tag": 4051, "value": "Generator" }, { "__type": "Text", "__tag": 4046, "value": " instance, then the provided instance is used. Specify " }, { "__type": "ParamRef", "__tag": 4071, "name": "rng" }, { "__type": "Text", "__tag": 4046, "value": " for repeatable interpolation." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If this argument " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " is passed by keyword, legacy behavior for the argument " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " applies:" } ] }, { "__type": "BulletList", "__tag": 4053, "ordered": false, "start": 1, "children": [ { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " is None (or " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": "), the " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random.RandomState", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random.mtrand:RandomState" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " singleton is used." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " is an int, a new " }, { "__type": "InlineCode", "__tag": 4051, "value": "RandomState" }, { "__type": "Text", "__tag": 4046, "value": " instance is used, seeded with " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ] }, { "__type": "ListItem", "__tag": 4054, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "If " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " is already a " }, { "__type": "InlineCode", "__tag": 4051, "value": "Generator" }, { "__type": "Text", "__tag": 4046, "value": " or " }, { "__type": "InlineCode", "__tag": 4051, "value": "RandomState" }, { "__type": "Text", "__tag": 4046, "value": " instance then that instance is used." } ] } ] } ] }, { "__type": "Admonition", "__tag": 4056, "kind": "versionchanged", "base_type": "neutral", "children": [ { "__type": "AdmonitionTitle", "__tag": 4055, "children": [ { "__type": "Text", "__tag": 4046, "value": "versionchanged 1.15.0" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "As part of the " }, { "__type": "Link", "__tag": 4049, "children": [ { "__type": "Text", "__tag": 4046, "value": "SPEC-007" } ], "url": "https://scientific-python.org/specs/spec-0007/", "title": "" }, { "__type": "Text", "__tag": 4046, "value": " transition from use of " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random.RandomState", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random.mtrand:RandomState" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " to " }, { "__type": "CrossRef", "__tag": 4002, "value": "numpy.random.Generator", "reference": { "__type": "RefInfo", "__tag": 4000, "module": "numpy", "version": "*", "kind": "api", "path": "numpy.random._generator:Generator" }, "kind": "module" }, { "__type": "Text", "__tag": 4046, "value": " this keyword was changed from " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " to " }, { "__type": "ParamRef", "__tag": 4071, "name": "rng" }, { "__type": "Text", "__tag": 4046, "value": ". For an interim period, both keywords will continue to work (only specify one of them). After the interim period using the " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " keyword will emit warnings. The behavior of the " }, { "__type": "ParamRef", "__tag": 4071, "name": "random_state" }, { "__type": "Text", "__tag": 4046, "value": " and " }, { "__type": "ParamRef", "__tag": 4071, "name": "rng" }, { "__type": "Text", "__tag": 4046, "value": " keywords is outlined above." } ] } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Extended Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Constructs a polynomial that passes through a given set of points. Allows evaluation of the polynomial and all its derivatives, efficient changing of the y-values to be interpolated, and updating by adding more x- and y-values. For numerical stability, a barycentric representation is used rather than computing the coefficients of the polynomial directly." } ] } ], "title": [], "level": 0, "target": null }, "Other Parameters": { "__type": "Section", "__tag": 4015, "children": [], "title": [], "level": 0, "target": null } }, "_ordered_sections": [ "Summary", "Extended Summary", "Parameters", "Attributes", "Methods", "Returns", "Yields", "Receives", "Other Parameters", "Raises", "Warns", "Warnings", "Notes" ], "item_file": "/scipy/interpolate/_polyint.py", "item_line": 537, "item_type": "class", "aliases": [ "scipy.interpolate.BarycentricInterpolator" ], "example_section_data": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Text", "__tag": 4046, "value": "To produce a quintic barycentric interpolant approximating the function\n:math:`\\sin x`, and its first four derivatives, using six randomly-spaced\nnodes in :math:`(0, \\pi/2)`:\n\n" }, { "__type": "Code", "__tag": 4050, "value": "import numpy as np\nimport matplotlib.pyplot as plt\nfrom scipy.interpolate import BarycentricInterpolator\nrng = np.random.default_rng()\nxi = rng.random(6) * np.pi/2\nf, f_d1, f_d2, f_d3, f_d4 = np.sin, np.cos, lambda x: -np.sin(x), lambda x: -np.cos(x), np.sin\nP = BarycentricInterpolator(xi, f(xi), random_state=rng)\nfig, axs = plt.subplots(5, 1, sharex=True, layout='constrained', figsize=(7,10))\nx = np.linspace(0, np.pi, 100)\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "axs[0].plot(x, P(x), 'r:', x, f(x), 'k--', xi, f(xi), 'xk')\naxs[1].plot(x, P.derivative(x), 'r:', x, f_d1(x), 'k--', xi, f_d1(xi), 'xk')\naxs[2].plot(x, P.derivative(x, 2), 'r:', x, f_d2(x), 'k--', xi, f_d2(xi), 'xk')\naxs[3].plot(x, P.derivative(x, 3), 'r:', x, f_d3(x), 'k--', xi, f_d3(xi), 'xk')\naxs[4].plot(x, P.derivative(x, 4), 'r:', x, f_d4(x), 'k--', xi, f_d4(xi), 'xk')\naxs[0].set_xlim(0, np.pi)\naxs[4].set_xlabel(r\"$x$\")\naxs[4].set_xticks([i * np.pi / 4 for i in range(5)],\n [\"0\", r\"$\\frac{\\pi}{4}$\", r\"$\\frac{\\pi}{2}$\", r\"$\\frac{3\\pi}{4}$\", r\"$\\pi$\"])\nfor ax, label in zip(axs, (\"$f(x)$\", \"$f'(x)$\", \"$f''(x)$\", \"$f^{(3)}(x)$\", \"$f^{(4)}(x)$\")):\n ax.set_ylabel(label)\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "labels = ['Interpolation nodes', 'True function $f$', 'Barycentric interpolation']\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "axs[0].legend(axs[0].get_lines()[::-1], labels, bbox_to_anchor=(0., 1.02, 1., .102),\n loc='lower left', ncols=3, mode=\"expand\", borderaxespad=0., frameon=False)\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-a96c7e332af9a9b8.png" } }, { "__type": "Text", "__tag": 4046, "value": "\nNext, we show how using Chebyshev points of the second kind avoids the\nRunge phenomenon. In this example, we also compute the weights explicitly.\n\n" }, { "__type": "Code", "__tag": 4050, "value": "n = 20\ndef f(x): return np.abs(x) + 0.5*x - x**2\ni = np.arange(n)\nx_cheb = np.cos(i*np.pi/(n - 1)) # Chebyshev points on [-1, 1]\nw_i_cheb = (-1.)**i # Explicit formula for weights of Chebyshev points\nw_i_cheb[[0, -1]] /= 2\np_cheb = BarycentricInterpolator(x_cheb, f(x_cheb), wi=w_i_cheb)\nx_equi = np.linspace(-1, 1, n)\np_equi = BarycentricInterpolator(x_equi, f(x_equi))\nxx = np.linspace(-1, 1, 1000)\nfig, ax = plt.subplots()\n", "execution_status": "success" }, { "__type": "Code", "__tag": 4050, "value": "ax.plot(xx, f(xx), label=\"Original Function\")\nax.plot(xx, p_cheb(xx), \"--\", label=\"Chebshev Points\")\nax.plot(xx, p_equi(xx), \"--\", label=\"Equally Spaced Points\")\nax.set(xlabel=\"$x$\", ylabel=\"$f(x)$\", xlim=[-1, 1])\nax.legend()\n", "execution_status": "failure" }, { "__type": "Code", "__tag": 4050, "value": "plt.show()\n", "execution_status": "success" }, { "__type": "Figure", "__tag": 4024, "value": { "__type": "RefInfo", "__tag": 4000, "module": "scipy", "version": "1.17.1", "kind": "assets", "path": "fig-5109b4a02cb7e58b.png" } } ], "title": [], "level": 0, "target": null }, "see_also": [], "signature": { "__type": "SignatureNode", "__tag": 4029, "kind": "function", "parameters": [ { "__type": "SigParam", "__tag": 4030, "name": "xi", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": { "__type": "Empty", "__tag": 4031 } }, { "__type": "SigParam", "__tag": 4030, "name": "yi", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "axis", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "POSITIONAL_OR_KEYWORD", "default": "0" }, { "__type": "SigParam", "__tag": 4030, "name": "wi", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "rng", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" }, { "__type": "SigParam", "__tag": 4030, "name": "random_state", "annotation": { "__type": "Empty", "__tag": 4031 }, "kind": "KEYWORD_ONLY", "default": "None" } ], "return_annotation": { "__type": "Empty", "__tag": 4031 }, "target_name": "BarycentricInterpolator" }, "references": [ ".. [1] https://en.wikipedia.org/wiki/Lagrange_polynomial", ".. [2] Jean-Paul Berrut and Lloyd N. Trefethen, \"Barycentric Lagrange", " Interpolation\", SIAM Review 2004 46:3, 501-517", " :doi:`10.1137/S0036144502417715`" ], "qa": "scipy.interpolate._polyint:BarycentricInterpolator", "arbitrary": [], "local_refs": [ "axis", "rng", "wi", "xi", "yi" ] }