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The rational function blends " }, { "__type": "InlineCode", "__tag": 4051, "value": "n - d" }, { "__type": "Text", "__tag": 4046, "value": " polynomials of degree " }, { "__type": "ParamRef", "__tag": 4071, "name": "d" }, { "__type": "Text", "__tag": 4046, "value": " together to produce a rational interpolant that contains no poles on the real axis, unlike " }, { "__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": ". The interpolant is given by" } ] }, { "__type": "Math", "__tag": 4058, "value": "r(x) = \\frac{\\sum_{i=0}^{n-d} \\lambda_i(x) p_i(x)}\n{\\sum_{i=0}^{n-d} \\lambda_i(x)}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where " }, { "__type": "InlineMath", "__tag": 4057, "value": "p_i(x)" }, { "__type": "Text", "__tag": 4046, "value": " is an interpolating polynomial of at most degree " }, { "__type": "ParamRef", "__tag": 4071, "name": "d" }, { "__type": "Text", "__tag": 4046, "value": " through the points " }, { "__type": "InlineMath", "__tag": 4057, "value": "(x_i,y_i),\\dots,(x_{i+d},y_{i+d})" }, { "__type": "Text", "__tag": 4046, "value": ", and " }, { "__type": "InlineMath", "__tag": 4057, "value": "\\lambda_i(z)" }, { "__type": "Text", "__tag": 4046, "value": " are blending functions defined by" } ] }, { "__type": "Math", "__tag": 4058, "value": "\\lambda_i(x) = \\frac{(-1)^i}{(x - x_i)\\cdots(x - x_{i+d})}." }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "When " }, { "__type": "InlineCode", "__tag": 4051, "value": "d = n - 1" }, { "__type": "Text", "__tag": 4046, "value": " this reduces to polynomial interpolation." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Due to its stability, the following barycentric representation of the above equation is used for computation" } ] }, { "__type": "Math", "__tag": 4058, "value": "r(z) = \\frac{\\sum_{k=1}^m\\ w_k f_k / (x - x_k)}{\\sum_{k=1}^m w_k / (x - x_k)}," }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "where the weights " }, { "__type": "InlineMath", "__tag": 4057, "value": "w_j" }, { "__type": "Text", "__tag": 4046, "value": " are computed as" } ] }, { "__type": "Math", "__tag": 4058, "value": "w_k &= (-1)^{k - d} \\sum_{i \\in J_k} \\prod_{j = i, j \\neq k}^{i + d}\n1/|x_k - x_j|, \\\\\nJ_k &= \\{ i \\in I: k - d \\leq i \\leq k\\},\\\\\nI &= \\{0, 1, \\dots, n - d\\}." } ], "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": "Floater-Hormann barycentric rational interpolator (C∞ smooth on real axis)." } ] } ], "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": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "weights", "annotation": "array", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Weights of the barycentric approximation." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Parameters": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "x", "annotation": "1D array_like, shape (n,)", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "1-D array containing values of the independent variable. 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[1] M.S. Floater and K. Hormann, \"Barycentric rational interpolation with no", " poles and high rates of approximation\", Numer. Math. 107, 315 (2007).", " :doi:`10.1007/s00211-007-0093-y`" ], "qa": "scipy.interpolate._bary_rational:FloaterHormannInterpolator", "arbitrary": [], "local_refs": [ "axis", "d", "weights", "x", "y" ] }