{ "__type": "IngestedDoc", "__tag": 4010, "_content": { "Notes": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "For values of " }, { "__type": "InlineCode", "__tag": 4051, "value": "ord < 1" }, { "__type": "Text", "__tag": 4046, "value": ", the result is, strictly speaking, not a mathematical 'norm', but it may still be useful for various numerical purposes." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The following norms can be calculated:" } ] }, { "__type": "Code", "__tag": 4050, "value": "===== ============================ ==========================\nord norm for matrices norm for vectors\n===== ============================ ==========================\nNone Frobenius norm 2-norm\n'fro' Frobenius norm --\n'nuc' nuclear norm --\ninf max(sum(abs(x), axis=1)) max(abs(x))\n-inf min(sum(abs(x), axis=1)) min(abs(x))\n0 -- sum(x != 0)\n1 max(sum(abs(x), axis=0)) as below\n-1 min(sum(abs(x), axis=0)) as below\n2 2-norm (largest sing. value) as below\n-2 smallest singular value as below\nother -- sum(abs(x)**ord)**(1./ord)\n===== ============================ ==========================", "execution_status": null }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The Frobenius norm is given by " }, { "__type": "FootnoteReference", "__tag": 4066, "label": "1" }, { "__type": "Text", "__tag": 4046, "value": ":" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "InlineMath", "__tag": 4057, "value": "||A||_F = [\\sum_{i,j} abs(a_{i,j})^2]^{1/2}" } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "The nuclear norm is the sum of the singular values." } ] }, { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Both the Frobenius and nuclear norm orders are only defined for matrices and raise a ValueError when " }, { "__type": "InlineCode", "__tag": 4051, "value": "x.ndim != 2" }, { "__type": "Text", "__tag": 4046, "value": "." } ] } ], "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": [ { "__type": "Parameters", "__tag": 4026, "children": [ { "__type": "DocParam", "__tag": 4016, "name": "n", "annotation": "float or ndarray", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Norm of the matrix or vector(s)." } ] } ] } ] } ], "title": [], "level": 0, "target": null }, "Summary": { "__type": "Section", "__tag": 4015, "children": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Matrix or vector norm." } ] } ], "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": "x", "annotation": "array_like", "desc": [ { "__type": "Paragraph", "__tag": 4045, "children": [ { "__type": "Text", "__tag": 4046, "value": "Input array. 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