"Lagrange's interpolation formula" meaning in All languages combined

{
  "etymology_text": "Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.",
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      "expansion": "Lagrange's interpolation formula (uncountable)",
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  "lang": "English",
  "lang_code": "en",
  "pos": "noun",
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          "name": "Mathematics",
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      "glosses": [
        "A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n."
      ],
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      "raw_glosses": [
        "(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n."
      ],
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      "wikipedia": [
        "Joseph Louis Lagrange",
        "Lagrange polynomial"
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  ],
  "word": "Lagrange's interpolation formula"
}

{
  "etymology_text": "Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.",
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  "lang_code": "en",
  "pos": "noun",
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      "glosses": [
        "A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n."
      ],
      "links": [
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      "raw_glosses": [
        "(mathematics) A formula which when given a set of n points (x_i,y_i), gives back the unique polynomial of degree (at most) n − 1 in one variable which describes a function passing through those points. The formula is a sum of products, like so: ∑ᵢⁿy_i∏_(j ne i)x-x_j/x_i-x_j. When x=x_i then all terms in the sum other than the iᵗʰ contain a factor x-x_i in the numerator, which becomes equal to zero, thus all terms in the sum other than the iᵗʰ vanish, and the iᵗʰ term has factors x_i-x_j both in the numerator and denominator, which simplify to yield 1, thus the polynomial should return y_i as the function of x_i for any i in the set 1,...,n."
      ],
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  "word": "Lagrange's interpolation formula"
}