Lagrange's interpolation formula

Meaning

Noun

• 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: $\backslash sum\_i^n\; y\_i\; \backslash prod\_\{j\; \backslash ne\; i\}\; \{x\; -\; x\_j\; \backslash over\; x\_i\; -\; x\_j\}$. When $x\; =\; x\_i$ then all terms in the sum other than the i ^th contain a factor $x\; -\; x\_i$ in the numerator, which becomes equal to zero, thus all terms in the sum other than the i ^th vanish, and the i ^th 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 $\backslash \{1,\; ...,\; n\backslash \}$.

Origin

• Named after Joseph Louis Lagrange (1736–1813), an Italian Enlightenment Era mathematician and astronomer.