Definitions from Data fitting and polynomial interpolation All definitions from Data fitting and polynomial interpolation.
[ DEFINITION ] Given n n n t 1 , … , t n t_1,\ldots,t_n t 1 , … , t n y 1 , … , y n y_1,\ldots,y_n y 1 , … , y n polynomial interpolation finds the polynomial f ∈ P n − 1 f\in P_{n-1} f ∈ P n − 1
f ( t i ) = y i , i = 1 , … , n . f(t_i)=y_i,\qquad i=1,\ldots,n. f ( t i ) = y i , i = 1 , … , n . Using the standard basis
B = ( 1 , t , t 2 , … , t n − 1 ) , B=(1,t,t^2,\ldots,t^{n-1}), B = ( 1 , t , t 2 , … , t n − 1 ) , write
f ( t ) = a 0 + a 1 t + ⋯ + a n − 1 t n − 1 . f(t)=a_0+a_1t+\cdots+a_{n-1}t^{n-1}. f ( t ) = a 0 + a 1 t + ⋯ + a n − 1 t n − 1 . The coefficients form the coordinate vector
[ f ] B = ( a 0 a 1 ⋮ a n − 1 ) . [f]_B=
\begin{pmatrix}
a_0\\a_1\\\vdots\\a_{n-1}
\end{pmatrix}. [ f ] B = a 0 a 1 ⋮ a n − 1 . The interpolation conditions become a linear system for this coordinate vector.
[RIGHT CLICK] [ASK AI] [ DEFINITION ] For distinct inputs t 1 , … , t n t_1,\ldots,t_n t 1 , … , t n j j j
ℓ j ( t ) = ∏ k = 1 k ≠ j n t − t k t j − t k , j = 1 , … , n . \ell_j(t)=\prod_{\substack{k=1\\k\ne j}}^n
\frac{t-t_k}{t_j-t_k},
\qquad j=1,\ldots,n. ℓ j ( t ) = k = 1 k = j ∏ n t j − t k t − t k , j = 1 , … , n . It has degree at most n − 1 n-1 n − 1
ℓ j ( t j ) = 1 , ℓ j ( t k ) = 0 when k ≠ j . \ell_j(t_j)=1,
\qquad
\ell_j(t_k)=0\quad\text{when }k\ne j. ℓ j ( t j ) = 1 , ℓ j ( t k ) = 0 when k = j . Each factor t − t k t-t_k t − t k ℓ j \ell_j ℓ j t j − t k t_j-t_k t j − t k 1 1 1 t j t_j t j
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