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A formula for obtaining a polynomial of degree  $  n $(
the Lagrange interpolation polynomial) that interpolates a given function  $  f ( x) $
at nodes  $  x _ {0} \dots x _ {n} $:

$$ \tag{1 }
L _ {n} ( x)  =  \sum_{i=0}^ { n }  f ( x _ {i} ) \prod _ {j \neq i }

\frac{x - x _ {j} }{x _ {i} - x _ {j} }
 .
$$

When the  $  x _ {i} $
are equidistant, that is,  $  x _ {1} - x _ {0} = \dots = x _ {n} - x _ {n-1} = h $, 
using the notation  $  ( x - x _ {0} ) / h = t $
one can reduce (1) to the form

$$ \tag{2 }
L _ {n} ( x)  =  L _ {n} ( x _ {0} + th )  =  ( - 1 )  ^ {n} 
\frac{t ( t- 1 )
{} \dots ( t- n ) }{n!}
 \times
$$

$$ 
\times 
\sum_{i=0}^ { n }  ( - 1 )  ^ {i} \left ( \begin{array}{c}
n
\\
 i 
\end{array}
 \right ) 
\frac{f ( x _ {i} ) }{t-i} .
$$

In the expression (2), called the Lagrange interpolation formula for equidistant nodes, the coefficients

$$ 
( - 1 )  ^ {n-i} \left ( \begin{array}{c}
n \\
 i 
\end{array}
 \right )

\frac{t ( t- 1 ) \dots ( t- n ) }{( t- i) n! }

$$

of the  $  f ( x _ {i} ) $
are called the Lagrange coefficients.

If  $  f $
has a derivative of order  $  n+ 1 $
on the interval  $  [ a , b ] $, 
if all interpolation nodes lie in this interval and if for any point  $  x \in [ a , b ] $
one defines

$$ 
\alpha _ {x}  =  \min \{ x _ {0} \dots x _ {n} , x
\} ,\  \beta _ {x}  =  \max \{ x _ {0} \dots x _ {n} , x \} ,
$$

then a point  $  \xi \in [ \alpha _ {x} , \beta _ {x} ] $
exists such that

$$ 
f ( x) - L _ {n} ( x)  =  
\frac{f ^ { ( n+ 1 ) } ( \xi ) \omega _ {n} ( x) }{( n+ 1) ! }
 ,
$$

where

$$ 
\omega _ {n} ( x)  =  \prod_{j=0}^ { n }  ( x - x _ {j} ) .
$$

If the absolute value of the derivative  $  f ^ { ( n+ 1 ) } $
is bounded on  $  [ a , b ] $
by a constant  $  M $
and if the interpolation nodes are chosen such that the roots of the Chebyshev polynomial of degree  $  n+ 1 $
are mapped into these points under a linear mapping from  $  [ - 1 , 1] $
onto  $  [ a , b ] $, 
then for any  $  x \in [ a , b ] $
one has

$$ 
| f ( x) - L _ {n} ( x) |  \leq   M 
\frac{( b- a )  ^ {n+1}}{( n+ 1 )!
2  ^ {2n+1} }
 .
$$

If the interpolation nodes are complex numbers  $  z _ {0} \dots z _ {n} $
and lie in some domain  $  G $
bounded by a piecewise-smooth contour  $  \gamma $, 
and if  $  f $
is a single-valued analytic function defined on the closure of  $  G $, 
then the Lagrange interpolation formula has the form

$$ 
L _ {n} ( z)  =  
\frac{1}{2 \pi i }
 \int\limits _  \gamma  
\frac{\omega ( \zeta ) -
\omega ( z) }{\omega ( \zeta ) ( \zeta - z ) }
 f ( \zeta )  d \zeta ,
$$

where

$$ 
f ( z) - L _ {n} ( z)  =  
\frac{\omega ( z) }{2 \pi i }
 \int\limits _  \gamma 

\frac{f ( \zeta ) }{\omega ( \zeta ) ( z - \zeta ) }
  d \zeta .
$$

The Lagrange interpolation formula for interpolation by means of trigonometric polynomials is:

$$ 
T _ {n} ( x)  =  \sum_{k=0}^ { n }  y _ {k} \prod _ {j \neq k } 
\frac{\sin
 ( x - x _ {j} ) / 2 }{\sin  ( x _ {k} - x _ {j} ) / 2 }
 ,
$$

which is a trigonometric polynomial of order  $  n $
having prescribed values  $  y _ {0} \dots y _ {n} $
at the given nodes  $  x _ {0} \dots x _ {n} $.

The formula was proposed by J.L. Lagrange in 1795.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  I.S. Berezin,   N.P. Zhidkov,   "Computing methods" , Pergamon  (1973)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  N.S. Bakhvalov,   "Numerical methods: analysis, algebra, ordinary differential equations" , MIR  (1977)  (Translated from Russian)</TD></TR>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.J. Davis,   "Interpolation and approximation" , Dover, reprint  (1975)  pp. 108–126</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  L.W. Johnson,   R.D. Riess,   "Numerical analysis" , Addison-Wesley  (1977)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  G.M. Phillips,   P.J. Taylor,   "Theory and applications of numerical analysis" , Acad. Press  (1973)</TD></TR></table>