( ) denote a polynomial of

( ) denote a polynomial of degree n which takes ( n + values. Let them be y y y y n ,... corresponding to the values x n ,... respectively.

The values of x ( x x x x n ,... ) are at equidistant. (i.e.) x h x h x nh … Then the value of f ( x ) at x = x + nh is given by nh n n ! !

n n f )( ! x ) ... (or) y n n x x nh ! !

n n )( ! ... where n = x Newton’s Gregory backward interpolation Formula. Newton’s forward interpolation formula is used when the value of y is required near the beginning of the table.

Note In general Newton’s forward interpolation formula not to be used when the value of y is required near the end of the table. For this we use another formula, called Newton’s Gregory backward interpolation formula. Then the value of f ( x ) at x = x n + nh is given by nh n n ! !

∇ n n f )( ! ∇ ∇ x n ) ... (or) y n n x x nh ! !

∇ ∇ n n y n )( ! ... ∇ when n = Newton’s backward interpolation formula is used when the value of y is required near the end of the table. Note Example .

Using Newton’s formula for interpolation estimate the population for the year from the table: Year Population , , , , , , , , , Solution To find the population for the year (i.e) the value of y at x = . Since the value of y is required near the beginning of the table, we use the Newton’s forward interpolation formula. n n x x nh ! !

n n )( ! ... To find y at x = ∴ nh , + n ( )= ⇒ n = . D y D y D y D y , , , , , , – , , , – , , , , , , , , , , y x = = 33533 ( .

+ . = , , . Example . The values of y