Question

Standard notation for divided difference is

Answer 1

Divided Differences

There are two disadvantages to using the Lagrangian polynomial or Neville's method for interpolation.

• It involves more arithmetic operations than does the divided- difference method.
• More importantly, if we desire to add or subtract a point from the set used to construct the polynomial, we essentially have to start over in the computations.
• Both the Lagrangian polynomials and Neville's method also must repeat all of the arithmetic if we must interpolate at a new $x$-value.
• The divided-difference method avoids all of this computation.
• Actually, we will not get a polynomial different from that obtained by Lagrange's technique.
• Every $n^{th}$-degree polynomial that passes through the same $n + 1$ points is identical.
• Only the way that the polynomial is expressed is different.
• We do not assume that the $x$'s are evenly spaced or even that the values are arranged in any particular order.
• The $a_i$'s are readily determined by using what are called the divided differences of the tabulated values.
• A special standard notation for divided differences is called the first divided difference between $x_0$ and $x_1$.

$f |x0.x1| = \frac{f1 - f0}{x1 - x0}$