Newton's Divided Differences Interpolation Polynomial Example

Newton interpolation maps x to f(x) using a table of known values.

|      | x1 | x2 | x3 | x4 |
|------+----+----+----+----|
| x    | -5 | -1 |  0 |  2 |
| f(x) | -2 |  6 |  1 |  3 |
|------+----+----+----+----|
|      | f1 | f2 | f3 | f4 |

• Divided differences table is a pyramid
• As you progress left-to-right on the differences table, the numerator does not change
• Further columns depend on values of the previous column (for the f component) and travel diagnally for the upper and lower x components

|  x | f(x) | f(x)[1] 1st     | f(x)[2] 2nd           | f(x)[3] 3rd           |
|----+------+-----------------+-----------------------+-----------------------|
| -5 |   -2 | (f2-f1)/(x2-x1) | (f2[1]-f1[1])/(x3-x1) | (f2[2]-f1[2])/(x4-x1) |
| -1 |    6 | (f3-f2)/(x3-x2) | (f3[1]-f2[1])/(x4-x2) |                       |
|  0 |    1 | (f4-f3)/(x4-x3) |                       |                       |
|  2 |    3 |                 |                       |                       |

$ f3(x)=b_1 + b_2(x-x1) + b_3(x-x1)(x-x2) + b_4(x-x1)(x-x2)(x-x3)

The topmost values from the differences table constitute the "b values."

| f(x) | f(x)[1] | f(x)[2] | f(x)[3] |
| b1   | b2      | b3      | b4      |