Hermite Interpolation Article Index for
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Hermite Interpolation
  CATEGORIES ABOUT HERMITE INTERPOLATION
   
interpolation
   
finite differences
   
factorial and binomial topics
   
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USAGE

The derivatives are treated as extra points, and in the divided difference table, the points are repeated. To avoid division by zero, the values where the division by zero would take place are replaced with the derivatives, multiplied by a constant, depending on the position in the table. For example, using the notation on the Newton Polynomial article, if point x_i is repeated n times, x_i, ..., x_i =f^{(n-1)}(x_i)/(n-1)!, e.g.

x_i, x_i, x_i =f^{(3)}(x_i)/3!

x_i, x_i =f^{(2)}(x_i)/2!

etc.

The table is calculated in the exact same fashion as before.


EXAMPLE

The example used here will be the polynomial x^8 + 1. The values, first, and second derivatives at the points x = -1, x = 0, and x = 1 will be used. This means that 9 pieces of data will be used, and so the polynomial discovered will be of degree 8.


-1	2

-1	2	-8

-1	2	-8	28

0	1	-1	7	-21

0	1	0	1	-6	15

0	1	0	0	-1	5	-10

1	2	1	1	1	1	-2	4

1	2	8	7	6	5	2	2	-1

1	2	8	28	21	15	10	4	1	1


P(x) = 2 - 8(x+1) + 28(x+1) ^2 - 21 (x+1)^3 + 15x(x+1)^3 - 10x^2(x+1)^3 + 4x^3(x+1)^3 -1x^3(x+1)^3(x-1)+x^3(x+1)^3(x-1)^2 =2 - 8 + 18 - 21 - 8x + 56x - 63x + 15x + 28x^2 - 63x^3 + 45x^2 - 10x^2 - 21x^3 + 45x^3 - 30x^3 + 4x^3 + x^3 + x^3 + 15x^4 - 30x^4 + 12x^4 + 2x^4 + x^4 - 10x^5 + 12x^5 - 2x^5 + 4x^5 - 2x^5 - 2x^5 - x^6 + x^6 - x^7 + x^7 + x^8 = x^8 + 1.



ERROR

  • \prod_{i=0}^n (x-x_i)^{N_i} where a is the number of pieces of data, n is the number of x-values minus 1, and N_i is the number of pieces of data used at x_i. This is because the function cannot change more quickly from the estimated Hermite interpolation polynomial than its a-th derivative divided by a! multiplied by the distance of the point of evaluation from the known points.


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