Iterative Linear Interpolation

4.2.4 Iterative Linear Interpolation

There are other recursive algorithms for interpolation. Of interest are those based on suc- cessive linear interpolations . The basic formula is given in the following theorem.

Theorem 4.2.7.

Assume that the two polynomials p n −1 (x) and q n −1 (x), both in P n −1 , interpolate f (x) at the points x 1 ,...,x n −1 and x 2 ,...,x n , respectively. If the n points x 1 ,x 2 ,...,x n −1 ,x n are distinct, then

is the unique polynomial in P n that interpolates f (x) at the m points x 1 ,x 2 ,...,x n −1 ,x n .

372 Chapter 4. Interpolation and Approximation Proof. Since q n −1 (x) and p n −1 (x) both interpolate f (x) at the points x 2 ,...,x n −1 and

it follows that p n (x) also interpolates f (x) at these points. Further, p n (x 1 ) =p n −1 (x 1 ) and hence interpolates f (x) at x 1 . A similar argument shows that p n (x) interpolates f (x) at x = x n . Hence p n (x) is the unique polynomial interpolating f (x) at the distinct points x 1 ,x 2 ,...,x n .

A variety of schemes use Theorem 4.2.7 to construct successively higher order inter- polation polynomials. Denote by P j,j +1,...,k (x) , k > j, the polynomial interpolating f (x) at the points x j ,x j +1 ,...,x k . The calculations in Neville’s algorithm can be arranged in a triangular table.

P 1,2,3 (x)

x 4 f (x 4 ) P 3,4 (x)

P 2,3,4 (x)

P 1,2,3,4 (x)

f (x k ) P k −1,k (x) P k −2,k−1,k (x) P k −3,k−2,k−1,k (x) ... P 1,2,3,...,k Any entry in this table is obtained as a linear combination of the entries to the left and

diagonally above in the preceding column. Note that it is easy to add a new interpolation point in this scheme. To proceed only the last row needs to be retained. This is convenient in applications where the function values are generated sequentially and it is not known in advance how many values are to

be generated. Neville’s algorithm is used, for example, in repeated Richardson extrapolation (see Sec. 3.4.6), where polynomial extrapolation to x = 0 is to be carried out. Another use of Neville’s algorithm is in iterative inverse interpolation; see Isaacson and Keller [208, Chapter 6.2].

If it is known in advance that a fixed number k of points are to be used, then one can instead generate the table column by column. When one column has been evaluated the preceding may be discarded.

Aitken’s algorithm uses another sequence of interpolants, as indicated in the table below. x 1 f (x 1 )

x 2 f (x 2 ) P 1,2 (x) x 3 f (x 3 ) P 1,3 (x) P 1,2,3 (x) x 4 f (x 4 ) P 1,4 (x) P 1,2,4 (x) P 1,2,3,4 (x)

f (x k ) P 1,k (x) P 1,2,k (x) P 1,2,3,k (x) ... P 1,2,3,...,k

4.2. Interpolation Formulas and Algorithms 373 For a fixed number k of points this table can be generated column by column. To add a new

point the upper diagonal f (x 1 ), P 1,2 (x), P 1,2,3 (x), . . . , P 1,2,...,k (x) needs to be saved. The basic difference between these two procedures is that in Aitken’s the interpolants in any row use points with subscripts near 1, whereas Neville’s algorithm uses rows with subscripts nearest n.

Neville’s and Aitken’s algorithms can easily be used in the case of multiple interpo- lation points also. The modification is similar to that in Newton’s interpolation method.