Divided differences

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In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions.[citation needed] Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in its operation.[1]

Divided differences is a recursive division process. Given a sequence of data points , the method calculates the coefficients of the interpolation polynomial of these points in the Newton form.

Definition[edit]

Given n + 1 data points

where the are assumed to be pairwise distinct, the forward divided differences are defined as:

To make the recursive process of computation clearer, the divided differences can be put in tabular form, where the columns correspond to the value of j above, and each entry in the table is computed from the difference of the entries to its immediate lower left and to its immediate upper left, divided by a difference of corresponding x-values:

Notation[edit]

Note that the divided difference depends on the values and , but the notation hides the dependency on the x-values. If the data points are given by a function ƒ,

one sometimes writes

for the divided difference

Several other notations for the divided difference of the function ƒ on the nodes x0, ..., xn are also used, for instance:

Example[edit]

Divided differences for and the first few values of :

Properties[edit]

  • Divided differences are symmetric: If is a permutation then
  • If is a polynomial function of degree , then
for a number in the open interval determined by the smallest and largest of the 's.

Matrix form[edit]

The divided difference scheme can be put into an upper triangular matrix:

.

Then it holds

  • if is a scalar
This follows from the Leibniz rule. It means that multiplication of such matrices is commutative. Summarised, the matrices of divided difference schemes with respect to the same set of nodes x form a commutative ring.
  • Since is a triangular matrix, its eigenvalues are obviously .
  • Let be a Kronecker delta-like function, that is
Obviously , thus is an eigenfunction of the pointwise function multiplication. That is is somehow an "eigenmatrix" of : . However, all columns of are multiples of each other, the matrix rank of is 1. So you can compose the matrix of all eigenvectors of from the -th column of each . Denote the matrix of eigenvectors with . Example
The diagonalization of can be written as
.

Polynomials and power series[edit]

The matrix

contains the divided difference scheme for the identity function with respect to the nodes , thus contains the divided differences for the power function with exponent . Consequently, you can obtain the divided differences for a polynomial function by applying to the matrix : If

and

then

This is known as Opitz' formula.[2][3]

Now consider increasing the degree of to infinity, i.e. turn the Taylor polynomial into a Taylor series. Let be a function which corresponds to a power series. You can compute the divided difference scheme for by applying the corresponding matrix series to : If

and

then

Alternative characterizations[edit]

Expanded form[edit]

With the help of the polynomial function this can be written as

Peano form[edit]

If and , the divided differences can be expressed as[4]

where is the -th derivative of the function and is a certain B-spline of degree for the data points , given by the formula

This is a consequence of the Peano kernel theorem; it is called the Peano form of the divided differences and is the Peano kernel for the divided differences, all named after Giuseppe Peano.

Forward differences[edit]

When the data points are equidistantly distributed we get the special case called forward differences. They are easier to calculate than the more general divided differences.

Given n+1 data points

with

the forward differences are defined as

The relationship between divided differences and forward differences is[5]

See also[edit]

References[edit]

  1. ^ Isaacson, Walter (2014). The Innovators. Simon & Schuster. p. 20. ISBN 978-1-4767-0869-0.
  2. ^ de Boor, Carl, Divided Differences, Surv. Approx. Theory 1 (2005), 46–69, [1]
  3. ^ Opitz, G. Steigungsmatrizen, Z. Angew. Math. Mech. (1964), 44, T52–T54
  4. ^ Skof, Fulvia (2011-04-30). Giuseppe Peano between Mathematics and Logic: Proceeding of the International Conference in honour of Giuseppe Peano on the 150th anniversary of his birth and the centennial of the Formulario Mathematico Torino (Italy) October 2-3, 2008. Springer Science & Business Media. p. 40. ISBN 978-88-470-1836-5.
  5. ^ Burden, Richard L.; Faires, J. Douglas (2011). Numerical Analysis (9th ed.). p. 129. ISBN 9780538733519.
  • Louis Melville Milne-Thomson (2000) [1933]. The Calculus of Finite Differences. American Mathematical Soc. Chapter 1: Divided Differences. ISBN 978-0-8218-2107-7.
  • Myron B. Allen; Eli L. Isaacson (1998). Numerical Analysis for Applied Science. John Wiley & Sons. Appendix A. ISBN 978-1-118-03027-1.
  • Ron Goldman (2002). Pyramid Algorithms: A Dynamic Programming Approach to Curves and Surfaces for Geometric Modeling. Morgan Kaufmann. Chapter 4:Newton Interpolation and Difference Triangles. ISBN 978-0-08-051547-2.

External links[edit]