Interpolation formula
A formula for the approximate calculation of values of a function by replacing it by a function
![]() |
that is simple in a certain sense and belongs to a certain class. The parameters ,
, are chosen in such a way that the values of
coincide with the known values of
on a given set of
distinct values of the argument:
![]() | (1) |
This method of approximately representing a function is called interpolation, and the points at which (1) should hold are called interpolation nodes. Instead of the simplest condition (1), the values of some quantity related to
may also be given, e.g. the values of a derivative of
at interpolation nodes.
The method of linear interpolation is the most widespread among the interpolation methods. The approximation is now looked for in the class of (generalized) polynomials
![]() | (2) |
in some fixed system of functions . In order for the interpolation polynomial (2) to exist for any function
defined on an interval
, and for any choice of
nodes
,
if
, it is necessary and sufficient that
is a Chebyshev system of functions on
. The interpolation polynomial will, moreover, be unique and its coefficients
can be found by directly solving (1).
For one often takes: the sequences of powers of
,
![]() |
the sequence of trigonometric functions,
![]() |
or the sequence of exponential functions,
![]() |
where is a sequence of distinct real numbers.
When interpolating by algebraic polynomials
![]() | (3) |
the system is
![]() | (4) |
while (1) has the form
![]() | (5) |
The system (4) is a Chebyshev system, which ensures the existence and uniqueness of the interpolation polynomial (3). A property of (4) is the possibility of obtaining an explicit representation of the interpolation polynomial (3) without immediately having to solve (5). One of the explicit forms of (3),
![]() | (6) |
is called the Lagrange interpolation polynomial (cf. Lagrange interpolation formula). If the derivative is continuous, the remainder of (6) can be written as
![]() | (7) |
![]() |
where ;
. The value of the remainder (7) depends, in particular, on the values of
. The choice of interpolation nodes for which
is minimal, is of interest. The distribution of the nodes is optimal in this sense if the roots
![]() |
of the polynomial
![]() |
which deviates least from zero on , are taken as the nodes. Here
is the Chebyshev polynomial of degree
(cf. Chebyshev polynomials).
There is a number of other explicit representations of (3) that are more useful for solving this or another practical interpolation problem (cf., e.g., Bessel interpolation formula; Gauss interpolation formula; Newton interpolation formula; Stirling interpolation formula; Steffensen interpolation formula; Everett interpolation formula). If it is difficult to estimate in advance the degree of the interpolation polynomial that is necessary for attaining the error desired (e.g., when interpolating a table), then one takes recourse to the Aitken scheme. In this scheme interpolation polynomials of increasing degrees are constructed sequentially, thus making it possible to control the accuracy in the computational process. Another approach to the construction of interpolation formulas can be found in Fraser diagram.
The Hermite interpolation formula gives the solution to the problem of the algebraic interpolation of the values of a function and its derivatives at interpolation nodes.
References
[1] | I.S. Berezin, N.P. Zhidkov, "Computing methods" , Pergamon (1973) (Translated from Russian) |
[2] | N.S. Bakhvalov, "Numerical methods: analysis, algebra, ordinary differential equations" , MIR (1977) (Translated from Russian) |
Comments
Many interpolation formulas are contained in [a1]–[a3].
Consider the interpolation problem of finding a polynomial of degree
satisfying the conditions
![]() | (a1) |
where the are
distinct knots, and there are precisely
equations in (a1). If for each
the orders of the derivatives occurring in (a1) form an unbroken series
, one has Hermite interpolation. (In case
for all
, i.e. if no interpolation conditions involving derivatives occur in (a1), one has Lagrange interpolation.) If gaps (lacunae) occur, one speaks of lacunary interpolation or Birkhoff interpolation. The pairs
which occur in (a1) are conveniently described in terms of an interpolation matrix
of size
,
,
,
, where
if
does occur in (a1) and
otherwise. The matrix
is called regular if (a1) is solvable for all choices of the
and
and singular otherwise.
More generally, let be a system of linearly independent
-times continuously-differentiable real-valued functions on an interval
or on the circle. Instead of polynomials now consider linear combinations
. A matrix of zeros and ones
,
,
, is an interpolation matrix if there are precisely
ones in
(and, usually, if there are no rows of zeros in
; this means that all knots do occur at least once in an interpolation condition). Let
be a set of knots, i.e.
distinct points of the interval or circle. Finally, for each
such that
let there be given a number
. These data
define a Birkhoff interpolation problem:
![]() | (a2) |
![]() |
The pair is called regular if (a2) is solvable for all choices of the
.
For each such that
, consider the row vector of length
,
![]() |
For varying such that
one thus finds
row vectors which together make up an
matrix. The pair
is regular if and only if this matrix is invertible. Its determinant, where the pairs
with
are ordered lexicographically, is denoted
.
Suppose that the are the values
of the derivatives of some function
at the knots. Then a simple formula for the solution of the interpolation problem (a2) follows from Cramer's rule. Indeed, if
denotes the determinant obtained by replacing
with
in the formula for
, then
![]() | (a3) |
See also Hermite interpolation formula.
References
[a1] | P.J. Davis, "Interpolation and approximation" , Dover, reprint (1975) pp. 108–126 |
[a2] | F.B. Hildebrand, "Introduction to numerical analysis" , McGraw-Hill (1974) |
[a3] | J.F. Steffenson, "Interpolation" , Chelsea, reprint (1950) |
[a4] | G.G. Lorentz, K. Jetter, S.D. Riemenschneider, "Birkhoff interpolation" , Addison-Wesley (1983) |
Interpolation formula. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Interpolation_formula&oldid=14575