Interpolation formula

A formula for the approximate calculation of values of a function $f ( x)$ by replacing it by a function

$$g ( x) \equiv g ( x ; a _ {0}, \dots, a _ {n} )$$

that is simple in a certain sense and belongs to a certain class. The parameters $a _ {i}$, $i = 0, \dots, n$, are chosen in such a way that the values of $g( x)$ coincide with the known values of $f ( x)$ on a given set of $n + 1$ distinct values of the argument:

$$\tag{1 } g ( x _ {k} ) = f ( x _ {k} ) ,\ \ k = 0, \dots, n .$$

This method of approximately representing a function is called interpolation, and the points $x _ {k}$ at which (1) should hold are called interpolation nodes. Instead of the simplest condition (1), the values of some quantity related to $f ( x)$ may also be given, e.g. the values of a derivative of $f ( x)$ 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

$$\tag{2 } g ( x ; a _ {0}, \dots, a _ {n} ) = \ \sum _ { i= 0} ^ { n } a _ {i} \phi _ {i} ( x)$$

in some fixed system of functions $\phi _ {0} ( x), \dots, \phi _ {n} ( x)$. In order for the interpolation polynomial (2) to exist for any function $f ( x)$ defined on an interval $[ a , b ]$, and for any choice of $n + 1$ nodes $x _ {0}, \dots, x _ {n} \in [ a , b ]$, $x _ {i} \neq x _ {j}$ if $i \neq j$, it is necessary and sufficient that $\{ \phi _ {i} ( x) \}$ is a Chebyshev system of functions on $[ a , b ]$. The interpolation polynomial will, moreover, be unique and its coefficients $a _ {i}$ can be found by directly solving (1).

For $\{ \phi _ {i} ( x) \}$ one often takes: the sequences of powers of $x$,

$$1 , x , x ^ {2}, \dots$$

the sequence of trigonometric functions,

$$1 , \sin x , \cos x ,\ \sin 2 x , \cos 2 x, \dots$$

or the sequence of exponential functions,

$$1 , e ^ {\alpha _ {1} x } ,\ e ^ {\alpha _ {2} x }, \dots$$

where $\{ \alpha _ {i} \}$ is a sequence of distinct real numbers.

When interpolating by algebraic polynomials

$$\tag{3 } \sum _ { i= 0} ^ { n } a _ {i} x ^ {i}$$

the system $\{ \phi _ {i} ( x) \}$ is

$$\tag{4 } \phi _ {i} ( x) = x ^ {i} ,\ \ i = 0, \dots, n ,$$

while (1) has the form

$$\tag{5 } \sum _ { i= 0} ^ { n } a _ {i} x _ {k} ^ {i} = \ f ( x _ {k} ) ,\ \ k = 0, \dots, n .$$

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),

$$\tag{6 } g _ {n} ( x) = L _ {n} ( x) = \ \sum _ { i= 0} ^ { n } f ( x _ {i} ) \prod _ {j \neq i } \frac{x - x _ {j} }{x _ {i} - x _ {j} } ,$$

is called the Lagrange interpolation polynomial (cf. Lagrange interpolation formula). If the derivative $f ^ { ( n+ 1) } ( x)$ is continuous, the remainder of (6) can be written as

$$\tag{7 } f ( x) - g _ {n} ( x) = \ \frac{f ^ { ( n + 1 ) } ( \xi ) }{( n + 1 ) ! } \omega _ {n} ( x) ,$$

$$\xi \in [ y _ {1} , y _ {2} ] ,\ \omega _ {n} ( x) = \prod _ { i= 0} ^ { n } ( x - x _ {i} ) ;$$

where $y _ {1} = \min ( x _ {0}, \dots, x _ {n} , x )$; $y _ {2} = \max ( x _ {0}, \dots, x _ {n} , x)$. The value of the remainder (7) depends, in particular, on the values of $\omega _ {n} ( x)$. The choice of interpolation nodes for which $\sup _ {[ a , b ] } | \omega _ {n} ( x) |$ is minimal, is of interest. The distribution of the nodes is optimal in this sense if the roots

$$x _ {k} = \ \frac{b + a }{2} + \frac{b - a }{2} \ \cos \frac{2 k + 1 }{2 n + 2 } \pi ,\ k = 0, \dots, n ,$$

of the polynomial

$$\overline{T} _ {n+1} ^ { [ a , b ] } ( x) = ( b - a ) ^ {n+1} 2 ^ {-1- 2n} T _ {n+1} \left ( \frac{2 x - ( b + a ) }{b - a } \right ) ,$$

which deviates least from zero on $[ a , b ]$, are taken as the nodes. Here $T _ {n+1} ( z)$ is the Chebyshev polynomial of degree $n+ 1$ (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)

Many interpolation formulas are contained in [a1][a3].

Consider the interpolation problem of finding a polynomial $P _ {N}$ of degree $\leq N$ satisfying the conditions

$$\tag{a1 } P _ {N} ^ {(k)} ( x _ {i} ) = c _ {i,k} ,$$

where the $x _ {1}, \dots, x _ {m}$ are $m$ distinct knots, and there are precisely $N + 1$ equations in (a1). If for each $i$ the orders of the derivatives occurring in (a1) form an unbroken series $k = 0, \dots, k _ {i}$, one has Hermite interpolation. (In case $k _ {i} = 0$ for all $i$, 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 $( i , k )$ which occur in (a1) are conveniently described in terms of an interpolation matrix $E$ of size $m \times ( n + 1 )$, $E = ( e _ {i,k} )$, $i = 1, \dots, m$, $k = 0, \dots, n$, where $e _ {ik} = 1$ if $( i , k )$ does occur in (a1) and $e _ {ik} = 0$ otherwise. The matrix $E$ is called regular if (a1) is solvable for all choices of the $x _ {i}$ and $c _ {i,k}$ and singular otherwise.

More generally, let $G = \{ g _ {0}, \dots, g _ {N} \}$ be a system of linearly independent $n$-times continuously-differentiable real-valued functions on an interval $[ a, b]$ or on the circle. Instead of polynomials now consider linear combinations $P = \sum _ {j=0} ^ {N} a _ {j} g _ {j}$. A matrix of zeros and ones $E = ( e _ {ik} )$, $i = 1, \dots, m$, $k = 0, \dots, n$, is an interpolation matrix if there are precisely $N + 1$ ones in $E$ (and, usually, if there are no rows of zeros in $E$; this means that all knots do occur at least once in an interpolation condition). Let $K = \{ x _ {1}, \dots, x _ {m} \}$ be a set of knots, i.e. $m$ distinct points of the interval or circle. Finally, for each $( i , k )$ such that $e _ {i,k} = 1$ let there be given a number $c _ {i,k}$. These data $( G , E , K , c _ {i,k} )$ define a Birkhoff interpolation problem:

$$\tag{a2 } P ^ {(k)} ( x _ {i} ) = c _ {i,k} \ \$$

$$\textrm{ for all } ( i , k ) \textrm{ such that } e _ {i,k} = 1 .$$

The pair $( E , K )$ is called regular if (a2) is solvable for all choices of the $c _ {i,k}$.

For each $( i , k )$ such that $e _ {i,k} = 1$, consider the row vector of length $N + 1$,

$$g _ {0} ^ {(k)} ( x _ {i} ), \dots, g _ {N} ^ {(k)} ( x _ {i} ) .$$

For varying $( i , k )$ such that $e _ {i,k} = 1$ one thus finds $N + 1$ row vectors which together make up an $( N + 1 ) \times ( N + 1 )$ matrix. The pair $( E , K )$ is regular if and only if this matrix is invertible. Its determinant, where the pairs $( i , k )$ with $e _ {i,k} = 1$ are ordered lexicographically, is denoted $D ( E , K )$.

Suppose that the $c _ {i,k}$ are the values $f ^ { ( k) } ( x _ {i} )$ of the derivatives of some function $f$ at the knots. Then a simple formula for the solution of the interpolation problem (a2) follows from Cramer's rule. Indeed, if $D ^ {(j)} ( E , K )$ denotes the determinant obtained by replacing $g _ {j}$ with $f$ in the formula for $D ( E , K )$, then

$$\tag{a3 } P ( t) = \sum _ { j= 0} ^ { N } \frac{D ^ {(j)} ( E , K ) }{D ( E , K ) } g _ {j} ( t)$$