Polynomial interpolation

In numerical analysis, polynomial interpolation is the interpolation of a given bivariate data set by the polynomial of lowest possible degree that passes through the points of the dataset.

Given a set of n + 1 data points $(x_{0},y_{0}),\ldots ,(x_{n},y_{n})$, with no two $x_{j}$ the same, a polynomial function $p(x)=a_{0}+a_{1}x+\cdots +a_{n}x^{n}$ is said to interpolate the data if Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_j)=y_j} for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\in\{0,1,\dotsc,n\}} .

There is always a unique such polynomial, commonly given by two explicit formulas, the Lagrange polynomials and Newton polynomials.

Applications

The original use of interpolation polynomials was to approximate values of important transcendental functions such as natural logarithm and trigonometric functions. Starting with a few accurately computed data points, the corresponding interpolation polynomial will approximate the function at an arbitrary nearby point. Polynomial interpolation also forms the basis for algorithms in numerical quadrature (Simpson's rule) and numerical ordinary differential equations (multigrid methods).

In computer graphics, polynomials can be used to approximate complicated plane curves given a few specified points, for example the shapes of letters in typography. This is usually done with Bézier curves, which are a simple generalization of interpolation polynomials (having specified tangents as well as specified points).

In numerical analysis, polynomial interpolation is essential to perform sub-quadratic multiplication and squaring, such as Karatsuba multiplication and Toom–Cook multiplication, where interpolation through points on a product polynomial yields the specific product required. For example, given a = f(x) = a₀x⁰ + a₁x¹ + ··· and b = g(x) = b₀x⁰ + b₁x¹ + ···, the product ab is a specific value of W(x) = f(x)g(x). One may easily find points along W(x) at small values of x, and interpolation based on those points will yield the terms of W(x) and the specific product ab. As fomulated in Karatsuba multiplication, this technique is substantially faster than quadratic multiplication, even for modest-sized inputs, especially on parallel hardware.

In computer science, polynomial interpolation also leads to algorithms for secure multi party computation and secret sharing.

Interpolation theorem

For any Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} bivariate data points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0),\dotsc,(x_n,y_n) \in \R^2} , where no two $x_{j}$ are the same, there exists a unique polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} of degree at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} that interpolates these points, i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_0)=y_0, \ldots, p(x_n)=y_n} .

Equivalently, for a fixed choice of interpolation nodes $x_{j}$, polynomial interpolation defines a linear bijection Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n} between the (n+1)-tuples of real-number values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (y_0,\ldots,y_n)\in \R^{n+1}} and the vector space $P(n)$ of real polynomials of degree at most n: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_n : \mathbb{R}^{n+1} \stackrel{\sim}{\longrightarrow}\, P(n).}

This is a type of unisolvence theorem. The theorem is also valid over any infinite field in place of the real numbers Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \R} , for example the rational or complex numbers.

First proof

Consider the Lagrange basis functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_1(x),\ldots,L_n(x)} given by:

$$L_{j}(x)=\prod _{i\neq j}{\frac {x-x_{i}}{x_{j}-x_{i}}}={\frac {(x-x_{0})\cdots (x-x_{j-1})(x-x_{j+1})\cdots (x-x_{n})}{(x_{j}-x_{0})\cdots (x_{j}-x_{j-1})(x_{j}-x_{j+1})\cdots (x_{j}-x_{n})}}.$$

Notice that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_j(x)} is a polynomial of degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , and we have $L_{j}(x_{k})=0$ for each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j\neq k} , while Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L_k(x_k)=1} . It follows that the linear combination: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = \sum_{j=0}^n y_j L_j(x)} has Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_k)=\sum_j y_j \,L_j(x_k) = y_k } , so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} is an interpolating polynomial of degree $n$.

To prove uniqueness, assume that there exists another interpolating polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q(x)} of degree at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , so that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_k)=q(x_k)} for all Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle k=0,\dotsc,n} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)-q(x)} is a polynomial of degree at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} which has $n+1$ distinct zeros (the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_k} ). But a non-zero polynomial of degree at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} can have at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} zeros, so Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)-q(x)} must be the zero polynomial, i.e. $p(x)=q(x)$.

Second proof

Write out the interpolation polynomial in the form

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_2 x^2 + a_1 x + a_0. } (1)

Substituting this into the interpolation equations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x_j) = y_j} , we get a system of linear equations in the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j} , which reads in matrix-vector form as the following multiplication: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} x_0^n & x_0^{n-1} & x_0^{n-2} & \ldots & x_0 & 1 \\ x_1^n & x_1^{n-1} & x_1^{n-2} & \ldots & x_1 & 1 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ x_n^n & x_n^{n-1} & x_n^{n-2} & \ldots & x_n & 1 \end{bmatrix} \begin{bmatrix} a_n \\ a_{n-1} \\ \vdots \\ a_0 \end{bmatrix} = \begin{bmatrix} y_0 \\ y_1 \\ \vdots \\ y_n \end{bmatrix}.}

An interpolant Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} corresponds to a solution $A=(a_{n},\ldots ,a_{0})$ of the above matrix equation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle X \cdot A = Y} . The matrix X on the left is a Vandermonde matrix, whose determinant is known to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textstyle \det(X) = \prod_{1 \le i < j \le n} (x_j - x_i), } which is non-zero since the nodes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} are all distinct. This ensures that the matrix is invertible and the equation has the unique solution Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A = X^{-1}\cdot Y} ; that is, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p(x)} exists and is unique.

Corollary

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} is a polynomial of degree at most Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} , then the interpolating polynomial of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} at Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} distinct points is Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} itself.

Constructing the interpolation polynomial

The red dots denote the data points (x_k, y_k), while the blue curve shows the interpolation polynomial.

Lagrange Interpolation

Main article: Lagrange polynomial

We may write down the polynomial immediately in terms of Lagrange polynomials as: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} p(x) &= \frac{(x-x_1)(x-x_2)\cdots(x-x_n)}{(x_0-x_1)(x_0-x_2)\cdots(x_0-x_n)} y_0 \\ [4pt] &+ \frac{(x-x_0)(x-x_2)\cdots(x-x_n)}{(x_1-x_0)(x_1-x_2) \cdots(x_1-x_n)}y_1 \\ [4pt] &+ \cdots\\ [4pt] &+\frac{(x-x_0)(x-x_1)\cdots(x-x_{n-1})}{(x_n-x_0)(x_n-x_1)\cdots(x_n-x_{n-1})}y_n \\ [7pt] &=\sum_{i=0}^n \Biggl( \prod_{\stackrel{\!0\,\leq\, j\,\leq\, n}{j\,\neq\, i}} \frac{x-x_j}{x_i-x_j} \Biggr) y_i =\sum_{i=0}^n \frac{p(x)}{p'(x_i)(x-x_i)}\,y_i \end{align}} For matrix arguments, this formula is called Sylvester's formula and the matrix-valued Lagrange polynomials are the Frobenius covariants.

Newton Interpolation

See also: Newton polynomial and Divided differences

Theorem

For a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n} of degree less than or equal to n, that interpolates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at the nodes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 0,1,2,3,\cdots,n} . Let Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}} be the polynomial of degree less than or equal to n+1 that interpolates Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} at the nodes Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i} where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 0,1,2,3,\cdots,n, n+1} . Then Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}} is given by:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}(x) = p_n(x) +a_{n+1}w_n(x) } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle w_n(x) := \prod_{i=0}^n (x-x_i) } also known as Newton basis and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_{n+1} :={f(x_{n+1})-p_n(x_{n+1}) \over w_n(x_{n+1})} } .

Proof:

This can be shown for the case where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = 0,1,2,3,\cdots,n} :Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}(x_i) = p_n(x_i) +a_{n+1}\prod_{j=0}^n (x_i-x_j) = p_n(x_i) } and when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i = n+1} :Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_{n+1}(x_{n+1}) = p_n(x_{n+1}) +{f(x_{n+1})-p_n(x_{n+1}) \over w_n(x_{n+1})} w_n(x_{n+1}) = f(x_{n+1}) } By the uniqueness of interpolated polynomials of degree less than Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} , Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle p_{n+1}(x) = p_n(x) +a_{n+1}w_n(x) } is the required polynomial interpolation. The function can thus be expressed as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle p_{n}(x) = a_0+a_1(x-x_0)+a_2(x-x_0)(x-x_1)+\cdots + a_n(x-x_0)\cdots(x-x_{n-1}) .}

Polynomial coefficients

To find Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_i} , we have to solve the lower triangular matrix formed by arranging Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle p_{n} (x_i)=f(x_i)=y_i } from above equation in matrix form:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{bmatrix} 1 & & \ldots & & 0 \\ 1 & x_1-x_0 & & & \\ 1 & x_2-x_0 & (x_2-x_0)(x_2-x_1) & & \vdots \\ \vdots & \vdots & & \ddots & \\ 1 & x_k-x_0 & \ldots & \ldots & \prod_{j=0}^{n-1}(x_n - x_j) \end{bmatrix} \begin{bmatrix} a_0 \\ \\ \vdots \\ \\ a_{n} \end{bmatrix} = \begin{bmatrix} y_0 \\ \\ \vdots \\ \\ y_{n} \end{bmatrix}}

The coefficients are derived as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_j := [y_0,\ldots,y_j]}

where

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [y_0,\ldots,y_j]}

is the notation for divided differences. Thus, Newton polynomials are used to provide a polynomial interpolation formula of n points.

Proof

The first few coefficients can be calculated using the system of equations. The form of n-th coefficient is assumed for proof by mathematical induction. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} a_0 &= y_0 = [y_0] \\ a_1 &= {y_1-y_0 \over x_1 - x_0}= [y_0,y_1] \\ \vdots\\ a_n &= [y_0,\cdots,y_n] \quad \text{(let)}\\ \end{align} } Let Q be polynomial interpolation of points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_1, y_1), \ldots, (x_n, y_n)} . Adding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,y_0)} to the polynomial Q: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(x)+ a'_n (x - x_1)\cdot\ldots\cdot(x - x_n) = P_{n}(x), } where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a'_n(x_{0} - x_1)\ldots(x_{0}-x_{n}) = y_{0} - Q(x_{0}) } . By uniqueness of the interpolating polynomial of the points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, y_0), \ldots, (x_n, y_n)} , equating the coefficients of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x^{n-1}} we get, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a'_n=[y_0, \ldots, y_{n}] } . Hence the polynomial can be expressed as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P_{n}(x)= Q(x)+ [y_0,\ldots,y_n](x - x_1)\cdot\ldots\cdot(x - x_n).} Adding Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_{n+1},y_{n+1})} to the polynomial Q, it has to satisfiy: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle [y_1, \ldots,y_{n+1}](x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1}-x_{n}) = y_{n+1} - Q(x_{n+1}) } where the formula for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_n } and interpolating polynomial are used. The Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle a_{n+1} } term for the polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle P_{n+1} } can be found by calculating:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} & [y_0,\ldots,y_{n+1}](x_{n+1} - x_0)\cdot\ldots\cdot(x_{n+1} - x_n)\\ &= \frac{[y_1,\ldots,y_{n+1}] - [y_0,\ldots,y_{n}]}{x_{n+1} - x_0}(x_{n+1} - x_0)\cdot\ldots\cdot(x_{n+1} - x_n) \\ &= \left([y_1,\ldots,y_{n+1}] - [y_0,\ldots,y_{n}]\right) (x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1} - x_n) \\ &= [y_1,\ldots,y_{n+1}](x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1} - x_n) - [y_0,\ldots,y_n](x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1} - x_n) \\ &= (y_{n+1} - Q(x_{n+1})) - [y_0,\ldots,y_n](x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1} - x_n) \\ &= y_{n+1} - (Q(x_{n+1}) + [y_0,\ldots,y_n](x_{n+1} - x_1)\cdot\ldots\cdot(x_{n+1} - x_n))\\ &=y_{n+1}-P(x_{n+1}). \end{align} } which implies that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a_{n+1}={y_{n+1}-P_n(x_{n+1}) \over w_n(x_{n+1})} = [y_0,\ldots,y_{n+1}]} . Hence it is proved by principle of mathematical induction.

Newton forward formula

The Newton polynomial can be expressed in a simplified form when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0, x_1, \dots, x_k} are arranged consecutively with equal spacing.

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0, x_1, \dots, x_k} are consecutively arranged and equally spaced with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}_{i}={x}_{0}+ih } for i = 0, 1, ..., k and some variable x is expressed as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}={x}_{0}+sh} , then the difference Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x-x_i} can be written as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (s-i)h} . So the Newton polynomial becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} N(x) &= [y_0] + [y_0,y_1]sh + \cdots + [y_0,\ldots,y_k] s (s-1) \cdots (s-k+1){h}^{k} \\ &= \sum_{i=0}^{k}s(s-1) \cdots (s-i+1){h}^{i}[y_0,\ldots,y_i] \\ &= \sum_{i=0}^{k}{s \choose i}i!{h}^{i}[y_0,\ldots,y_i]. \end{align}}

Since the relationship between divided differences and forward differences is given as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [y_j, y_{j+1}, \ldots , y_{j+n}] = \frac{1}{n!h^n}\Delta^{(n)}y_j,} Taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i=f(x_i)} , if the representation of x in the previous sections was instead taken to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_j+sh} , the Newton forward interpolation formula is expressed as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \approx N(x)=N(x_j+sh) = \sum_{i=0}^{k}{s \choose i}\Delta^{(i)} f(x_j) } which is the interpolation of all points after Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} . It is expanded as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_j+sh)=f(x_j)+\frac{s}{1!}\Delta f(x_j)+ \frac{s(s-1)}{2!}\Delta^2 f(x_j)+\frac{s(s-1)(s-2)}{3!}\Delta^3 f(x_j)+\frac{s(s-1)(s-2)(s-3)}{4!}\Delta^4 f(x_j)+\cdots }

Newton backward formula

If the nodes are reordered as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}_{k},{x}_{k-1},\dots,{x}_{0}} , the Newton polynomial becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle N(x)=[y_k]+[{y}_{k}, {y}_{k-1}](x-{x}_{k})+\cdots+[{y}_{k},\ldots,{y}_{0}](x-{x}_{k})(x-{x}_{k-1})\cdots(x-{x}_{1}).}

If Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}_{k},\;{x}_{k-1},\;\dots,\;{x}_{0}} are equally spaced with Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}_{i}={x}_{k}-(k-i)h} for i = 0, 1, ..., k and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {x}={x}_{k}+sh} , then,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} N(x) &= [{y}_{k}]+ [{y}_{k}, {y}_{k-1}]sh+\cdots+[{y}_{k},\ldots,{y}_{0}]s(s+1)\cdots(s+k-1){h}^{k} \\ &=\sum_{i=0}^{k}{(-1)}^{i}{-s \choose i}i!{h}^{i}[{y}_{k},\ldots,{y}_{k-i}]. \end{align}}

Since the relationship between divided differences and backward differences is given as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle [{y}_{j}, y_{j-1},\ldots,{y}_{j-n}] = \frac{1}{n!h^n}\nabla^{(n)}y_j, } taking Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i=f(x_i)} , if the representation of x in the previous sections was instead taken to be Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=x_j+sh} , the Newton backward interpolation formula is expressed as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) \approx N(x) =N(x_j+sh)=\sum_{i=0}^{k}{(-1)}^{i}{-s \choose i}\nabla^{(i)} f(x_j). } which is the interpolation of all points before Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} . It is expanded as:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x_j+sh)=f(x_j)+\frac{s}{1!}\nabla f(x_j)+ \frac{s(s+1)}{2!}\nabla^2 f(x_j)+\frac{s(s+1)(s+2)}{3!}\nabla^3 f(x_j)+\frac{s(s+1)(s+2)(s+3)}{4!}\nabla^4 f(x_j)+\cdots }

Lozenge Diagram

A Lozenge diagram is a diagram that is used to describe different interpolation formulas that can be constructed for a given data set. A line starting on the left edge and tracing across the diagram to the right can be used to represent an interpolation formula if the following rules are followed:

Lozenge Diagram: geometric representation of polynomial interpolations.

1. Left to right steps indicate addition whereas right to left steps indicate subtraction
2. If the slope of a step is positive, the term to be used is the product of the difference and the factor immediately below it. If the slope of a step is negative, the term to be used is the product of the difference and the factor immediately above it.
3. If a step is horizontal and passes through a factor, use the product of the factor and the average of the two terms immediately above and below it. If a step is horizontal and passes through a difference, use the product of the difference and the average of the two terms immediately above and below it.

The factors are expressed using the formula:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(u+k,n)=\frac{(u+k)(u+k-1)\cdots(u+k-n+1)}{n!} }

Proof of equivalence

If a path goes from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^{n-1}y_s } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^{n+1}y_{s-1} } , it can connect through three intermediate steps, (a) through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^{n}y_{s-1} } , (b) through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle C(u-s ,n) } or (c) through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^{n}y_s } . Proving the equivalence of these three two-step paths should prove that all (n-step) paths can be morphed with the same starting and ending, all of which represents the same formula.

Path (a):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(u-s, n) \Delta^n y_{s-1}+C(u-s+1, n+1) \Delta^{n+1} y_{s-1} }

Path (b):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(u-s, n) \Delta^n y_s + C(u-s, n+1) \Delta^{n+1} y_{s-1} }

Path (c):

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C(u-s, n) \frac{\Delta^n y_{s-1}+\Delta^n y_{s}}{2} \quad+\frac{C(u-s+1, n+1)+C(u-s, n+1)}{2} \Delta^{n+1} y_{s-1} }

Subtracting contributions from path a and b:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} \text{Path a - Path b}= & C(u-s, n)(\Delta^n y_{s-1}-\Delta^n y_s) +(C(u-s+1, n+1)-C(u-s, n-1)) \Delta^{n+1} y_{s-1} \\ = & - C(u-s, n)\Delta^{n+1} y_{s-1} + C(u-s, n) \frac{(u-s+1)-(u-s-n)}{n+1} \Delta^{n+1} y_{s-1} \\ = & C(u-s, n)(-\Delta^{n+1} y_{s-1}+\Delta^{n+1} y_{s-1} )=0 \\ \end{aligned} }

Thus, the contribution of either path (a) or path (b) is the same. Since path (c) is the average of path (a) and (b), it also contributes identical function to the polynomial. Hence the equivalence of paths with same starting and ending points is shown. To check if the paths can be shifted to different values in the leftmost corner, taking only two step paths is sufficient: (a) Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s+1} } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} } through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta y_{s} } or (b) factor between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s+1} } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} } , to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} } through Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta y_{s} } or (c) starting from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} } .

Path (a)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s+1}+C(u-s-1,1) \Delta y_s - C(u-s, 1) \Delta y_s }

Path (b)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{y_{s+1}+y_s}{2}+\frac{C(u-s-1,1)+C(u-s, 1)}{2} \Delta y_s - C(u-s, 1) \Delta y_s }

Path (c)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} }

Since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta y_{s} = y_{s+1}-y_s } , substituting in the above equations shows that all the above terms reduce to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_{s} } and are hence equivalent. Hence these paths can be morphed to start from the leftmost corner and end in a common point.

Newton formula

Taking negative slope transversal from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0 } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^n y_0 } gives the interpolation formula of all the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1 } consecutively arranged points, equivalent to Newton's forward interpolation formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} y(s) &=y_0+C(s, 1) \Delta y_0+C(s, 2) \Delta^2 y_0+C(s, 3) \Delta^3 y_0+\cdots \\ & =y_0+s \Delta y_0+\frac{s(s-1)}{2} \Delta^2 y_0+\frac{s(s-1)(s-2)}{3 !} \Delta^3 y_0+\frac{s(s-1)(s-2)(s-3)}{4 !} \Delta^4 y_0+\cdots \end{aligned} }

whereas, taking positive slope transversal from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_n } to Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \nabla^n y_n = \Delta^n y_0 } , gives the interpolation formula of all the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1 } consecutively arranged points, equivalent to Newton's backward interpolation formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} y(u) & = y_k+C(u-k, 1) \Delta y_{k-1}+C(u-k+1,2) \Delta^2 y_{k-2} +C(u - k+2,3) \Delta^3 y_{k-3}+\cdots \\ & = y_k+(u-k) \Delta y_{k-1} +\frac{(u-k+1) (u-k)}{2} \Delta^2 y_{k-2}+\frac{(u-k+2)(u-k+1)(u-k)}{3 !} \Delta^3 y_{k-3}+\cdots \\ y(k+s) & = y_k+(s) \nabla y_{k} +\frac{(s+1) s}{2} \nabla^2 y_{k}+\frac{(s+2)(s+1) s}{3 !} \nabla^3 y_{k}+\frac{(s+3)(s+2)(s+1) s}{4 !} \nabla^4 y_{k}+\cdots \\ \end{aligned} }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=u-k } is the number corresponding to that introduced in Newton interpolation.

Gauss formula

Taking a zigzag line towards the right starting from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0 } with negative slope, we get Gauss forward formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(u)=y_0+u \Delta y_0+\frac{u(u-1)}{2} \Delta^2 y_{-1} +\frac{(u+1)u\left(u-1\right)}{3 !} \Delta^3 y_{-1}+ \frac{(u+1)u\left(u-1\right)(u-2)}{4 !} \Delta^4 y_{-2} + \cdots }

whereas starting from $y_{0}$ with positive slope, we get Gauss backward formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(u)=y_0+u \Delta y_{-1}+\frac{(u+1)u}{2} \Delta^2 y_{-1} +\frac{(u+1)u\left(u-1\right)}{3 !} \Delta^3 y_{-2}+ \frac{(u+2)(u+1)u\left(u-1\right)}{4 !} \Delta^4 y_{-2} + \cdots }

Stirling formula

By taking a horizontal path towards the right starting from Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0 } , we get Stirling formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{aligned} y(u)&= y_0 +u \frac{\Delta y_0+\Delta y_{-1}}{2}+\frac{C(u+1,2)+C(u, 2)}{2} \Delta^2 y_{-1} +C(u+1,3) \frac{\Delta^3 y_{-2}+\Delta^3 y_{-1}}{2}+\cdots \\ & = y_0+u \frac{\Delta y_0+\Delta y_{-1}}{2}+\frac{u^2}{2} \Delta^2 y_{-1}+\frac{u(u^2-1)}{3 !} \frac{\Delta^3 y_{-2}+\Delta^3 y_{-1}}{2}+\frac{u^2(u^2-1)}{4!}\Delta^4 y_{-2}+\cdots \end{aligned} }

Stirling formula is the average of Gauss forward and Gauss backward formulas.

Bessel formula

By taking a horizontal path towards the right starting from factor between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_1 } , we get Stirling formula:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y(u)&=1{\frac{y_{0}+y_{1}}{2}}+{\frac{C(u,1)+C(u-1,1)}{2}}\Delta y_{0}+C(u,2){\frac{\Delta^{2}y_{-1}+\Delta^{2}y_{0}}{2}}+\cdots\\ &= \frac{y_{0}+y_{1}}{2}+\left(u-{\frac{1}{2}}\right)\Delta y_{0}+\frac{u(u-1)}{2}\frac{\Delta^{2}y_{-1} +\Delta^{2}y_{0}}{2}+\frac{\left(u-{\frac{1}{2}}\right)u\left( u-1\right)}{3!}\Delta^{3} y_{0} + \frac{(u+1)u(u-1)(u-2)}{4!}\frac{\Delta^{4}y_{-1}+\Delta^{4}y_{-2}}{2}+\cdots\\ \end{align} }

Vandermonde Algorithms

The Vandermonde matrix in the second proof above may have large condition number, causing large errors when computing the coefficients a_i if the system of equations is solved using Gaussian elimination.

Several authors have therefore proposed algorithms which exploit the structure of the Vandermonde matrix to compute numerically stable solutions in O(n²) operations instead of the O(n³) required by Gaussian elimination. These methods rely on constructing first a Newton interpolation of the polynomial and then converting it to a monomial form.

Non-Vandermonde algorithms

To find the interpolation polynomial p(x) in the vector space P(n) of polynomials of degree n, we may use the usual monomial basis for P(n) and invert the Vandermonde matrix by Gaussian elimination, giving a computational cost of O(n³) operations. To improve this algorithm, a more convenient basis for P(n) can simplify the calculation of the coefficients, which must then be translated back in terms of the monomial basis.

One method is to write the interpolation polynomial in the Newton form (i.e. using Newton basis) and use the method of divided differences to construct the coefficients, e.g. Neville's algorithm. The cost is O(n²) operations. Furthermore, you only need to do O(n) extra work if an extra point is added to the data set, while for the other methods, you have to redo the whole computation.

Another method is preferred when the aim is not to compute the coefficients of p(x), but only a single value p(a) at a point x = a not in the original data set. The Lagrange form computes the value p(a) with complexity O(n²).

The Bernstein form was used in a constructive proof of the Weierstrass approximation theorem by Bernstein and has gained great importance in computer graphics in the form of Bézier curves.

Interpolations as linear combinations of values

Given a set of (position, value) data points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, y_0), \ldots, (x_j, y_j), \ldots, (x_n, y_n)} where no two positions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} are the same, the interpolating polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x)} may be considered as a linear combination of the values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j} , using coefficients which are polynomials in Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} depending on the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} . For example, the interpolation polynomial in the Lagrange form is the linear combination Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x) := \sum_{j=0}^{k} y_j c_j(x)} with each coefficient Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_j(x)} given by the corresponding Lagrange basis polynomial on the given positions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} : Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_j(x) = L_j(x_0,\ldots,x_n;x) = \prod_{ 0 \le i \le n \atop i \neq j } \frac{x-x_i}{x_j-x_i} = \frac{(x-x_0)}{(x_j-x_0)} \cdots \frac{(x-x_{j-1})}{(x_j-x_{j-1})} \frac{(x-x_{j+1})}{(x_j-x_{j+1})} \cdots \frac{(x-x_n)}{(x_j-x_n)}.}

Since the coefficients depend only on the positions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_j} , not the values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j} , we can use the same coefficients to find the interpolating polynomial for a second set of data points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0, v_0), \ldots, (x_n, v_n)} at the same positions: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v(x) := \sum_{j=0}^{k} v_j c_j(x).}

Furthermore, the coefficients Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle c_j(x)} only depend on the relative spaces Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i-x_j} between the positions. Thus, given a third set of data whose points are given by the new variable Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t = ax+b} (an affine transformation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x} , inverted by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x=\tfrac{t-b}{a}} ): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (t_0, w_0), \ldots, (t_j, w_j) \ldots, (t_n, w_n) \qquad \text{with}\qquad t_j = ax_j + b,}

we can use a transformed version of the previous coefficient polynomials:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \tilde c_j(t) := c_j(\tfrac{t-b}{a}) = c_j(x),}

and write the interpolation polynomial as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle w(t) := \sum_{j=0}^{k} w_j \tilde c_j(t).}

Data points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_j,y_j)} often have equally spaced positions, which may be normalized by an affine transformation to $x_{j}=j$. For example, consider the data points

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (0,y_0), (1,y_1), (2,y_2)} .

The interpolation polynomial in the Lagrange form is the linear combination

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} y(x) := \sum_{j=0}^2 y_j c_j(x) &= y_0 \frac{(x-1)(x-2)}{(0-1)(0-2)} + y_1 \frac{(x-0)(x-2)}{(1-0)(1-2)} + y_2 \frac{(x-0)(x-1)}{(2-0)(2-1)} \\ &= \tfrac12 y_0 (x-1)(x-2) - y_1 (x-0)(x-2) + \tfrac12 y_2 (x-0)(x-1). \end{align} }

For example, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(3) = y_3 = y_0 - 3y_1 + 3y_2 } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(1.5) = y_{1.5} = \tfrac18 (-y_0 + 6y_1 + 3y_2)} .

The case of equally spaced points can also be treated by the method of finite differences. The first difference of a sequence of values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle v=\{v_j\}_{j=0}^\infty} is the sequence Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta v = u = \{u_j\}_{j=0}^\infty } defined by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_j = v_{j+1}-v_j } . Iterating this operation gives the n^th difference operation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^n v = u} , defined explicitly by:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle u_j = \sum_{k=0}^n (-1)^{n-k} {n\choose k} v_{j+k}, } where the coefficients form a signed version of Pascal's triangle, the triangle of binomial transform coefficients:

								1									Row n = 0
							1		−1								Row n = 1 or d = 0
						1		−2		1							Row n = 2 or d = 1
					1		−3		3		−1						Row n = 3 or d = 2
				1		−4		6		−4		1					Row n = 4 or d = 3
			1		−5		10		−10		5		−1				Row n = 5 or d = 4
		1		−6		15		−20		15		−6		1			Row n = 6 or d = 5
	1		−7		21		−35		35		−21		7		−1		Row n = 7 or d = 6

A polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x)} of degree d defines a sequence of values at positive integer points, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_j = y(j)} , and the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (d+1)^{\text{th}}} difference of this sequence is identically zero:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \Delta^{d+1} y = 0} .

Thus, given values Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0,\ldots,y_n} at equally spaced points, where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n=d+1 } , we have: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (-1)^n y_0 + (-1)^{n-1} \binom{n}{1} y_1 + \cdots - \binom n{n-1} y_{n-1} + y_n= 0. } For example, 4 equally spaced data points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_0,y_1,y_2,y_3 } of a quadratic Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y(x)} obey Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0 = -y_0 + 3y_1 - 3y_2 + y_3} , and solving for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_3} gives the same interpolation equation obtained above using the Lagrange method.

Interpolation error: Lagrange remainder formula

When interpolating a given function f by a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n} of degree n at the nodes x₀,..., x_n we get the error Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) - p_n(x) = f[x_0,\ldots,x_n,x] \prod_{i=0}^n (x-x_i) }

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle f[x_0,\ldots,x_n,x]} is the (n+1)^st divided difference of the data points

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (x_0,f(x_0)),\ldots,(x_n,f(x_n)),(x,f(x)) } .

Furthermore, there is a Lagrange remainder form of the error, for a function f which is n + 1 times continuously differentiable on a closed interval Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle I} , and a polynomial Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)} of degree at most n that interpolates f at n + 1 distinct points Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_0,\ldots,x_n\in I} . For each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x\in I} there exists Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \xi\in I} such that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) - p_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x-x_i). }

This error bound suggests choosing the interpolation points x_i to minimize the product Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle \left | \prod (x - x_i) \right |} , which is achieved by the Chebyshev nodes.

Proof of Lagrange remainder

Set the error term as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R_n(x) = f(x) - p_n(x) } , and define an auxiliary function:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(t) = R_n(t) - \frac{R_n(x)}{W(x)} W(t) \qquad\text{where}\qquad W(t) = \prod_{i=0}^n (t-x_i). } Thus:Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^{(n+1)}(t) = R_n^{(n+1)}(t) - \frac{R_n(x)}{W(x)} \ (n+1)! }

But since Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)} is a polynomial of degree at most n, we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\textstyle R_n^{(n+1)}(t) = f^{(n+1)}(t) } , and: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^{(n+1)}(t) = f^{(n+1)}(t) - \frac{R_n(x)}{W(x)} \ (n+1)! }

Now, since x_i are roots of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_n(t)} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle W(t)} , we have Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y(x)=Y(x_j)=0 } , which means Y has at least n + 2 roots. From Rolle's theorem, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^\prime(t)} has at least n + 1 roots, and iteratively Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^{(n+1)}(t)} has at least one root ξ in the interval I. Thus: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Y^{(n+1)}(\xi) = f^{(n+1)}(\xi) - \frac{R_n(x)}{W(x)} \ (n+1)! = 0 }

and: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle R_n(x) = f(x) - p_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \prod_{i=0}^n (x-x_i) .}

This parallels the reasoning behind the Lagrange remainder term in the Taylor theorem; in fact, the Taylor remainder is a special case of interpolation error when all interpolation nodes x_i are identical. Note that the error will be zero when Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x = x_i} for any i. Thus, the maximum error will occur at some point in the interval between two successive nodes.

Equally spaced intervals

In the case of equally spaced interpolation nodes where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle x_i = a + ih} , for Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=0,1,\ldots,n,} and where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h = (b-a)/n,} the product term in the interpolation error formula can be bound as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|\prod_{i=0}^n (x-x_i)\right| = \prod_{i=0}^n \left|x-x_i\right| \leq \frac{n!}{4} h^{n+1}.}

Thus the error bound can be given as Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left|R_n(x)\right| \leq \frac{h^{n+1}}{4(n+1)} \max_{\xi\in[a,b]} \left|f^{(n+1)}(\xi)\right| }

However, this assumes that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n+1)}(\xi)} is dominated by Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle h^{n+1}} , i.e. Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f^{(n+1)}(\xi) h^{n+1} \ll 1} . In several cases, this is not true and the error actually increases as n → ∞ (see Runge's phenomenon). That question is treated in the section Convergence properties.

Lebesgue constants

Main article: Lebesgue constant

We fix the interpolation nodes x₀, ..., x_n and an interval [a, b] containing all the interpolation nodes. The process of interpolation maps the function f to a polynomial p. This defines a mapping X from the space C([a, b]) of all continuous functions on [a, b] to itself. The map X is linear and it is a projection on the subspace Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(n) } of polynomials of degree n or less.

The Lebesgue constant L is defined as the operator norm of X. One has (a special case of Lebesgue's lemma): Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left\|f-X(f)\right\| \le (L+1) \left\|f-p^*\right\|. }

In other words, the interpolation polynomial is at most a factor (L + 1) worse than the best possible approximation. This suggests that we look for a set of interpolation nodes that makes L small. In particular, we have for Chebyshev nodes: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L \le \frac2\pi \log(n+1) + 1.}

We conclude again that Chebyshev nodes are a very good choice for polynomial interpolation, as the growth in n is exponential for equidistant nodes. However, those nodes are not optimal.

Convergence properties

It is natural to ask, for which classes of functions and for which interpolation nodes the sequence of interpolating polynomials converges to the interpolated function as n → ∞? Convergence may be understood in different ways, e.g. pointwise, uniform or in some integral norm.

The situation is rather bad for equidistant nodes, in that uniform convergence is not even guaranteed for infinitely differentiable functions. One classical example, due to Carl Runge, is the function f(x) = 1 / (1 + x²) on the interval [−5, 5]. The interpolation error || f − p_n||_∞ grows without bound as n → ∞. Another example is the function f(x) = |x| on the interval [−1, 1], for which the interpolating polynomials do not even converge pointwise except at the three points x = ±1, 0.

One might think that better convergence properties may be obtained by choosing different interpolation nodes. The following result seems to give a rather encouraging answer:

Theorem — For any function f(x) continuous on an interval [a,b] there exists a table of nodes for which the sequence of interpolating polynomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p_n(x)} converges to f(x) uniformly on [a,b].

Proof

It is clear that the sequence of polynomials of best approximation Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^*_n(x)} converges to f(x) uniformly (due to the Weierstrass approximation theorem). Now we have only to show that each Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle p^*_n(x)} may be obtained by means of interpolation on certain nodes. But this is true due to a special property of polynomials of best approximation known from the equioscillation theorem. Specifically, we know that such polynomials should intersect f(x) at least n + 1 times. Choosing the points of intersection as interpolation nodes we obtain the interpolating polynomial coinciding with the best approximation polynomial.

The defect of this method, however, is that interpolation nodes should be calculated anew for each new function f(x), but the algorithm is hard to be implemented numerically. Does there exist a single table of nodes for which the sequence of interpolating polynomials converge to any continuous function f(x)? The answer is unfortunately negative:

Theorem — For any table of nodes there is a continuous function f(x) on an interval [a, b] for which the sequence of interpolating polynomials diverges on [a,b].

The proof essentially uses the lower bound estimation of the Lebesgue constant, which we defined above to be the operator norm of X_n (where X_n is the projection operator on Π_n). Now we seek a table of nodes for which

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \lim_{n \to \infty} X_n f = f,\text{ for every }f \in C([a,b]).}

Due to the Banach–Steinhaus theorem, this is only possible when norms of X_n are uniformly bounded, which cannot be true since we know that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \|X_n\|\geq \tfrac{2}{\pi} \log(n+1)+C.}

For example, if equidistant points are chosen as interpolation nodes, the function from Runge's phenomenon demonstrates divergence of such interpolation. Note that this function is not only continuous but even infinitely differentiable on [−1, 1]. For better Chebyshev nodes, however, such an example is much harder to find due to the following result:

Theorem — For every absolutely continuous function on [−1, 1] the sequence of interpolating polynomials constructed on Chebyshev nodes converges to f(x) uniformly.

Related concepts

Runge's phenomenon shows that for high values of n, the interpolation polynomial may oscillate wildly between the data points. This problem is commonly resolved by the use of spline interpolation. Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree.

Interpolation of periodic functions by harmonic functions is accomplished by Fourier transform. This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and trigonometric polynomial.

Hermite interpolation problems are those where not only the values of the polynomial p at the nodes are given, but also all derivatives up to a given order. This turns out to be equivalent to a system of simultaneous polynomial congruences, and may be solved by means of the Chinese remainder theorem for polynomials. Birkhoff interpolation is a further generalization where only derivatives of some orders are prescribed, not necessarily all orders from 0 to a k.

Collocation methods for the solution of differential and integral equations are based on polynomial interpolation.

The technique of rational function modeling is a generalization that considers ratios of polynomial functions.

At last, multivariate interpolation for higher dimensions.

See also

• Newton series
• Polynomial regression

References

• Bernstein, Sergei N. (1912). "Sur l'ordre de la meilleure approximation des fonctions continues par les polynômes de degré donné" [On the order of the best approximation of continuous functions by polynomials of a given degree]. Mem. Acad. Roy. Belg. (in français). 4: 1–104.
• Faber, Georg (1914). "Über die interpolatorische Darstellung stetiger Funktionen" [On the Interpolation of Continuous Functions]. Deutsche Math. Jahr. (in Deutsch). 23: 192–210.
• Watson, G. Alistair (1980). Approximation Theory and Numerical Methods. John Wiley. ISBN 0-471-27706-1.

Further reading

• Atkinson, Kendell A. (1988). "Chapter 3.". An Introduction to Numerical Analysis (2nd ed.). John Wiley and Sons. ISBN 0-471-50023-2.
• Brutman, L. (1997). "Lebesgue functions for polynomial interpolation — a survey". Ann. Numer. Math. 4: 111–127.
• Powell, M. J. D. (1981). "Chapter 4". Approximation Theory and Methods. Cambridge University Press. ISBN 0-521-29514-9.
• Schatzman, Michelle (2002). "Chapter 4". Numerical Analysis: A Mathematical Introduction. Oxford: Clarendon Press. ISBN 0-19-850279-6.
• Süli, Endre; Mayers, David (2003). "Chapter 6". An Introduction to Numerical Analysis. Cambridge University Press. ISBN 0-521-00794-1.

External links

• "Interpolation process", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• ALGLIB has an implementations in C++ / C#.
• GSL has a polynomial interpolation code in C
• Polynomial Interpolation demonstration.