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In
numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). Numerical analysis ...
, polynomial interpolation is the
interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

interpolation
of a given
data set A data set (or dataset) is a collection of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more technical sense, data are a set of values of qualitative property, qualitative or quantity, quantit ...
by the
polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

polynomial
of lowest possible degree that passes through the points of the dataset.


Applications

Polynomials can be used to approximate complicated curves, for example, the shapes of letters in
typography Typography is the art and technique of arranging type to make written language A written language is the representation of a spoken or gestural language A language is a structured system of communication used by humans, including ...

typography
, given a few points. A relevant application is the evaluation of the
natural logarithm The natural logarithm of a number is its logarithm In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained ( ...
and
trigonometric function In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

trigonometric function
s: pick a few known data points, create a
lookup table In computer science, a lookup table (LUT) is an array data structure, array that replaces runtime (program lifecycle phase), runtime computation with a simpler array indexing operation. The process is termed as "direct addressing" and LUTs differ fr ...
, and interpolate between those data points. This results in significantly faster computations. Polynomial interpolation also forms the basis for algorithms in
numerical quadrature In analysis Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics Mathematics (from Ancient Greek, ...
and
numerical ordinary differential equations Numerical may refer to: * Number A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in l ...
and Secure Multi Party Computation,
Secret Sharing Secret sharing (also called secret splitting) refers to methods for distributing a '' secret'' among a group of participants, each of whom is allocated a ''share'' of the secret. The secret can be reconstructed only when a sufficient number, of p ...
schemes. Polynomial interpolation is also essential to perform sub-quadratic multiplication and squaring such as
Karatsuba multiplication The Karatsuba algorithm is a fast multiplication algorithm. It was discovered by Anatoly Karatsuba in 1960 and published in 1962. Knuth D.E. (1969) ''The Art of Computer Programming. v.2.'' Addison-Wesley Publ.Co., 724 pp. It reduces the ...

Karatsuba multiplication
and
Toom–Cook multiplicationToom–Cook, sometimes known as Toom-3, named after Andrei Toom, who introduced the new algorithm with its low complexity, and Stephen Cook, who cleaned the description of it, is a multiplication algorithm for large integers. Given two large integer ...
, where an interpolation through points on a polynomial which defines the product yields the product itself. For example, given ''a'' = ''f''(''x'') = ''a''0''x''0 + ''a''1''x''1 + ... and ''b'' = ''g''(''x'') = ''b''0''x''0 + ''b''1''x''1 + ..., the product ''ab'' is equivalent to ''W''(''x'') = ''f''(''x'')''g''(''x''). Finding points along ''W''(''x'') by substituting ''x'' for small values in ''f''(''x'') and ''g''(''x'') yields points on the curve. Interpolation based on those points will yield the terms of ''W''(''x'') and subsequently the product ''ab''. In the case of Karatsuba multiplication this technique is substantially faster than quadratic multiplication, even for modest-sized inputs. This is especially true when implemented in parallel hardware.


Definition

Given a set of data points where no two are the same, a polynomial p:\mathbb\rightarrow\mathbb is said to ''interpolate'' the data if p(x_j)=y_j for each j\in\.


Interpolation theorem

Given n+1 distinct points x_0,x_1,\dotsc,x_n and corresponding values y_0,y_1,\dotsc,y_n, there exists a unique polynomial of degree at most n that interpolates the data \.


Proof

Consider the Lagrange basis functions given by : L_(x)=\prod_\frac. Notice that L_ is a polynomial of degree n. Furthermore, for each x_k we have L_(x_k)=\delta_, where \delta_ is the
Kronecker delta In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
. It follows that the linear combination : p(x)=\sum_^n y_j L_(x) is an interpolating polynomial of degree n. To prove uniqueness, assume that there exists another interpolating polynomial q of degree at most n. Since p(x_k)=q(x_k) for all k=0,\dotsc,n, it follows that the polynomial p-q has n+1 distinct zeros. However, p-q is of degree at most n and, by the
fundamental theorem of algebra The fundamental theorem of algebra states that every non- constant single-variable polynomial In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, s ...
, can have at most n zeros; therefore, p=q.


Corollary

An interesting corollary to the interpolation theorem is that if f is a polynomial of degree at most n, then the interpolating polynomial of f at n+1 distinct points is f itself.


Unisolvence theorem

Given a set of data points where no two are the same, one is looking for a polynomial of degree at most with the property :p(x_i) = y_i, \qquad i=0,\ldots,n. The unisolvence theorem states that such a polynomial ''p'' exists and is unique, and can be proved by the
Vandermonde matrixIn linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix (math), matrix with the terms of a geometric progression in each row, i.e., an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \ ...
, as described below. The theorem states that for interpolation nodes , polynomial interpolation defines a linear
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

bijection
:L_n:\mathbb^ \to \Pi_n where Π''n'' is the
vector space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...
of polynomials (defined on any interval containing the nodes) of degree at most .


Constructing the interpolation polynomial

Suppose that the interpolation polynomial is in the form :p(x) = a_n x^n + a_ x^ + \cdots + a_2 x^2 + a_1 x + a_0. \qquad (1) The statement that ''p'' interpolates the data points means that :p(x_i) = y_i \qquad\mbox i \in \left\. If we substitute equation (1) in here, we get a
system of linear equations In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
in the coefficients . The system in matrix-vector form reads the following
multiplication Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four Elementary arithmetic, elementary Operation (mathematics), mathematical operations ...

multiplication
: :\begin x_0^n & x_0^ & x_0^ & \ldots & x_0 & 1 \\ x_1^n & x_1^ & x_1^ & \ldots & x_1 & 1 \\ \vdots & \vdots & \vdots & & \vdots & \vdots \\ x_n^n & x_n^ & x_n^ & \ldots & x_n & 1 \end \begin a_n \\ a_ \\ \vdots \\ a_0 \end = \begin y_0 \\ y_1 \\ \vdots \\ y_n \end. We have to solve this system for to construct the interpolant ''p''(''x''). The matrix on the left is commonly referred to as a
Vandermonde matrixIn linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix (math), matrix with the terms of a geometric progression in each row, i.e., an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \ ...
. The
condition number In the field of numerical analysis Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic computation, symbolic manipulations) for the problems of mathematical analysis (as distinguished from discre ...
of the Vandermonde matrix may be large, causing large errors when computing the coefficients 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(''n''2) operations instead of the O(''n''3) required by Gaussian elimination. These methods rely on constructing first a Newton interpolation of the polynomial and then converting it to the monomial form above. Alternatively, we may write down the polynomial immediately in terms of
Lagrange polynomial Image:Lagrange polynomial.svg, upright=2, This image shows, for four points ((−9, 5), (−4, 2), (−1, −2), (7, 9)), the (cubic) interpolation polynomial ''L''(''x'') (dashed, black), which is the sum of the ...

Lagrange polynomial
s: :\begin p(x) &= \frac y_0 + \fracy_1 +\cdots+\fracy_n \\
pt
pt
&=\sum_^n \Bigg( \prod_ \frac \Bigg) y_i \end For matrix arguments, this formula is called
Sylvester's formulaIn matrix theory In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analy ...
and the matrix-valued Lagrange polynomials are the Frobenius covariants.


Uniqueness of the interpolating polynomial


Proof 1

Suppose we interpolate through data points with an at-most degree polynomial ''p''(''x'') (we need at least datapoints or else the polynomial cannot be fully solved for). Suppose also another polynomial exists also of degree at most that also interpolates the points; call it ''q''(''x''). Consider r(x) = p(x) - q(x). We know, # ''r''(''x'') is a polynomial # ''r''(''x'') has degree at most , since ''p''(''x'') and ''q''(''x'') are no higher than this and we are just subtracting them. # At the data points, r(x_i) = p(x_i) - q(x_i) = y_i - y_i = 0. Therefore, ''r''(''x'') has roots. But ''r''(''x'') is a polynomial of degree . It has one root too many. Formally, if ''r''(''x'') is any non-zero polynomial, it must be writable as r(x) = A(x-x_0)(x-x_1)\cdots(x-x_n), for some constant ''A''. By distributivity, the ''xs multiply together to give leading term Ax^, i.e. one degree higher than the maximum we set. So the only way ''r''(''x'') can exist is if , or equivalently, . : r(x) = 0 = p(x) - q(x) \implies p(x) = q(x) So ''q''(''x'') (which could be any polynomial, so long as it interpolates the points) is identical with ''p''(''x''), and ''q''(''x'') is unique.


Proof 2

Given the
Vandermonde matrixIn linear algebra, a Vandermonde matrix, named after Alexandre-Théophile Vandermonde, is a matrix (math), matrix with the terms of a geometric progression in each row, i.e., an matrix :V=\begin 1 & x_1 & x_1^2 & \dots & x_1^\\ 1 & x_2 & x_2^2 & \ ...
used above to construct the interpolant, we can set up the system : V a = y To prove that V is
nonsingularIn linear algebra, an ''n''-by-''n'' square matrix In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calc ...
we use the Vandermonde determinant formula: : \det(V) = \prod_^n (x_i - x_j) since the points are distinct, the
determinant In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

determinant
can't be zero as x_i - x_j is never zero, therefore ''V'' is nonsingular and the system has a unique solution. Either way this means that no matter what method we use to do our interpolation: direct,
Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi LagrangiaRoman_Forum.html" ;"title="Curia Julia in the Roman Forum">Curia Julia in the Roman Forum A senate is a deliberative assembly, often the upper house or Debating chamber, chamber of a bicame ...

Lagrange
etc., (assuming we can do all our calculations perfectly) we will always get the same polynomial.


Non-Vandermonde solutions

We are trying to construct our unique interpolation polynomial in the vector space Π''n'' of polynomials of degree . When using a
monomial basis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
for Π''n'' we have to solve the Vandermonde matrix to construct the coefficients for the interpolation polynomial. This can be a very costly operation (as counted in clock cycles of a computer trying to do the job). By choosing another basis for Π''n'' we can simplify the calculation of the coefficients but then we have to do additional calculations when we want to express the interpolation polynomial in terms of a
monomial basis In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
. One method is to write the interpolation polynomial in the Newton form and use the method of divided differences to construct the coefficients, e.g.
Neville's algorithm In mathematics, Neville's algorithm is an algorithm used for polynomial interpolation In numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal fi ...
. The cost is O(''n''2) operations, while Gaussian elimination costs O(''n''3) 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 to use the Lagrange form of the interpolation polynomial. The resulting formula immediately shows that the interpolation polynomial exists under the conditions stated in the above theorem. Lagrange formula is to be preferred to Vandermonde formula when we are not interested in computing the coefficients of the polynomial, but in computing the value of ''p''(''x'') in a given ''x'' not in the original data set. In this case, we can reduce complexity to O(''n''2). The Bernstein form was used in a constructive proof of the
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
by
Bernstein Bernstein is a common surname in the German language The German language (, ) is a West Germanic language mainly spoken in Central Europe. It is the most widely spoken and official or co-official language in Germany, Austria, Switzerland, Li ...
and has gained great importance in computer graphics in the form of
Bézier curveBézier can refer to: *Pierre Bézier, French engineer and creator of Bézier curves *Bézier curve *Bézier triangle *Bézier spline (disambiguation) *Bézier surface * The town of Béziers in France * AS Béziers Hérault, a French rugby union tea ...

Bézier curve
s.


Linear combination of the given values

The Lagrange form of the interpolating polynomial is a linear combination of the given values. In many scenarios, an efficient and convenient polynomial interpolation is a linear combination of the given values, using previously known coefficients. Given a set of k+1 data points (x_0, y_0),\ldots,(x_j, y_j),\ldots,(x_k, y_k) where each data point is a (position, value) pair and where no two positions x_j are the same, the interpolation polynomial in the Lagrange form is a
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
:y(x) := \sum_^ y_j c_j(x) of the given values y_j with each coefficient c _j(x) given by evaluating the corresponding Lagrange basis polynomial using the given k+1 positions x_j. :c_j(x) = \ell_j(x,x_0,x_1,\ldots,x_k) := \prod_ \frac = \frac \cdots \frac \frac \cdots \frac. Each coefficient c _j(x) in the linear combination depends on the given positions x_j and the desired position x, but not on the given values y_j. For each coefficient, inserting the values of the given positions x_j and simplifying yields an expression c _j(x), which depends only on x. Thus the same coefficient expressions c_j(x) can be used in a polynomial interpolation of a given second set of k+1 data points (x_0, v_0),\ldots,(x_j, v_j),\ldots,(x_k, v_k) at the same given positions x_j, where the second given values v_j differ from the first given values y_j. Using the same coefficient expressions c_j(x) as for the first set of data points, the interpolation polynomial of the second set of data points is the linear combination :v(x) := \sum_^ v_j c_j(x). For each coefficient c _j(x) in the linear combination, the expression resulting from the Lagrange basis polynomial \ell_j(x,x_0,x_1,\ldots,x_k) only depends on the relative spaces between the given positions, not on the individual value of any position. Thus the same coefficient expressions c_j(x) can be used in a polynomial interpolation of a given third set of k+1 data points :(t_0, w_0),\ldots,(t_j, w_j),\ldots,(t_k, w_k) where each position t_j is related to the corresponding position x_j in the first set by t_i = ax_i + b and the desired positions are related by t = ax + b, for a constant scaling factor ''a'' and a constant shift ''b'' for all positions. Using the same coefficient expressions c_j(t) as for the first set of data points, the interpolation polynomial of the third set of data points is the linear combination :w(t) := \sum_^ w_j c_j(t). In many applications of polynomial interpolation, the given set of k+1 data points is at equally spaced positions. In this case, it can be convenient to define the ''x''-axis of the positions such that x_i = i. For example, a given set of 3 equally-spaced data points (x_0,y_0),(x_1,y_1),(x_2,y_2) is then (0,y_0),(1,y_1),(2,y_2). The interpolation polynomial in the Lagrange form is the
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
: \begin y(x) := \sum_^2 y_j c_j(x) = y_0 \frac + y_1 \frac + y_2 \frac \\ = y_0 \frac + y_1 \frac + y_2 \frac. \end This quadratic interpolation is valid for any position ''x'', near or far from the given positions. So, given 3 equally-spaced data points at x=0,1,2 defining a quadratic polynomial, at an example desired position x=1.5, the interpolated value after simplification is given by y(1.5)=y_= (-y_0 + 6y_1 + 3y_2)/8 This is a quadratic interpolation typically used in the Multigrid method. Again given 3 equally-spaced data points at x=0,1,2 defining a quadratic polynomial, at the next equally spaced position x=3, the interpolated value after simplification is given by :y(3)=y_3 = y_0 - 3y_1 + 3y_2. In the above polynomial interpolations using a linear combination of the given values, the coefficients were determined using the Lagrange method. In some scenarios, the coefficients can be more easily determined using other methods. Examples follow. According to the method of finite differences, for any polynomial of degree ''d'' or less, any sequence of d+2 values at equally spaced positions has a (d+1)th difference exactly equal to 0. The element ''s''''d+1'' of the
Binomial transform In combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ar ...
is such a (d+1)th difference. This area is surveyed here. The
binomial transform In combinatorics Combinatorics is an area of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they ar ...
, ''T'', of a sequence of values , is the sequence defined by :s_n = \sum_^n (-1)^k v_k. Ignoring the sign term (-1)^k, the n+1 coefficients of the element ''s''''n'' are the respective n+1 elements of the row ''n'' of Pascal's Triangle. The triangle of binomial transform coefficients is like Pascal's triangle. The entry in the ''n''th row and ''k''th column of the BTC triangle is (-1)^k\tbinom for any non-negative integer ''n'' and any integer ''k'' between 0 and ''n''. This results in the following example rows ''n'' = 0 through ''n'' = 7, top to bottom, for the BTC triangle: For convenience, each row ''n'' of the above example BTC triangle also has a label d=n-1. Thus for any polynomial of degree ''d'' or less, any sequence of d+2 values at equally spaced positions has a
linear combination In mathematics, a linear combination is an Expression (mathematics), expression constructed from a Set (mathematics), set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be ...
result of 0, when using the d+2 elements of row ''d'' as the corresponding linear coefficients. For example, 4 equally spaced data points of a quadratic polynomial obey the
linear equation In mathematics, a linear equation is an equation that may be put in the form :a_1x_1+\cdots +a_nx_n+b=0, where x_1, \ldots, x_n are the variable (mathematics), variables (or unknown (mathematics), unknowns), and b, a_1, \ldots, a_n are the coeffi ...

linear equation
given by row d=2 of the BTC triangle. 0 = y_0 - 3y_1 + 3y_2 -y_3 This is the same linear equation as obtained above using the Lagrange method. The BTC triangle can also be used to derive other polynomial interpolations. For example, the above quadratic interpolation :y(1.5)=y_= (-y_0 + 6y_1 + 3y_2)/8 can be derived in 3 simple steps as follows. The equally spaced points of a quadratic polynomial obey the rows of the BTC triangle with d=2 or higher. First, the row d=3 spans the given and desired data points y_0, y_1, y_, y_2 with the linear equation :0 = 1y_0 - 4y_ + 6y_1 - 4y_ + 1y_2 Second, the unwanted data point y_ is replaced by an expression in terms of wanted data points. The row d=2 provides a linear equation with a term 1y_, which results in a term 4y_ by multiplying both sides of the linear equation by 4. 0 = 1y_ - 3y_1 + 3y_ - 1 y_2 = 4y_ -12y_1 +12y_ - 4y_2 Third, the above two linear equations are added to yield a linear equation equivalent to the above quadratic interpolation for y_. 0 = (1+0)y_0 + (-4+4)y_ + (6-12)y_1 + (-4+12)y_ + (1-4)y_2 = y_0 -6y_1 + 8y_ - 3y_2 Similar to other uses of linear equations, the above derivation scales and adds vectors of coefficients. In polynomial interpolation as a linear combination of values, the elements of a vector correspond to a contiguous sequence of regularly spaced positions. The ''p'' non-zero elements of a vector are the ''p'' coefficients in a linear equation obeyed by any sequence of ''p'' data points from any degree ''d'' polynomial on any regularly spaced grid, where ''d'' is noted by the subscript of the vector. For any vector of coefficients, the subscript obeys d\leq p-2. When adding vectors with various subscript values, the lowest subscript applies for the resulting vector. So, starting with the vector of row d=3 and the vector of row d=2 of the BTC triangle, the above quadratic interpolation for y_ is derived by the vector calculation :(1,-4,6,-4,1)_3 +4(0,1,-3,3,-1)_2=(1,0,-6,+8,-3)_2 Similarly, the cubic interpolation typical in the
Multigrid methodIn numerical analysis, a multigrid method (MG method) is an algorithm for solving differential equations using a hierarchy of discretizations. They are an example of a class of techniques called Multiresolution analysis, multiresolution methods, very ...
, :y_= (-y_0 + 9y_1 + 9y_2 - y_3)/16, can be derived by a vector calculation starting with the vector of row d=5 and the vector of row d=3 of the BTC triangle. : (1,-6,15,-20,15,-6,1)_5 + 6(0, 1,-4, 6,-4, 1,0)_3 = (1, 0,-9, 16,-9, 0,1)_3


Interpolation error

When interpolating a given function ''f'' by a polynomial of degree at the nodes ''x''0,...,''x''''n'' we get the error :f(x) - p_n(x) = f _0,\ldots,x_n,x\prod_^n (x-x_i) where :f _0,\ldots,x_n,x/math> is the notation for divided differences. If ''f'' is times continuously differentiable on a closed interval ''I'' and p_n(x) is a polynomial of degree at most that interpolates ''f'' at distinct points (''i''=0,1,...,n) in that interval, then for each x in the interval there exists in that interval such that : f(x) - p_n(x) = \frac \prod_^n (x-x_i). The above error bound suggests choosing the interpolation points such that the product \left , \prod (x - x_i) \right , , is as small as possible. The
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial ...
achieve this.


Proof

Set the error term as : R_n(x) = f(x) - p_n(x) and set up an auxiliary function: : Y(t) = R_n(t) - \frac W(t) where : W(t) = \prod_^n (t-x_i) Since are roots of R_n(t) and W(t), we have , which means has at least roots. From
Rolle's theorem In calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. ...

Rolle's theorem
, Y^\prime(t) has at least roots, then Y^(t) has at least one root , where is in the interval . So we can get : Y^(t) = R_n^(t) - \frac \ (n+1)! Since p_n(x) is a polynomial of degree at most , then : R_n^(t) = f^(t) Thus : Y^(t) = f^(t) - \frac \ (n+1)! Since is the root of Y^(t), so : Y^(\xi) = f^(\xi) - \frac \ (n+1)! = 0 Therefore, : R_n(x) = f(x) - p_n(x) = \frac \prod_^n (x-x_i) . Thus the remainder term in the Lagrange form of the Taylor theorem is a special case of interpolation error when all interpolation nodes are identical. Note that the error will be zero when x = x_i for any ''i''. Thus, the maximum error will occur at some point in the interval between two successive nodes.


For equally spaced intervals

In the case of equally spaced interpolation nodes where x_i = a + ih, for i=0,1,\ldots,n, and where h = (b-a)/n, the product term in the interpolation error formula can be bound as :\left, \prod_^n (x-x_i)\ = \prod_^n \left, x-x_i\ \leq \frac h^. Thus the error bound can be given as : \left, R_n(x)\ \leq \frac \max_ \left, f^(\xi)\ However, this assumes that f^(\xi) is dominated by h^, i.e. f^(\xi) h^ \ll 1. In several cases, this is not true and the error actually increases as (see
Runge's phenomenon In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
). That question is treated in the section ''Convergence properties''.


Lebesgue constants

:''See the main article: Lebesgue constant.'' We fix the interpolation nodes ''x''0, ..., ''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 Π''n'' of polynomials of degree ''n'' or less. The Lebesgue constant ''L'' is defined as the
operator norm In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...
of ''X''. One has (a special case of Lebesgue's lemma): : \, f-X(f)\, \le (L+1) \, f-p^*\, . 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 In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial ...
: : 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 ? 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 + ''x''2) on the interval . The interpolation error grows without bound as . Another example is the function ''f''(''x'') = , ''x'', on the interval , 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 p_n(x) converges to ''f''(''x'') uniformly on 'a'',''b'' Proof. It's clear that the sequence of polynomials of best approximation p^*_n(x) converges to ''f''(''x'') uniformly (due to
Weierstrass approximation theorem Karl Theodor Wilhelm Weierstrass (german: link=no, Weierstraß ; 31 October 1815 – 19 February 1897) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, G ...
). Now we have only to show that each 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 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 :\lim_ X_n f = f,\textf \in C( ,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 :\, X_n\, \geq \tfrac \log(n+1)+C. For example, if equidistant points are chosen as interpolation nodes, the function from
Runge's phenomenon In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
demonstrates divergence of such interpolation. Note that this function is not only continuous but even infinitely differentiable on . For better
Chebyshev nodes In numerical analysis, Chebyshev nodes are specific real algebraic numbers, namely the roots of the Chebyshev polynomials of the first kind. They are often used as nodes in polynomial interpolation because the resulting interpolation polynomial ...
, however, such an example is much harder to find due to the following result: :Theorem. For every
absolutely continuousIn calculus, absolute continuity is a smoothness property of function (mathematics), functions that is stronger than continuous function, continuity and uniform continuity. The notion of absolute continuity allows one to obtain generalizations of the ...
function on the sequence of interpolating polynomials constructed on Chebyshev nodes converges to ''f''(''x'') uniformly. MR 18-32.


Related concepts

Runge's phenomenon In the mathematical Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...
shows that for high values of , the interpolation polynomial may oscillate wildly between the data points. This problem is commonly resolved by the use of
spline interpolation In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its populatio ...

spline interpolation
. Here, the interpolant is not a polynomial but a spline: a chain of several polynomials of a lower degree. Interpolation of
periodic function A periodic function is a Function (mathematics), function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used th ...

periodic function
s by
harmonic A harmonic is any member of the harmonic series Harmonic series may refer to either of two related concepts: *Harmonic series (mathematics) *Harmonic series (music) {{Disambig .... The term is employed in various disciplines, including music ...
functions is accomplished by
Fourier transform#REDIRECT Fourier transform In mathematics, a Fourier transform (FT) is a Integral transform, mathematical transform that decomposes function (mathematics), functions depending on space or time into functions depending on spatial or temporal frequenc ...
. This can be seen as a form of polynomial interpolation with harmonic base functions, see trigonometric interpolation and
trigonometric polynomialIn the mathematical subfields of numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numeri ...
.
Hermite interpolationIn numerical analysis (c. 1800–1600 BC) with annotations. The approximation of the square root of 2 is four sexagesimal figures, which is about six decimal figures. 1 + 24/60 + 51/602 + 10/603 = 1.41421296... Numerical analysis is the study of al ...

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 In number theory, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer An integer (from the Latin wikt:integer#Latin, ''integer'' meaning "whole") is colloquially defined as a number ...
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 methodIn mathematics, a collocation method is a method for the numerical analysis, numerical solution of ordinary differential equations, partial differential equations and integral equations. The idea is to choose a finite-dimensional space of candidate s ...
s 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 A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient In single-variable calculus, the difference quotient is usually the name for the expression : \frac which whe ...
*
Polynomial regression In statistics Statistics is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data Data (; ) are individual facts, statistics, or items of information, often numeric. In a more ...


Notes


References

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Further reading

* * * * *


External links

* {{springer, title=Interpolation process, id=p/i051970
ALGLIB
has an implementations in C++ / C#.
GSL
has a polynomial interpolation code in C
Polynomial Interpolation demonstration
Interpolation Polynomials Articles containing proofs