Mathematical expression
In the mathematical field of numerical analysis, a Newton polynomial, named after its inventor Isaac Newton, is an interpolation polynomial for a given set of data points. The Newton polynomial is sometimes called Newton's divided differences interpolation polynomial because the coefficients of the polynomial are calculated using Newton's divided differences method.
Definition.
Given a set of "k" + 1 data points
formula_1
where no two "x""j" are the same, the Newton interpolation polynomial is a linear combination of Newton basis polynomials
formula_2
with the Newton basis polynomials defined as
formula_3
for "j" ; 0 and formula_4.
The coefficients are defined as
formula_5
where
formula_6
is the notation for divided differences.
Thus the Newton polynomial can be written as
formula_7
Newton forward divided difference formula.
The Newton polynomial can be expressed in a simplified form when formula_8 are arranged consecutively with equal spacing.
If formula_8 are consecutively arranged and equally spaced with formula_10 for "i" = 0, 1, ..., "k" and some variable x is expressed as formula_11, then the difference formula_12 can be written as formula_13. So the Newton polynomial becomes
formula_14
This is called the Newton forward divided difference formula.
Newton backward divided difference formula.
If the nodes are reordered as formula_15, the Newton polynomial becomes
formula_16
If formula_17 are equally spaced with formula_18 for "i" = 0, 1, ..., "k" and formula_19, then,
formula_20
This is called the Newton backward divided difference formula.
Significance.
Newton's formula is of interest because it is the straightforward and natural differences-version of Taylor's polynomial. Taylor's polynomial tells where a function will go, based on its "y" value, and its derivatives (its rate of change, and the rate of change of its rate of change, etc.) at one particular "x" value. Newton's formula is Taylor's polynomial based on finite differences instead of instantaneous rates of change.
Polynomial interpolation.
For a polynomial formula_21 of degree less than or equal to n, that interpolates formula_22 at the nodes formula_23 where formula_24. Let formula_25 be the polynomial of degree less than or equal to n+1 that interpolates formula_22 at the nodes formula_23 where formula_28. Then formula_25 is given by:
formula_30
where formula_31 and formula_32.
Proof:
This can be shown for the case where formula_24:formula_34and when formula_35:
formula_36
By the uniqueness of interpolated polynomials of degree less than formula_37, formula_38 is the required polynomial interpolation. The function can thus be expressed as:
formula_39
where the factors formula_40 are divided differences. Thus, Newton polynomials are used to provide polynomial interpolation formula of n points.
Taking formula_41 for some unknown function in Newton divided difference formulas, if the representation of x in the previous sections was instead taken to be formula_42, in terms of forward differences, the Newton forward interpolation formula is expressed as:formula_43whereas for the same in terms of backward differences, the Newton backward interpolation formula is expressed as:formula_44This follows since relationship between divided differences and forward differences is given as:formula_45whereas for backward differences, it is given as:formula_46
Addition of new points.
As with other difference formulas, the degree of a Newton interpolating polynomial can be increased by adding more terms and points without discarding existing ones. Newton's form has the simplicity that the new points are always added at one end: Newton's forward formula can add new points to the right, and Newton's backward formula can add new points to the left.
The accuracy of polynomial interpolation depends on how close the interpolated point is to the middle of the "x" values of the set of points used. Obviously, as new points are added at one end, that middle becomes farther and farther from the first data point. Therefore, if it isn't known how many points will be needed for the desired accuracy, the middle of the x-values might be far from where the interpolation is done.
Gauss, Stirling, and Bessel all developed formulae to remedy that problem.
Gauss's formula alternately adds new points at the left and right ends, thereby keeping the set of points centered near the same place (near the evaluated point). When so doing, it uses terms from Newton's formula, with data points and "x" values renamed in keeping with one's choice of what data point is designated as the "x"0 data point.
Stirling's formula remains centered about a particular data point, for use when the evaluated point is nearer to a data point than to a middle of two data points.
Bessel's formula remains centered about a particular middle between two data points, for use when the evaluated point is nearer to a middle than to a data point.
Bessel and Stirling achieve that by sometimes using the average of two differences, and sometimes using the average of two products of binomials in "x", where Newton's or Gauss's would use just one difference or product. Stirling's uses an average difference in odd-degree terms (whose difference uses an even number of data points); Bessel's uses an average difference in even-degree terms (whose difference uses an odd number of data points).
Strengths and weaknesses of various formulae.
For any given finite set of data points, there is only one polynomial of least possible degree that passes through all of them. Thus, it is appropriate to speak of the "Newton form", or Lagrange form, etc., of the interpolation polynomial. However, different methods of computing this polynomial can have differing computational efficiency. There are several similar methods, such as those of Gauss, Bessel and Stirling. They can be derived from Newton's by renaming the "x"-values of the data points, but in practice they are important.
Bessel vs. Stirling.
The choice between Bessel and Stirling depends on whether the interpolated point is closer to a data point, or closer to a middle between two data points.
A polynomial interpolation's error approaches zero, as the interpolation point approaches a data-point. Therefore, Stirling's formula brings its accuracy improvement where it is least needed and Bessel brings its accuracy improvement where it is most needed.
So, Bessel's formula could be said to be the most consistently accurate difference formula, and, in general, the most consistently accurate of the familiar polynomial interpolation formulas.
Divided-Difference Methods vs. Lagrange.
Lagrange is sometimes said to require less work, and is sometimes recommended for problems in which it is known, in advance, from previous experience, how many terms are needed for sufficient accuracy.
The divided difference methods have the advantage that more data points can be added, for improved accuracy. The terms based on the previous data points can continue to be used. With the ordinary Lagrange formula, to do the problem with more data points would require re-doing the whole problem.
There is a "barycentric" version of Lagrange that avoids the need to re-do the entire calculation when adding a new data point. But it requires that the values of each term be recorded.
But the ability, of Gauss, Bessel and Stirling, to keep the data points centered close to the interpolated point gives them an advantage over Lagrange, when it isn't known, in advance, how many data points will be needed.
Additionally, suppose that one wants to find out if, for some particular type of problem, linear interpolation is sufficiently accurate. That can be determined by evaluating the quadratic term of a divided difference formula. If the quadratic term is negligible—meaning that the linear term is sufficiently accurate without adding the quadratic term—then linear interpolation is sufficiently accurate. If the problem is sufficiently important, or if the quadratic term is nearly big enough to matter, then one might want to determine whether the sum of the quadratic and cubic terms is large enough to matter in the problem.
Of course, only a divided-difference method can be used for such a determination.
For that purpose, the divided-difference formula and/or its "x"0 point should be chosen so that the formula will use, for its linear term, the two data points between which the linear interpolation of interest would be done.
The divided difference formulas are more versatile, useful in more kinds of problems.
The Lagrange formula is at its best when all the interpolation will be done at one "x" value, with only the data points' "y" values varying from one problem to another, and when it is known, from past experience, how many terms are needed for sufficient accuracy.
With the Newton form of the interpolating polynomial a compact and effective algorithm exists for combining the terms to find the coefficients of the polynomial.
Accuracy.
When, with Stirling's or Bessel's, the last term used includes the average of two differences, then one more point is being used than Newton's or other polynomial interpolations would use for the same polynomial degree. So, in that instance, Stirling's or Bessel's is not putting an "N"−1 degree polynomial through "N" points, but is, instead, trading equivalence with Newton's for better centering and accuracy, giving those methods sometimes potentially greater accuracy, for a given polynomial degree, than other polynomial interpolations.
General case.
For the special case of "xi" = "i", there is a closely related set of polynomials, also called the Newton polynomials, that are simply the binomial coefficients for general argument. That is, one also has the Newton polynomials formula_47 given by
formula_48
In this form, the Newton polynomials generate the Newton series. These are in turn a special case of the general difference polynomials which allow the representation of analytic functions through generalized difference equations.
Main idea.
Solving an interpolation problem leads to a problem in linear algebra where we have to solve a system of linear equations. Using a standard monomial basis for our interpolation polynomial we get the very complicated Vandermonde matrix. By choosing another basis, the Newton basis, we get a system of linear equations with a much simpler lower triangular matrix which can be solved faster.
For "k" + 1 data points we construct the Newton basis as
formula_49
Using these polynomials as a basis for formula_50 we have to solve
formula_51
to solve the polynomial interpolation problem.
This system of equations can be solved iteratively by solving
formula_52
Derivation.
While the interpolation formula can be found by solving a linear system of equations, there is a loss of intuition in what the formula is showing and why Newton's interpolation formula works is not readily apparent. To begin, we will need to establish two facts first:
Fact 1. Reversing the terms of a divided difference leaves it unchanged: formula_53 
The proof of this is an easy induction: for formula_54 we compute formula_55 
Induction step: Suppose the result holds for any divided difference involving at most formula_37 terms. 
Then using the induction hypothesis in the following 2nd equality we see that for a divided difference involving formula_57 terms we have 
formula_58 
We formulate next Fact 2 which for purposes of induction and clarity we also call Statement formula_59
(formula_60) :
Fact 2. (formula_60) : If formula_62 are any formula_59 points with distinct formula_64-coordinates and formula_65 is the unique polynomial of degree (at most) 
formula_66 whose graph passes through these formula_59 points then there holds the relation
formula_68 
Proof. (It will be helpful for fluent reading of the proof to have the precise statement and its subtlety in mind: formula_69 is defined by passing through formula_70 but the formula also speaks at both sides of an additional arbitrary point formula_71 with formula_64-coordinate distinct from the other formula_73.) 
We again prove these statements by induction.
To show formula_74 let formula_75 be any one point and let formula_76 be the unique polynomial of degree 0 passing through formula_77. Then evidently formula_78 and we can write formula_79 as wanted.
Proof of formula_80 assuming formula_81 already established: Let formula_76 be the polynomial of degree (at most) formula_59 passing through formula_84
With formula_85 being the unique polynomial of degree (at most) formula_66 passing through the points formula_87, we can write the following chain of equalities, where we use in the 
penultimate equality that Stmformula_88 applies to formula_89: 
formula_90
The induction hypothesis for formula_89 also applies to the second equality in the following computation, where 
formula_75 is added to the points defining formula_89 :
formula_94 
Now look at formula_95 By the definition of formula_89 
this polynomial passes through formula_97 and, as we have just shown, it also passes 
through formula_98 Thus it is the unique polynomial of degree formula_99 which passes through these points. Therefore this polynomial is formula_100 i.e.: formula_101
Thus we can write the last line in the first chain of equalities as `formula_102' and have thus established that
formula_103 So we 
established formula_104, and hence completed the proof of Fact 2. 
Now look at Fact 2: It can be formulated this way: If formula_69 is the unique polynomial of degree at most formula_66 whose graph passes through the points formula_107 then formula_108 is the unique polynomial of degree at most formula_59 passing 
through points formula_110 So we see Newton interpolation permits indeed to add new interpolation points without destroying what