Bases for Polynomial Interpolation

4.1.2 Bases for Polynomial Interpolation

A set of polynomials {p 1 (x), p 2 (x), . . . , p n (x) } such that any polynomial p ∈ P n can be expressed as a linear combination

p(x) =

c j p j (x)

j =1

is called a basis in P n . The column vector c = (c T 1 ,c 2 ,...,c n ) can be viewed as a coordinate vector of p in the space P n , with respect to this basis. The interpolation problem

(4.1.1) leads to a linear system of equations

c 1 p 1 (x i ) +c 2 p 2 (x i ) +···+c n p n (x i ) = f (x i ), i = 1 : n. (4.1.2) If we introduce the matrix

n = [p j (x i ) ] i,j =1 ,

(4.1.3) and the column vector f = (f (x 1 ), f (x 2 ), . . . , f (x n )) T , then the linear system becomes

(4.1.4) Mathematically, the choice of basis (for a finite-dimensional space) makes no differ-

P n c = f.

ence. Computationally, working with rounded values of the coefficients, the choice of basis can make a great difference. If the purpose is to compute derivatives or integrals of the inter- polation polynomial, the power basis or the shifted power basis, where p j (x)

= (x −c) j −1 , i.e.,

is convenient although not always the best. If a shifted power basis is to be used for polynomial approximation on an interval [a, b], it is often best to choose c = (a + b)/2,

i.e., equal to the midpoint of the interval. For the power basis p j (x)

=x j −1 , the coefficients of the interpolation polynomial are given by the solution of the linear system V T n c = f , where V n is the Vandermonde

1 n 2 ···x

n −1

By Theorem 3.3.4 this matrix is nonsingular, since the Vandermonde determinant equals (see (3.3.12))

det(V n ) =

(x i −x j ).

1≤i<j≤n

Let {p 1 (x), p 2 (x), . . . , p n (x) } and {q 1 (x), q 2 (x), . . . , q n (x) } be two bases for P n . Then the q j must be linear combinations of the p k , k = 1 : n. This can be expressed in

vector-matrix form: q 1 (x), q 2 (x), . . . , q n (x) = p 1 (x), p 2 (x), . . . , p n (x) S ∀x,

4.1. The Interpolation Problem 353 where S is a constant matrix. S must be nonsingular, since if S were singular, then there

would exist a nontrivial vector v such that Sv = 0, hence (q 1 (x), q 2 (x), . . . , q n (x))v = (p 1 (x), p 2 (x), . . . , p n (x))Sv = 0 ∀x, and (q 1 (x), q 2 (x), . . . , q n (x)) would thus not be a basis.

j (x i ) ] n i,j =1 = [q ] i,j =1 . By putting x = x i , i = 1 : m, into (4.1.6), we see that Q n =P n S , and Q n is nonsingular for every basis. If we set p(x) =

Let P n = [p j (x i ) n

and Q n

d j q j (x) , the system (4.1.2) becomes for this basis Q n d = f , and then

c =P n −1 f = Sd. (4.1.7) The matrix S for the transformation between representations is thus like a coordinate trans-

P n c =f=Q n d =P n Sd,

formation in geometry. Matrix representations of various common bases transformations are given by Gander [130].

The power basis has a bad reputation which is related to the ill-conditioning of the corresponding Vandermonde matrix; see Sec. 4.1.3. There are other bases in P n which are often more advantageous to use. By a triangle family of polynomials we mean a sequence of polynomials

q n (x) =s 1n +s 2n x +s 3n x 2 +···+s nn x n −1 , where s jj

Conversely, for any j, p j (x) =x j −1 can be expressed recursively and uniquely as linear combinations of q 1 (x), . . . , q j (x) . We obtain a triangular scheme also for the inverse

transformation:

1=t 11 q 1 (x), x =t 12 q 1 +t 22 q 2 ,

x n =t 1n q 1 +t 2n q 2 +t 3n q 3 +···+t nn q n , where t jj

−1 . Thus every triangle family is a basis for P m . (Recall the well-known fact that the inverse of a

triangular matrix with nonzero diagonal exists and is triangular.) Among interesting triangle families are the shifted power basis (x − c) j , the Chebyshev polynomials T j (x) , and many

other families of orthogonal polynomials.

A triangle family which is often very convenient for solving the interpolation problem is the family of Newton polynomials

p 1 (x) = 1, p j (x) = (x − x 1 )(x −x 2 ) . . . (x −x j −1 ), j = 2 : n, (4.1.10)

354 Chapter 4. Interpolation and Approximation which has unit leading coefficients. Since p j (x k ) = 0, if k < j we obtain, using the

representation p(x) =c 1 p 1 +c 2 p 2 (x) +c 3 p 3 (x) +···+c n p n (x),

(4.1.11) lower triangular system Lc = f for the coefficients, where

j −1 =1 (x n −x j ) Hence the coefficients can be computed by forward substitution. In the next section we

1 (x

n −x 1 ) (x n −x 1 )(x n −x 2 ) ···

shall see how this basis leads to Newton’s interpolation formula. This is one of the best interpolation formulas with respect to flexibility, computational economy, and numerical stability.

If a polynomial p(x) is given in the form (4.1.11), then it can be evaluated using only n multiplications and 2n additions for a given numerical value x using the nested form

p(x) = (· · · (c n (x −x n −1 ) +c n −1 )(x −x n −2 ) +···+c 3 )(x −x 2 ) +c 2 )(x −x 1 ) +c 1 .

This can be evaluated by a recursion formula similar to Horner’s rule (see Sec. 1.2.2). Another basis of P n that has many advantages is the Lagrange basis of polynomials ℓ j (x) . If x i , i = 1 : n, are distinct interpolation points, these are

These bases polynomials of degree n − 1 satisfy

1 if i = j,

(4.1.14) From this property directly follows Lagrange’s interpolation formula, which directly

ℓ j (x i ) =δ ij =

displays the solution of the interpolation problem for n distinct points. 127

Theorem 4.1.1.

The unique interpolation polynomial p ∈ P n interpolating the function f at the distinct points x i , i = 1 : n, can be written

n & (x −x

p(x) =

127 Lagrange published this formula in 1794.

4.1. The Interpolation Problem 355 It is an easy exercise to show that by l’Hôpital’s rule the Lagrange polynomials can

be written

(x −x i ). (4.1.16)

(x −x j )M n (x j )

i =1

This property characterizes what, in a more general context, is known as a cardinal basis. Lagrange’s interpolation formula has been widely regarded as being more suitable for deriving theoretical results than for practical computation. However, in Sec. 4.2.3 two modified forms of Lagrange’s interpolation formula will be given, which are also very attractive computationally.

A natural extension of the interpolation problem is to determine a polynomial p(x) = j =1 c j p j (x) ∈P n that, in some sense, best fits to the data (x i , f (x i )) , i = 1 : m, where m>n . Since the number of data points is larger than the number of parameters, the corresponding linear system P c = f is overdetermined and can typically be satisfied only approximately. Overdetermination can be used to attain two different types of smoothing:

(i) to reduce the effect of random or other irregular errors in the values of the function; (ii) to give the polynomial a smoother behavior between the grid points. In least squares approximation one determines the coefficient vector c so that the sum

of squared residuals

S(c) =

(p(x i ) − f (x i )) 2 (4.1.17)

i =1

is minimized; see Sec. 1.4. This can in many applications be motivated by statistical ar- guments; see the Gauss–Markov theorem (Theorem 1.4.1). It also leads to rather simple computations. The conditions for the minimization are

∂S(c) m =2

p k (x i ) p(x i ) − f (x i ) = 0, k = 1 : n. ∂c k

i =1

A stable method for discrete least squares polynomial approximation, based on using a basis of orthogonal polynomials, will be given in Sec. 4.5.5. We mention here that a large part of the theory of polynomial interpolation and ap- proximation is valid also for generalized polynomials

u(x) =

a k u k (x),

k =1

where {u 1 ,u 2 ,...,u n } are continuous real-valued functions that form a Chebyshev system on a closed finite interval [a, b]; see Sec. 4.5.7.