Heath - Scientific Computing (523150), страница 62
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We illustrate by building theNewton interpolant for the previous example incrementally as new data points are added.We begin with the first data point, (t1 , y1 ) = (−2, −27), which is interpolated by theconstant polynomialp1 (t) = y1 = −27.Adding the second data point, (t2 , y2 ) = (0, −1), we modify the previous polynomial so thatit interpolates the new data point as well:p2 (t) = p1 (t) + x2 φ2 (t) = p1 (t) += p1 (t) +y2 − p1 (t2 )φ2 (t)φ2 (t2 )y2 − y1(t − t1 ) = −27 + 13(t + 2).t2 − t1228CHAPTER 7. INTERPOLATIONFinally, we add the third data point, (t3 , y3 ) = (1, 0), modifying the previous polynomial sothat it interpolates the new data point as well:p3 (t) = p2 (t) + x3 φ3 (t) = p2 (t) +y3 − p2 (t3 )φ3 (t)φ3 (t3 )y3 − p2 (t3 )(t − t1 )(t − t2 )(t3 − t1 )(t3 − t2 )= −27 + 13(t + 2) − 4(t + 2)t.= p2 (t) +Given a set of data points (ti , yi ), i = 1, . .
. , n, an alternative method for computingthe coefficients xk of the Newton polynomial interpolant is via quantities known as divideddifferences, which are usually denoted by f [ ] and are defined recursively by the formulaf [t1 , t2 , . . . , tk ] =f [t2 , t3 , . . . , tk ] − f [t1 , t2 , . .
. , tk−1 ],tk − t 1where the recursion begins with f [tk ] = yk , k = 1, . . . , n. It turns out that the coefficientof the jth basis function in the Newton interpolant is given by xj = f [t1 , t2 , . . . , tj ]. Likeforward-substitution, use of this recursion also requires O(n2 ) arithmetic operations tocompute the coefficients of the Newton interpolant, but it is less prone to overflow orunderflow than is direct formation of the entries of the triangular Newton basis matrix.Example 7.5 Divided Differences.
We illustrate divided differences by using thisapproach to derive the Newton interpolant for the same data points as in the previousexamples.f [t1 ] = y1 = −27, f [t2 ] = y2 = −1, f [t3 ] = y3 = 0,f [t2 ] − f [t1 ]−1 − (−27)f [t1 , t2 ] === 13,t2 − t10 − (−2)f [t3 ] − f [t2 ]0 − (−1)f [t2 , t3 ] === 1,t3 − t21−0f [t2 , t3 ] − f [t1 , t2 ]1 − 13f [t1 , t2 , t3 ] === −4.t3 − t 11 − (−2)Thus, the Newton polynomial is given byp(t) = f [t1 ]φ1 (t) + f [t1 , t2 ]φ2 (t) + f [t1 , t2 , t3 ]φ3 (t)= f [t1 ] + f [t1 , t2 ](t − t1 ) + f [t1 , t2 , t3 ](t − t1 )(t − t2 )= −27 + 13(t + 2) − 4(t + 2)t.Note that the validity of Newton interpolation does not depend on any particular ordering of the points t1 , .
. . , tn : in principle any ordering gives the same polynomial. However,the conditioning of the triangular basis matrix A does depend on the ordering of the points.Thus, the sensitivity of the coefficients to perturbations in the data depends on the particular ordering chosen, and the “left-to-right” ordering is not necessarily the best. Forexample, it is often better to take the points in order of their distances from their mean ortheir distances from a specific point at which the resulting interpolant will be evaluated.7.2. POLYNOMIAL INTERPOLATION7.2.4229Orthogonal PolynomialsOrthogonal polynomials are yet another useful type of basis functions for polynomials. Aninner product can be defined on the space of polynomials on an interval [a, b] by takingZ b(p, q) =p(t)q(t)w(t) dt,awhere w(t) is a nonnegative weight function.
Two polynomials p and q are said to beorthogonal if (p, q) = 0. A set of polynomials {pi } is said to be orthonormal if1 for i = j(pi , pj ) =.0 for i 6= jGiven a set of polynomials, the Gram-Schmidt orthogonalization process (see Section 3.4.6) can be used to generate an orthonormal set spanning the same space. Forexample, with the inner product given by the weight function w(t) ≡ 1 on the interval[−1, 1], if we apply the Gram-Schmidt process to the set of monomials, 1, t, t2 , t3 , .
. ., andscale the results so that pk (1) = 1 for each k, we obtain the Legendre polynomials1, t, (3t2 − 1)/2, (5t3 − 3t)/2, (35t4 − 30t2 + 3)/8, (63t5 − 70t3 + 15t)/8, . . . ,the first n of which form an orthogonal basis for the set of polynomials of degree at mostn − 1. The first few Legendre polynomials are plotted in Fig. 7.4. Other choices of intervaland weight function similarly yield other well-known sets of orthogonal polynomials, someof which are listed in Table 7.1.1.0 ..............................0.5.........................................................................................................................................................................................................0..........
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.............................................. .... ............................ .... ...............3... .... ......... ............. ...... ....................... ....................... 4. ......... ................... ...................................... ...... ...... ............ ............. ....... ....................... .......... ... 5. ........... ...................................................................... ................ ..
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.................ppp0.0−0.5−1.0−1.0p−0.50.00.5pp1.0Figure 7.4: The first six Legendre polynomials.Orthogonal polynomials have many useful properties and are the subject of an eleganttheory. One of their most important properties is that they satisfy a three-term recurrenceof the formpk+1 (t) = (αk t + βk )pk (t) − γk pk−1 (t),which makes them very efficient to generate and evaluate. For example, the Legendrepolynomials satisfy the recurrence(k + 1)pk+1 (t) = (2k + 1)tpk (t) − kpk−1 (t).230CHAPTER 7.
INTERPOLATIONTable 7.1: Some commonly occurringNameIntervalLegendre[−1, 1]Chebyshev, first kind[−1, 1]Chebyshev, second kind[−1, 1]Jacobi[−1, 1]Laguerre[0, ∞]Hermite[−∞, ∞]sets of orthogonal polynomialsWeight function1(1 − t2 )−1/2(1 − t2 )1/2(1 − t)α (1 + t)β , α, β > −1e−t2e−tOrthogonality makes such polynomials very convenient for least squares approximation ofa given function by a polynomial of any desired degree, since the matrix of the resultingsystem of normal equations is diagonal.
Orthogonal polynomials are also useful in generatingGaussian quadrature rules, a topic considered in Section 8.3.7.2.5Interpolating a FunctionThus far we have thought only in terms of interpolating a discrete set of data points, solittle could be said about the behavior of the interpolant between the data points. If thedata points represent a discrete sample of an underlying continuous function, however, thenwe may wish to know how closely the interpolant approximates the given function betweenthe sample points.For polynomial interpolation, an answer to this question is given by the following relationship, where f is a sufficiently smooth function, pn−1 is the unique polynomial of degreeat most n − 1 that interpolates f at n points t1 , .
. . , tn , and θ is some (unknown) point inthe interval [t1 , tn ]:f (t) − pn−1 (t) =f (n) (θ)(t − t1 )(t − t2 ) · · · (t − tn ).n!Since the point θ is unknown, this result is not particularly useful unless we have a boundon the appropriate derivative of f , but it still provides some insight into the factors affectingthe accuracy of polynomial interpolation.Another useful form of polynomial interpolation for an underlying smooth function f isthe polynomial given by the truncated Taylor seriespn (t) = f (a) + f 0 (a)(t − a) +f 00 (a)f (n) (a)(t − a)2 + · · · +(t − a)n .2n!This Taylor polynomial interpolates f in the sense that the values of pn and its first nderivatives match those of f and its first n derivatives evaluated at t = a, so that pn (t)is a good approximation to f (t) for t near a.
We have seen the usefulness of this typeof polynomial interpolant in Newton’s method for root finding (where we used a linearpolynomial) and for minimization (where we used a quadratic polynomial).7.2. POLYNOMIAL INTERPOLATION7.2.6231High-Degree Polynomial InterpolationHigh-degree interpolating polynomials are expensive to determine and evaluate. Moreover,in some bases the coefficients of the polynomial may be poorly determined as a result ofill-conditioning of the linear system to be solved.
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