266 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Hermite’s interpolation formula is H(x)= () () () () 11 2 2 00 [1 2 ( )] [ ] ii i i i i i ii xxLxL x y xx Lx y == ′′ −− + −   ∑∑ = [1–2 (x – x 0 ) L 0 ’ (x 0 )] [L 0 (x)] 2 y 0 + [1 – 2(x – x i ) L 1 ’ (x 1 )] [L 1 (x)] 2 y 1 + (x – x 0 ) [L 0 (x)] 2 y’ 0 + (x – x 1 ) [L 1 (x)] 2 1 y ′ (1) Now, L 0 (x)= 1 01 xx xb xx ab −− = −− L 1 (x)= 0 10 xx xa xx ba −− = −− ∴ () 0 Lx ′ = 1 ab − and L′ 1 (x) = 1 ba − Hence, () 00 Lx ′ = 1 ab − and L′ 1 (x 1 ) = 1 ba − Therefore from equation (1) H(x)= () () () () 22 12 12 xa xb xb xa fa fb ab ab ba ba  −−   −− −+−     −− −−     () () () () 22 xb xa xa fa xb fb ab ba   −− ′′ +− +−   −−   H 2 ab+    = () () 22 22 22 12 12 ab ab ab ab ab ba fa fb ab ab ba ba ++     ++   −− −−           −+−   −− −−             () () ++   −− ++     +− +− ′′       −−     22 22 221 ab ab ba ab ab afabfb ab b = 1 2 f(a) + 1 2 f(b) + () 8 ba − f ′ (a) – () 8 ba − f ’(b) = () () () () () 28 ba fa fb fa fb ′′ −−  +  + . Hence Proved. PROBLEM SET 5.4 1. Apply Hermite formula to find a polynomial which meets the following specifications: 012 010 000 k k k x y y ′ [Ans. x 4 – 4x 3 + 4x 2 ] INTERPOLATION WITH UNEQUAL INTERVAL 267 2. Apply Hermite’s interpolation to find f(1.05) given: 11.0 0.5 1.1 1.04881 0.47673 xf f ′ [Ans. 1.02470] 3. Apply Hermite’s interpolation to find log 2.05 given that: 1 log 2.0 0.69315 0.5 2.1 0.74194 0.47619 xx x [Ans. 0.71784] 4. Determine the Hermite polynomial of degree 5 which fits the following data and hence find an approximate value of log e 2.7 1 log 2.0 0.69315 0.5 2.5 0.91629 0.4 3.0 1.09861 0.33333 e xyxy x ′ += [Ans. 0.993252] 5. Find y = f(x) by Hermite’s interpolation from the table: 11 5 011 137 iii xyy ′ −− Computer y 2 and y 2 ′. [Ans. 1 + x – x 2 + 2x 4 , y 2 = 31, y′ 2 = 61] 6. Compute e by Hermite’s formula for the function f(x) = e x at the points 0 and 1. Compare the value with the value obtained by using Lagrange’s interpolation. [Ans. (1 + 3x) (1 – x) 2 + (2 – x) ex 2 ; 1.644, 1.859] 268 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES 7. Apply Hermite’s formula to find a polynomial which meets the following specifications: 110 000 110 iii xyy ′ −− () 35 1 53 2 xx  −   Ans. 8. Apply osculating interpolation formula to find a polynomial which meets the following requirements: 010 100 290 iii xyy ′ [Ans. x 4 – 4x 3 + 4x 2 ] 9. Apply Hermite’s interpolation formula to find f(x) at x = 0.5 which meets the following requirements: () () 11 5 01 1 13 7 ii i xfx fx ′ −− Also find f(–0.5). 42 11 3 21;, 88 xxx  −++   Ans. 10. Construct the Hermite interpolation polynomial that fits the data: () () 1 7.389 14.778 2 54.598 109.196 xfx fx ′ Estimate the value of f(1.5). [Ans. 29.556x 3 – 85.793x 2 + 97.696x – 34.07; 19.19125] 11. (i) Construct the Hermite interpolation polynomial that fits the data: () () 00 1 0.5 0.4794 0.8776 1.0 0.8415 0.5403 xfx fx ′ Estimate the value of f(0.75). INTERPOLATION WITH UNEQUAL INTERVAL 269 (ii) Construct the Hermite interpolation polynomial that fits the data () () 04 5 16 14 222 17 xyx yx ′ − −− −− Interpolate y(x) at x = 0.5 and 1.5. 12. Obtain the unique polynomial p(x) of degree 3 or less corresponding to a funcion f(x) where f(0) = 1, f’(0) = 2, f(1) = 5, f’(1) = 4. 13. (i) Construct the Hermite interpolation polynomial that fits the data () () 229 50 3 105 105 xfx fx ′ Interpolate f(x) at x = 2.5 (ii) Fit the cubic polynomial P(x) = c 0 + c 1 x + c 2 x 2 + c 3 x 3 to the data given in problem 13 (i). Are these polynomials same? 5.7. SOME RELATED TERMS 5.7.1 Some Remarkable Points about Chosen Different Interpolation Formulae We have derived some central difference interpolation formulae. Obviously here a question arise that which one of these formulae gives the most accurate or approximate, nearest result? (1) If interpolation is required near the beginning or end of a given data, there is only alternative to apply Newton’s Forward and backward difference formulae. (2) For interpolation near the center of a given data, Stirling’s formula gives the best or most accurate result for – 1 4 < u < 1 4 and Bessel’s formula is most efficient near u = 1 2 or 1 4 ≤ u ≤ 3 4 . (3) But in the case where a series of calculations have to be made, it would be inconvenient to use both these (Stirling’s & Bessel’s) formulae i.e., the choice depends on the order of the highest differences that could be neglected so that contributions from it and further differences would be less than half a unit in the last decimal place. If this highest difference is of odd order. Stirling’s formula is recommended; if it is even order Bessel’s formula might be preferred. (4) It is known from algebra that the n th degree polynomial which passes through (n + 1) points is unique. Hence the various interpolaion formulae derived here are actually, only different forms of the same polynomial. Therefore all the interpolation formulae should give the same functional value. 270 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (5) Here we discussed several interpolation formulae for equispaced argument value. The most important thing about these formulae is that, the co-efficients in the central difference formulae are smaller and converges faster than those in Newton’s formulae. After a few terms, the co-efficients in the stirling’s formula decrease more rapidly than those of the Bessel’s formulae and the co-efficient of Bessel’s formula decreases more rapidly than those of Newton’s formula. Therefore, whenever possible central difference formulae should be used in preference to Newton’s formulae. However, the right choice depends on the position of the interpolated value in the given pairs of values. The Zig-Zag paths for various formulae y ∆ ∆ 2 ∆ 3 ∆ 4 ∆ 5 ∆ 6 Newton’s Backward Newton’s Forward Gauss Backward Gauss Forward Stirling Bessel’s Laplace Evertt’s FIG. 5.1 5.7.2 Approximation of Function To evaluate most mathematical functions, we must first produce computable approximations to them. Functions are defined in a variety of ways in applications, with integrals and infinite series being the most common types of formulas used for the definition. Such a definition is useful in establishing the properties of the function, but it is generally not an efficient way to evaluate the function. In this part we examine the use of polynomials as approximation to a given function. For evaluating a function f(x) on a computer it is generally more efficient of space and time to have an analytic approximation to f(x) rather than to store a table and use interpolation i.e., function evaluation through interpolation techniques over stored table of values has been found to be quite costlier when compared to the use of efficient function approximations. It is also INTERPOLATION WITH UNEQUAL INTERVAL 271 desirable to use the lowest possible degree of polynomial that will give the desired accuracy in approximating f(x). The amount of time and effort expended on producing an approximation should be directly proportional to how much the approximation will be used. If it is only to be used a few times, a truncated Taylor series will often suffice. But if an approximation is to be used millions of times by many people, then much care should be used in producing the approximation. There are forms of approximating function other than polynomials. Let f 1 , f 2 f n be the values of given function and 12 , n φφ φ be the corresponding values of the approximating function. Then the error vector is e where the components of e are given by e i = f i – i φ . So the approximation may be chosen in two ways. One is, to find the approximation such that the quantity 22 2 12 n ee e ++ is minimum. This leads us to the least square approximation. Second is, choose the approximation such that the maximum components of e is minimized. This leads to Chebyshev polynomials which have found important applications in the approximation of functions. (i) Approximation of function by Taylors series method: Taylor’s series approximation is one of the most useful series expressions of a function. If a function f(x) has upto (n + 1) th derivatives in an interval [a, b], near x = x 0 then it can be expressed as, f (x) = f(x 0 ) + f ’ (x 0 ) (x – x 0 ) + f ‘’(x 0 ) () 2 0 2! xx − + f n (x 0 ) () 0 ! n xx n − + f n+1 (s) × () () 1 0 1! n xx n + − + (1) In the above expansion f ’(x 0 ), f ’’(x 0 ) etc., are the first, second derivatives of f(x) evaluated at x 0 . The term () () () 1 1 0 1! n n fsxx n + + − + is called the remainder term. The quantity s is a number which is a function of x and lies between x and x 0 . The remainder term gives the truncation error if only the first n terms in the Taylor series are used to represent the function. The truncation error is thus: Truncation error = () () () 1 1 0 1! n n fsxx n + + − + (2) or T ∈ = () () 1 0 1! n xx M n + − + (3) where M = max. () 1n fs + for x in [a, b]. Obviously, the Taylor’s series is a polynomial with base function 1, (x – x 0 ), (x – x 0 ) 2 , (x – x 0 ) n . The co-efficients are constants given by f(x 0 ), f’(x 0 ), f ’’(x 0 ) f ’’(x 0 )/2! etc. Thus the series can be written in the rested form. (ii) Approximation of function by Chebyshev polynomial: The polynomials are linear combination of the monomials 1, x, x 2 x n . An examination of the monomial in the interval (–1, + 1) shows that each achieves its maximum magnitude 1 at x = ± 1 and minimum magnitude O at x = 0. y(x) = a 0 + a 1 x + a 2 x 2 + + a n x n dropping the higher order terms or modification of the co-efficients a 1 , a 2 a n will produce little error for small x near zero. But probably substantial error near the ends of the interval 272 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (x near ± 1). In particular, it seems reasonable to look for other sets of simple related functions that have their extreme values well distributed on their interval (–1, 1). We want to find approximations which are fairly easy to generate and which reduce the maximum error to minimum value. The cosine functions cos θ , cos 2θ, cos nθ appear to be good candidates. The set of polynomials T n (x) = cos n θ , n = 0, 1 generates from the sequence of cosine functions using the transformation. θ = cos –1 x is known as Chebyshev polynomial. These polynomial are used in the theory of approximation of function. Chebyshev polynomials: Chebyshev polynomial T n (x) of the first kind of degree n over the interval [–1, 1] is defined by the relation T n (x) = cos [n cos –1 (x)] (1) Let cos –1 x = θ , so that x = cos θ ⇒ T n (x) = cos n θ for n = 0, T 0 (x) = 1 for n = 1, T 1 (x) = x The Chebyshev polynomials satisfy the recurrence relation T n+1 (x) = 2x T n (x) – T n–1 (x) (2) which can be obtained easily using the following trigonometric identity. cos (n + 1) θ + cos (n – 1) θ = 2 cos θ cos n θ Above recurrence relation can be used to generate successively all T n (x), as well as to express the powers of x in terms of the Chebyshev polynomials. Some of the Chebyshev polynomials and the expansion for powers of x in terms of T n (x) are given as follows: T 0 (x)= 1, 1= T 0 , (x), T 1 (x)= x, x = T 1 (x), T 2 (x)= 2x 2 – 1, x 2 = 1 2 (T 0 (x) + T 2 (x)) T 3 (x)= 4x 3 – 3x, x 3 = () () () 13 1 3, 4 Tx Tx + T 4 (x)= 8x 4 – 8x 2 + 1, x 4 = () () () () 024 1 34 , 8 Tx Tx Tx ++ T 5 (x)= 16x 5 – 20x 3 + 5x, x 5 = () () () () 135 , 1 10 5 16 Tx Tx Tx ++ T 6 (x)= 32x 6 – 48x 4 + 18x 2 – 1, x 6 = () () () () () 0246 1 10 15 6 , 32 Tx Tx Tx Tx +++ T 7 (x)= 64x 7 – 112x 5 + 56x 3 – 7x, T 8 (x) = 128x 8 – 256x 6 + 160x 4 – 32x 2 + 1, T 9 (x) = 256x 9 – 576x 7 + 432x 5 – 120x 3 + 9x, T 10 (x) = 512x 10 – 1280x 8 + 1120x 6 – 400x 4 + 50x 2 – 1 (3) INTERPOLATION WITH UNEQUAL INTERVAL 273 Note that the co-efficient of x n in T n (x) is always 2 n–1 , and expression for x n i.e., 1, x, x 2 x n will be useful in the economization of power series. Further, these polynomials satisfy the differential equation. () 2 22 2 10 dy dy xxny dx dx −−+= where, y = T n (x), We also have () [] 1, 1,1 n Tx x ≤∈− (4) Also, the Chebyshev polynomials satisfy the orthogonality relation () () 1 2 1 0, if ; 1 ,if 0; 1 ,if 0 2 nm mn TxT xdx mn x mn −   ≠  =π = =  −  π  =≠  ∫ (5) Another important property of these polynomials, is that, of all polynomials of degree n where the co-efficient of x n is unity, the polynomial 2 1–n T n (x) has the smallest least upper bound to its magnitude in the interval [–1, 1], i.e., () () 1 11 11 max 2 max n nn xx Tx Px − −≤≤ −≤≤ ≤ (6) This is called the minimax property. Here, p n (x) is any polynomial of degree n with leading co-efficient unity, T n (x) is defined by T n (x) = cos(n cos –1 x) = 2 n–1 x n – (7) Because the maximum magnitude of T n (x) is one, the upper bound referred to is 1/2 n–1 i.e., 2 1–n . This is important because we will be able to write power-series representations of functions whose maximum errors are given in terms of this upper bound. Thus in Chebyshev approximation, the maximum error is kept down to a minimum. This is called as minimax principle and the polynomial P n (x) = 2 1–n T n (x); (n ≥ 1) is called the minimax polynomial. By this process we can obtain the best lower order approximation called the minimax approximation. Example 1. Find the best lower-order approximation to the cubic 2x 3 + 3x 2 . Sol. Using the relation gives in equation (3), we have 2x 3 + 3x 2 = 2 () () {} 2 13 1 33 4 Tx Tx x  ++   = 3x 2 + 3 2 T 1 (x) + 1 2 T 3 (x) = 3x 2 + 31 22 x + T 3 (x), Since T 1 (x) = x The polynomial 3x 2 + 3 2 x is the required lower order approximation to the given cubic with a maximum error ± 1 2 in the range [–1, 1]. 274 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES Example 2. Obtain the best lower degree approximation to the cubic (x 3 + 2x 2 ), on the interval [–1, 1]. Sol. We write x 3 + 2x 2 = 1 4 [3 T 1 (x) + T 3 (x)] + 2x 2 , = 2x 2 + 3 4 T 1 (x) + 1 4 T 3 (x), = 2x 2 + 3 4 x + 1 4 T 3 (x). Hence, the polynomial 2 3 2 4 xx  +   is the required lower order approximation to the given cubic. The error of this approximation on the interval [–1, 1] is () 3 11 11 max 44 x Tx −≤≤ = . Example 3. Use Chebyshev polynomials to find the best uniform approximation of degree 4 or less to x 5 on [–1, 1]. Sol. x 5 in terms of Chebyshev polynomials can be written as x 5 = 135 551 81616 TTT++ Now T 5 being polynomial of degree five therefore we omit the term 5 16 T and approximate f(x) = x 5 by 13 55 816 TT  +   . Thus the uniform polynomial approximation of degree four or less to x 5 is given by x 5 = 5 8 T 1 + 5 16 T 3 = 5 8 x + 5 16 [4x 3 – 3x] = 3 55 16 4 xx  −+   and the error of this approximation on [–1, 1] is 5 11 1 max 16 16 x T −≤ ≤  =   . Example 4. Find the best lower order approximation to the polynomial. () 234 5 xxx x 11 yx 1 x ,x , 2 6 24 120 2 2  =+++++ ∈−   . Sol. On substituting x = 2 ξ , we get () ξξ ξ ξ ξ ξ= + + + + + − <ξ< 23 4 5 1,11 2 8 48 384 3840 y . INTERPOLATION WITH UNEQUAL INTERVAL 275 Above equation can be written in Chebyshev polynomals y( ξ )= T 0 ( ξ ) + 1 2 T 1 ( ξ ) + 1 16 [T 0 ( ξ ) + T 2 ( ξ )] () () 13 1 3 192 TT +ξ+ξ   [][] 024 135 11 3 ( ) + 4T ( ) + ( ) 10 ( ) + 5T ( ) + T ( ) 3072 61440 TT T + ξ ξξ+ ξ ξξ = 1.063477T 0 () ξ + 0.515788T 1 () ξ + 0.063802T 2 () ξ + 0.00529T 3 () ξ + 0.000326T 4 () ξ + 0.000052T 5 () ξ dropping the term containing T 5 () ξ , we get y () ξ = 1 + 0.4999186 ξ + 0.125ξ 2 + 0.0211589ξ 3 + 0.0026041ξ 4 . Hence, y(x) = 1 + 0.999837x + 0.5x 2 + 0.0211589x 3 + 0.041667x 4 . Properties of Chebyshev polynomial T n (x) 1. T n (x) is a polynomial of degree n. 2. T n (–x) = (–1) n T n (x), which show that T n (x) is an odd function of x if n is odd and an even function of x if n is even. 3. () [] ,1,1 n Tx x ≤∈− 4. T n (x) assumes extreme values at (n + 1) points x m = cos () m π , m = 0, 1, 2, ,n and the extreme value of x m is (–1) m . 5. () () 1 1 0, /2, 0 ,0 mn if m n TxTxdx ifmn if m n − ≠   =π = ≠   π==  ∫ which can be proved easily by putting x = cos θ. Also T n (x) are orthogonal on the interval [–1, 1] with respect to the weight function w(x) = 1/ () 2 1 x − 6. If P n (x) is a monic polynomial of degree n then () () 1 11 11 max 2 max n nn xx Tx Px − −≤ ≤ −≤ ≤ ≤ is known as minimax property since () 1 n Tx ≤ . Chebyshev polynomial approximation: Let f(x) be a continuous function defined on the interval [–1, 1] and let B 0 + B 1 x + Bx 2 + B n x n be the required minimax polynomial approximation for f(x). Suppose f(x) = 0 2 a 1 ii i aT ∞ = ∑ (x) is Chebyshev series expansion for f(x). Then the truncated series of the partial sum. . is generally more efficient of space and time to have an analytic approximation to f(x) rather than to store a table and use interpolation i.e., function evaluation through interpolation techniques. co-efficients a 1 , a 2 a n will produce little error for small x near zero. But probably substantial error near the ends of the interval 272 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (x near. polynomial. Therefore all the interpolation formulae should give the same functional value. 270 COMPUTER BASED NUMERICAL AND STATISTICAL TECHNIQUES (5) Here we discussed several interpolation formulae