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Errors in Polynomial Interpolation

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LECTURE 6 Errors i Polyomial Iterpolatio As with ay approximate method, the utility of polyomial iterpolatio ca ot be stretched too far. I this lecture we shall quatify the errors that ca occur i polyomial
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LECTURE 6 Errors i Polyomial Iterpolatio As with ay approximate method, the utility of polyomial iterpolatio ca ot be stretched too far. I this lecture we shall quatify the errors that ca occur i polyomial iterpolatio ad develop techiques to miimize such errors. We shall begi with a easy theorem. Theorem 6.. Let f be a fuctio i C + [a, b], ad let P be a polyomial of degree that iterpolates the fuctio f at + distict poits x 0,x,...,x [a, b]. The to each x [a, b] there exists a poit ξ x [a, b] such that fx) px) = + )! f +) ξ x ) Remark 6.2. Although this formula for the error is somewhat remiiscet of the error term associated with a th order Taylor expasio, this theorem has little to do with Taylor expasios. Proof. If x = x i, oe of the odes of the iterpolatio, the the statemet is certaily true sice both sides vaish idetically. Suppose ow that x x i, i =0,,...,. Put wt) = t x i ) φ x t) = ft) P t) fx) P x) wt) The φ x t) C + [a, b], ad φ x t) vaishes at + 2 distict poits: i.e., whe t = x 0,x,...,x, or x. Now from Calculus I, we have Rolle s Theorem which states that if a differetiable fuctio fx) has distict zeros, the its derivative must have at least zeros these beig the poits where the graph of the fuctio fx) turs aroud to re-cross the x-axis). Hece, φ x t) has at least + distict zeros, φ x t) has at least distict zeros, ad so o util we ca coclude that φ +) x t) has at least oe distict zero i [a, b]; call it ξ x. Now fx) P x) wt) = f +) t) P +) fx) P x) d + t) wt)) dt +) = f +) t) P +) fx) P x) d + t) dt +) t x 0)t x ) t x ) = f +) t) P +) fx) P x) t) +)! φ +) t) = d+ ft) P t) x dt +) ) Hece, 0=φ +) ξ x x )=f +) ξ x ) P +) ξ x ) fx) P x) ) + )! . CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 2 Now, because P x) is a polyomial of degree, P +) x) = 0. Hece we have 0=f +) fx) P x) ξ x ) + )! or fx) P x) = + )! f +) ξ x ) = +)! f +) ξ x ). Example 6.3. If P x) is the polyomial that iterpolates the fuctio fx) = six) at0poitsothe iterval [0, ], what is the greatest possible error? I this example, we have +=0 ad f +) x) =f 0) x) = six) so the largest possible error would be the maximal value of 0! f 0) ξ x ) for x, x 0,x,...,x,ξ x [0, ]. Clearly, o the iterval [0, ] max x x i = max f +) ξ x ) = max siξ x ) =, so the maximal error would be 0! )) Chebyshev Polyomials ad the Miimalizatio of Error The theorem i the precedig system ot oly tells us how large the error could be whe a give fuctio is replaced by a iterpolatig polyomial; it also gives us a clue as to how we might arrge thigs to make the error as small as possible. To see this, let me write dow agai the expressio for the error term: Ex) fx) P x) = + )! f +) ξ x ), for some ξ x [a, b] Now we do t eve kow what ξ x is except that it s some poit i the iterval [a, b] that depeds o x; so there s ot much we ca do with the term f +) ξ x ) particularly, because, i physical applicatios, we do t eve kow what f is). However, we ca try to make the term as small as possible by pickig a suitable choice of odes {x i }. Now oe simple choice for the x i would be to set x = a + b x i = a + i x) However, this is a case where the simplest choice of the x i turs out ot to be the best choice. We ca see this with a simple example. Cosider the case where a =, b =, ad x i = +0.5)i, i =0,, 2, 3, 4 . CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 3 The if we set =x )x 0.5)x 0)x 0.5)x ) ad plot it we fid that has a maximum value of about 0.. Suppose istead we, for some strage reaso, choose the poits x0 = x = x2 = 0.0 x3 = x4 = ad plot =x )x )x 0)x )x ) We the fid . CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 4 which has a maximum value of about 0.06, which is about half the value that we obtaied for the simpler choice of poits x i. Thus, by choosig a special set of poits x i it is possible to reduce the cotributio of the factor to the error term, ad thus miimize the overall error of the iterpolatio polyomial. So ow the questio becomes: how to choose a good set of poits to sample data, so that a polyomial iterpolatio is as accurate as possible? This is where Chebyshev polyomials will come ito play... Chebyshev Polyomials. The Chebyshev polyomials are defied recursively, via the formula T 0 x) = T x) = x The first six Chebyshev polyomials are thus T+x) = 2xTx) T x), =, 2, 3, 4,... T 0 x) = T x) = x T 2 x) = 2x 2 T 3 x) = 4x 3 3x T 4 x) = 8x 4 8x 2 + T 5 x) = 6x 5 20x 3 +5x T 6 x) = 32x 6 48x 4 +8x 2 Note that the leadig term of the Chebyshev polyomial Tx) is2 x. Theorem 6.4. For x [, ] we have. Tx) =cos cos x) ) This theorem is proved by showig that fx) cos cos x)), the ad the usig the trig idetity to demostrate that f 0 x) = f x) = x cosa + B) = cosa)cosb) sia) sib) f+x) =2xfx) f x) Hece, fx) satisfies the defiig properties of the Chebyshev polyomials. Corollary 6.5. We have T cos T x), x [, ] ) jπ T cos = ) j, j =0,..., 2j 2 π )) = 0, j =,..., . CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 5 First, however we shall give a egative result. Defiitio 6.6. A polyomial P of degree is called moic if the coefficiet x is. Note that expressios of the form = are moic polyomials, as are the polyomials obtaied from the Chebyshev polyomials by dividig through by the leadig coefficiet. Q x) = 2 T x) Our first applicatio of Chebyshev polyomials will be to prove a lower boud for maximum value of a moic polyomial o the iterval [, ]. Theorem 6.7. If P is a moic polyomial of degree, the P x) max P x) 2 x Proof. Suppose that P x) is a moic polyomial of degree ad that Set P x) 2, x [, ] Q x) = 2 T x) iπ x i = cos, i =0,,..., The by costructio Q x) is a moic polyomial of degree, ad we ll have ) iπ ) i Q x i )= ) i 2 T cos =2 ) i ) i =2 Sice P x) ad Q x) both have leadig coefficiet, their differece Q x) P x) will be a polyomial of degree. O the other had, Hece, ) i P x i ) P x i ) 2 = ) i Q x i ), i =0,, 2,..., ) i [Q x i ) P x i )] 0, i =0,, 2,..., Thus, the fuctio Q x) P x) must oscillate i sigs at least + times over the iterval [, ]. But this is ot possible sice Q x) P x) is a polyomial of degree at most. Hece, we have a cotradictio if P x) 2, x [, ]. Thus, the opposite iequality must hold. Lemma 6.8. If Q x) is the moic polyomial defied by Q x) =2 T x) the the maximal value of Q x) o the iterval [, ] is 2. Proof. This is easy sice o the iterval [, ] ad cosθ) cos0) = for all θ. Q x) =2 T x) =2 cos cos ) x) . CHEBYSHEV POLYNOMIALS AND THE MINIMALIZATION OF ERROR 6 Lemma 6.9. Let The 6.) 2i + x i =cos 2 +2 π, i =,..., x x ) x x )=Q x) 2 T x) Proof. This follows from the Fudametal Theorem of Algebra. By costructio each of the x i is a distict root of the moic polyomial Q x), which is of degree. The Fudametal Theorem of Algebra tells us that Q x) must therefore factorize asq x) =x x ) x x ). We ow have Theorem 6.0. If the odes x i are chose as the roots of the Chebyshev polyomial T + x) 2i + x i =cos 2 +2 π, i =0,,..., the the error term for polyomial iterpolatio usig the odes x i is Ex) = fx) P x) max f +) t) 2 + )! t Moreover, this is the best upper boud we ca achieve by varyig the choice of the x i..2. Pickig Optimal Nodes o More Geeral Itervals. The results of the precedig sectio ca be summarized as follows: if we wat a polyomial iterpolatig a fuctio f at + poits x i i the iterval [, ] to be as accurate as possible, the we should choose the data poits x i so that they are the zeros of the Chebyshev polyomial T + x). Put more practically: suppose we have a experimet that measures a quatity Q that depeds o a parameter x [, ]. If we are to fid the polyomial Px) of degree that most accurately represets the actual fuctio Qx), by iterpolatig the data take at + poits x i, the we should choose the x i so that they are the zeros of the Chebyshev polyomial T + x). What do we do if a experimetal parameter x is allowed to rage through some other itevarl [a, b] [, ]? The aswer is quite easy. To fid a optimal set of + data poits x i i a iterval [a, b] wesimply rescale the + zeros of T + x) topoitsi[a, b]. More precisely, let s be the liear map that maps a poit x [, ] to a poit sx) [a, b], such that s ) = a ad s ) = b. These properties actually fix s uiquely, b a) sx) =a + x +)= b + a + b a x The optimal set of + data poits x i for iterpolatig a fuctio Qx) o the iterval [a, b] willthebe the image uder s of the +zerosof T + x): x i = b + a) 2 + b a 2 cos 2i π, i =0,...,
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