Taylor s Series and Interpolation. Interpolation & Curve-fitting. CIS Interpolation. Basic Scenario. Taylor Series interpolates at a specific

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1 CIS 54 - Iterpolato Roger Crawfs Basc Scearo We are able to prod some fucto, but do ot kow what t really s. Ths gves us a lst of data pots: [x,f ] f(x) f f + x x + August 2, 25 OSU/CIS 54 3 Taylor s Seres ad Iterpolato Taylor Seres terpolates at a specfc pot: The fucto Its frst dervatve It may ot terpolate at other pots. We wat a terpolat at several f(c) s. August 2, 25 OSU/CIS 54 2 Iterpolato & Curve-fttg Ofte, we have data sets from expermetal/observatoal measuremets Typcally, fd that the data/depedet varable/output vares As the cotrol parameter/depedet varable/put vares. Examples: Classc gravty drop: locato chages wth tme Pressure vares wth depth Wd speed vares wth tme Temperature vares wth locato Scetfc method: Gve data detfy uderlyg relatoshp Process kow as curve fttg: August 2, 25 OSU/CIS 54 4

2 Iterpolato & Curve-fttg Gve a data set of + pots (x,y ) detfy a fucto f(x) (the curve), that s some (well-defed) sese the best ft to the data Used for: Idetfcato of uderlyg relatoshp (modellg/predcto) Iterpolato (fllg the gaps) Extrapolato (predctg outsde the rage of the data) August 2, 25 OSU/CIS 54 5 Iterpolato Cocetrate frst o the case where we beleve there s o error the data (ad roud-off s assumed to be eglgble). So we have y f(x ) at + pots x,x x, x : x j > x j- (Ofte but ot always evely spaced) I geeral, we do ot kow the uderlyg fucto f(x) Coceptually, terpolato cossts of two stages: Develop a smple fucto g(x) that Approxmates f(x) Passes through all the pots x Evaluate f(x t ) where x < x t < x August 2, 25 OSU/CIS 54 7 Iterpolato Vs Regresso Dstctly dfferet approaches depedg o the qualty of the data Cosder the pctures below: extrapolate terpolate extrapolate Pretty cofdet: there s a polyomal relatoshp Lttle/o scatter Wat to fd a expresso that passes exactly through all the pots Usure what the relatoshp s Clear scatter Wat to fd a expresso that captures the tred: mmze some measure of the error Of all the pots August 2, 25 OSU/CIS 54 6 Iterpolato Clearly, the crucal questo s the selecto of the smple fuctos g(x) Types are: Polyomals Sples Trgoometrc fuctos Spectral fuctos Ratoal fuctos etc August 2, 25 OSU/CIS 54 8

3 Curve Approxmato We wll look at three possble approxmatos (tme permttg): Polyomal terpolato Sple (polyomal) terpolato Least-squares (polyomal) approxmato If you kow your fucto s perodc, the trgoometrc fuctos may work better. Fourer Trasform ad represetatos August 2, 25 OSU/CIS 54 9 Polyomal Iterpolato There are a varety of ways of expressg the same polyomal Lagrage terpolatg polyomals Newto s dvded dfferece terpolatg polyomals We wll look at both forms August 2, 25 OSU/CIS 54 Polyomal Iterpolato Cosder our data set of + pots y f(x ) at + pots x,x x, x : x j > x j- I geeral, gve + pots, there s a uque polyomal g (x) of order : g ( x) a + a x+ a x + K + a x 2 2 That passes through all + pots August 2, 25 OSU/CIS 54 Polyomal Iterpolato Exstece does there exst a polyomal that exactly passes through the data pots? Uqueess Is there more tha oe such polyomal? We wll assume uqueess for ow ad prove t latter. August 2, 25 OSU/CIS 54 2

4 Lagrage Polyomals Summato of terms, such that: Equal to f() at a data pot. Equal to zero at all other data pots. Each term s a th - degree polyomal Exstece!!! p ( x) L( x) f( x ) L( x) L( x ) j j ( x xk ) ( ) k, k k δ j j August 2, 25 OSU/CIS 54 3 Lagrage Polyomals 2 d Order Case > quadratc polyomals Addg The frst thrd secod them quadratc all together, has has roots at we roots x get ad at the x x 2 ad terpolatg x a 2 ad value a equal quadratc value to equal the polyomal, to fucto the fucto data such that: data x. at x 2. P(x ) f P(x ) f P(x 2 ) f 2 x x x 2 August 2, 25 OSU/CIS 54 5 Lear Iterpolato Summato of two les: p ( x) L( x) f( x ) x x x x f( x) + f( x) x x x x x x Remember ths whe we talk about pecewse-lear sples August 2, 25 OSU/CIS 54 4 Lagrage Polyomals Sum must be a uque 2 d order polyomal through all the data pots. What s a effcet mplemetato? August 2, 25 OSU/CIS 54 6

5 Newto Iterpolato Cosder our data set of + pots y f(x ) at x,x x, x : x > x Sce p (x) s the uque polyomal p (x) of order, wrte t: p ( x) b + b( x x) + b2( x x)( x x) + K+ b( x x)( x x) L( x x ) b f( x) f( x) f( x) b f [ x x] x x f[ x2, x] f[ x, x] b2 f [ x2, x, x] M [,, K, ] b f x x x 2 [, K, ] [, K, ] f x x f x x f[x,x j ] s a frst dvded dfferece f[x 2,x,x ] s a secod dvded dfferece, etc. August 2, 25 OSU/CIS 54 7 Lear Iterpolato Smple lear terpolato results from havg oly 2 data pots. f( x) f( x) p( x) f ( x) + ( x x) slope x x August 2, 25 OSU/CIS 54 9 Ivarace Theorem Note, that the order of the data pots does ot matter. All that s requred s that the data pots are dstct. Hece, the dvded dfferece f[x, x,, x k ] s varat uder all permutatos of the x s. August 2, 25 OSU/CIS 54 8 Quadratc Iterpolato Three data pots: p2( x) f( x) + f( x) f( x) x x ( x x) + f[ x, x, x2]( x x)( x x) f ( x ) f ( x ) f( x ) f( x ) 2 f( x ) f( x ) f( x ) + ( x x ) + ( x x )( x x ) 2 2 f( x ) f( x ) f( x ) + ( x x ) + f( x2) f( x) ( ( ) ( ) x x) ( ) f x f x x x ( x x) ( x x) x2 x x x 2 August 2, 25 OSU/CIS 54 2,

6 Newto Iterpolato Let s look at the recurso formula: f[ x ] f( x ) [,, K, ] b f x x x where [, K, ] [, K, ] f x x f x x For the quadratc term: 2 f( x2) f( x) f( x) f( x) f[ x2, x] f[ x, x] x2 x x x b2 f [ x2, x, x] x x x x f( x2) f( x) x2 x b 2 2 August 2, 25 OSU/CIS 54 2 Example: l(x) Iterpolato of l(2): gve l(); l(4) ad l(6) Data pots: {(,), (4,.3863), (6,.7976)} Lear Iterpolato: + {(.3863-)/(4-)}(x-).462(x-) Quadratc Iterpolato:.462(x-)+(( )/2)(x-)(x-4).462(x-) -.49(x-)(x-4) Note the dvergece for values outsde of the data rage. August 2, 25 OSU/CIS Evaluatg for x 2 f( x ) b + b + b x x x x f f f + b + b x x f + b x x + f2 f f f f + + f f f 2 x x August 2, 25 OSU/CIS Example: l(x) Quadratc terpolato catches some of the curvature Improves the result somewhat Not always a good dea: see later August 2, 25 OSU/CIS 54 24

7 Calculatg the Dvded-Dffereces A dvded-dfferece table ca easly be costructed cremetally. Cosder the fucto l(x). x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS Calculatg the Dvded-Dffereces x l(x) f[i,i+] f[i,i+,,i+7] f[. +, + 2] [ +, + 2 ] [, + ] f f x x x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS Calculatg the Dvded-Dffereces x l(x) f[i,i+] f[i,i+,,i+7] f ( x ) f ( x ) f[. + ] ( x x) x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS Calculatg the Dvded-Dffereces f[, K, + 3] [ +, + 2, + 3 ] [, +, + 2 ] ( ) f f + 3 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS 54 28

8 Calculatg the Dvded-Dffereces f[, K, + 4] [ +, K, + 4 ] [, K, + 3 ] ( ) f f + 4 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS Calculatg the Dvded-Dffereces f[, K, + 6] [ +, K, + 6 ] [, K, + 5 ] ( ) f f + 6 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS 54 3 Calculatg the Dvded-Dffereces f[, K, + 5] [ +, K, + 5 ] [, K, + 4 ] ( ) f f + 5 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS 54 3 Calculatg the Dvded-Dffereces Fally, we ca calculate the last coeffcet. f[, K, + 7] [ +, K, + 7 ] [, K, + 6 ] ( ) f f + 7 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS 54 32

9 Calculatg the Dvded-Dffereces All of the coeffcets for the resultg polyomal are bold. b b 4 x l(x) f[i,i+] f[i,i+,,i+7] x l(x) b-b9)/(a-a9(c-c9)/(a-a8)d-d9)/(a-a7d-d9)/(a-a6e-e9)/(a-a5f-f9)/(a-a4 (g-g9)/(a-a3) August 2, 25 OSU/CIS Add a slde that talks about the frst colum beg a set of lear sples. The secod colum gves us quadratcs thru three adjacet pots. There are two such quadratcs for each terval. August 2, 25 OSU/CIS b 7 Add a slde that lsts the polyomal Here, as well as after slde 35. August 2, 25 OSU/CIS Addg a Addtoal Data Pot Addg a addtoal data pot, smply adds a addtoal term to the exstg polyomal. Hece, oly addtoal dvded-dffereces eed to be calculated for the + st data pots. x l(x) f[i,i+] f[i,i+,,i+7] August 2, 25 OSU/CIS b 8

10 Addg More Data Pots Quadratc terpolato: does lear terpolato The add hgher-order correcto to catch the curvature Cubc, Cosder the case where the data pots are orgazed such the the frst two are the edpots, the ext pot s the md-pot, followed by successve md-pots of the half-tervals. Worksheet: f(x)x 2 from - to 3. August 2, 25 OSU/CIS Uqueess Sce p ad q both terpolate the + data pots, Ths polyomal r, has at least + roots!!! Ths ca ot be! A polyomal of degree- ca oly have at most roots. Therefore, r(x) p ( x) a ( x r ) p ( x) a ( x r) + + August 2, 25 OSU/CIS Uqueess Suppose that two polyomals of degree (or less) exsted that terpolated to the + data pots. Subtractg these two polyomals from each other also leads to a polyomal of at most degree. r( x) p( x) q( x) August 2, 25 OSU/CIS Example Suppose f was a polyomal of degree m, where m<. Ex: f(x) 3x-2 We have evaluatos of f(x) at fve locatos: (-2,-8), (-,-5), (,-2), (,), (2,4) August 2, 25 OSU/CIS 54 4

11 Error Defe the error term as: ε ( x) f( x) p( x) If f(x) s a th order polyomal p (x) s of course exact. Otherwse, sce there s a perfect match at x, x,,x Ths fucto has at least + roots at the terpolato pots. ε ( x) ( x x )( x x ) L ( x x ) h( x) August 2, 25 OSU/CIS 54 4 Iterpolato Errors Use the pot x, to expad the polyomal. ε x { x, x, K x } Pot s, we ca take a arbtrary pot x, ad create a (+) th polyomal that goes thru the pot x. ( x) f( x) p ( x) f[ x, x, K x, x] x x August 2, 25 OSU/CIS Iterpolato Errors ε x f x p x f ξ x x ( + ) ( + )! x [ a, b], ξ a, b Proof s the book. Itutvely, the frst + terms of the Taylor Seres s also a th degree polyomal. August 2, 25 OSU/CIS Iterpolato Errors Combg the last two statemets, we ca also get a feel for what these dvded dffereces represet. ( ) f[ x, x, K x ] f ξ Corollary book If f(x) s a polyomal of degree m<, the all (m+) th dvded dffereces ad hgher are zero.! August 2, 25 OSU/CIS 54 44

12 Problems wth Iterpolato Is t always a good dea to use hgher ad hgher order polyomals? Certaly ot: 3-4 pots usually good: 5-6 ok: See tedecy of polyomal to wggle Partcularly for sharp edges: see fgures August 2, 25 OSU/CIS Chebyshev odes Let s look at these for 4. Spreads the pots out the ceter. x x x x x 3 4 cos π 4 2 cos π cos π cos π cos π 4 August 2, 25 OSU/CIS Chebyshev odes Equally dstrbuted pots may ot be the optmal soluto. If you could select the x s, what would they be? Wat to mmze the ( x x ) term. These are the Chebyshev odes. For x- to : x cos, ( ) π August 2, 25 OSU/CIS Polyomal Iterpolato Two-Dmesos Cosder the case hgher-dmesos. August 2, 25 OSU/CIS 54 48

13 Fdg the Iverse of a Fucto What f I am after the verse of the fucto f(x)? For example arccos(x). Smply reverse the role of the x ad the f. August 2, 25 OSU/CIS 54 49