( ) [ ( k) ( k) ( x) ( ) ( ) ( ) [ ] ξ [ ] [ ] [ ] ( )( ) i ( ) ( )( ) 2! ( ) = ( ) 3 Interpolation. Polynomial Approximation.

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1 3 Interpolaton {( y } Gven:,,,,,, [ ] Fnd: y for some Mn, Ma Polynomal Appromaton Theorem (Weerstrass Appromaton Theorem --- estence ε [ ab] f( P( <ε [ a, b]. If f Cab, and gven >, then there ests a polynomal P defned on, wth the property that for all {( y } Gven:,,,,,, smplest method: pecewse lnear nterpolaton Soluton: fnd the unque polynomal of degree, P (, passng through all these ( ponts. P a a a a Defnton: Lagrange nterpolatng polynomal L, ( ( ( ( ( ( ( ( ( ( ( ( ( polynomal of degree f j L, ( j δ j f j Consequently, j j j j P y L, P y L y δ y, for all j ξ Theorem If,,, are dstrnct numbers n ab, and f C ab,, then for each a, b, a, b such that ( f ( ξ ( ( (! f P pecewse lnear nterpolaton ~ use P ( between two adjacent ponts,.e. for [, ] Lagrange error ( ( f ξ! ( f ma Ma ( ( ma 4 f ~ O( Δ

2 th Polynomal appromaton Lagrange error [ ab] [ ] Consder,, ( f ( ξ (! ( dependng on the choces of nodes ( unformly spaced nodes: constant;,,,, T ( th ( Chebychev nodes: roots of the Chebychev polynomal T cos cos π cos π,,,, The Chebychev nodes mnmze the mamum absolute value of ( ( all sets of nodes a b Mn Ma evll s method --- computng hgher-order polynomal based on lower-order polynomals Defne Pm, m,, m the polynomal of degree passng through the ponts,,,,,, 3 { m } { } e.g.,,4 ( m ym ( m ym ( m ym P,,4 ( the polynomal of degree passng through ( y ( y ( y the ponts,,,,, 4 4

3 Construct the th order polynomal usng the ( th polynomals P,,, ( ( j P,,, j, j, ( ( P,,,,, ( ( j P y P y P, P y P P,,, P y P P P 3 3,3,,3,,,3 P y P P P P 4 4 3,4,3,4,,3,4,,,3,4 P,,, ( ( j P,,, j, j, ( ( P,,,,, ( ( j Because P y for j,, j, j, P y fo r,,,, ( ( ( j j y y P,,, ( y for, j,,, ( P ( ( j y ( P,,,,, j ( P,,, j, j, ( j ( ( j y y j j j j,,, j yj P ewton form of nterpolaton polynomal Gven: Fnd: P ( Wrte: P ( a ( a a a a 3 a ( 3( ( ( ( ( ( P a a a a a a a a3 a ( a ( { ( [ a a3( a ( ( ]} a a ( ( a a P ( ( a a 3 a ( nested multplcaton ( a a (

4 ewton form of nterpolaton polynomal a a P ( a ( a 3 a ( ( a a ( STEP: compute all a,,,, STEP : b a ( ( b a a a b b a b b a b,,,,, ( P b STEP: compute all a,,,, Defne ewton s dvded dfference : f [, ] f y [,, ] f f [ ] f [,,, ] f [, ] [, ] f f [,, ] [,,, ] f f ( 3( ( ( ( ( ( P a a a a a ( P a y f ( ( P a a y f a f [ ] f ( P a a a y [, ]( ( ( f f a f ( ( [, ]( a f f f f [ ] f a f [, ] f a [, ] f [, ] f [,, ] f [, ] f f f f [,,, ] a f y y y y 3 3 a [, ] f [, ] f [, ] f 3 ( P a b a a [,, ] f a f [,, 3] f [,,, ] b a b,,,,, 3 a 3 P b

5 Hermte nterpolaton Gven: f and f,,,, Fnd: a polynomal passng all the ponts wth the gven slops Theorem: If f C a, b and,,, are dstnct, the unque polynomal of least degree agreeng f and f at,,, s the polynomalof degree gven by Theorem: If,, then, f C a b ξ a b ( f ( ξ (! f ( H ( H ( s called the Hermte polynomal. ˆ H f H f H j, j j, j j j { ( } H L L, j j, j j, j ( ˆ, j j, j H L ewton s form of Hermte polynomal { } { } Defne ˆ ˆ,.e. ˆ,,,,,, ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ ( ˆ H a a a a3 a a a a a 3 4 ( ˆ { ( ˆ { ( ˆ { ( ˆ ( ( ˆ } a a a a a a [ ab] Gven:, Want: f H? STEP : fnd a,,,,, STEP : b a b a b for,,,, b H

6 ewton s form of Hermte polynomal { } { } Defne ˆ ˆ,.e. ˆ,,,,,, [ ˆ ] [ ˆ ] Defne f f f ˆ : ˆ ˆ ˆ ˆ ˆ ˆ ˆ : st : f f f f f f Defne [ ˆ, ˆ ] f ( f ( [ ˆ, ˆ ] f f f [ ˆ ˆ ˆ ] Defne f,,, [ ˆ, ˆ,, ˆ ] [ ˆ, ˆ,, ˆ ] f f for 3 ˆ ˆ nd : 3rd : f DD f DD f DD DD DD DD [ ˆ, ˆ,, ˆ ] a f DD: ewton dvded dfference Cubc splne nterpolaton ~ pecewse cubc polynomals ( Gven:, y f,,,, Use a cubc polynomal to ft for each subnterval, S ( for [, ] hgh-order polynomals > oscllatng nature Alternatve choce: pecewse polynomal appromaton Pecewse lnear nterpolaton: dsadvantage: dscontnuous frst dervatves at nodes n general Pecewse cubc nterpolaton: Goal: contnuous st and nd dervatves at nodes C : S y S y S y S y C : S S C : S S 6 constrants 8 degrees of freedom

7 Cubc Splne Interpolaton: Let a < < < < b. A cubc splne nterpolant S for f s a functon that satsfes the followng condtons: ( S s a cubc polynomal on the nterval,, denoted by S, for,,,, ( S y and S y, for,,,, ( S S, for,,,, ( v S S, for,,,, 4 degrees of freedom! boundary condtons: ( a free or natural splne: S S ( b clamped splne: S f and S f for,,,, ( ( ( 3 wrte S a b c d 3 S b c d 6 S c d defne h for,,, : C S a y S a b h c h d h y 3 for,,, C : for,,, 3 S b c d 3 S b c d S b c h 3d h S b b c h 3d h b C : for,,, 6 S c d 6 S c d c 6d h c

8 Summary: ( ( y a,,, ( 3 ( y a bh ch dh,,, ( ( b ch 3 dh b,,, ( ( v c 6d h c,,, From ( v : d d c ( v Substtute ( v nto ( : b b a, c ( v Substtute ( v( v nto ( : c c a, c For,,, 3 3 h c h h c h c a a a a h h a S y a b h c h d h 3 c S c 6d h ( ( { c } ~ equatons for unnowns boundary condtons: ( a free or natural splne: S S c c ( b f (, 3 ( b clamped splne: S f and S f S S b c h d h f ( a free or natural splne: c c c h ( h h h c r h ( h h h c r h- ( h- h h h- ( h- h- h r - c 3 3 r a a a a h h ~ trdagonal matr ~ ( ( ( 3 ( f a a h h c c ( a clamped splne: f a a h h c c h 3 h 3 h ( h h h h ( h h h h- ( h- h h h- ( h- h- h- h 3 h 3 r f a a h r f a a h 3 ~ trdagonal matr ~

9 Tragonal lnear system: A r a a b a a a3 b a3 a33 a34 a, a, a, a, a, a, b a, a, b Gven: Want: an appromaton of f ( Step : fnd the subnterval that belongs to, say, ( ( ( f ( 3 Step : compute S a b c d v f C a b Ma f M S 4 Theorem: Let, wth. If s the unque a b clamped cubc splne nterpolant to f wth respect to the nodes a < < < b. Then 5M Ma f ( S( Ma ( 4 j j a b 384 j cf.. natural splne: 4th order too. Boundary condtons for splne nterpolatons ( atural splne ( f ( f ( Parabolc runout f f f f ( clamped splne b Ths choce mnmzes the value of. a mplyng f f (parabolc endng curves f, f, or f nown f, f, or f nown f d (v Ft a cubc polynomal C( to the four endng ponts. f C f C (v Perodc boundary condton f f f f h f f h f f f f

10 Tenson splne nterpolaton Let a < < < < b. A tenson splne nterpolant S for f s a functon that satsfes the followng condtons: ( S s a cubc polynomal on the nterval,, denoted by S, for,,,, ( S y and S y, for,,,, ( S S, for,,,, ( v S S, for,,,, ( v s a soluton of τ. v S S S ( v s a soluton of τ. v S S S a specal case τ cubc polynomals cubc splne! τ another specal case appromately lnear fttng (stll C ( v S for, τ S S y C S y S z C S z nown unnown Tenson splne nterpolaton { snh[ τ( ] snh[ τ( ]} τ snh ( τ S z z h ( τ ( ( τ ( y z h y z h C S S : α z β β z α z γ γ ( h α h τ snh τ β τcosh τh snh τh h γ τ ( y y h for,,, { } ( equatons for unnowns z S,,,,, Two dmensonal nterpolatons Cartesan Product and Grd Gven: f f, y, M, j j j Soluton: D nterpolaton > two-layer D nterpolatons

11 Frst, nterpolate n one drecton M (, (, M, LM, ( f y f y L L j, j, j( y j (, (, f y f y L y M ( ( ( y y ( yj y j f, y f, y L y f L y ( ( j, j j, j j j M (, f y f L yl j, j M, j Irregular Grd Gven: f f, y, for,,,, Possble soluton: a polynomal to ft and nterpolate. Π m (R the set consstng of all real-coeffcent polynomals wth two varables and y of degree at most m. r If P (, y Π ( R, P (, y C y m m m rs r s m ( m ( m ( m degrees of freedom 3 Wsh f y, P y, m want f, y f P, y for,,,, m unque choce f m m s eample: 3 m m or m (, P y C C C y (, P y C C C y f for,, y C f y C f y C f eample: 6 m m or m (, P y C C C y C C y C y y y y C f y y y C f y y y C f 3 y3 3 3y3 y C 3 f3 4 y C 4 4 4y4 y 4 f4 C f5 5 y5 5 5y5 y 5 Matr nvertable? ot always! Theorem: Interpolaton of arbtray data by the subspace Π(R s possble on a set of (m(m/ nodes f these nodes le on lnes L, L,,L m n such a way that for each L contans eactly ( nodes.

12 Shepard nterpolaton method Wrte P(,y represents a pont n the R space. Q s another pont. Let Φ(P,Q be a real-valued functon on R R whch satsfes Φ(P,Q f and only f P Q ( PQ P Q ( ( y y e.g. Φ, p q p q Defne D Lagrange nterpolaton functon as or wrtten as L L,, ( P (, y Φ Φ ( PP, ( P, P Φ Φ (, y ;, y (, y ;, y L ( P ( Pj, P Φ δ Φ ( P, P, j j (, (, (, f y Py f L y,, P, y f L, y f δ f j j j j j j References: Kncad & Cheny Lancaster & Salausas Curve & Surface Fttng (98