Approximation by Splines →Motivation ●Linear Splines ●Quadratic Splines ●Cubic Splines ●Summary Copyright©2011NA⊙Yin 1
Approximation by Splines ⇒ Motivation • Linear Splines • Quadratic Splines • Cubic Splines • Summary Copyright c 2011 NA Yin 1
Motivation ● -40 -4-2 0 2 Given:set of many points,or perhaps very involved function Want:simple representative function for analysis or manufacturing Any suggestions? Copyright©2011NA⊙Yin 2
Motivation • Given:set of many points,or perhaps very involved function • Want:simple representative function for analysis or manufacturing Any suggestions? Copyright c 2011 NA Yin 2
Let's Try Interpolation -40 -4 -2 0 Disadvantages: Values outside x-range diverge quickly (interp(10)=-1592) Numerical instabilities of high-degree polynomials Copyright©2011NA⊙Yin 3
Let’s Try Interpolation Disadvantages: • Values outside x−range diverge quickly (interp(10)=-1592) • Numerical instabilities of high-degree polynomials Copyright c 2011 NA Yin 3
Runge Function-Two Interpolations 1 uniform Chebyshev anjen 8.7650、c3c2C0 (=-5) (x4=c4) (=5) More disadvantages: .Within x-range,often high oscillations Even Chebyshev points=often uncharacteristic oscillations Copyright©2011NA⊙Yin 4
Runge Function – Two Interpolations More disadvantages: • Within x−range,often high oscillations • Even Chebyshev points ⇒ often uncharacteristic oscillations Copyright c 2011 NA Yin 4
Splines Given domain [a,b],a spline S(x) Is defined on entire domain Provides a certain amount of smoothness 3partition of"knots"(=where spline can change form) {a to,t1,t2;...:tn=bh such that So(x),x∈[to,t], S(x)= S(x),x∈[t1,t2], Sn-1(x),x∈[tn-1,tn] is piecewise polynomial Copyright 2011 NAOYin 5
Splines Given domain [a, b],a spline S(x) • Is defined on entire domain • Provides a certain amount of smoothness • ∃ partition of ”knots”(=where spline can change form) {a = t0, t1, t2, . . . , tn = b} such that S(x) = S0(x), x ∈ [t0, t1], S1(x), x ∈ [t1, t2], ... Sn−1(x), x ∈ [tn−1, tn] is piecewise polynomial Copyright c 2011 NA Yin 5
Interpolatory Splines Note:splines split up range [a, *opposite of CTR-→CSR→GQ development ."Spline"implies no interpolation,not even any y-values ●If given points {(to,o),(t1,y1),(t22),…,(tn,yn)} "interpolatory spline"traverses these as well Splines=nice,analytical functions Copyright©2011NA⊙Yin 6
Interpolatory Splines • Note:splines split up range [a, b] ∗ opposite of CTR→CSR→GQ development • ”Spline” implies no interpolation,not even any y−values • If given points {(t0, y0),(t1, y1),(t2, y2), . . . ,(tn, yn)} ”interpolatory spline” traverses these as well Splines=nice,analytical functions Copyright c 2011 NA Yin 6
Approximation by Splines ●Motivation →Linear Splines ●Quadratic Splines ●Cubic Splines ●Summary Copyright 2011 NAOYin 7
Approximation by Splines • Motivation ⇒ Linear Splines • Quadratic Splines • Cubic Splines • Summary Copyright c 2011 NA Yin 7
Linear Splines Given domain a,b,a spline S(x) Is defined on entire domain .Provides continuity,i.e.,is Co[a,b] partition of "knots" ta=to,ti,t2,...,tn=b} such that S(x)=ax+b∈P([t,ti+1]),i=0,,n-1 Recall:no y-values or interpolation yet Copyright©2011NA⊙Yin 8
Linear Splines Given domain [a, b],a spline S(x) • Is defined on entire domain • Provides continuity,i.e.,is C 0 [a, b] • ∃ partition of ”knots” {a = t0, t1, t2, . . . , tn = b} such that Si(x) = aix + bi ∈ P1([ti , ti+1]), i = 0, . . . , n − 1 Recall:no y−values or interpolation yet Copyright c 2011 NA Yin 8
Linear Splines Examples undefined part discontinuous nonlinear part linear spline a Definition outside of a,b is arbitrary Copyright 2011 NAOYin g
Linear Splines – Examples • Definition outside of [a, b] is arbitrary Copyright c 2011 NA Yin 9
Interpolatory Linear Splines ●Given points {(to,y0),(t1,y1),(t2,y2,,(tn,yn)} spline must interpolate as well .Are the Si(z)(with no additional knots)unique? Coefficients:aix+bi,i=0,...,n-1 =total=2n Conditions:2 prescribed interpolation points for Si(),i=0,...,n-1(includes continuity condition)=total=2n ●Obtain S(）=ax+(5-at,a=班+1-，i=0,.,n-1 titi-ti Copyright©2011NA⊙Yin 10
Interpolatory Linear Splines • Given points {(t0, y0),(t1, y1),(t2, y2), . . . ,(tn, yn)} spline must interpolate as well • Are the Si(x)(with no additional knots)unique? ∗ Coefficients:aix + bi , i = 0, . . . , n − 1 ⇒total= 2n ∗ Conditions:2 prescribed interpolation points for Si(x), i = 0, . . . , n − 1(includes continuity condition)⇒total= 2n • Obtain Si(x) = aix + (yi − aiti), ai = yi+1 − yi ti+1 − ti , i = 0, . . . , n − 1 Copyright c 2011 NA Yin 10