MA4229 Lectures 15, 16Week 13 Nov 1,2 2010

Chapter 18 Interpolation by piecewise polynomials

Interlacing Roots of Orthog. Poly.http://en.wikipedia.org/wiki/Orthogonal_polynomials

http://en.wikipedia.org/wiki/Orthogonal_polynomials#Interlacing_of_roots

Disadvantages of Polynomial Approx.

Question Discuss other types of approximation.

Question Discuss how polynomial approximation byinterpolation works.

Question What are some of its disadvantages?

Question What are some of their disadvantages?

Spaces of Piecewise Polynomials

Example 1.Piecewise linear

is a piecewise polynomial ofA function

bxxa m 1if the restrictions

],[ baCsdegree n with respect to data points

1,...,1,| ],[ 1

mjPs nxx jj

Rcxxcxss jjjjxx jj

),()(| ],[ 1

)()()( 11 jjjjj xxcxsxs

)/())()(( 11 jjjjj xxxsxsc

and are determined by the values )(),...,( 1 mxsxs

Spaces of Piecewise PolynomialsExample 2. Piecewise cubic for

1,...,1,| 3],[ 1

mjPs

jj xx

22 mj

m4

determined from values at four consecutive data points

for use data points

211 ,,, jjjj xxxxfor 1j use data points 4321 ,,, xxxxfor 1mj use data points mmmm xxxx ,,, 123

Spaces of Piecewise PolynomialsExample 3. Piecewise cubic for

1,...,1,| 3],[ 1

mjPs

jj xx

m4

by the values of

)(),(),(),( 11 jjjj xsxsxsxsdetermined

at five consecutive points near

Question What is the smoothness of the functions in

each of these three spaces of piecewise polynomials ?

For each )( jxsmj ,...,1 the value is determined

q where 4Pq interpolates sjx

Then are

Interpolation by Piecewise Polynomials

in any of these spaces of piecewise polynomials is

of a function by piecewise polynomials

obtained by setting

],[ baCf

mjxfxs jj ,...,1),()( This provides three interpolation methods that are

Question What is the accuracy of each method ?

described in detail in Section18.1 in Jackson’s book.

Cardinal BasesEach of these three spaces admits a basis consisting of

functions mjj ,...,1, that satisfy

in one of these spaces satisfiesAny function

jiij x )( ji if 1ji if 0

sFigure 18.1 shows cardinal functions for each space.

m

j jjxss1

)( Question How does this compare with the Lagrange basis for nP described by equation (4.3) on page 33 ?

Interpolation Using Cardinal Basesof a function ],[ baCf easy. The function

at the points

Question Derive an upper bound for the operator norm

is the unique function in the space that interpolates

m

j jjxffX1

)(

f

mxx ,...,1

of the interpolation operator X (with respect to the

norm |||| ) using these cardinal functions.

Suggestion Review Section 4.4.

Spline Spaces

,11 ba n For a set of knots

Read Section 4.4 Piecewise poly. Approx., pp 28-31

and integer aa n ,0Set.0k and define

}1,...,0,|:],[{ ],[)1(

1

njPsbaCs kk

jj

),...,,,( 10 nkS the spline space

of spline functions of degree k with knots 11,..., n

Spline Spaces},0max{)( jj xx

Question Show that

Define

})()({ 1

1!1

0

n

jk

jjkk

jj

j xdxcxs

),...,,,( 10 nkS

where the coefficients .,...,,,..., 110 Rddcc nk

Suggestion : show that .1,...,1,)( njds jjk

Question What is ?),...,,,(dim 10 nkS

Cubic Spline InterpolationLet ),...,,,3( 10 nSs

so it has the form

33

22 )()())(()( jjjjj xcxcxsf

Question Show that

3],[ 1| Ps

jj

211 )/())()((6 jjjj ff

interpolate ],[ baCf

)/())(4)(2( 11 jjjj ss

321''

],[ 32)()|(1

ccs jjj

Cubic Spline InterpolationQuestion Why does

Question Use these equations to derive equation 18.11

?1,...,1),()|()()|( 1''

],[1''

],[ 211

mjss jj jjjj

on page 216 of Jackson.

Question Show that if the knot spacings are equal,

,1,...,1,1 mjhjj then

,1,...,1)),()(()()(4)( 113

11 mjffsss jjhjjj

This is the three term recursion for cubic splines.

Cardinal Cubic SplinesNow we consider an infinite number of uniformly

cubic splines that satisfy

Then

Zjjhj ,

))()(()()(4)( 113

11 kjkjhkjkjkj

Question Derive equation 18.18 on page 217

spaced knots and construct cardinal

Zjjhj , jiij )(

)( 1,1,3

kjkjh 1,3 jkh

1,3 jkh}1,1{,0 jjk

Question Why must 0 if j is bounded ?

Question Why does 0 imply )(xj decaysdecays exponentially.

Tutorial 8 Due Tuesday 9 NovemberAnswer all questions in these vufoils and write your

Read Section 18.3 and prepare to explain the material

answers up in sufficient detail to present them tothe class during the tutorial session.

during the tutorial session.