Chapter 5 Curve Fitting

8/8/2019 Chapter 5 Curve Fitting

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Chapter 5 Curve Fitting : Splines5.1 Introduction

Fig. 5.1


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Fig. 5.25.2 Linear Splineso Piecewise interpolating polynomial :

Fig. 5.3 Notation used to drive splines.(n-1) intervals and n data points

o Piecewise linear interpolating funtionsn set data points :

Piecewise linear polynomial in interval:

, i = 1, 2, , n-1


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Fig. 5.4 (a) linear spline, (b) quadratic spline, (c) cubic spline

5.3 Quadratic SplinesPiecewise quadratic polynomial in interval :

(5.5)


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In each interval we need 3 conditions. So total 3(n-1) conditions.1. : (n-1) eqns2. : (n-1) eqns3. : (n-2) eqnsThe 3

rdcondition is not satisfied for , :

4. or : 2 eqnsTotal number of equations : 3(n-1)

By 1st

condition:

(5.6),

By 2nd condition:

(5.7)Let ,

(5.8),

By 3rd condition:

Then(5.9)

i=2,3,,n-1;By 4th condition:


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[Ex 15.2]

1 3.0 2.52 4.5 1.03 7.0 2.54 9.0 0.5

Unknowns (i

b ) :2

stcondition:

3nd

condition :

,,

,

5.4 Piecewise cubic splinen data points

(5-10)


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There are 4(n-1) unknowns. We need 4(n-1) conditions.1)

(5-11)(5-12)

2)

(5-13)3)1st derivative conditions: (n-2).

(5-14)(5-15)

4) 2nd derivative conditions: (n-2).

(5-16), (5-17)

(5-18)(5-18) is substituted in (5-13):

(5-19)(5-18) is substituted in (5-15):

(5-20)(5-23)

Rewrite the equation (5-19):


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(5-21)

(5-22)(5-21) and (5-22) are substituted in (5-23):

(5-24)for

In equation (15-24), there are (n-3) equations and (n-1) unknowns. Hencewe need an extra 2 conditions:

How to give extra conditions?Type:1) natural spline :

( See equation (5-17))In equation (5-24), Take the interior points (i=2, , n-2):

i = 2 :

i = 3 :


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i = n-2 :

Matrix form :

[Ex 5.3]

1 3.0 2.52 4.5 1.03 7.0 2.54 9.0 0.5


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Suppose that ,211

xzx ,21 nn xzx and suppose that ),( 1zf )( 2zf are

known. Then use the extra condition

)()( 11 zfzs , )()( 22 zfzs (not-a-knot interpolation boundary conditions)

Periodic spline: Complete spline:

[ Matlab function ] lookfor spliney=spline(x_nodes, y_nodes, x)

If one uses the statements

pp=spline(x_nodes, y_nodes)

[breaks, coefs, l, k, d]=unmkpp(pp)

yi = interp1(x, y, xi, 'method')

yi = interp1(x, y, xi, 'spline')yi = interp1(x, y, xi, 'linear')yi = interp1(x, y, xi, 'cubic')[Ex] x=0:10;y=sin(x);xi=0:0.25:1yi=interp1(x,y,xi);plot(x,y,'o',xi,yi)


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yi=interp1(x,y,xi.'spline');


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Homework:

Is there a choice of coefficients {a, b,c,} for which the following function is a cubic

spline?

Chapter 5 Curve Fitting

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