Chapter 16 Curve Fitting: Splines. Spline Interpolation zThere are cases where polynomial interpolation is bad zNoisy data zSharp Corners (slope discontinuity)

Chapter 16Chapter 16

Curve Fitting:Curve Fitting:SplinesSplines

Spline InterpolationSpline Interpolation

There are cases where polynomial interpolation is bad

Noisy dataSharp Corners (slope discontinuity)Humped or Flat data

Overshoot

Oscillations

ExampleExample: : ff((xx) = ) = sqrtsqrt((absabs((xx))))

Interpolation at 4, 3, 2, 1, 0, 1, 2, 3, 4

Cubic interpolation

5th-order

7th-order

Linear spline



Idea behind splines

• Use lower order polynomials to connect subsets of data points

• Make connections between adjacent splines smooth

• Lower order polynomials avoid oscillations and overshoots


Use “piecewise” polynomials instead of a single polynomial

Spline -- a thin, flexible metal or wooden lathBent the lath around pegs at the required pointsSpline curves -- curves of minimum strain energyPiecewise Linear InterpolationPiecewise Quadratic InterpolationPiecewise Cubic Interpolation (cubic splines)

Drafting SplineDrafting SplineContinuous function and derivatives

Zero curvatures at end points

SplinesSplines

There are n-1 intervals and n data points

si(x) is a piecewise low-order polynomial

Spline fits of a set of 4 points

Example: Ship LinesExample: Ship Lines

waterline

Bow

Stern

Original databaseOriginal database

Spline interpolation Spline interpolation for submerged hull for submerged hull (below waterline)(below waterline)

1n1n1n1n

iiii

2222

1111

xxbaxs

xxbaxs

xxbaxs

xxbaxs

b is the slope between points

i1i

i1i

i1i

i1ii

ii

xx

ff

xx

xfxfb

fa

Linear SplinesLinear Splines Connect each two points with straight line

functions connecting each pair of points

Linear SplinesLinear Splines

n1n1n322211

nn,33,22,11,

xxI xxI xxI interval

y(xy(xy(xy(x points data

,,,,,,:

),),),),:

x1 x2 x3 x4

si (x) = ai + bi (x xi)


n1-nn

1nn

1n1n

n1n

n

32323

22

32

3

21212

11

21

2

i

xxx , xf xx

xxxf

xx

xx

xxx ,xf xx

xxxf

xx

xx

xxx , xf xx

xxxf

xx

xx

xs

)()(

)()(

)()(

)(

Identical to Lagrange interpolating polynomials

n1n1n322211

nn,33,22,11,



,,,,,,:

),),),),:

Connect each two points with straight line Functions connecting each pair of points are

slope i1i

i1ii xx

ffb

Linear splinesLinear splines

1 1 1 1 1 2

2 2 2 2 2 3

1

1 1 1 1 1

;

;

;

;

i i i i i i

n n n n n n

s x a b x x x x x

s x a b x x x x x

s x a b x x x x x

s x a b x x x x x

Linear splines are exactly the same as linear interpolation!

Example:

586

39

195

7

22

3

40

bxfx i


Problem with linear splines -- not smooth at data points (or knots)

First derivative (slope) is not continuousUse higher-order splines to get continuous

derivativesEquate derivatives at neighboring splinesContinuous functional values and derivatives

Quadratic splines - continuous first derivatives

May have discontinuous second and higher derivatives

Derive second order polynomial between each pair of points

For n points (i=1,…,n): (n-1) intervals & 3(n-1) unknown parameters (a’s, b’s, and c’s)

Need 3(n-1) equations

ii2

ii cxbxaxf

Quadratic SplinesQuadratic Splines

Quadratic SplinesQuadratic Splines

x1 x2 x3 x4 x5

s 1 (x)

s2 (x)s 4

(x)

si(x) = ai + bi (xxi) + ci(xxi)2 3(n-1)3(n-1) unknownsunknowns

Interval 1 Interval 2 Interval 3 Interval 4

f(x1)

f(x2) f(x3)f(x4)

f(x5)

s3 (x)

n1n1n322211

nn,33,22,11,



,,,,,,:

),),),),:

Piecewise Quadratic SplinesPiecewise Quadratic Splines

244444454

233333343

222222232

211111121

xxcxxbaxs xx on

xxcxxbaxs xx on

xxcxxbaxs xx on

xxcxxbaxs xx on

)()()(:,

)()()(:,

)()()(:,

)()()(:,

1. The functional must pass through all the points xi (continuity condition)

2. The function values of adjacent polynomials must be equal at all nodes (identical to condition 1.) 2(n-2) + 2 = 2(n-1)

3. The 1st derivatives are continuous at interior nodes xi, (n-2) 4. Assume that the second derivatives is zero at the first points

2(n1) + (n2) + (1) = 3(n1) 3(n1) equations for 3(n1) unknowns

Example with n = 5

Function must pass through all the points x = xi (2n 2 eqns)

This also ensure the same function values of adjacent polynomials at the interior knots (hi = xi+1 – xi)

Apply to every interval (4 intervals, 8 equations)


2iiiii

2i1iii1iii1i1ii

iiii

hchbfxxcxxbafxs

afxs

)()()(

)(

2444445

2333334

2222223

2111112

44

33

22

11

hchbff

hchbff

hchbff

hchbff

&

fa

fa

fa

fa

Continuous slope at interior knots x = xi (n2 equations)

Apply to interior knots x2 , x3 and x4 (3 equations)

Zero curvatures at x = x1 (1 equation)


5444

4333

3222

bhc2b

bhc2b

bhc2b

0c 0c2xs 1111 )(

1iiii

1i1i1i1ii1iii

1i1i1ii

bhc2b

xxc2bxxc2b

xsxs

)()(

)()(

Example: Quadratic SplineExample: Quadratic Spline

0

0

0

0

ff

f

ff

f

ff

f

ff

f

c

b

a

c

b

a

c

b

a

c

b

a

100

0102h10

0102h10

0102h10

hh0

001

hh0

001

hh0

001

hh0

001

45

4

34

3

23

2

12

1

4

4

4

3

3

3

2

2

2

1

1

1

3

2

1

244

233

222

211

function [A, b] = quadratic(x, f)

% exact solution f(x)=x^3-5x^2+3x+4

% x=[-1 0 2 5 6]; f=[-5 4 -2 19 58];

h1=x(2)-x(1); h2=x(3)-x(2); h3=x(4)-x(3); h4=x(5)-x(4);

f1=f(1); f2=f(2); f3=f(3); f4=f(4); f5=f(5);

A=[1 0 0 0 0 0 0 0 0 0 0 0

0 h1 h1^2 0 0 0 0 0 0 0 0 0

0 0 0 1 0 0 0 0 0 0 0 0

0 0 0 0 h2 h2^2 0 0 0 0 0 0

0 0 0 0 0 0 1 0 0 0 0 0

0 0 0 0 0 0 0 h3 h3^2 0 0 0

0 0 0 0 0 0 0 0 0 1 0 0

0 0 0 0 0 0 0 0 0 0 h4 h4^2

0 1 2*h1 0 -1 0 0 0 0 0 0 0

0 0 0 0 1 2*h2 0 -1 0 0 0 0

0 0 0 0 0 0 0 1 2*h3 0 -1 0

0 0 1 0 0 0 0 0 0 0 0 0];

b=[f1; f2-f1; f2; f3-f2; f3; f4-f3; f4; f5-f4; 0; 0; 0; 0];

Quadratic Spline InterpolationQuadratic Spline Interpolation

>> x=[-1 0 2 5 6]; f=[-5 4 -2 19 58];>> [A,b]=quadratic(x,f)p = A\b -5.0000 9.0000 0 4.0000 9.0000 -6.0000 -2.0000 -15.0000 7.3333 19.0000 29.0000 10.0000

dx = 0.1; z1=x(1):dx:x(2); s1=p(1)+p(2)*(z1-x(1))+p(3)*(z1-x(1)).^2;z2=x(2):dx:x(3); s2=p(4)+p(5)*(z2-x(2))+p(6)*(z2-x(2)).^2;z3=x(3):dx:x(4); s3=p(7)+p(8)*(z3-x(3))+p(9)*(z3-x(3)).^2;z4=x(4):dx:x(5); s4=p(10)+p(11)*(z4-x(4))+p(12)*(z4-x(4)).^2;H=plot(z1,s1,'r',z2,s2,'b',z3,s3,'m',z4,s4,'g');xx=-1:0.1:6; fexact=xx.^3-5*xx.^2+3*xx+4;hold on; G=plot(xx,fexact,'c',x,f,'ro');set(H,'LineWidth',3); set(G,'LineWidth',3,'MarkerSize',8);H1=xlabel('x'); set(H1,'FontSize',15);H2=ylabel('f(x)'); set(H2,'FontSize',15);H3=title('f(x)=x^3-5x^2+3x+4'); set(H3,'FontSize',15);

Quadratic Spline InterpolationQuadratic Spline Interpolation

4

4

4

3

3

3

2

2

2

1

1

1

c

b

a

c

b

a

c

b

a

c

b

a

p

s1(x)

s2(x)

s3(x)

s4(x)Quadratic Spline

Exact function

Cubic splines avoid the straight line and the over-swing

3ii

2iiiiii xxdxxcxxbaf )()()(

Can develop method like we did for quadratic – 4(n–1) unknowns – 4(n–1) equations

interior knot equality end point fixed interior knot first derivative equality assume derivative value if needed

Cubic SplinesCubic Splines

Piecewise Cubic SplinesPiecewise Cubic Splines

x1 x2 x3 xn

s 1 (x

)

s2 (x)s n-1

(x)

44((nn11)) unknownsunknowns

Continuous Continuous slopes and slopes and curvaturescurvatures

xn-1

n1n1n322211

nn,33,22,11,



,,,,,,:

),),),),:

3ii

2iiiiii xxdxxcxxbaf )()()(

Piecewise Cubic SplinesPiecewise Cubic Splines

x0 x1 x2 xn

ssi i (x)(x) - piecewise cubic polynomials - piecewise cubic polynomials

ssii’(x)’(x) - piecewise quadratic polynomials (slope) - piecewise quadratic polynomials (slope)

ssii”(x) ”(x) - piecewise linear polynomials (curvatures)- piecewise linear polynomials (curvatures)

xn-1

s 1” (x)

s2”(x)

s3 ”(x)

S n-1” (x

)

Reduce to (n-1) unknowns and (n-1) equations for si’’

n1n1n322211

nn,33,22,11,



,,,,,,:

),),),),:

Cubic SplinesCubic Splines Piecewise cubic polynomial with continuous

derivatives up to order 2

1. The function must pass through all the data points

gives 2(n-1) equations

2 3

2 31 1 1 1 1 1

: ( ) ( ) ( ) ( )

: ( ) ( ) ( ) ( )

i i i i i i i i i i i i i i i

i i i i i i i i i i i i i i

x x s x f a b x x c x x d x x a

x x s x f a b x x c x x d x x

2 31

i i

i i i i i i i i

a f

f b h c h d h f

i = 1,2,…, n-1

Cubic SplinesCubic Splines2. First derivatives at the interior nodes must be equal:

(n-2) equations

3. Second derivatives at the interior nodes must be equal:

(n-2) equations

1i2iiiii

1i1i1ii

2iiiiii

bhd3hc2b

xsxs

xxd3xxc2bxs

)()(

)()()(

1iiii

1i1i1ii

iiii

chd3c

xsxs

xxd6c2xs

)()(

)()(


4. Two additional conditions are needed (arbitrary)

The last two equations

Total equations: 2(n-1) + (n-2) + (n-2) +2 = 4(n-1)

1n1n1nn1n

111

hd6c20xs

c20xs

)(

)(

0hd3cc

0c

0xs

0xs

1n1n1nn

1

n1n

11

)(

)(


3ii

2iiiiii xxdxxcxxbaxs )()()()(

Solve for (ai, bi, ci, di) – see textbook for details

Tridiagonal system with boundary conditions c1 = cn= 0

]),[],[(

)(

1iii1i

1i

1ii

i

i1i1iiii1i1i1i

xxfxxf3

h

ff3

h

ff3chchh2ch

1iii

i

i1ii

i

i1iiii cc2

3

h

h

ffb

h3

ccd fa

,,


0

xxfxxf3

xxfxxf3

xxfxxf3

0

c

c

c

c

c

1

hhh2h

hhh2h

hhh2h

1

2n1n1nn

2334

1223

n

1n

3

2

1

1n1n2n2n

3322

2211

],[],[

],[],[

],[],[

)(

)(

)(

Tridiagonal matrix

Hand CalculationsHand Calculations

f(4)estimate 581924)x(f

6520x

0f(4) ,4x3x5x f(x) :solution Exact 23

391

1958

h

ffxxf

73

219

h

ffxxf

32

42

h

ffxxf

58f 19f 2f 4f

156xxh 325xxh 202xxh

3

3434

2

2323

1

1212

4321

343232121

],[

)(],[

],[

,,,

,,

Hand CalculationsHand Calculations

0

96

30

0

0

7393

373

0

c

c

c

c

1

183

3102

1

0

xxfxxf3

xxfxxf3

0

c

c

c

c

1

hhh2h

hhh2h

1

4

3

2

1

2334

1223

4

3

2

1

3322

2211

)(

],[],[

],[],[

)(

)(

can be further simplified since c1 = c4 = 0 (natural spline)

25352112

6760560

c

c

96

30

c

c

83

310

3

2

3

2

.

.

Cubic Spline InterpolationCubic Spline Interpolation

1iii

i

i1ii

i

i1iiii

3ii

2iiiiii

cc23

h

h

ffb

h3

ccd fa

xxdxxcxxbaxs

,,

)()()()(

08309863cc23

h

h

ffb

9014083cc23

h

h

ffb

5492962cc23

h

h

ffb

08450704h3

ccd

43661971h3

ccd

1126760h3

ccd

0c

25352112c

6760560c

0c

58fa

19fa

2fa

4fa

433

3

343

322

2

232

211

1

121

3

343

2

232

1

121

4

3

2

1

44

33

22

11

.)(

.)(

.)(

.

.

.

.

.


1iii

i

i1ii

i

i1iiii

3ii

2iiiiii

cc23

h

h

ffb

h3

ccd fa

xxdxxcxxbaxs

,,

)()()()(

32

333

2333333

32

322

2222222

3

311

2111111

5x084507045x253521125x8309863019

xxdxxcxxbaxs

2x4366197212x67605602x90140832

xxdxxcxxbaxs

x1126760x54929624

xxdxxcxxbaxs

)(.)(.)(.

)()()()(

)(.)(.)(.

)()()()(

..

)()()()(

Piecewise cubic splines (cubic polynomials)


The exact solution is a cubic function Why cubic spline interpolation does not give the exact

solution for a cubic polynomial?

Because the conditions on the end knots are different! In general, f (x0) 0 and f (xn) 0 !!

014114s4f :ioninterpolatspline Cubic

0f(4) 4x3x5x f(x) :solution Exact

2

23

.)()(

,


>> xx=[-1 0 2 5 6]; f = [-5 4 -2 19 58];>> Cubic_spline(xx,f);

Resulting piecewise function:s1 =(-5.000000)+(11.081218)*(x-(-1.000000))s2 =(0.000000)*(x-(-1.000000)).^2+(-2.081218)*(x-(-1.000000)).^3

Resulting piecewise function:s1 =(4.000000)+(4.837563)*(x-(0.000000))s2 =(-6.243655)*(x-(0.000000)).^2+(1.162437)*(x-(0.000000)).^3

Resulting piecewise function:s1 =(-2.000000)+(-6.187817)*(x-(2.000000))s2 =(0.730964)*(x-(2.000000)).^2+(1.221658)*(x-(2.000000)).^3

Resulting piecewise function:s1 =(19.000000)+(31.182741)*(x-(5.000000))s2 =(11.725888)*(x-(5.000000)).^2+(-3.908629)*(x-(5.000000)).^3

(i = 1)

(i = 2)

(i = 3)

(i = 4)

3ii

2iiiiii xxdxxcxxbaxs )()()()(


Piecewise cubic Continuous slope Continuous curvature

zero curvatures at end knots

exact solution

06f )(

01f )(

Natural spline : zero second derivative (curvature) at end points Clamped spline: prescribed first derivative (clamped) at end points Not-a-Knot spline: continuous third derivative at x2 and xn-1

End Conditions of SplinesEnd Conditions of Splines

Comparison of Runge’s function (dashed red line) with a 9-point not-a-knot spline fit generated with MATLAB (solid green line)

>> x = linspace(-1,1,9); y = 1./(1+25*x.^2);>> xx = linspace(-1,1,100); yy = spline(x,y,xx);>> yr=1./(1+25*xx.^2);>> H=plot(x,y,'o',xx,yy,xx,yr,'--');

Continuous third derivatives

at x2 and xn-1

Note that first derivatives of 1 and 4 are specified at the left and right boundaries, respectively.

>> yc = [1 y –4];% 1 and -4 are the 1st-order derivatives (or slopes)at 1st & last point, respectively.>> yyc = spline(x,yc,xx);>> >> H=plot(x,y,'o',xx,yyc,xx,yr,'--');

Clamped End Spline - Use 11 values including slopes at end pointsClamped End Spline - Use 11 values including slopes at end points

MATLAB FunctionsMATLAB Functions

One-dimensional interpolationsyy = spline (x, y, xx)

yi = interp1 (x, y, xi, ‘method’)yi = interp1 (x, y, xi, ‘linear’) yi = interp1 (x, y, xi, ‘spline’) – not-a-knot splineyi = interp1 (x, y,, xi, ‘pchip’) – cubic Hermite yi = interp1 (x, y,, xi, ‘cubic’) – cubic Hermiteyi = interp1 (x, y,, xi, ‘nearest’) – nearest neighbor

Piecewise polynomial interpolation on velocity time series for an automobile

MATLAB FunctionMATLAB Function: : interp1interp1

>> x=[1 2 4 7 9 10]

x =

1 2 4 7 9 10

>> y=[3 -2 5 4 9 2]

y =

3 -2 5 4 9 2

>> xi=1:0.1:10;

>> y1=interp1(x,y,xi,'linear');

>> y2=interp1(x,y,xi,'spline');

>> y3=interp1(x,y,xi,'cubic');

>> plot(x,y,'ro',xi,y1,xi,y2,xi,y3,'LineWidth',2,'MarkerSize',12)

>> print -djpeg spline00.jpg

Spline InterpolationsSpline Interpolations

Spline InterpolationsSpline Interpolations

Linear

Not-a-knot

Cubic

MATLAB’s FunctionsMATLAB’s Functions

Two-dimensional interpolationszi = interp2 (x, y, z, xi, yi)

))(),((),(

),(,),,(),,(

),(,),,(),,(

),(,),,(),,(

jyixfjiz

yxyxyx

yxyxyx

yxyxyx

mn2n1n

m22212

m12111

MATLAB FunctionMATLAB Function: : interp2interp2

>> [x,y,z]=peaks(100); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);>> zi = interp2(x,y,z,xi,yi); surf(xi,yi,zi)>> print -djpeg075 peaks1.jpg>> [x,y,z]=peaks(10); [xi,yi]=meshgrid(-3:0.1:3,-3:0.1:3);>> zi = interp2(x,y,z,xi,yi); surf(xi,yi,zi)>> print -djpeg075 peaks2.jpg

100 100 data base 10 10 data base

CVEN 302-501CVEN 302-501Homework No. 11Homework No. 11

Chapter 16 Problem 16.1(a) (30) – Hand Calculation

Problem 16.1(b) (20)– MATLAB program not-a-knot spline: yy = spline(x, y, xx)

Problem 16.1(c)(20) – MATLAB program

Chapter 19 Problem 19.2 (20)– Hand Calculation

Problem 19.4 (20)– Hand Calculation

Due on Wed. 11/05/08 at the beginning of the periodDue on Wed. 11/05/08 at the beginning of the period

Documents

Chapter 16 Curve Fitting: Splines. Spline Interpolation zThere are cases where polynomial interpolation is bad zNoisy data zSharp Corners (slope discontinuity)