Chapter 15 Chapter 15 Curve Fitting : SplinesCurve Fitting : Splines

Gab Byung ChaeGab Byung Chae

Oscillation in a higher order Oscillation in a higher order interpolationinterpolation

The alternative : splineThe alternative : spline

Notation used to derive splinesNotation used to derive splinesn-1 intervals in n data points

Splines in 1Splines in 1stst, 2, 2ndnd, 3, 3rdrd order orderLinear splineLinear spline

Quadratic splineQuadratic spline

Cubic splineCubic spline

15.2 Linear Splines15.2 Linear Splineslinear splines passing thrulinear splines passing thru

Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)

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15.3 Quadratic Splines15.3 Quadratic SplinesThe objective in quadratic splines is to derive The objective in quadratic splines is to derive a second-order polynomial for each interval a second-order polynomial for each interval between data points between data points

The polynomial :The polynomial :

ssii(x) =a(x) =ai i + b+ bii(x-x(x-xii)+ c)+ cii(x-x(x-xii))22

For n data points, there are n-1 intervals and, For n data points, there are n-1 intervals and, consequently, 3(n-1) unknown constants to consequently, 3(n-1) unknown constants to evaluate. evaluate. 3(n-1) equations or conditions 3(n-1) equations or conditions are required to evaluate the unknowns.are required to evaluate the unknowns.

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,1 iii xxh Let for all i=1,,..,n-1 1. Continuity condition : the function must pass through all the points.

2. The function values of adjacent polynomials must be equal at the knots.

for all i=1,,..,n-1 : (n-1)

equations

for all i=1,,..,n-1 : (n-1)

equations

--- (15.8)

--- (15.6)

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Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)Example : (3,2.5),(4.5,1),(7,2.5),(9,.5) -> (5,?)

,1 iii xxh Let for all i=1,,..,n-1 3. The first derivatives at the interior nodes must be equal.

4. Assume that the second derivative is zero at the first point.

--- (15.9)

Example 15.2 Example 15.2 Problem] Fit quadratic splines to the data Problem] Fit quadratic splines to the data in Table 15.1. Use the results to estimate in Table 15.1. Use the results to estimate the value at x=5.the value at x=5.

Solution] Equation (15.8) is written for i=1 Solution] Equation (15.8) is written for i=1 through 3 (with cthrough 3 (with c11=0) to give=0) to give

ff11+ b+ b11hh11 = f = f22

ff22+ b+ b22hh22 + c + c22hh2222 = f = f33

ff33+ b+ b33hh33 + c + c33hh3322 = f = f44

Equation (15.9) creates 2 conditions with Equation (15.9) creates 2 conditions with

cc11=0 :=0 :

bb11 = b = b22

bb22+ 2c+ 2c22hh22 = b = b33

ff1 1 = 2.5 h= 2.5 h1 1 = 4.5-3.0 = 1.5= 4.5-3.0 = 1.5

ff2 2 = 1.0 h= 1.0 h2 2 = 7.0-4.5 = 2.5= 7.0-4.5 = 2.5

ff3 3 = 2.5 h= 2.5 h3 3 = 9.0-7.0 = 2.0 = 9.0-7.0 = 2.0

ff4 4 = 0.5= 0.5

2.5 + 1.5 b1 = + 1.5 b1 = 1.01.0 1.0 + 2.5 b2 + 6.25 c2 1.0 + 2.5 b2 + 6.25 c2 = 2.5= 2.5 2.5 + 2.0 b3 + 4c3 2.5 + 2.0 b3 + 4c3 = 0.5= 0.5 b1 - b2 = b1 - b2 = 00 b2 - b3 + 5c2b2 - b3 + 5c2 = = 00

bb11 = -1 = -1

bb2 2 = -1 c= -1 c2 2 = 0.64= 0.64

bb33= 2.2 c= 2.2 c3 3 = -1.6= -1.6

ss11(x) = 2.5 –(x-3)(x) = 2.5 –(x-3)

ss22(x) = 1.0 –(x-4.5)+0.64(x-4.5)(x) = 1.0 –(x-4.5)+0.64(x-4.5)22

ss33(x) = 2.5 + 2.2(x-7.0)-1.6(5-7.0)(x) = 2.5 + 2.2(x-7.0)-1.6(5-7.0)22

x=5 x=5 이므로 이므로 ss22(5) = 1.0 –(5-4.5)+0.64(5-4.5)(5) = 1.0 –(5-4.5)+0.64(5-4.5)2 2 =0.66=0.66

15.4 Cubic Splines15.4 Cubic Splines

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Solving Cubic SplinesSolving Cubic Splines

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Example 15.3 Example 15.3 Fit cubic splines to the data . Utilize the results to estimate the value at x =5

Solution : employ Eq.(15.27)Solution : employ Eq.(15.27)

ff1 1 = 2.5 h= 2.5 h1 1 = 4.5-3.0 = 1.5= 4.5-3.0 = 1.5

ff2 2 = 1.0 h= 1.0 h2 2 = 7.0-4.5 = 2.5= 7.0-4.5 = 2.5

ff3 3 = 2.5 h= 2.5 h3 3 = 9.0-7.0 = 2.0 = 9.0-7.0 = 2.0

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Using MATLAB the results areUsing MATLAB the results are

cc11 = 0 c = 0 c22 = 0.839543726 = 0.839543726

cc33 = -0.766539924 c = -0.766539924 c44 = 0 = 0

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Using Eq. (15.21) and (15.18), Using Eq. (15.21) and (15.18),

bb11 = -1.419771863 d = -1.419771863 d11 = 0.186565272 = 0.186565272

bb22 = -0.160456274 d = -0.160456274 d22 = -0.214144487 = -0.214144487

bb33 = 0.022053232 d = 0.022053232 d33 = 0.127756654 = 0.127756654

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15.4.2 End Conditions15.4.2 End ConditionsClamped End Condition : Specifying the Clamped End Condition : Specifying the first derivatives at the first and last nodes. first derivatives at the first and last nodes.

Not-a-Knot End Condition : Force Not-a-Knot End Condition : Force continuity of the third derivative at the continuity of the third derivative at the second and the next-to-last knots. – same second and the next-to-last knots. – same cubic functions will apply to each of the cubic functions will apply to each of the first and last two adjacent segments, first and last two adjacent segments,

15.5 Piecewise interpolation in 15.5 Piecewise interpolation in MatlabMatlab

15.5.1 MATLAB function : spline15.5.1 MATLAB function : spline

Syntax :Syntax :

yy = spline(x, y, xx)yy = spline(x, y, xx)

Example 15.4 Splines in Example 15.4 Splines in MATLABMATLAB

Use MATLAB to fit 9 equally spaced data points Use MATLAB to fit 9 equally spaced data points sampled from this function in the interval [-1, 1].sampled from this function in the interval [-1, 1].

a) a not-a-knot spline b) a clamped spline with a) a not-a-knot spline b) a clamped spline with end slopes of fend slopes of f11’=1 and f’’=1 and f’n-1n-1 = -4 = -4

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Solution :Solution :

a)a)

>> x = linspace(-1,1,9);>> x = linspace(-1,1,9);

>> y = 1./(1+25*x.^2);>> y = 1./(1+25*x.^2);

>> xx = linspace(-1,1);>> xx = linspace(-1,1);

>> yy = spline(x,y,xx); >> yy = spline(x,y,xx);

>> yr = 1./(1+25*xx.^2);>> yr = 1./(1+25*xx.^2);

>> plot(x,y,’o’,xx,yy,xx,yr,’—’)>> plot(x,y,’o’,xx,yy,xx,yr,’—’)

Cubic spline of the Runge Cubic spline of the Runge functionfunction

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b)b)

>> yc =[1 y -4];>> yc =[1 y -4];

>> yyc =spline(x,yx,xx);>> yyc =spline(x,yx,xx);

>> plot(x,y,’o’,xx,yyc,xx,yr,’—’)>> plot(x,y,’o’,xx,yyc,xx,yr,’—’)

15.5.2 MATLAB function : interp115.5.2 MATLAB function : interp1

Implement a number of different types of Implement a number of different types of piecewise one-dimensional interpolation. piecewise one-dimensional interpolation.

Syntax :Syntax :

yi = interp1(x,y,xi, ‘method’)yi = interp1(x,y,xi, ‘method’)

yi = a vector containing the results of the yi = a vector containing the results of the interpolation as evaluated at the points in interpolation as evaluated at the points in the vector xi , and ‘method’ = the desired the vector xi , and ‘method’ = the desired method. method.

methodmethodNearest – nearest neighbor interpolation. Nearest – nearest neighbor interpolation.

Linear – linear interpolationLinear – linear interpolation

Spline – piecewise cubic spline Spline – piecewise cubic spline interpolation(identical to the spline interpolation(identical to the spline function)function)

pchip and cubic – piecewise cubic Hermite pchip and cubic – piecewise cubic Hermite interpolation.interpolation.

The default is linear.The default is linear.

MATLAB’s methodsMATLAB’s methods

>> yy=spline(x,y,xx); >> yy=spline(x,y,xx); cubic spilnecubic spilne

>> yy=interp1(x,y,xx): >> yy=interp1(x,y,xx): (piecewise) linear spline (piecewise) linear spline

>> yy=interp1(x,y,xx,’spline’)>> yy=interp1(x,y,xx,’spline’): cubic spline: cubic spline

>> yy=interp1(x,y,xx,’nearest’>> yy=interp1(x,y,xx,’nearest’) : nearest ) : nearest neighborneighbor methodmethod

>> >> yy=interp1(x,y,xx,’pchip’)yy=interp1(x,y,xx,’pchip’): piecewise cubic : piecewise cubic Hermite interpolationHermite interpolation

>> zz=interp2(x,y,z,xx,yy)>> zz=interp2(x,y,z,xx,yy): 2 dimensional : 2 dimensional interpolationinterpolation

Example 15.5 Trade-offs Using Example 15.5 Trade-offs Using interpl interpl

Linear, nearest, cubic spline, piecewise Linear, nearest, cubic spline, piecewise cubic Hermite interpolationcubic Hermite interpolation

>> t = [ 0 20 40 56 68 80 84 96 104 110];>> t = [ 0 20 40 56 68 80 84 96 104 110];

>> v = [0 20 20 38 80 80 100 100 125 125];>> v = [0 20 20 38 80 80 100 100 125 125];

>> tt = linspace(0,110);>> tt = linspace(0,110);

>> vl=interp1(t,v,tt); %(piecewise) linear spline >> vl=interp1(t,v,tt); %(piecewise) linear spline

>> plot(t,v,’o’,tt,vl)>> plot(t,v,’o’,tt,vl)

>> vn=interp1(t,v,tt,’nearest’); % nearest>> vn=interp1(t,v,tt,’nearest’); % nearest

>> plot(t,v,’o’,tt,vn)>> plot(t,v,’o’,tt,vn)

>> vs=interp1(t,v,tt,’spline’); % cubic spline>> vs=interp1(t,v,tt,’spline’); % cubic spline

>> plot(t,v,’o’,tt,vs)>> plot(t,v,’o’,tt,vs)

>> vh=interp1(t,v,tt,’pchip’); % piecewise cubic >> vh=interp1(t,v,tt,’pchip’); % piecewise cubic Hermite interpolationHermite interpolation

>> plot(t,v,’o’,tt,vh)>> plot(t,v,’o’,tt,vh)