l08 interpolation

7/30/2019 L08 Interpolation

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Interpolation

- Linear Interpolation

- Quadratic Interpolation

- Polynomial Interpolation

- Piecewise Polynomial Interpolation

Numerical Methods forNumerical Methods forCivilCivilEngineersEngineers

Lecture 8

ByBy MongkolMongkol JIRAVACHARADETJIRAVACHARADET

School of Civil EngineeringSchool of Civil EngineeringSuranareeSuranaree University of TechnologyUniversity of Technology


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Visual Interpolation

Vehicle speed is approximately 49 km/h

0

20

4060

80

100

120

km/h


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BASIC IDEAS

From the known data ( xi , yi ), interpolate ( ) for iy F x x x=

Determining coefficient a1, a2, . . . , an of basis function (x)

F(x) = a11(x) + a22(x) + . . . + ann(x)

Polynomials are often used as the basis functions.

F(x) = a1 + a2x + a3x2 + . . . + an x

n-1


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Interpolation v.s. Curve Fitting

known datay

x

interpolation

curve fit

Curve fitting: fit function & data not exactly agree

Interpolation: function passes exactly through known data


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Interpolation & Extrapolation

Interpolation approximate within the range of independent variable

of the given data set.

Extrapolation approximate outside the range of independent variable

of the given data set.

xx1 x2

y


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Linear Interpolation

Connect two data points with a straight line

1 0 1 0

0 1 0

( ) ( ) ( ) ( )x f x f x f x

x x x x

=

Using similar triangles:

x0 x1

f(x0)

f(x1)

x

f1(x)

1 01 0 0

1 0

( ) ( )( ) ( ) ( )

f x f xf x f x x x

x x

= +


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Quadratic Interpolation

Second-order polynomial interpolation using 3 data points

Convenient form:

2 0 1 0 2 0 1( ) ( ) ( )( )f x b b x x b x x x x= + + 1

2

2 0 1 1 0 2 2 0 1 2 0 2 1( )f x b b x b x b x b x x b xx b xx= + + +

2

2 0 1 2( )f x a a x a x= + +

where 0 0 1 0 2 0 1

1 1 2 0 2 1

2 2

a b b x b x xa b b x b x

a b

= +=

=


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Example: Quadratic Interpolation atx= 2

0

1

2

1

4

6

x

x

x

==

=

0

1

2

( ) 0

( ) 1.3863

( ) 1.7918

f x

f x

f x

==

=

Solution:

0

1

2

0

1.3863 00.4621

4 1

1.7918 1.38630.46216 4 0.05187

6 1

b

b

b

=

= =

= =


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Substitute b0, b1 and b2 into equation

f2(x) = 0 + 0.4621(x - 1) - 0.05187(x - 1)(x - 4)

which can be evaluated atx= 2 for

f2(2) = 0 + 0.4621(2 - 1) - 0.05187(2 - 1)(2 - 4)

f2(2) = 0.5658


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Polynomial Interpolation

FindingPn-1(x) of degree n-1 that passes through n known data pairs

Pn-1(x) = c1xn-1 + c2x

n-2 + . . . + cn-1x + cn

Vandermonde Systems n pairs of (x, y)

n equations

n unknowns

Polynomial pass through each of data points


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Example: Construct a quadratic interpolating function

y = c1x2 + c2x + c3

that pass through (x, y) support points (-2, -2), (-1, 1), and (2, -1)

Substitute known points into equation:

- 2 = c1 (-2)2 + c2 (-2) + c3

1 = c1 (-1)2 + c2 (-1) + c3

- 1 = c1 (2)2 + c2 (2) + c3

Rewritten in matrix form:

1

2

3

4 2 1 2

1 1 1 1

2 2 1 1

c

c

c

=


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Vandermonde Matrix

2

1 1 1 1

2

2 2 2 2

2

3 3 3 3

1

1

1

x x c y

x x c y

x x c y

=

MATLAB statements:

>> x = [-2 -1 2];

>> A = [x.^2 x ones(size(x))];

or use the built-in vander function>> A = vander([-2 -1 2]);

>> y = [-2 1 -1];

>> c = A\y

c =-0.9167

0.2500

2.1667


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Polynomials Wiggle

2nd-order

3rd-order

4th-order

5th-order

y

x

xi 1 2 3 4 5 6 7 8 9 10

yi 3.5 3.0 2.5 2.0 1.5 -2.4 -2.8 -3.2 -3.6 -4.0


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MATLABs Command Lines to Demonstrate Polynomials Wiggle

>> x = [1 2 3 4 5 6 7 8 9 10];

>> y = [3.5 3.0 2.5 2.0 1.5 -2.4 -2.8 -3.2 -3.6 -4.0];

>> x0 = 1:0.1:10;

>> y2 = polyval(polyfit(x(4:6), y(4:6), 2), x0);>> y3 = polyval(polyfit(x(4:7), y(4:7), 3), x0);

>> y4 = polyval(polyfit(x(3:7), y(3:7), 4), x0);

>> y5 = polyval(polyfit(x(3:8), y(3:8), 5), x0);

>> axis([0 10 -5 5])

>> plot(x, y, o)

>> hold on

>> plot(x0, y2)

>> plot(x0, y3)

>> plot(x0, y4)

>> plot(x0, y5)


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Piecewise Polynomial Interpolation

Using a set of lower degree interpolants on subinterval of

the whole domain= breakpoint or knot

y

x

Piecewise-linear interpolation

Piecewise-quadratic interpolation

Piecewise-cubic interpolation

f(x) and f(x) continuous at breakpoint = cubic spline


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MATLABs Built-in Interpolation Functions

Funnction Description

interp1 1-D interpolation with piecewise polynomials.

interp2 2-D interpolation with nearest neighbor, bilinear,

or bicubic interpolants.

interp3 3-D interpolation with nearest neighbor, bilinear,

or bicubic interpolants.

interpft 1-D interpolation of uniformly spaced data using

Fourier Series (FFT).

interpn n-D extension of methods used by interp3.

spline 1-D interpolation with cubic-splines using

not-a-knot or fixed-slope end conditions.


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interp1 Built-in Function

1-D interpolation with one of the following 4 methods:

1. Nearest-neighbor uses piecewise-constant function.

Interpolant discontinue at midpoint between knots

2. Linear interpolation uses piecewise-linear polynomials.

3. Cubic interpolation uses piecewise-cubic polynomials.

Interpolant and f(x) are continuous.

4. Spline interpolation uses cubic splines. This option performs

the same interpolation as built-in function spline.


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How to use interp1 ?

>> yhat = interp1(y, xhat)

>> yhat = interp1(x, y, xhat)>> yhat = interp1(x, y, xhat, method)

where y = tabulated values to be interpolated.

x = independent values. If not given x=1:length(y).

xhat = values at which interpolant be evaluated.

method = nearest, linear, cubic orspline


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>> x = [1 2 3 4 5 6 7 8 9 10];

>> y = [3.5 3.0 2.5 2.0 1.5 -2.4 -2.8 -3.2 -3.6 -4.0];

>> xhat = 1:0.1:10; % eval interpolant at xhat

>> yn = interp1(x, y, xhat, nearest);

>> plot(x, y, o, xhat, yn); pause;

>> yl = interp1(x, y, xhat, linear);

>> plot(x, y, o, xhat, yl); pause;

>> yc = interp1(x, y, xhat, cubic);

>> plot(x, y, o, xhat, yc); pause;

>> ys = interp1(x, y, xhat, spline);

or >> ys = spline(x, y, xhat);>> plot(x, y, o, xhat, ys);

Example: Interpolation with piecewise-polynomials

l08 interpolation

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