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Lagrangian Interpolation

Civil Engineering Majors

Authors: Autar Kaw, Jai Paul

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Transforming Numerical Methods Education for STEM Undergraduates

Lagrange Method of Interpolation

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What is Interpolation ?

Given (x0,y0), (x1,y1), …… (xn,yn), find the value of ‘y’ at a value of ‘x’ that is not given.

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Interpolants

Polynomials are the most common choice of interpolants because they are easy to:

EvaluateDifferentiate, and Integrate.

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Lagrangian Interpolation

Lagrangian interpolating polynomial is given by

n

iiin xfxLxf

0

)()()(

where ‘ n ’ in )(xf n stands for the thn order polynomial that approximates the function )(xfy

given at )1( n data points as nnnn yxyxyxyx ,,,,......,,,, 111100 , and

n

ijj ji

ji xx

xxxL

0

)(

)(xLi is a weighting function that includes a product of )1( n terms with terms of ij

omitted.

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ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for linear interpolation.

Temperature vs. depth of a lake

Temperature Depth

T (oC) z (m)19.1 019.1 -119 -2

18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10

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Linear Interpolation

15 10 5 011

12

13

14

15

16

17

1817.6

11.7

y s

f range( )

f x desired

x s1

10x s0

10 x s range x desired

1

0

)()()(i

ii zTzLzT

)()()()( 1100 zTzLzTzL

7.11,8 00 zTz

6.17,7 11 zTz

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Linear Interpolation (contd)

10

11

00 0

0 )(zz

zz

zz

zzzL

jj j

j

01

01

10 1

1 )(zz

zz

zz

zzzL

jj j

j

78),6.17(87

8)7.11(

78

7)()()( 1

01

00

10

1

zzz

zTzz

zzzT

zz

zzzT

C65.14

)6.17(5.0)7.11(5.0)6.17(87

85.7)7.11(

78

75.7)5.7(

T

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Quadratic InterpolationFor the second order polynomial interpolation (also called quadratic interpolation), we

choose the velocity given by

2

0

)()()(i

ii tvtLtv

)()()()()()( 221100 tvtLtvtLtvtL

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ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for quadratic interpolation.

Temperature vs. depth of a lake

Temperature Depth

T (oC) z (m)19.1 019.1 -119 -2

18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10

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Quadratic Interpolation (contd)

9 8.5 8 7.5 78

10

12

14

16

1817.6

9.89238

y s

f range( )

f x desired

79 x s range x desired

9.9,9 oo zTz

7.11,8 11 zTz

6.17,7 22 zTz

2

00 0

0 )(

jj j

j

zz

zzzL

20

2

10

1

zz

zz

zz

zz

2

10 1

1 )(

jj j

j

zz

zzzL

21

2

01

0

zz

zz

zz

zz

2

20 2

2 )(

jj j

j

zz

zzzL

12

1

02

0

zz

zz

zz

zz

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Quadratic Interpolation (contd)

212

1

02

01

21

2

01

00

20

2

10

1)( zTzz

zz

zz

zzzT

zz

zz

zz

zzzT

zz

zz

zz

zzzT

C138.14

6.17375.07.1175.09.9125.0

6.178797

85.795.77.11

7898

75.795.79.9

7989

75.785.75.7

T

The absolute relative approximate error a obtained between the results from the first and second

order polynomial is

100138.14

65.14138.14

a

%6251.3

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Cubic InterpolationFor the third order polynomial (also called cubic interpolation), we choose the

temperature given by

3

0

)()()(i

ii zTzLzT

)()()()()()()()( 33221100 zTzLzTzLzTzLzTzL

9 8.5 8 7.5 7 6.5 68

10

12

14

16

18

2019.19774

9.44745

y s

f range( )

f x desired

69 x s range x desired

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ExampleTo maximize a catch of bass in a lake, it is suggested to throw the line to the depth of the thermocline. The characteristic feature of this area is the sudden change in temperature. We are given the temperature vs. depth plot for a lake. Determine the value of the temperature at z = −7.5 using the Lagragian method for cubic interpolation.

Temperature vs. depth of a lake

Temperature Depth

T (oC) z (m)19.1 019.1 -119 -2

18.8 -318.7 -418.3 -518.2 -617.6 -711.7 -89.9 -99.1 -10

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Cubic Interpolation (contd)

9 8.5 8 7.5 7 6.5 68

10

12

14

16

18

2019.19774

9.44745

y s

f range( )

f x desired

69 x s range x desired

9.9,9 oo zTz 7.11,8 11 zTz

6.17,7 22 zTz 2.18,6 33 zTz

3

00 0

0 )(

jj j

j

zz

zzzL

30

3

20

2

10

1

zz

zz

zz

zz

zz

zz

3

10 1

1 )(

jj j

j

zz

zzzL

31

3

21

2

01

0

zz

zz

zz

zz

zz

zz

3

20 2

2 )(

jj j

j

zz

zzzL

32

3

12

1

02

0

zz

zz

zz

zz

zz

zz

3

30 3

3 )(

jj j

j

zz

zzzL

23

2

13

1

03

0

zz

zz

zz

zz

zz

zz

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Cubic Interpolation (contd)

323

2

13

1

03

02

30

3

10

1

02

0

131

3

21

2

01

00

30

3

20

2

10

1

zTzz

zz

zz

zz

zz

zzzT

zz

zz

zz

zz

zz

zz

zTzz

zz

zz

zz

zz

zzzT

zz

zz

zz

zz

zz

zzzT

30 zzz

2.18768696

75.785.795.76.17

678797

65.785.795.7

7.11687898

65.775.795.79.9

697989

65.775.785.75.7

T

2.180625.06.175625.07.115625.09.90625.0

C 725.14

The absolute relative approximate error a obtained between the results from the second and

third order polynomial is

100725.14

138.14725.14

a

%9898.3

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Comparison Table

Order of Polynomial

1 2 3

Temperature °C 65.14 138.14 725.14

Absolute Relative Approximate Error

---------- %6251.3 %9898.3

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ThermoclineWhat is the value of depth at which the thermocline exists?

The position where the thermocline exists is given where 02

2

dz

Td

69,5667.155.3558.2629.615

2.18768696

7896.17

678797

689

7.11687898

6799.9

697989

678

32

zzzz

zzzzzz

zzzzzzzT

69,4.91.71

69,7.41.7158.262

2

2

2

zzdz

Td

zzzdz

dT

Simply setting this expression equal to zero, we get

m5638.7

69,4.91.710

z

zz

Additional ResourcesFor all resources on this topic such as digital audiovisual lectures, primers, textbook chapters, multiple-choice tests, worksheets in MATLAB, MATHEMATICA, MathCad and MAPLE, blogs, related physical problems, please visit

http://numericalmethods.eng.usf.edu/topics/lagrange_method.html

THE END

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