interpolare lagrange

7/29/2019 interpolare lagrange

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05.04.1

Chapter 05.04Lagrangian Interpolation More ExamplesCivil Engineering

Example 1

To 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 data for a lake in Table 1.

Table 1 Temperature vs. depth for a lake.

Temperature, C T Depth, mz

19.1 0

19.1 1

19 2

18.8 3

18.7 4

18.3 5

18.2 6

17.6 7

11.7 8

9.9 9

9.1 10


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05.04.2 Chapter 05.0

Figure 1 Temperature vs. depth of a lake.

Using the given data, we see the largest change in temperature is between m8z and

m7z . Determine the value of the temperature at m5.7z using a first order

Lagrange polynomial.

SolutionFor first order Lagrange polynomial interpolation (also called linear interpolation), the

temperature is given by

1

0

)()()(i

ii zTzLzT

)()()()( 1100 zTzLzTzL


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Lagrange Method of Interpolation-More Examples: Civil Engineering 05.04.3

Figure 2 Linear interpolation.

Since we want to find the temperature at m5.7z , we need to choose the two data points

that are closest to m5.7z that also bracket m5.7z to evaluate it. The two points are

80 z and 71 z .

Then

7.11,8 00 zTz 6.17,7 11 zTz

gives

1

00 0

0 )(

jj j

j

zzzzzL

10

1

zz

zz

1

10 1

1 )(

jj j

j

zz

zzzL

01

0

zz

zz

Hence

)()()( 101

0

0

10

1 zTzz

zzzT

zz

zzzT

78),6.17(87

8)7.11(

78

7

z

zz

)6.17(87

85.7)7.11(

78

75.7)5.7(

T

(x0,y0)

(x1,y1)

1(x)


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05.04.4 Chapter 05.0

)6.17(5.0)7.11(5.0

C65.14

You can see that 5.0)(0 zL and 5.0)(1 zL are like weightages given to the temperatures

at m8z and m7z to calculate the temperature at m5.7z .

Example 2


thermocline. The characteristic feature of this area is the sudden change in temperature. We

are given the temperature vs. depth data for a lake in Table 2.



19.1 0

19.1 1

19 218.8 3

18.7 4

18.3 5

18.2 6

17.6 7

11.7 8

9.9 9

9.1 10


m7z . Determine the value of the temperature at m5.7z using a second order

Lagrange polynomial. Find the absolute relative approximate error for the second order

polynomial approximation.

Solution

For second order Lagrange polynomial interpolation (also called quadratic interpolation), the

temperature is given by

2

0

)()()(i

ii zTzLzT

)()()()()()( 221100 zTzLzTzLzTzL


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Figure 3 Quadratic interpolation.

Since we want to find the temperature at ,5.7z we need to choose the three data points

that are closest to 5.7z that also bracket 5.7z to evaluate it. These three points are,90 z 81 z and 72 z . (Choosing the three points as 80 z , 71 z and 62 z

is equally valid.)

Then

9.9,9 00 zTz 7.11,8 11 zTz

6.17,7 22

zTz gives

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

zzzz

zzzz

2

20 2

2 )(

jj j

j

zz

zzzL

(x0,y0)

(x1,y1) (x2,y2)

2(x)


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05.04.6 Chapter 05.0

12

1

02

0

zz

zz

zz

zz

Hence

20

2

12

1

02

01

21

2

01

00

20

2

10

1 ),()()()(

zzz

zT

zz

zz

zz

zzzT

zz

zz

zz

zzzT

zz

zz

zz

zzzT

)6.17()87)(97(

)85.7)(95.7(

)7.11()78)(98(

)75.7)(95.7()9.9(

)79)(89(

)75.7)(85.7()5.7(

T

)6.17)(375.0()7.11)(75.0()9.9)(125.0(

C138.14

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

Example 3


thermocline. The characteristic feature of this area is the sudden change in temperature. Weare given the temperature vs. depth data for a lake in Table 3.



19.1 0

19.1 1

19 2

18.8 3

18.7 4

18.3 5

18.2 6

17.6 7

11.7 8

9.9 9

9.1 10


m7z .


7/9


a) Determine the value of the temperature at m5.7z using a third order Lagrangepolynomial. Find the absolute relative approximate error for the third order

polynomial approximation.

b) The position where the thermocline exists is given where 02

2

dz

Td. Using the

expression from part (a), what is the value of the depth at which the thermoclineexists?

Solution

a) For third order Lagrange polynomial interpolation (also called cubic interpolation), we

choose the temperature given by

3

0

)()()(i

ii zTzLzT

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

Figure 4 Cubic interpolation.

Since we want to find the temperature at ,5.7z we need to choose the four data points

that are closest to 5.7z that also bracket 5.7z to evaluate it. The four data points are

,90 z ,81 z 72 z and 63 z .Then

9.9,9 00 zTz 7.11,8 11 zTz 6.17,7 22 zTz 2.18,6 33 zTz

y

x

3(x)

(x0,y0)

(x1,y1)

(x2,y2)

(x3,y3)


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05.04.8 Chapter 05.0

gives

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

zzzz

zzzz

zzzz

3

30 3

3 )(

jj j

j

zz

zzzL

23

2

13

1

03

0

zz

zz

zz

zz

zz

zz

Hence

)()(

)()()(

3

23

2

13

1

03

0

2

32

3

12

1

02

0

1

31

3

21

2

01

0

0

30

3

20

2

10

1

zTzz

zz

zz

zz

zz

zzzT

zz

zz

zz

zz

zz

zz

zT

zz

zz

zz

zz

zz

zzzT

zz

zz

zz

zz

zz

zzzT

30 zzz

)2.18()76)(86)(96(

)75.7)(85.7)(95.7()6.17(

)67)(87)(97(

)65.7)(85.7)(95.7(

)7.11()68)(78)(98(

)65.7)(75.7)(95.7()9.9(

)69)(79)(89(

)65.7)(75.7)(85.7()5.7(

T

)2.18)(0625.0()6.17)(5625.0()7.11)(5625.0()9.9)(0625.0(

C725.14 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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b) To find the position of the thermocline, we must find the points of inflection of the third

order polynomial, given by .02

2

dz

Td

)2.18()76)(86)(96(

)7)(8)(9()6.17(

)67)(87)(97(

)6)(8)(9(

)7.11()68)(78)(98(

)6)(7)(9()9.9(

)69)(79)(89(

)6)(7)(8()(

zzzzzz

zzzzzzzT

69 z 69,5667.155.3558.2629.615 32 zzzz

69,7.41.7158.262 2 zzzdz

dT

69,4.91.712

2

zzdz

Td

Simply setting this expression equal to zero, we get

69,4.91.710 zz m5638.7z

This answer can be verified due to the fact that it falls within the specified range of the third

order polynomial and it also falls within the region of the greatest temperature change in the

collected data from the lake.

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