Chem 302 - Math 252 Chapter 3 Interpolation / Extrapolation.

Chem 302 - Math 252

Chapter 3Interpolation / Extrapolation

Interpolation / Extrapolation

• Experimental data at discrete points• Need to know the dependent variable at a value of the independent

variable that was not measured• Need to know what value of the independent variable gives a particular

value of the dependent variable• Point is within range of experimental data then called interpolation• Point is outside range of experimental data then call extrapolation• Same techniques• Extrapolation more risky

Linear Interpolation

• Assume data varies linearly between 2 points

• Connect-the-dots 1 1, , ,k k k kx y x y

0 1y a a x

0 1k ky a a x

1 0 1 1k ky a a x 1

1

kk k k

k k

x xy y y y

x x

Linear Interpolation – Viscosity of WaterT / °C / mP

0 17.921

10 13.077

20 10.050

30 8.007

40 6.560

50 5.494

Find at 25 °C

11

kk k k

k k

x xy y y y

x x

20 30 20

20

30 2010.050 0.2043 20

T

T

25 10.050 0.2043 25 20

9.029

Exp 25 =8.937 mP

Linear Interpolation – Viscosity of Water

0

2

4

6

8

10

12

14

16

18

20

0 10 20 30 40 50

T / °C

/

mP

ln() vs 1/T nearly linear

Linear Interpolation – Viscosity of Water

1.5

1.7

1.9

2.1

2.3

2.5

2.7

2.9

3.1

0.003 0.0031 0.0032 0.0033 0.0034 0.0035 0.0036 0.0037

1/T ×K

ln(

/ m

P)

Linear Interpolation – Viscosity of Water1/T × K ln( / mP)

0.003661 2.88597

0.003532 2.57086

0.003411 2.30757

0.003299 2.08032

0.003193 1.88099

0.003095 1.70366

Find at 25 °C

11

kk k k

k k

x xy y y y

x x

2020 30 20

30 20

1/

1/ 298.15 0.0034112.30757 2.08032 2.30757

0.003299 .00034112.19204

T xy y y y

x x

2.1920425 8.953e

Exp 25 =8.937 mP

Linear Interpolation – Heat Capacity of BenzeneT / K C / (J K-1)

200.00 83.7

240.00 104.1

260.00 116.1

278.69 128.7

Find C at 220, 250 & 270 K

11

kk k k

k k

x xy y y y

x x

220

220 20083.7 104.1 83.7 93.9

240 200C

250

250 240104.1 116.1 104.1 110.1

260 240C

270

270 260116.1 128.7 116.1 122.8

278.69 260C

Find C at 20 K

20

20 20083.7 104.1 83.7 8.1

240 200C

Exp C20 = 8.4 J/K

Quadratic Interpolation

• Assume data is quadratic between 3 points

1 1 1 1, , , , ,k k k k k kx y x y x y

20 1 2y a a x a x

21 0 1 1 2 1k k ky a a x a x

20 1 2k k ky a a x a x

21 0 1 1 2 1k k ky a a x a x

21 1 0 1

21

21 1 2 1

1

1

1

k k k

k k k

k k k

x x a y

x x a y

x x a y

Quadratic Interpolation

1 1k k kx x x x x

21 1

2

21 1

1

1 0

1

k k

k k

k k

x x

x x

x x

Do it !

1 1k k kx x x x x

1 1k k kx x x x x

1 1 1 1 10 0 0k k k k k kx x x

1 10 0 0k k k k k kx x x

1 1 1 1 10 0 0k k k k k kx x x

1 1 1 1k k k k k ky b x b x b x

Quadratic Interpolation 1 1 1 1k k k k k ky b x b x b x

1 1 1 1 1 1 1 1

1 1 1

k k k k k k k k k k

k k k

y b x b x b x

b x

1

11 1

kk

k k

yb

x

k

kk k

yb

x

1

11 1

kk

k k

yb

x

1 11 1

1 1 1 1

k k kk k k

k k k k k k

x x xy y y y

x x x

Quadratic Interpolation – Viscosity of WaterT / °C / mP

0 17.921

10 13.077

20 10.050

30 8.007

40 6.560

50 5.494

Find at 25 °C

Exp 25 =8.937 mP

1 11 1

1 1 1 1

k k kk k k

k k k k k k

x x xy y y y

x x x

(25 20) 25 30 (25 10) 25 30 (25 10) 25 2013.077 10.050 8.007

10 20 10 30 20 10 20 30 30 10 30 20

8.906

y

(25 30) 25 40 (25 20) 25 40 (25 20) 25 3010.050 8.007 6.560

20 30 20 40 30 20 30 40 40 20 40 30

8.954

y

Quadratic Interpolation – Viscosity of Water1/T × K ln( / mP)

0.003661 2.88597

0.003532 2.57086

0.003411 2.30757

0.003299 2.08032

0.003193 1.88099

0.003095 1.70366

Find at 25 °C

Exp 25 =8.937 mP

1 11 1

1 1 1 1

k k kk k k

k k k k k k

x x xy y y y

x x x

(0.003354 0.003411) 0.003354 0.0032992.57086

0.003532 0.003411 0.003532 0.003299

(0.003354 0.003532) 0.003354 0.0032992.30757

0.003411 0.003532 0.003411 0.003299

(0.003354 0.003532) 0.003354 0.003412.08032

y

1

0.003299 0.003532 0.003299 0.003411

2.18979

=8.933

2.1902

=8.937

y

Using points 2,3,4

Using points 3,4,5

Quadratic Interpolation – Heat Capacity of BenzeneT / K C / (J K-1)

200.00 83.7

240.00 104.1

260.00 116.1

278.69 128.7

Find C at 20, 220, 250 & 270 K

Lagrangian Interpolation

• Generalization of linear & quadratic interpolations• Uses nth order polynomial & n+1 points

1 1 2 2 1 1, , , , , ,n nx y x y x y

20 1 2

0

nn i

n ii

y p x a a x a x a x a x

0 11 1

1 22 2

11 1

1

1

1

n

n

nn nn n

a yx x

a yx x

a yx x

1 1

2 2

1 1

1

10

1

n

n

nn n

x x

x x

x x

Unique solution


1 2 3 1nx x x x x x x

1

1

n

k kk

p x b x

2 1 3 1nx x x x x x x

1 2 1 1 1k k k nx x x x x x x x x x x

1 1 2n nx x x x x x x

1

1

1,2, , 1n

k iii k

x x x k n


k

kk k

yb

x

1

1

n

k kk

p x b x

0kk x

0ik x i k

1

1

nk

kk k k

xp x y

x

Other Interpolation Functions

• Does not have to be a power series

0

n

i ii

p x a f x

• Methods are same as Lagrangian Interpolation– Usually 2nd order (quadratic) or 3rd order (cubic) Lagrangian

interpolation is sufficient

Lorentzian Interpolation

• Uses Lorentzian lineshape

2 2

01

AL x

x x B

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

x -x 0

y/ A

Peak height – A

Peak Position – x0

Full Width at Half Height (FWHH) – 2/B

3 three points (usually three at top of peak)


1 1 2 2

1 01

Ay L x

x x B

2 2 2 2

2 01

Ay L x

x x B

3 3 2 2

3 01

Ay L x

x x B

2 20

12 20

12 2 2

0 0

120 1 2

20 1 2

1

1

1 2

1

AL x

x x B

x x B

A

x x x x B

A

a a x a x

a a x a xL x

2 20

0

20

1

2

2

1

2

B xa

A

x Ba

A

Ba

A


20 1 1 2 1

1

1a a x a x

y

22

0 2 1

22 2

20 2 1

10

2

4

4

4

4

2

aA

a a a

aB

a a a

ax

a

20 1 2 2 2

2

1a a x a x

y

20 1 3 2 3

3

1a a x a x

y

Quadratic interpolation on 1/y

Magnitude-Lorentzian Interpolation

Uses square root of Lorentzian lineshape

1/ 2

2 201

AML x

x x B

Peak height – A1/2


Full Width at Half Height (FWHH) –

3 three points (usually three at top of peak)0

0.2

0.4

0.6

0.8

1

-10 -8 -6 -4 -2 0 2 4 6 8 10

x -x 0

y/ A

1/2

2 3 / B


20 1 1 2 1

1

1a a x a x

y

22

0 2 1

22 2

20 2 1

10

2

4

4

4

4

2

aA

a a a

aB

a a a

ax

a

20 1 2 2 2

2

1a a x a x

y

20 1 3 2 3

3

1a a x a x

y

Quadratic interpolation on 1/y

KCe Interpolation

Based on Lorentzian & Magnitude-Lorentzian

20 1 2

e

eKC x a a x a x

e = 1 – quadratic

e = -1 – Lorentzian

e = -1/2 – Magnitude-Lorentzian

Optimized e for different lineshapes (mostly used in FTICR-MS)Keefe, Comisarow, App. Spectrosc. 44, 600 (1990)


21 0 1 1 2 1ey a a x a x

22 0 1 2 2 2ey a a x a x

23 0 1 3 2 3ey a a x a x

Quadratic interpolation on y-e

Gaussian Interpolation

Based on Gaussian lineshape

220B x xG x Ae

Peak height – A


Full Width at Half Height (FWHH) – 2 ln 2 / B

0

0.2

0.4

0.6

0.8

1

-10 -8 -6 -4 -2 0 2 4 6 8 10

x -x 0

y/ A

Gaussian Interpolation

21 0 1 1 2 1ln y a a x a x

21 0 1 2 2 2ln y a a x a x

21 0 1 3 2 3ln y a a x a x

Quadratic interpolation on lny

Can be converted to form

Find Peak Position & Height

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

1010 1015 1020 1025 1030 1035 1040 1045 1050

wavenumber / cm-1

imag

inar

y m

olar

pol

ariz

abili

ty/(

cm3 m

ol-1

)

0.69

0.7

0.71

0.72

0.73

0.74

0.75

0.76

0.77

0.78

1034.5 1035.5 1036.5 1037.5

Find Peak Position & HeightSeven Highest points -1 / cm 3 -1 / (cm mol )m

1034.6 0.73050

1035.1 0.75487

1035.5 0.76894

1036.0 0.77183

1036.5 0.75979

1037.0 0.73585

1037.5 0.69860

Use various interpolation functions to find peak position and height

1. Determine interpolation function (top 3, 5 or 7 points)

2. Differentiate interpolation function and find root (i.e. find location max) - position

3.Evaluate interpolation function at peak position (height)

Find Peak Position & HeightQuadratic Interpolation

1 11 1

1 1 1 1

k k kk k k

k k k k k k

x x xy y y y

x x x

2

1036.0 1036.50.76894

1035.5 1036.0 1035.5 1036.5

1035.5 1036.50.77183

1036.0 1035.5 1036.0 1036.5

1035.5 1036.00.75979

1036.5 1035.5 1036.5 1036.0

32038.368 61.86077 0.02986

x xy

x x

x x

x x

61.86077 0.05972y x 61.86077 0.05972 0

61.860771035.847

0.05972

peak

peak

y x

x

232038.368 61.86077 0.02986

0.77253

peak peak peaky x x

Find Peak Position & HeightComparison of Interpolation Methods

Interpolation Method Peak Position /cm-1 Peak Height /(cm3 mol-1)

No Interpolation 1036.0 0.77183

Quadratic 1035.847 0.77253

Lorentzian 1035.846 0.77255

Magnitude-Lorentzian 1035.846 0.77255

Gaussian 1035.846 0.77254

KC2 1035.847 0.77254

KC4 1035.846 0.77254

5-point Lagrangian 1035.840 0.77263

7-point Lagrangian 1035.834 0.77270

• So far methods have used a moving window of subset of data– May be discontinuous at edges of windows– Causes jagged plots

• Spline interpolation forces slopes (and in some cases higher derivatives) to match at edges of windows– Creates smooth plots

Spline Interpolation

Cubic Spline with Slope Matching

Value of x such that x2 < x < x3

2 30 1 2 3p x a a x a x a x

p(x) forced to pass through (x2,y2) & (x3,y3)

p(x) forced to match slopes at (x2,y2) & (x3,y3)2 3

2 0 1 2 2 2 3 2y a a x a x a x 2 3

3 0 1 3 2 3 3 3y a a x a x a x 2

2 1 2 2 3 22 3y a a x a x 2

3 1 2 3 3 32 3y a a x a x


Approximate slopes

3 12

3 1

y yy

x x

2 30 22 2 2

2 31 33 3 3

22 22 2

23 33 3

1

1

0 1 2 3

0 1 2 3

a yx x x

a yx x x

a yx x

a yx x

4 23

4 2

y yy

x x


• Between 1st and last pair of points– Can set slopes = 0

• Natural spline

• Good if data is flat at extremes

– Can set • Useful if slope is basically constant

– Can extrapolate using closest region– Can set

1 2y y 1n ny y

2 11

2 1

y yy

x x

1

1

n nn

n n

y yy

x x

0.5

0.6

0.7

0.8

0.9

1

1.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4x

y

Experimental

Cubic Spline Interpolation

Cubic Interpolation

6th Order Polynomial

1.3

1.4

1.5

1.6

1.7

1.8

1.9

2

2.5 2.7 2.9 3.1 3.3 3.5 3.7 3.9x

y

Experimental

Cubic Spline Interpolation

Cubic Interpolation

6th Order Polynomial

Chem 302 - Math 252 Chapter 3 Interpolation / Extrapolation.

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