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1. Example on Quadratic Fit using Maxima (Posted to Forum by Martin)

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t1 0 time :

Data 1 0.9 2 2.4 374 17.4 5 30.5 6 48.3 62 mat :

Assumed Quadratic Function x f A B x * + C x 2^* + :

non-linear curve fit,  based on Maxima's package lsquares L Data x y 21 sys y x f ≡ A B C 31 sys 11131 sys Fit : (

Assignment using boolean 
equations  based on Maxima L Assign 3.83 4.9604 - 2.0625 31 sys

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 Data points color red ≡ x f x 0 6 explicit grid true ≡ 4 1 sys plot.png Draw2D

A 3.83

B 4.9604 -

C 2.0625

X Data 1 col :

Y Data 2 col :

Y 0.9 2.4 7 17.4 30.5 48.3 61 mat

Y.fit X f vectorize : 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

Y Y.fit - 0.0321 - 0.2407 0.5114 - 0.4114 0.0907 - 0.0179 - 61 mat

cleanup MaximaControl Cleanup complete.

0 time t1 - 2.968 s *

2. Above Example on Quadratic Fit using Native SMath

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t1 0 time :

x X :

y Y :

X Y A B C Clear 1

x 12345661 mat

y 0.9 2.4 7 17.4 30.5 48.3 61 mat

Just to show what 
equation we are fitting. 
We have to find 3x1 matrix "b" y.fit b 1 el b 2 el x * + b 3 el x 2^* + ≡

The data to be fitted

u φ 1 u u 2^31 mat :

List of functions to make X matrix

n x length : 6

k x φ length : 3

j 1 k range i 1 n range X i j el x i el φ j el : for for

X 11112413914161525163663 mat

Quadratic Fit

Column 1: Coeffs. of C ( All ones )

Column 2: Coeffs. of x

Column 3: Coeffs. of x²

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i 6

j 3

k 3

Compute the two matrices that will be multiplied to give matrix B

M1 X transpose X * : 62191219144191441227533 mat

M2 X transpose y * : 106.5 538.6 2853.2 31 mat

B M1 1 -^M2 * : 3.83 4.9604 - 2.0625 31 mat

M1 1 -^3.2 1.95 - 0.25 1.95 - 1.3696 0.1875 - 0.25 0.1875 - 0.0268 33 mat

Alternatively, form the Quadratic Eqn r F B 1 el B 2 el r * + B 3 el r 2^* + :

x F vectorize 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

3 φ 13931 mat

x 1 el F 0.9321

0.5 F 1.8654

x rows 6

0 F 3.83

8 F 96.1471

Alternayive;y, We can use following simple

method, instead of using φ (x) function

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Compute the fitted values and plot

i 1 n range yfit i el B x i el φ * : for

value of y when   x = 1 B sum 0.9321

c1 x x / vectorize : 11111161 mat

b1 x : 12345661 mat

a1 x x * vectorize : 14916253661 mat

SMath Fit yfit 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

Maxima Fit Y.fit 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

Same: compare

Expt x y augment :

Fit x yfit augment :

XX1 c1 b1 a1 augment : 11112413914161525163663 mat

X 11112413914161525163663 mat

Compare

Expt Fit 2 1 sys

X value at min x.min r F r diff 0 ≡ r solve : 1.2025

B 3.83 4.9604 - 2.0625 31 mat

x.min F 0.8476

r F r 2 diff 4.125

cordinates of minimum point Pt_min x.min x.min F augment : 1.2025 0.8476 12 mat

Expt r F Pt_min 3 1 sys

0 time t1 - 0.047 s *

------------------------------------------------------------------------------------------------------------------------------------------------------

3. Example on Linear Fit using Native SMath

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x y X B r F n k j Expt yfit u φ x.min Clear 1

t1 0 time :

x 0.5 1.1 1.5 2.1 2.3 51 mat :

y 3233 34.2 35.1 35.7 51 mat :

Just to show what 
equation we are fitting. 
We have to find 2x1 matrix "b" y.fit b 1 el b 2 el x * + ≡

The data to be fitted

u φ 1 u 21 mat :

List of functions to make X matrix

n x length : 5

k x φ length : 2

j 1 k range i 1 n range X i j el x i el φ j el : for for

X 1 0.5 1 1.1 1 1.5 1 2.1 1 2.3 52 mat

Linear fit

Column 1: Coeffs. of C ( All ones )

Column 2: Coeffs. of x

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i 5

j 2

k 2

Compute the two matrices that will be multiplied to give matrix B

M1 X transpose X * : 5 7.5 7.5 13.41 22 mat

M2 X transpose y * : 170 259.42 21 mat

B M1 1 -^eval M2 * : 30.9306 2.0463 21 mat

Alternatively, form the Quadratic Eqn r F B 1 el B 2 el r * + :

x 1 el F 31.9537

0.5 F 31.9537

x F vectorize 31.9537 33.1815 34 35.2278 35.637 51 mat

0 F 30.9306

8 F 47.3009

Compute the fitted values and plot

i 1 n range yfit i el B x i el φ * : for

value of y when   x = 1 B sum 32.9769

1 F 32.9769

B 30.9306 2.0463 21 mat

SMath Fit yfit 31.9537 33.1815 34 35.2278 35.637 51 mat

y 3233 34.2 35.1 35.7 51 mat

Expt x y augment : 0.5 32 1.1 33 1.5 34.2 2.1 35.1 2.3 35.7 52 mat

Expt r F 2 1 sys

0 time t1 - 0

x y X B r F n k j Expt yfit u φ a1 b1 c1 M1 M2 Clear 1

4. Program for Linear & Polynomial Fit using Native SMath

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Program  for  Linear and Polynomial Fit   Linear: n = 1 ,  Poly:  n = 2  3, 4, ... x# y# n Find_Cf j 1 n 1 + (range j 1 ≡ X x# x# / vectorize : X X x# j 1 -^vectorize augment : if 11 line for M1 X transpose X * : M2 X transpose y# * : B M1 1 -^eval M2 * : X B * (B 12 mat 51 line :

t1 0 time :

xL 0.5 1.1 1.5 2.1 2.3 51 mat :

yL 3233 34.2 35.1 35.7 51 mat :

xP 12345661 mat :

yP 0.9 2.4 7 17.4 30.5 48.3 61 mat :

y1 xL yL 1 Find_Cf 1 el : B1 xL yL 1 Find_Cf 2 el : 12 mat Linear ≡ y2 xP yP 2 Find_Cf 1 el : B2 xP yP 2 Find_Cf 2 el : 12 mat 2nd\0020\order ≡ y3 xP yP 3 Find_Cf 1 el : B3 xP yP 3 Find_Cf 2 el : 12 mat 3rd order ≡ y4 xP yP 4 Find_Cf 1 el : B4 xP yP 4 Find_Cf 2 el : 12 mat 4th order ≡ y5 xP yP 5 Find_Cf 1 el : B5 xP yP 5 Find_Cf 2 el : 12 mat 5th order ≡ 51 line

Linear Fit : x2, y2 y1 31.9537 33.1815 34 35.2278 35.637 51 mat

(2 nd  order) :  x1, y1 y2 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

(3 rd  order) :  x1, y1 y3 0.9683 2.1087 7.4825 17.0175 30.6413 48.2817 61 mat

(4 th  order) :  x1, y1 y4 0.9433 2.1837 7.4325 16.9675 30.7163 48.2567 61 mat

(5 th  order) :  x1, y1 y5 0.9 2.4 7 17.4 30.5 48.3 61 mat

Linear Coeffs. B1 30.9306 2.0463 21 mat

2 nd  order:  Coeffs B2 3.83 4.9604 - 2.0625 31 mat

3 rd  order:  Coeffs B3 4.1333 5.3419 - 2.1889 0.012 - 41 mat

4 th  order:  Coeffs B4 3.0833 3.5336 - 1.216 0.1921 0.0146 - 51 mat

5 th  order:  Coeffs B5 16.9 - 38.7717 30.0458 - 10.5875 1.6042 - 0.0908 61 mat

0 time t1 - 0.047 s *

EQUATIONS x F1 B1 1 el B1 2 el x * + : x F2 B2 1 el B2 2 el x * + B2 3 el x 2^* + : x F3 B3 1 el B3 2 el x * + B3 3 el x 2^* + B3 4 el x 3^* + : x F4 B4 1 el B4 2 el x * + B4 3 el x 2^* + B4 4 el x 3^* + B4 5 el x 4^* + : x F5 B5 1 el B5 2 el x * + B5 3 el x 2^* + B5 4 el x 3^* + B5 5 el x 4^* + B5 6 el x 5^* + : 51 line

t1 0 time :

xL F1 vectorize 31.9537 33.1815 34 35.2278 35.637 51 mat

xP F2 vectorize 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

xP F3 vectorize 0.9683 2.1087 7.4825 17.0175 30.6413 48.2817 61 mat

xP F4 vectorize 0.9433 2.1837 7.4325 16.9675 30.7163 48.2567 61 mat

xP F5 vectorize 0.9 2.4 7 17.4 30.5 48.3 61 mat

xL yL augment x F1 xP yP augment x F2 4 1 sys

Ex.1: Quadratic Fit (Maxima) Y.fit 0.9321 2.1593 7.5114 16.9886 30.5907 48.3179 61 mat

X value at function minomum x.min x F2 x diff 0 ≡ x solve : 1.2025

Minimum value of F2 x.min F2 0.8476

Alternatively B2 - 2 el 2 B2 3 el * / 1.2025

2nd derivative indicates max or min y'' x F2 x 2 diff : 4.125

y'' 0 > Function has a Min. black light green 1 3 mat function has a Max. blue pink 1 3 mat if

0 time t1 - 0.015 s *