Least Square Fit In Fortran 1l4r5h

1

Least square fit of a straight line to data:

The equation of a straight line is y = mx + b Consider the data points ( x1 , y1 ) , ( x 2 , y 2 )…….etc. n

2 Error is defined as ε (m, b) = ∑ (mxi + b − y i ) . i =1

For the best fit, this error should be minimum. ∂ε ∂ε = 0 and =0 . Therefore, ∂m

∂b

n ∂ε ∂  ∂ 2 (mxi + b − y i ) 2 = ( mx + b − y ) = ∑ ∑ i i   ∂ m ∂m ∂m  i =1 i =1  n ∂ (mx i + b − y i ) = ∑ 2.( mxi + b − y i ) ∂m i =1

n

Now,

n

n

i =1

i =1

n

n

i =1

i =1

= 2∑ (mx i + b − y i ) xi = 2m∑ xi + 2b∑ xi − ∑ y i xi = 0 Similarly,

2

(1)

n n ∂ε = 0 = m∑ xi + nb = ∑ yi ∂b i =1 i =1

(2) From (1) and (2),

  n   n  ∑ xi  i =1 n

Slope, m =

n

  n  ∑i=1 xi  b   ∑i=1 yi    = n n   2  m  ∑i=1 xi   ∑ y i xi   i =1  n

n

n ∑ xi y i − ∑ y i ∑ xi i =1

i =1

n

n ∑ x i − ( ∑ xi ) i =1

2

and Intercept, b =

i =1

n

n

n

i =1

i =1 n

n

n

i =1 n

i =1

∑ y i ∑ x i − ∑ xi ∑ xi y i

2

i =1

2

n ∑ x i − (∑ xi ) i =1

2

.

2

i =1

Example: For the data points (1,2), (2,3), (3,4), (4,5) n

n =4,

∑ xi = 1 + 2 + 3 + 4 = 10 , i =1

n

∑x y i =1

i

i

n

∑y i =1

i

= 1 × 2 + 2 × 3 + 3 × 4 + 4 × 5 = 40 ,

= 2 + 3 + 4 + 5 = 14 n

∑x i =1

∴m = 4 × 40 − 14 ×210 = 160 − 140 = 20 = 1 , 4 × 30 − 10

120 − 100

20

2 i

= 1 × 1 + 2 × 2 + 3 × 3 + 4 × 4 = 30

b=

14 × 30 − 10 × 40 420 − 400 20 = = = 1. 120 − 100 20 4 × 30 − 10 2

2

FORTRAN PROGRAM FOR LEAST SQUARE FITTING:

C

Least square fitting of a straight line to data points write(*,*)'Give the Number of Points' read(*,*)n write(*,*)'Write the data points: x,y' sumx=0.0 sumy=0.0 sumsqx=0.0 sumxy=0.0 do i=1,n read(*,*)x,y sumx=sumx+x sumy=sumy+y sumsqx=sumsqx+x*x sumxy=sumxy+x*y enddo deno=n*sumsqx-sumx*sumx slope=(n*sumxy-sumx*sumy)/deno b=(sumsqx*sumy-sumx*sumxy)/deno write(*,*)'Slope, Intercept= ',slope,b stop end