Least-squares method : Approximation in 3rd order polynomial equation

Find the approximation in the 3rd order polynomial equation for the heat capacity of carbon dioxide shown in the literature data below.
/* Approximation in 3rd order polynomial (hear capacity
 of carbon dioxide) */

VAR  x(8)    "Temperature [K]"                ..
    ,y(8)    "Heat capacity (lit.) [J/mol.K]" ..
    ,ycal(8) "Heat capacity (calc.)[J/mol.K]" ..
    ,e(8)    "Marginal error"

x = ( 250,   300,   400,   500,   700,   1000, ..
     1200,  1500  )
y = ( 35.44, 37.52, 41.44, 44.68, 49.59, 54.32, ..
      56.34, 58.38 )

ycal = a + b*x + c*x^2 + d*x^3
e = y - ycal

FIND (a#1,b#1,c#1,d#1) LEAST e

OUTPUT a, b, c, d
Explanation

X, y, ycal and e are arrays with 8 elements. Array variables are required to be specified in the VAR statement in advance.

The least-squares method is specified in the FIND statement. The coefficients a,b,c and d that minimize the sum of the residual error e square between the literature data y and the calculation data ycal are to be solved. The number after " # " is an initial value.

Enter equations and click on the [Solve] button.

/* Approximation in 3rd order polynomial (hear capacity of carbon dioxide) */ VAR x(8) "Temperature [K]" .. ,y(8) "Heat capacity (lit.) [J/mol.K]" .. ,ycal(8) "Heat capacity (calc.)[J/mol.K]" .. ,e(8) "Marginal error" x = ( 250, 300, 400, 500, 700, 1000, 1200, 1500 ) y = ( 35.44, 37.52, 41.44, 44.68, 49.59, 54.32, 56.34, 58.38 ) ycal = a + b*x + c*x^2 + d*x^3 e = y - ycal FIND (a#1,b#1,c#1,d#1) LEAST e OUTPUT a, b, c, d

Sample problem

The following sample problems will help you understand how EQUATRAN-G works.

Linear equations: Tsuru-Kame-Zan (Japanese traditional problem, means problem of cranes and tortoises); Nutrients problem
Non-linear equations: Root of 3rd order polynomial equation; Loan calculation
Ordinary differential equations: 2nd order ordinary differential equation
Optimization and least square problems: Approximation in 3rd order polynomial equation