Rational Function Regression Calculator
You can use this rational function regression calculator to determine the rational function equation that is best fitted to a given set of points.
Simply follow the two steps outlined below to use the calculator.
- Complete the respective fields with the known X and Y variables.
- Click on the "Calculate" button to generate the rational function regression equation.
If you wish to clear the results to compute the results for a different set of values, click on the "Reset" button.
Rational Function Regression: y = (ax + c)/(x − b)
The ratio of the two linear functions represents one of the most straightforward rational functions. Rational functions that take the form y = (ax + c)/(x − b) represent a good method of modeling any data that levels off after a given time period without any oscillations. The horizontal asymptote of a rational function is y = a, while the vertical asymptote is x = b, and the y-intercept is −c/b.
When a function takes the form y = (ax + c)/(x − b), the a, b, and c parameters are not linear. However, it is possible to transform the equation through the use of simple algebra:
y = (ax + c)/(x − b)
(x − b)y = ax + c
xy − by = ax + c
xy = ax + by + c
This equation does not incorporate linear a, b, and c variables; as such, it is not possible to apply the least-squares method to identify the "best fit" values a, b, and c values; i.e., you can minimize the equation.
F(a, b, c) = ∑(xiyi − axi − byi − c)2,
This is tantamount to solving the system as follows:
∂F/∂a = 0, ∂F/∂b = 0, and ∂F/∂c = 0
The solution can be determined using matrix math.
Using Matrices to Determine a, b, and c
The matrix equation that can be employed for simple rational regression is as follows:
where n is the number of data pairs (xi, yi), and ∑ the summation sign.
Providing the three-by-three matrix presented on the left is invertible, a unique solution (a, b, c) can be employed to minimize the function F(a, b, c) and delivers the parameters for the best fit rational function.
Identify the equation of a rational function that fits the points (x, y):
(4, 3), (2, 4), (3, 6)
Through the use of this rational function regression calculator, you can delineate the following equation:
y = (3.6x − 12)/(x − 3.2)
This equation represents a reasonably good fit for the data.
You may also be interested in our Linear Regression Calculator or Quadratic Regression Calculator