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Approximation Functions using The Method of Least Squares>>

It is not knowledge, but the act of learning,
not possession but the act of getting there,
not being but the act of becoming,
which grants the greatest enjoyment.

C. F. Gauss

On the first day of the 19th century a new asteroid, CERES, had been discovered in the sky, but a few weeks later the celestial body was lost to sight.

C.F. Gauss (1777-1855) realized that he had the necessary computational ability and "the method of least squares" (by him discovered), and he took up the challenge to calculate, from the few recorded observations of the asteroid, the orbit in which it moved.
The result was a prediction of astonishing accuracy, the asteroid was rediscovered at the end of the year in the exact position indicated by his calculations.

The Method of Least Squares is used to approximate data. As functions of approximation, depending on the graphical character of the points of function f (See Figure), are usually used the following ones:

  1. y = ax + b (Linear Regression).
  2. y = ax2 + bx + c (Regression of Degree Two).
  3. y = axm (Power Function or Geometric Regression).
  4. y = a.emx (Exponential Function).
  5. y = 1/(ax + b) (Linear Rational Function).
  6. y = a.ln x + b (Logarithmic Function).
  7. y = (a/x) + b (Hyperbola).
  8. y = x/(ax + b) (Rational Function).

Where a, b, c, m are parameters. When the type of function of approximation is established, the problem is reduced to finding the values of these parameters.


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