PiMath.de Planetary Systems of the Earth 1
Classisc Systems
  Copyright © Klaus Piontzik  
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7.4 - Determination of the approximation line

There are two ways to determine an approximate line from the logarithmic (and linearized) data.
a) through linear regression (without prior linearization)
b) by the existing minimum-maximum values

Case b) will be discussed below, since case a) can be handled using a commercially available calculation program.

We are now looking for the approximation line y = ax + b for the logarithmic values. In the following figure, the approximation line is shown as a dashed line.


approximation line

Illustration 7.2 approximation line


There are n values ??given, namely: y0, y1, y2, ... yk, ... yn

with yk = ln wk

There is a minimum ymin and a maximum ymax


The slope a of the approximation line can be determined from the min-max values ??and the new approximate numbering. The following applies:


7.4.1 - Equation:



Δ y is the difference between the minimum and maximum values:
7.4.2 - Equation:



Δ x is the maximum value of the new numbering:
7.4.3 - Equation:

new numbering


The additive constant of the function you are looking for results from the smallest value:
7.4.4 - Equation:

additive constant


The following applies to the approximation line:
7.4.5 - Equation:

approximate line logarithmed


Inserting all terms gives:
7.4.6 - Equation:

Approximate line total

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 Planetary Systems 1

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Der Autor - Klaus Piontzik