Alpha Max Plus Beta Min Calculation
I read about the Alpha Max Plus Beta Min algorithm described here. Here is a screenshot from the wikipedia page:
I think understand what the algorithm is supposed to do. It makes an approximation of the value of the square root of two squares. In other words, at least the way I understand it, it is another way to calculate the Pythagorean theorem. Alpha and Beta are just constants.
I am having trouble understanding what the max and min values are. How can you have the maximum or minimum value of a or b? For example, if I am calculating the Pythagorean theorem with a=2 and b=4, there is no maximum or minimum value of either number. I am just making the calculation with those two numbers. There isn't a max of 2 or a min of 2.
Can someone explain what it means by the maximum/ minimum value of a and b? Or is it the max/min of the two numbers. So, in my example min = 2 and max = 4. If that is the case, does someone have a link to a proof that shows how this algorithm equates to the Pythagorean theorem?
Thanks for any advice in advance!
$\endgroup$ 42 Answers
$\begingroup$I know this question is old, but it seems to need a clearer and simpler answer with example, so here it is. There are two questions asked here:
Q1: What does Max and Min refer to in this algorithm?
A1: "Max" is the greater of those two positive numbers, and "Min" is the other one.
Q2: Is there a proof that shows how this algorithm equates to the Pythagorean theorem?
A2: No, because they do not "equate" - they are not the same. The algorithm gives only an approximation that is faster to compute on many systems, and similar enough for some purposes.
Example: Calculate the distance between origo and coordinate (-4, 2).
(Note how the sign is ignored and removed.)
Approximation: $4 \times 0.960433870103 + 2 \times 0.397824734759 = 4.63738494993$
Truth: $\sqrt{4 \times 4 + 2 \times 2} = 4.472135955$
$\endgroup$ $\begingroup$min is the smaller $a$ and $b$ and max is the bigger of the two, so the algorithm says $\sqrt{a^2+b^2}\approx \alpha\max(a,b)+\beta\min(a,b)$, where $\alpha\approx 0.96$ and $\beta\approx 0.40$.
example: 3-4-5 triangle. the algorithm says $5=\sqrt{3^2+4^2}\approx 0.96\times 4+0.40\times 3=5.04$.
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