Valid Estimation of Square Roots?

In summary, the conversation discusses a formulaic representation of a known square roots estimation method and its properties. The main concern is whether using SQRT(x) in the equation, which is what is being estimated, is valid. It is also mentioned that there is no standard for approximation descriptions. The formula's error approaches 0 as x approaches infinity and a graph shows its comparison to SQRT(x). The formula also has some interesting properties, including a relationship between the integral and the largest difference for k-ke in a given interval. It is noted that the formula is similar to one iteration of Newton's method, but is less accurate and more complex computationally.
  • #1
clabulis
2
0
I've been playing around with a forumlaic representation of a known square roots estimation method and I came up with this:

th_squareroot.gif



This estimation has several interesting properties to it which I've been looking into.

My main concern is the fact that I'm using SQRT(x) in the equation which is exactly what we're estimating. Obviously the exact value of SQRT(x) does not need to be known in order to perform this equation since the FLOOR and CEILING functions are being applied to it, but is this valid?
 
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  • #2
clabulis said:
My main concern is the fact that I'm using SQRT(x) in the equation which is exactly what we're estimating. Obviously the exact value of SQRT(x) does not need to be known in order to perform this equation since the FLOOR and CEILING functions are being applied to it, but is this valid?
Valid according to what standards?

If you're wondering whether it's actually easier to calculate the right hand side, then the answer is yes assuming you are not completely naive (naive approach: calculate sqrt(x), then apply ceil or floor).
To calculate [itex]\lfloor \sqrt{x}\rfloor[/itex] simply iterate the integers k=1,2,... until k^2 > x which gives a quick way of calculating it.

As I see it under a normal computation model your approximation takes O(sqrt(x)) time to calculate.

However there is no standard of approximation descriptions that says exactly what you're allowed to do. I consider:
[tex]\sqrt{x+1}^2[/tex]
to be a perfectly valid way to approximate a positive real number x. For large x it's fairly accurate. However from a practical standpoint I probably wouldn't use it since I have access to the precise value.
 
  • #3
This is like a weight combo of up and down Bahkshali, right?Here's your relative error:

It has an exponential approach curve

8ee08e26f56f95d6dc7084b1d23177692g.jpg


I think its a smart idea but computationally its as efficient as Bahkshali... and there are more efficient methods than Bahkshali. Mathematica isn't cooperating with me to show the error for B right now. :(
 
  • #4
I think you might've entered the forumla incorrectly into Mathematica. Here's what my relative error graphs look like. The first one shows 0<=x<=2 (when the error is at its greatest). The second one shows 0<=x<=100,000.

squarerootgraph.gif



sqarerootgraph2.gif



The formula's error approaches 0 as x approaches ∞.

In fact, here is a graph of SQRT(x) in comparison to the estimating formula:

squarerootgraph3a.gif


There are some very interesting properties with this:

1.)
sqrtpic.jpg


So this also says that the integral from [0,n] = SQRT(n)/6

2.) The largest difference for k-ke for each interval [m,n] can be found at the point: (2SQRT(m)+1/4, 1/4(SQRT(m)+SQRT(n)))

3.) The x-value for the largest relative error in the interval [m,n] = m + SQRT(m)

Overall, the largest relative error that my formula will give on the interval [1,∞] is at x=2. The relative error at that value is approximately 0.05719
 
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  • #5
Right: I just meant the ratio, not the relative error.

I just did [tex]\frac{yours}{actual}[/tex]I don't understand that third statement: the largest error in [m,n] is at m + sqrt(m)? This isn't always in the interval. I'm also not sure what you mean by 2) your largest error is at two points? The error is definitely "parabolic" between zeros, so only one max: and 2*sqrt(m) + 1/4 is always less than m for m > 4, so its not in the interval. Likewise unless n is large enough the second value isn't in there.
 
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  • #6
For the record, your formula looks very much like one iteration of Newton's method for approximating a root of
f(y) = y2 - x​
 
  • #7
right, just approximating x by floors and ceilings
 
  • #8
Hurkyl said:
For the record, your formula looks very much like one iteration of Newton's method for approximating a root of
f(y) = y2 - x​
Almost, but this new expression is less accurate and is more complex computationally.

It would be even more like one iteration of that method by using only floor rather than floor and ceiling in the denominator of the second term:

[tex]\sqrt x \approx \lfloor \sqrt x \rfloor +
\frac{a - \lfloor \sqrt x \rfloor^2}{2\lfloor \sqrt x \rfloor}[/tex]

This is more accurate than the floor+ceiling method. It is in fact the first iteration of Newton's method starting with an initial guess of [itex]\lfloor \sqrt x\rfloor[/itex].

Except it still has a higher computational cost compared to Newton's method.
 

1. What is the concept of valid estimation of square roots?

The concept of valid estimation of square roots involves finding an approximate value for the square root of a given number. It is a way of estimating the value of a square root without using a calculator or performing complex mathematical calculations.

2. Why is it important to be able to estimate square roots?

Estimating square roots allows us to quickly and easily approximate the value of a square root without having to perform time-consuming calculations. This is useful in everyday life, as well as in scientific and mathematical fields where quick estimations are needed.

3. What are the different methods for estimating square roots?

The most common methods for estimating square roots include the prime factorization method, the repeated subtraction method, and the estimation by rounding method. These methods involve using different techniques to find an approximate value for a square root.

4. How accurate are estimations of square roots?

The accuracy of an estimation of a square root depends on the method used and the level of precision required. Generally, estimations can be accurate up to a few decimal places, but may not be as precise as finding the exact value using a calculator or mathematical formula.

5. Can estimations of square roots be used in real-world applications?

Yes, estimations of square roots are commonly used in real-world applications such as engineering, finance, and statistics. They are also used in everyday situations, such as calculating the area of a square or estimating the distance between two points on a map.

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