Seed values for estimating square roots.

In summary, when calculating square roots of a positive real number S, an initial seed value is needed. To avoid slowing down the calculation, it is helpful to have a rough estimate, even if it may be inaccurate. The estimation is based on the number of digits to the left or right of the decimal point, and depending on whether it is odd or even, the estimation will use either two or six. These numbers are chosen because they result in a nice round number that can be easily iterated.
  • #1
aarciga
6
0
This is taken from Wikipedia:

Many of the methods for calculating square roots of a positive real number S require an initial seed value. If the initial value is too far from the actual square root, the calculation will be slowed down. It is therefore useful to have a rough estimate, which may be very inaccurate but easy to calculate. If S ≥ 1, let D be the number of digits to the left of the decimal point. If S < 1, let D be the negative of the number of zeros to the immediate right of the decimal point. Then the rough estimation is this:

If D is odd, D = 2n + 1, then use [tex]\sqrt{S}\approx2\cdot10^{n}[/tex]
If D is even, D = 2n + 2, then use [tex]\sqrt{S}\approx6\cdot10^{n}[/tex]

Two and six are used because

[tex]\sqrt{\sqrt{1\cdot10}}=\sqrt[4]{10}\approx2[/tex] and [tex]\sqrt{\sqrt{10\cdot100}}=\sqrt[4]{1000}\approx6[/tex]

Im just wondering if anyone could elaborate further as to why 2 and 6 are used.

I see why i works in that it gives you an estimate with the same number of digits as [tex]\sqrt{S}[/tex]

but I am confused as to what 4th root of 1*10 and 10*100 represent. where did those numbers come from?

any clarification is appreciated.
 
Mathematics news on Phys.org
  • #2
aarciga said:
This is taken from Wikipedia:

Many of the methods for calculating square roots of a positive real number S require an initial seed value. If the initial value is too far from the actual square root, the calculation will be slowed down. It is therefore useful to have a rough estimate, which may be very inaccurate but easy to calculate. If S ≥ 1, let D be the number of digits to the left of the decimal point. If S < 1, let D be the negative of the number of zeros to the immediate right of the decimal point. Then the rough estimation is this:

If D is odd, D = 2n + 1, then use [tex]\sqrt{S}\approx2\cdot10^{n}[/tex]
If D is even, D = 2n + 2, then use [tex]\sqrt{S}\approx6\cdot10^{n}[/tex]

Two and six are used because

[tex]\sqrt{\sqrt{1\cdot10}}=\sqrt[4]{10}\approx2[/tex] and [tex]\sqrt{\sqrt{10\cdot100}}=\sqrt[4]{1000}\approx6[/tex]

Im just wondering if anyone could elaborate further as to why 2 and 6 are used.

I see why i works in that it gives you an estimate with the same number of digits as [tex]\sqrt{S}[/tex]

but I am confused as to what 4th root of 1*10 and 10*100 represent. where did those numbers come from?

any clarification is appreciated.

I'm just guessing here, but it might come from something like this for D even:

[tex]10^{2n+2} = 10^210^{2n}\approx 10^{\frac 3 2}10^{2n}=(10^{\frac 3 4}10^n)^2\approx (6\cdot 10^n)^2[/tex]

which gives a nice round number to start the iteration.
 

1. What are seed values used for when estimating square roots?

Seed values are initial estimates that are used to determine the closest possible square root of a given number. They help in the process of finding the actual square root by providing a starting point for calculations.

2. How are seed values determined for estimating square roots?

Seed values are typically chosen based on the number being evaluated. They can be chosen based on personal preference or through a systematic process such as trial and error or using a formula.

3. Why is it important to use seed values when estimating square roots?

Seed values help in providing a starting point for calculations and can improve the accuracy of the estimated square root. Without seed values, it would be more difficult to determine the closest possible square root.

4. Can seed values be negative when estimating square roots?

No, seed values should be positive when estimating square roots. Negative values can lead to incorrect estimates and make the process more complicated.

5. How many seed values should be used when estimating square roots?

It is recommended to use at least two seed values when estimating square roots, but more can be used for increased accuracy. The more seed values used, the closer the estimated square root will be to the actual square root.

Similar threads

Replies
15
Views
1K
  • General Math
Replies
2
Views
1K
Replies
1
Views
708
Replies
3
Views
898
Replies
2
Views
727
Replies
1
Views
1K
Replies
12
Views
3K
  • Calculus
Replies
3
Views
3K
  • Differential Equations
Replies
1
Views
609
Replies
7
Views
2K
Back
Top