In math, we always need to derive different algorithms to get a tighter error bound. It may be correct almost surely that: the error bound of B is better than that of A by a factor x (x is bigger than 1). However, I am not sure if it is right to say: the error bound of A is weaker than that of B by a factor x (x is samller than 1)?


If B is better than A by a factor of x, then A is weaker than B by a factor of x.

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Not only is it correct, but it is equivalent to this statement:

If B == A * x, then A == B / x

There should be no restriction on x though. No need to say (x bigger than 1) or (x smaller than 1). Whatever x is in one statement, it should be the same in the other.

Take the starting statement: B is better than A by a factor of x. What "better" means can be confusing since it is unique to your exact problem, so I"ll just use the value (rather than the algorithm) of B and compare that with the value of A.

Now we have: B > A by a factor of X.

The phrase "by a factor of, means when either increasing or decreasing a value by multiplying it by some factor. Since we are measuring how "good" an algorithm is, we aren"t just comparing two numbers on the normal scale.If we have a logarithmic scale (base 10) where 5 is the worst an algorithm can perform, and 8 is the best an algorithm can perform, then the values 6 is better than the value 5 by a factor of 10. Similarly, 5 is worse than 6 by a factor of 10.

Below I provide links and show details about how to find this equation, but in this section I"ll only solve it.

If we have some initial value, i, and we increase it by a factor of some unknown, x, and we end at some final value, f, we can solve for the factor:

(f / i) == fIf we had decreased i by a factor of x and ended at f, we could solve for the factor like this:

(f / i) == (1 / f)See the inverse relationship?

We can use this to show that A is weaker than B by a factor of X.

Initial value = AFinal value = BFactor = X(A / B) = X(B / A) = 1 / XSo to get the factor, when we increase A by a factor of X to get to B, we find the factor is X. And if we decrease by a factor, i.e. go from B to A, we take the inverse and see the factor is 1/X.

If we give X a number, let"s say 5, we can see the property in action:

(A / B) = 5 => A = 5B(B / A) = (1 / 5) => B = 1/5AYou can plug in either value of A or B into the other equation to see they are both true.

The usage of saying that some value is decreased by a factor of some value isn"t too common in pure mathematics (excluding grammar school math), but you do see it in branches like economics and physics.

See the definition of factor according to the Cambridge Dictionary"s listing:

a ​number or ​variable (= ​letter or ​symbol) that is being ​multiplied in a ​product (= ​result of ​multiplying):

According to this definition, a factor is simply some number or variable being multiplied to another.

We can use the definition of "factor" to see how the following examples make sense:

Suppose you invested $100, and after a time, your investment was worth $300. The final value ($300) would be three (3) times the initial value. We would say that your investment had increased by a factor of 3.

On the other hand, if you made a poor investment, and the value decreased from $100 to $25, then the final value would be a quarter (1/4) of the initial value. We would say the investment had decreased by a factor of 4.

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This short article goes on to show how the final value divided by the initial value is equal to the factor when increasing a value by a factor. It also shows the factor is the reciprocal of that value when decreasing by a factor.