Lipschitz Condition: Finding the Lipschitz Constant

In summary, the Lipschitz condition is a form of uniform continuity that is written as |f(x)-f(x')| <= M*|x-x'| and the minimum M is called the Lipschitz constant. The value of M is not always necessary to know, but sometimes it is important, such as in convergence theorems. However, there is no general method for determining the value of M, as it depends on the specific function involved. Some methods, like the mean value theorem, can be used in certain cases to estimate M, but for general nonlinear functions, it may require iterations like the Newton method.
  • #1
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Hi, as I see Lipschitz condition is written as:

|f(x)-f(x')| <= M*|x-x'|

and minimum M is called Lipschitz constant. I would like to ask how the minimum M is found out? For instance for many convergence theorem include Lipschitz condition and no say something about value of M but how M is derived how we can reach out its value??
 
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  • #2
mertcan said:
Hi, as I see Lipschitz condition is written as:

|f(x)-f(x')| <= M*|x-x'|

and minimum M is called Lipschitz constant. I would like to ask how the minimum M is found out? For instance for many convergence theorem include Lipschitz condition and no say something about value of M but how M is derived how we can reach out its value??
There is usually no need to know the value of ##M##, only that it's uniformly continuous. Sometimes, it's important to know, whether ##M<1## in which case we get a fixed point, but in general, only the fact, that ##M## is independent of the point is important, at which continuity is examined Otherwise, it depends on the case, so there is no general answer possible, except, that it isn't important, except eventually in numeric algrithms.
 
  • #3
fresh_42 said:
There is usually no need to know the value of ##M##, only that it's uniformly continuous. Sometimes, it's important to know, whether ##M<1## in which case we get a fixed point, but in general, only the fact, that ##M## is independent of the point is important, at which continuity is examined Otherwise, it depends on the case, so there is no general answer possible, except, that it isn't important, except eventually in numeric algrithms.
In some situations where we need convergence, for instance |f(x)-f(x')| <= M*b*|x-x'| may be obtained for relevant situation and we can say M*b < 1 by the way b is a variable we need to determine in order to obtain convergence. Thus if we want to know b and we want M*b < 1 then we need to know the range of M but in that case how do we know M?? without M we can not determine range of b to ensure convergence...
 
  • #4
Again, there is no general answer which applies to all cases. You can try to estimate ##M## by the approximation ##\dfrac{f(x)-f(x_0)}{x-x_0}=f\,'(x_0)+r(x_0)##. But in general, you only have continuity, so it depends on the example.
 
  • #5
fresh_42 said:
There is usually no need to know the value of M, only that it's uniformly continuous.
To be crystal clear here, "it" in the "it's uniformly continuous" part refers to the function f, not the constant M.

BTW, I edited the thread level from A to I.
 
  • #6
Mark44 said:
To be crystal clear here, "it" in the "it's uniformly continuous" part refers to the function f, not the constant M.

BTW, I edited the thread level from A to I.
fresh_42 said:
Again, there is no general answer which applies to all cases. You can try to estimate ##M## by the approximation ##\dfrac{f(x)-f(x_0)}{x-x_0}=f\,'(x_0)+r(x_0)##. But in general, you only have continuity, so it depends on the example.

For instance in that link http://www.mast.queensu.ca/~speicher/Section7.pdf and at the bottom of page 4, you will see that in order to ensure the contraction mapping then Lipschitz constant multiplied by (b-a) must be less than 1 = L*(b-a)<1, so in order to decide on how to pick the b and a to ensure contraction mapping we have to know Lipschitz constant "L". Here in that case we have to definitely know the L but how we again determine it ??
 
  • #7
mertcan said:
For instance in that link http://www.mast.queensu.ca/~speicher/Section7.pdf and at the bottom of page 4, you will see that in order to ensure the contraction mapping then Lipschitz constant multiplied by (b-a) must be less than 1 = L*(b-a)<1, so in order to decide on how to pick the b and a to ensure contraction mapping we have to know Lipschitz constant "L". Here in that case we have to definitely know the L ...
Yes.
fresh_42 said:
Sometimes, it's important to know, whether ##1<M##
... but how we again determine it ??
That depends on the example. In yours: What is ##f\,?##
 
  • #8
mertcan said:
Here in that case we have to definitely know the L but how we again determine it ??

fresh_42 said:
That depends on the example. In yours: What is ##f\,##?
What he said -- you can't determine the constant without knowing the function involved.
 
  • #9
Mark44 said:
What he said -- you can't determine the constant without knowing the function involved.
fresh_42 said:
Yes.That depends on the example. In yours: What is ##f\,?##
I have not seen that "L" is determining for a given specific "f"
So could you explain to me how to find L for a specific "f" you improvise for this time? Also if there is a General way to find L for arbitrary function could you lead the way for me for instance explaining the process of finding constant "L"?
 
  • #10
mertcan said:
I have not seen that "L" is determining for a given specific "f"

mertcan said:
So could you explain to me how to find L for a specific "f" you improvise for this time?
Here's an easy one: f(x) = 2x on the interval [0, 4]. See if you can figure out a value for L.
 
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  • #11
Mark44 said:
Here's an easy one: f(x) = 2x on the interval [0, 4]. See if you can figure out a value for L.
... and then compare it with ##g(x) := \dfrac{1}{x}## on ##(0,1]##.
 
  • #12
fresh_42 said:
... and then compare it with ##g(x) := \dfrac{1}{x}## on ##(0,1]##.
Mark44 said:
Here's an easy one: f(x) = 2x on the interval [0, 4]. See if you can figure out a value for L.
Let say by the mean value theorem ( f (x)- f (y) )/ (x-y) <= f'(z) for x < z < y and we need to find bound L with | f'(z)|<=L, so in order to find maximum of f' we need to equate f'' to 0 and f''' must be < 0 BUT for GENERAL NONLINEAR f I consider we should use some kind of iterations like Newton methods. For instance in the equation f'' = 0 let's say we reached some local point BUT the value just found from f" = 0 using iteration may not satisfy f"' < 0 so then what happens?? We iterate and until satisfy 2 conditions ?? We iterate until almost infinity??
 
  • #13
Don't use these theorems, simply look for ##|f(x)-f(y)|< L\cdot |x-y|## with ##f(x)=2x## which should be quite easy, and then for ##f(x)=\frac{1}{x}## which might be a bit of a problem. What does Lipschitz continuity mean in words?
 
  • #14
fresh_42 said:
Don't use these theorems, simply look for ##|f(x)-f(y)|< L\cdot |x-y|## with ##f(x)=2x## which should be quite easy, and then for ##f(x)=\frac{1}{x}## which might be a bit of a problem. What does Lipschitz continuity mean in words?
My previous question relates to how to find constant L in general nonlinear cases of f (for instance x(t)^22 + x* dx/dt)?? What would you do @fresh_42 ? I try to grasp the main logic that is why asking
 
  • #15
mertcan said:
My previous question relates to how to find constant L in general nonlinear cases of f (for instance x(t)^22 + x* dx/dt)?? What would you do @fresh_42 ? I try to grasp the main logic that is why asking
Again, there is no general way. Given an example, a function might or might not be Lipschitz continuous. And if, it depends on the function and eventually the interval which constant can be chosen.

Your differential equation is ambiguous, a) because you left out brackets which are necessary in a linear notation and that's why we use LaTeX, cp. https://www.physicsforums.com/help/latexhelp/, and b) because it is no equation, and thus defines no function(s).

It is hard to answer, if there is none.
 
  • #16
mertcan said:
for instance x(t)^22 + x* dx/dt)
In addition to what @fresh_42 said about this not being an equation, this sure looks like a typo - ##x(t)^{22}## seems odd. Also, why do you have x(t) (to the power 22) in one place, and just x in the other term?
 
  • #17
fresh_42 said:
Again, there is no general way. Given an example, a function might or might not be Lipschitz continuous. And if, it depends on the function and eventually the interval which constant can be chosen.

Your differential equation is ambiguous, a) because you left out brackets which are necessary in a linear notation and that's why we use LaTeX, cp. https://www.physicsforums.com/help/latexhelp/, and b) because it is no equation, and thus defines no function(s).

It is hard to answer, if there is none.

Mark44 said:
In addition to what @fresh_42 said about this not being an equation, this sure looks like a typo - ##x(t)^{22}## seems odd. Also, why do you have x(t) (to the power 22) in one place, and just x in the other term?
I just wrote the equation x(t)^22 + x(t)* dx/dt = 0 to give weird nonlinear case while finding lipschitz constant. By the way, I believe we do not have a chance to determine lipschitz constant when we have weird nonlinear equations. Thus we may not know whether or not we reach the exactly converged point for those equations because of the lack of determined Lipschitz constant. So what should we do to ensure convergence exactly?? what kind of tool should we use to make ourselves know the iterations are going to converge when we have weird nonlinear equations??
I mean lipschitz condition measures the convergence is going to be provided in future but when lipschitz can not be determined for complicated function then what are the other methods to see the convergence is going to be provided in future??
 
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  • #18
mertcan said:
I mean lipschitz condition measures the convergence is going to be provided in future but when lipschitz can not be determined for complicated function then what are the other methods to see the convergence is going to be provided in future??
Attend a lecture in numerical analysis.
 

1. What is the Lipschitz condition?

The Lipschitz condition is a mathematical concept used to describe the smoothness or regularity of a function. It states that the rate of change of a function over a given interval is bounded by a constant value.

2. Why is the Lipschitz condition important?

The Lipschitz condition is important because it guarantees the existence and uniqueness of solutions to certain mathematical problems. It also allows for the use of efficient numerical methods for solving these problems.

3. How is the Lipschitz constant found?

The Lipschitz constant is found by calculating the maximum absolute value of the derivative of a function over a given interval. This value represents the bound on the rate of change of the function and is denoted by the symbol "L".

4. What are the applications of the Lipschitz condition?

The Lipschitz condition has applications in various fields such as optimization, control theory, and differential equations. It is also used in the analysis of algorithms and machine learning.

5. Can the Lipschitz constant change for different intervals?

Yes, the Lipschitz constant can change for different intervals. It depends on the behavior of the function over each interval. In general, the larger the interval, the larger the Lipschitz constant will be.

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