I have the following problem to code using python:
I have 7 parameters: x, y, z, t, HF, M1F, and M2F. The user should input any of these 3 and the program should calculate the rest.
The relations that I have are:
HF = -xyt
M1F = -2xzt + 4yzt - xyt + 4tz^2
M2F = 2yzt - xyt
1 = -2xt + 2yt + 4zt
Attempt to solve the problem:
I have 7 parameters and the user should input 3 => I will be left with 4 parameters. So it's all about solving a system of 4 nonlinear equations with 4 unknowns.
I read online that scipy.optimize could be used to solve a system of nonlinear equations.
But I need an initial guess.
Going back to the physics of the problem I have the following initial conditions:
x > 0
y > 0
z < 0
HF > 0
M1F > 0
M2F > 0
M2F > M1F (solving this inequality from the above equations I get: -x + y + 2z < 0)
HF > M1F + d (solving this inequality from the above equations I get: -x + 2y + 2z < 0)
How can these initial conditions help me get the initial guess so that I can solve my problem using scipy.optimize?
I'm not sure optimization is the right way to go here. I think personally I'd start with the three variables given and algebraically solve the rest. There are a lot of combinations, but all things considered the analytic solution is usually best if it's obtainable.
Related
I've been wrestling with this issue for a week and I just need some guidance on the math part of it. If I could just understand the math behind it I could piece together the functions to make it work. The assignment is;
Design and develop a C++ program for Calculating e(n) when delta <= 0.000001
e(n-1) = 1 + 1/1! + 1/2! + 1/3! + 1/4! + … + 1/(n-1)!
e(n) = 1 + 1/1! + 1/2! + 1/3! + 1/4! + … + 1/(n)!
delta = e(n) – e(n-1)
You do not have any input to the program. Your output should be something like this:
N = 2 e(1) = 2 e(2) = 2.5 delta = 0.5
N = 3 e(2) = 2.5 e(3) = 2.565 delta = 0.065
...
You must use recursive function calls.
My first issue is the math and the variables that would contain them.
the delta, e(n), and e(n-1) variable must doubles
if e(n) = 1 + 1 / 1! = 2 then e(n-1) must equal 1, which means delta = 1 (that's my thinking anyway) I'm just not sure of the math behind the .5 delta the first time and the 0.065 in the second iteration.
Can someone point me in the right direction on this problem?
Thank you,
T
From the wikipedia link, you can see that
I will not explain the notion of limits here, but what this basically means is that, if we define a function e where e(n) = 1 + 1/1! + 1/2! + 1/3! + 1/4! + … + 1/(n)! (which is the function given in your problem), we are able to approximate the real value of the constant e.
The higher n is, the closer we get from e.
If you look closely at the function, you can see that each time, we add a term which is smaller than the previous one: 1 >= 1/1! >= 1/2! >= .... >= 1/(n)!
That basically means that, every time we increase n we are getting closer to e but we are slowing down in the way.
The real value of e is 2.71828...
In our first step e(1) = 1, we are 1.71828... too far from the real value
In the second step e(2) = 2, we are at 0.71828..., 1 distance closer
In the third step e(3) = 2.5, we are now at 0.21828..., 0.5 distance closer
As you can see, we are getting there, but the closer we get, the slower we move. Now let's say that at each step, we want to know how close we have moved compared to the previous value.
We then do simply e(n) - e(n-1). This is basically what the delta means.
At some point, we are moving so slow that it does no longer make any sense to keep going. We are almost staying put. At this point, we decide that our approximation is close enough from e.
In your case, the problem defines the minimum progression speed to 0.000001
here is a solution :-
delta = e(n) - e(n-1)
delta = 1/n!
delta < 0.000001
n! > 1000000
n >= 10 as 10! = 3628800
I am trying to write a linear program and need a variable z that equals the sign of x-c, where x is another variable, and c is a constant.
I considered z = (x-c)/|x-c|. Unfortunately, if x=c, then this creates division by 0.
I cannot use z=x-c, because I don't want to weight it by the magnitude of the difference between x and c.
Does anyone know of a good way to express z so that it is the sign of x-c?
Thank you for any help and suggestions!
You can't model z = sign(x-c) exactly with a linear program (because the constraints in an LP are restricted to linear combinations of variables).
However, you can model sign if you are willing to convert your linear program into a mixed integer program, you can model this with the following two constraints:
L*b <= x - c <= U*(1-b)
z = 1 - 2*b
Where b is a binary variable, and L and U are lower and upper bounds on the quantity x-c. If b = 0, we have 0 <= x - c <= U and z = 1. If b = 1, we have L <= x - c <= 0 and z = 1 - 2*1 = -1.
You can use a solver like Gurobi to solve mixed integer programs.
For k » 1 this is a smooth approximation of the sign function:
Also
when ε → 0
These two approximations haven't the division by 0 issue but now you must tune a parameter.
In some languages (e.g. C++ / C) you can simply write something like this:
double sgn(double x)
{
return (x > 0.0) - (x < 0.0);
}
Anyway consider that many environments / languages already have a sign function, e.g.
Sign[x] in Mathematica
sign(x) in Matlab
Math.signum(x) in Java
sign(1, x) in Fortran
sign(x) in R
Pay close attention to what happens when x is equal to 0 (e.g. the Fortran function will return 1, with other languages you'll get 0).
Lets have two points, (x1, y1) and (x2,y2)
dx = |x1 - x2|
dy = |y1 - y2|
D_manhattan = dx + dy where dx,dy >= 0
I am a bit stuck with how to get x1 - x2 positive for |x1 - x2|, presumably I introduce a binary variable representing the polarity, but I am not allowed multiplying a polarity switch to x1 - x2 as they are all unknown variables and that would result in a quadratic.
If you are minimizing an increasing function of |x| (or maximizing a decreasing function, of course),
you can always have the aboslute value of any quantity x in a lp as a variable absx such as:
absx >= x
absx >= -x
It works because the value absx will 'tend' to its lower bound, so it will either reach x or -x.
On the other hand, if you are minimizing a decreasing function of |x|, your problem is not convex and cannot be modelled as a lp.
For all those kind of questions, it would be much better to add a simplified version of your problem with the objective, as this it often usefull for all those modelling techniques.
Edit
What I meant is that there is no general solution to this kind of problem: you cannot in general represent an absolute value in a linear problem, although in practical cases it is often possible.
For example, consider the problem:
max y
y <= | x |
-1 <= x <= 2
0 <= y
it is bounded and has an obvious solution (2, 2), but it cannot be modelled as a lp because the domain is not convex (it looks like the shape under a 'M' if you draw it).
Without your model, it is not possible to answer the question I'm afraid.
Edit 2
I am sorry, I did not read the question correctly. If you can use binary variables and if all your distances are bounded by some constant M, you can do something.
We use:
a continuous variable ax to represent the absolute value of the quantity x
a binary variable sx to represent the sign of x (sx = 1 if x >= 0)
Those constraints are always verified if x < 0, and enforce ax = x otherwise:
ax <= x + M * (1 - sx)
ax >= x - M * (1 - sx)
Those constraints are always verified if x >= 0, and enforce ax = -x otherwise:
ax <= -x + M * sx
ax >= -x - M * sx
This is a variation of the "big M" method that is often used to linearize quadratic terms. Of course you need to have an upper bound of all the possible values of x (which, in your case, will be the value of your distance: this will typically be the case if your points are in some bounded area)
So i'm implementing a heuristic algorithm, and i've come across this function.
I have an array of 1 to n (0 to n-1 on C, w/e). I want to choose a number of elements i'll copy to another array. Given a parameter y, (0 < y <= 1), i want to have a distribution of numbers whose average is (y * n). That means that whenever i call this function, it gives me a number, between 0 and n, and the average of these numbers is y*n.
According to the author, "l" is a random number: 0 < l < n . On my test code its currently generating 0 <= l <= n. And i had the right code, but i'm messing with this for hours now, and i'm lazy to code it back.
So i coded the first part of the function, for y <= 0.5
I set y to 0.2, and n to 100. That means it had to return a number between 0 and 99, with average 20.
And the results aren't between 0 and n, but some floats. And the bigger n is, smaller this float is.
This is the C test code. "x" is the "l" parameter.
//hate how code tag works, it's not even working now
int n = 100;
float y = 0.2;
float n_copy;
for(int i = 0 ; i < 20 ; i++)
{
float x = (float) (rand()/(float)RAND_MAX); // 0 <= x <= 1
x = x * n; // 0 <= x <= n
float p1 = (1 - y) / (n*y);
float p2 = (1 - ( x / n ));
float exp = (1 - (2*y)) / y;
p2 = pow(p2, exp);
n_copy = p1 * p2;
printf("%.5f\n", n_copy);
}
And here are some results (5 decimals truncated):
0.03354
0.00484
0.00003
0.00029
0.00020
0.00028
0.00263
0.01619
0.00032
0.00000
0.03598
0.03975
0.00704
0.00176
0.00001
0.01333
0.03396
0.02795
0.00005
0.00860
The article is:
http://www.scribd.com/doc/3097936/cAS-The-Cunning-Ant-System
pages 6 and 7.
or search "cAS: cunning ant system" on google.
So what am i doing wrong? i don't believe the author is wrong, because there are more than 5 papers describing this same function.
all my internets to whoever helps me. This is important to my work.
Thanks :)
You may misunderstand what is expected of you.
Given a (properly normalized) PDF, and wanting to throw a random distribution consistent with it, you form the Cumulative Probability Distribution (CDF) by integrating the PDF, then invert the CDF, and use a uniform random predicate as the argument of the inverted function.
A little more detail.
f_s(l) is the PDF, and has been normalized on [0,n).
Now you integrate it to form the CDF
g_s(l') = \int_0^{l'} dl f_s(l)
Note that this is a definite integral to an unspecified endpoint which I have called l'. The CDF is accordingly a function of l'. Assuming we have the normalization right, g_s(N) = 1.0. If this is not so we apply a simple coefficient to fix it.
Next invert the CDF and call the result G^{-1}(x). For this you'll probably want to choose a particular value of gamma.
Then throw uniform random number on [0,n), and use those as the argument, x, to G^{-1}. The result should lie between [0,1), and should be distributed according to f_s.
Like Justin said, you can use a computer algebra system for the math.
dmckee is actually correct, but I thought that I would elaborate more and try to explain away some of the confusion here. I could definitely fail. f_s(l), the function you have in your pretty formula above, is the probability distribution function. It tells you, for a given input l between 0 and n, the probability that l is the segment length. The sum (integral) for all values between 0 and n should be equal to 1.
The graph at the top of page 7 confuses this point. It plots l vs. f_s(l), but you have to watch out for the stray factors it puts on the side. You notice that the values on the bottom go from 0 to 1, but there is a factor of x n on the side, which means that the l values actually go from 0 to n. Also, on the y-axis there is a x 1/n which means these values don't actually go up to about 3, they go to 3/n.
So what do you do now? Well, you need to solve for the cumulative distribution function by integrating the probability distribution function over l which actually turns out to be not too bad (I did it with the Wolfram Mathematica Online Integrator by using x for l and using only the equation for y <= .5). That however was using an indefinite integral and you are really integration along x from 0 to l. If we set the resulting equation equal to some variable (z for instance), the goal now is to solve for l as a function of z. z here is a random number between 0 and 1. You can try using a symbolic solver for this part if you would like (I would). Then you have not only achieved your goal of being able to pick random ls from this distribution, you have also achieved nirvana.
A little more work done
I'll help a little bit more. I tried doing what I said about for y <= .5, but the symbolic algebra system I was using wasn't able to do the inversion (some other system might be able to). However, then I decided to try using the equation for .5 < y <= 1. This turns out to be much easier. If I change l to x in f_s(l) I get
y / n / (1 - y) * (x / n)^((2 * y - 1) / (1 - y))
Integrating this over x from 0 to l I got (using Mathematica's Online Integrator):
(l / n)^(y / (1 - y))
It doesn't get much nicer than that with this sort of thing. If I set this equal to z and solve for l I get:
l = n * z^(1 / y - 1) for .5 < y <= 1
One quick check is for y = 1. In this case, we get l = n no matter what z is. So far so good. Now, you just generate z (a random number between 0 and 1) and you get an l that is distributed as you desired for .5 < y <= 1. But wait, looking at the graph on page 7 you notice that the probability distribution function is symmetric. That means that we can use the above result to find the value for 0 < y <= .5. We just change l -> n-l and y -> 1-y and get
n - l = n * z^(1 / (1 - y) - 1)
l = n * (1 - z^(1 / (1 - y) - 1)) for 0 < y <= .5
Anyway, that should solve your problem unless I made some error somewhere. Good luck.
Given that for any values l, y, n as described, the terms you call p1 and p2 are both in [0,1) and exp is in [1,..) making pow(p2, exp) also in [0,1) thus I don't see how you'd ever get an output with the range [0,n)
I'm kind of stuck here, I guess it's a bit of a brain teaser. If I have numbers in the range between 0.5 to 1 how can I normalize it to be between 0 to 1?
Thanks for any help, maybe I'm just a bit slow since I've been working for the past 24 hours straight O_O
Others have provided you the formula, but not the work. Here's how you approach a problem like this. You might find this far more valuable than just knowning the answer.
To map [0.5, 1] to [0, 1] we will seek a linear map of the form x -> ax + b. We will require that endpoints are mapped to endpoints and that order is preserved.
Method one: The requirement that endpoints are mapped to endpoints and that order is preserved implies that 0.5 is mapped to 0 and 1 is mapped to 1
a * (0.5) + b = 0 (1)
a * 1 + b = 1 (2)
This is a simultaneous system of linear equations and can be solved by multiplying equation (1) by -2 and adding equation (1) to equation (2). Upon doing this we obtain b = -1 and substituting this back into equation (2) we obtain that a = 2. Thus the map x -> 2x - 1 will do the trick.
Method two: The slope of a line passing through two points (x1, y1) and (x2, y2) is
(y2 - y1) / (x2 - x1).
Here we will use the points (0.5, 0) and (1, 1) to meet the requirement that endpoints are mapped to endpoints and that the map is order-preserving. Therefore the slope is
m = (1 - 0) / (1 - 0.5) = 1 / 0.5 = 2.
We have that (1, 1) is a point on the line and therefore by the point-slope form of an equation of a line we have that
y - 1 = 2 * (x - 1) = 2x - 2
so that
y = 2x - 1.
Once again we see that x -> 2x - 1 is a map that will do the trick.
Subtract 0.5 (giving you a new range of 0 - 0.5) then multiply by 2.
double normalize( double x )
{
// I'll leave range validation up to you
return (x - 0.5) * 2;
}
To add another generic answer.
If you want to map the linear range [A..B] to [C..D], you can apply the following steps:
Shift the range so the lower bound is 0. (subract A from both bounds:
[A..B] -> [0..B-A]
Scale the range so it is [0..1]. (divide by the upper bound):
[0..B-A] -> [0..1]
Scale the range so it has the length of the new range which is D-C. (multiply with D-C):
[0..1] -> [0..D-C]
Shift the range so the lower bound is C. (add C to the bounds):
[0..D-C] -> [C..D]
Combining this to a single formula, we get:
(D-C)*(X-A)
X' = ----------- + C
(B-A)
In your case, A=0.5, B=1, C=0, D=1 you get:
(X-0.5)
X' = ------- = 2X-1
(0.5)
Note, if you have to convert a lot of X to X', you can change the formula to:
(D-C) C*B - A*D
X' = ----- * X + ---------
(B-A) (B-A)
It is also interesting to take a look at non linear ranges. You can take the same steps, but you need an extra step to transform the linear range to a nonlinear range.
Lazyweb answer: To convert a value x from [minimum..maximum] to [floor..ceil]:
General case:
normalized_x = ((ceil - floor) * (x - minimum))/(maximum - minimum) + floor
To normalize to [0..255]:
normalized_x = (255 * (x - minimum))/(maximum - minimum)
To normalize to [0..1]:
normalized_x = (x - minimum)/(maximum - minimum)
× 2 − 1
should do the trick
You could always use clamp or saturate within your math to make sure your final value is between 0-1. Some saturate at the end, but I've seen it done during a computation, too.