I have a system of non-linear equations, which may or may not have a unique solution. The default GAMS mixed complementary problem (MCP) solver returns one solution to this system (when it exists).
For example, the system
x-y=0
x-y2=0
has two solution, one at x=y=0, and one at x=y=1, but GAMS code
Variables
x
y;
Equations
eq1
eq2;
eq1 .. x =e= y;
eq2 .. x =e= y**2;
model test /eq1,eq2/;
solve test using mcp;
Display x.l,y.l;
only returns the first of solution.
Is there a way to use GAMS to give an exhaustive list of solutions?
I have not managed to find any suitable methods in the GAMS documentation or on the web.
Related
I am trying to solve a set of PDEs with FiPy, but not sure how to represent the equations: see the PDEs here.
The terms give me trouble are those in red rectangle and blue rectangle. The one in red rectangle is a convection term times a function depending on x, T1 and T2; The one in blue rectangle is a transient term times another function depending on x, T1 and T2.
I couldn't find any example including this kind of terms. Could you please give me some suggestions, or share me an example?
I appreciate any help. Thanks in advance.
This question is similar to cellvariable*Diffusion in fipy, but for convection instead of diffusion. The solution is the same though. Use,
to convert the red terms into a convection term with a coefficient of f and a source term.
For the blue terms do exactly the same to get a transient term with a coefficient and a source term.
Edit: if we assume that then we can still use
and approximate explicitly (with f as a variable in FiPy and use grad). However, we can take it further and use,
and approximate the final explicitly.
Again, we can go even further with,
Again the last term can be solved explicitly. Depending on the form of f the last them should become more insignificant and thus, the explicitness less of an issue.
I'm implementing a library which makes use of the GSL for matrix operations. I am now at a point where I need to raise things to any power, including imaginary ones. I already have the code in place to handle negative powers of a matrix, but now I need to handle imaginary powers, since all numbers in my library are complex.
Does the GSL have facilities for doing this already, or am I about to enter loop hell trying to create an algorithm for this? I need to be able to raise not only to imaginary but also complex numbers, such as 3+2i. Having limited experience with matrices as a whole, I'm not even certain on the process for doing this by hand, much less with a computer.
Hmm I never thought the electrical engineering classes I went through would help me on here, but what do you know. So the process for raising something to a complex power is not that complex and I believe you could write something fairly easily (I am not too familiar with the library your using, but this should still work with any library that has some basic complex number functions).
First your going to need to change the number to polar coordinates (i.e 3 + 3i would become (3^2 + 3^2) ^(1/2) angle 45 degrees. Pardon the awful notation. If you are confused on the process of changing the numbers over just do some quick googling on converting from cartesian to polar.
So now that you have changed it to polar coordinates you will have some radius r at an angle a. Lets raise it to the nth power now. You will then get r^n * e^(jan).
If you need more examples on this, research the "general power rule for complex numbers." Best of luck. I hope this helps!
Just reread the question and I see you need to raise to complex as well as imaginary. Well complex and imaginary are going to be the same just with one extra step using the exponent rule. This link will quickly explain how to raise something to a complex http://boards.straightdope.com/sdmb/showthread.php?t=261399
One approach would be to compute (if possible) the logarithm of your matrix, multiply that by your (complex) exponent, and then exponentiate.
That is you could have
mat_pow( M, z) = mat_exp( z * mat_log( M));
However mat_log and even mat_exp are tricky.
In case it is still relevant to you, I have extended the capabilities of my package so that now you can raise any diagonalizable matrix to any power (including, in particular, complex powers). The name of the function is 'Matpow', and can be found in package 'powerplus'. Also, this package is for the R language, not C, but perhaps you could do your calculations in R if needed.
Edit: Version 3.0 extends capabilities to (some) non-diagonalizable matrices too.
I hope it helps!
I have a large sparse linear system generated as a part of PDE solution for flows in the form Ax=b. The condition number of matrix A is very bad - of the order 3000!. But I get expected solutions with direct solvers. So, now I want to precondition the matrix so that I can use iterative solvers and use the sparseness. I have tried Jacobi preconditioner, but it does not work well as the matrix is not diagonally dominant. I need some help in proceeding further:
1) Imagine I get an approximate solution for x (generated by one run of biconjugate gradient solver). Now can I get "inverse of A" (for preconditioning) from this, seems like it must be possible but I am unable to figure out how! i.e knowing x and b can I calculate the A inverse (which may be used as preconditioner!).
2) Any other way of preconditioning which you feel would be worth a try?
3) Any way to circumvent pre-conditioning for iterative schemes for bad condition number systems?
Thanks a lot in advance for any help. Any comments are welcome.
As far as my understanding is, it's possible to replace a look-up-table for Q-values (state-action-pair-evaluation) by a neural network for estimating these state-action pairs. I programmed a small library, which is able to propagate and backpropagate through a self-built neural network for learning wanted target-values for a certain in-out-put.
So I also found this site while googling, and googling through the whole web (as it felt for me): http://www.cs.indiana.edu/~gasser/Salsa/nn.html where the Q-learning combined with a neural network is shortly explained.
For each action, there's an extra output neuron, and the activation-value of one of these output-"units" tells me, the estimated Q-value. (One question: Is the activation value the same as the "output" of the neuron or something different?)
I used the standard sigmoid-function as activation-function, so the range of the function-values x is
0<x<1
So I thought, my target value should always be from 0.0 to 1.0 -> Question: Is that point of my understanding correct? Or did I missunderstand something about that?
If yes, there comes following problem:
The equation for calculating the target-reward / new Q-value is:
q(s,a) = q(s,a) + learningrate * (reward + discountfactor * q'(s,a) - q(s,a))
So how do I perform this equation to get the right target for the neural network, if targets should be from 0.0 to 1.0?!
How do I calculate good reward-values? Is moving toward the aim more worth it, than going away from it? (more +reward when nearing the aim than -reward for bigger distance to aim?)
I think there are some missunderstandings of mine. I hope, you can help me to answer that questions. Thank you very much!
Using a neural-network to store q-value is a good extension of table lookup. This makes it possible to use q-learning when the state space is continuous.
input layer ......
|/ \ | \|
output layer a1 a2 a3
0.1 0.2 0.9
Suppose you have 3 actions available. Above shows the outputs from the neural network using current state and learned weights. So you know a3 is the best action to go with.
Now the questions you have:
One question: Is the activation value the same as the "output" of the neuron or something different?
Yes, I think so. In the referred link, the author said:
Some of the units may also be designated output units; their activations represent the network's response.
So I thought, my target value should always be from 0.0 to 1.0 -> Question: Is that point of my understanding correct? Or did I missunderstand something about that?
If you choose sigmoid as your activation function, for sure you output will be from 0.0 to 1.0. There are different choices of activation functions, e.g., here. Sigmoid is one of the most popular choices though. I think the output value being from 0.0 to 1.0 is not a problem here. if at current time, you have only two available actions, Q(s,a1) = 0.1, Q(s,a2) = 0.9, you know that action a2 is much better than a1 with respective to q-value.
So how do I perform this equation to get the right target for the neural network, if targets should be from 0.0 to 1.0?! How do I calculate good reward-values?
I am not sure for this, but you can try to clamp the new target q-value to be between 0.0 and 1.0, i.e.,
q(s,a) = min(max(0.0, q(s,a) + learningrate * (reward + discountfactor * q'(s,a) - q(s,a))), 1.0)
Try to do some experiments for finding a proper reward value.
Is moving toward the aim more worth it, than going away from it? (more +reward when nearing the aim than -reward for bigger distance to aim?)
Normally you should give more reward when it's close to the aim if you use the classical update equation, so that the new q-value gets increased.
I have a database of items. They are for cars and similar parts (eg cam/pistons) work better than others in different combinations (eg one product will work well with another, while another combination of 2 parts may not).
There are so many possible permutations, what solutions apply to this problem?
So far, I feel that these are possible approaches (Where I have question marks, something tells me these are solutions but I am not 100% confident they are).
Neural networks (?)
Collection-based approach (selection of parts in a collection for cam, and likewise for pistons in another collection, all work well with each other)
Business rules engine (?)
What are good ways to tackle this sort of problem?
Thanks
The answer largely depends on how do you calculate 'works better'?
1) Independent values
Assuming that 'works better' function f of x combination of items x=(a,b,c,d,...) and(!) that there are no regularities that can be used to decide if f(x') is bigger or smaller then f(x) knowing only x, f(x) and x' (which could allow to find the xmax faster) you will have to calculate f for all combinations at least once.
Once you calculate it for all combinations you can sort. If you will need to look up data in a partitioned way, using SQL/RDBMS might be a good approach (for example, finding top 5 best solutions but without such and such part).
For extra points after calculating all of the results and storing them you could analyze them statistically and try to establish patterns
2) Dependent values
If you can establish some regularities (and maybe you can) regarding the values the search for the max value can be simplified and speeded up.
For example if you know that function that you try to maximize is linear combination of all the parameters then you could look into linear programming
If it is not...