Hello I have a problem using Scipy's fsolve function when I have an equation system.
My code only let's me have as much variables as there are equations in my system, but we all know that in practice, you can have more equations than variables, especially when the solution is not a concrete number but a range of numbers, an additional equation can help narrow down the "haystack" so to speak.
How to insert more equations than variables is my question?
Suppose I have the following non-linear system:
A/B=0.4583(3)
A/C=0.25
A/D=0.72(2)
B/C=0.54(54)
So I have the following code for this:
from scipy.optimize import *
from numpy import *
def FUNC(arg):
A,B,C,D=arg
UNK=empty((4))
UNK[0]= A/B-0.458333333333333
UNK[1]= A/C-0.25
UNK[2]= A/D-0.722222222222222
UNK[3]= B/C-0.545454545454546
return UNK
SOLVED= fsolve(FUNC, [1.0]*4)
print (SOLVED)
The problem is that I also know the following information:
B/D=1.57(57)
C/D=2.8(8)
How can I insert these 2 additional equations into my equation system?
Also how can I display the range of solutions instead of just 1 solution, it seems like fsolve only displays the 1st solution it encounters, not the entire range of possible solutions.
Note that scipy's fsolve is just a Wrapper for the MINPACK's hybrd routine:
The purpose of HYBRD is to find a zero of a system of N non-linear functions in N variables by a modification of the Powell hybrid method. The user must provide a subroutine which calculates the functions. The Jacobian is then calculated by a forward-difference approximation.
If you want to solve a system of 6 equations, your function FUNC needs to be a function of 6 variables. We could do it like this:
import numpy as np
from scipy.optimize import fsolve
def FUNC(arg):
A, B, C, D, E, F = arg
UNK=np.empty((6))
UNK[0]= A/B-0.458333333333333
UNK[1]= A/C-0.25
UNK[2]= A/D-0.722222222222222
UNK[3]= B/C-0.545454545454546
UNK[4]= B/D-1.575757575757575
UNK[5]= C/D-2.888888888888888
return UNK
fsolve(FUNC, x0=[1.0]*6)
It isn't possible to get all solutions with fsolve. But you could try different initial points x0 with the hope to get different solutions.
Related
I want to solve the following optimization problem using Gekko in python 3.7 window version.
Original Problem
Here, x_s are continuous variables, D and Epsilon are deterministic and they are also parameters.
However, since minimization function exists in the objective function, I remove it using binary variables(z1, z2) and then the problem becomes MINLP as follows.
Modified problem
With Gekko,
(1) Can both original problem & modified problem be solved?
(2) How can I code summation in the objective function and also D & epsilon which are parameters in Gekko?
Thanks in advance.
Both problems should be feasible with Gekko but the original appears easier to solve. Here are a few suggestions for the original problem:
Use m.Maximize() for the objective
Use sum() for the inner summation and m.sum() for outer summation for the objective function. I switch to m.sum() when the summation would create an expression that is over 15,000 characters. Using sum() creates one long expression and m.sum() breaks the summation into pieces but takes longer to compile.
Use m.min3() for the min(Dt,xs) terms or slack variables s with x[i]+s[i]=D[i]. It appears that Dt (size 30) is an upper bound, but it has different dimensions that xs (size 100). Slack variables are much more efficient than using binary variables.
D = np.array(100)
x = m.Array(m.Var,100,lb=0,ub=2000000)
The modified problem has 6000 binary variables and 100 continuous variables. There are 2^6000 potential combinations of those variables so it may take a while to solve, even with the efficient branch and bound method of APOPT. Here are a few suggestions for the modified problem:
Use matrix multiplications when possible. Below is an example of matrix operations with Gekko.
from gekko import GEKKO
import numpy as np
m = GEKKO(remote=False)
ni = 3; nj = 2; nk = 4
# solve AX=B
A = m.Array(m.Var,(ni,nj),lb=0)
X = m.Array(m.Var,(nj,nk),lb=0)
AX = np.dot(A,X)
B = m.Array(m.Var,(ni,nk),lb=0)
# equality constraints
m.Equations([AX[i,j]==B[i,j] for i in range(ni) \
for j in range(nk)])
m.Equation(5==m.sum([m.sum([A[i][j] for i in range(ni)]) \
for j in range(nj)]))
m.Equation(2==m.sum([m.sum([X[i][j] for i in range(nj)]) \
for j in range(nk)]))
# objective function
m.Minimize(m.sum([m.sum([B[i][j] for i in range(ni)]) \
for j in range(nk)]))
m.solve()
print(A)
print(X)
print(B)
Declare z1 and z2 variables as integer type with integer=True. Here is more information on using the integer type.
Solve locally with m=GEKKO(remote=False). The processing time will be large and the public server resets connections and deletes jobs every day. Switch to local mode to avoid a potential disruption.
While using integrate in the scipy module, I notice that the values of the integral do not match with the expected value (which I computed by hand).
I have a simple function as follows:
import math
def integrand(x, L):
return (math.cos(sc.pi * x / L))**2
L=10
I = quad(integrand, 0, 1, args=(L))
I = 0.9677446418943196 # from scipy
But theoretically, the result should be 0.5519233944327677. I am quite confused as to why discrepancy exists.
The integral I am trying to find is as follows:
The scipy is correct. Your calculation is incorrect:
Note that the Decimal given matches the one obtained by scipy. This question is best suited for stackexchange.
I'm trying to investigate the behavior of the following Delayed Differential Equation using Python:
y''(t) = -y(t)/τ^2 - 2y'(t)/τ - Nd*f(y(t-T))/τ^2,
where f is a cut-off function which is essentially equal to the identity when the absolute value of its argument is between 1 and 10 and otherwise is equal to 0 (see figure 1), and Nd, τ and T are constants.
For this I'm using the package JiTCDDE. This provides a reasonable solution to the above equation. Nevertheless, when I try to add a noise on the right hand side of the equation, I obtain a solution which stabilize to a non-zero constant after a few oscillations. This is not a mathematical solution of the equation (the only possible constant solution being equal to zero). I don't understand why this problem arises and if it is possible to solve it.
I reproduce my code below. Here, for the sake of simplicity, I substituted the noise with an high-frequency cosine, which is introduced in the system of equation as the initial condition for a dummy variable (the cosine could have been introduced directly in the system, but for a general noise this doesn't seem possible). To simplify further the problem, I removed also the term involving the f function, as the problem arises also without it. Figure 2 shows the plot of the function given by the code.
from jitcdde import jitcdde, y, t
import numpy as np
from matplotlib import pyplot as plt
import math
from chspy import CubicHermiteSpline
# Definition of function f:
def functionf(x):
return x/4*(1+symengine.erf(x**2-Bmin**2))*(1-symengine.erf(x**2-Bmax**2))
#parameters:
τ = 42.9
T = 35.33
Nd = 8.32
# Definition of the initial conditions:
dt = .01 # Time step.
totT = 10000. # Total time.
Nmax = int(totT / dt) # Number of time steps.
Vt = np.linspace(0., totT, Nmax) # Vector of times.
# Definition of the "noise"
X = np.zeros(Nmax)
for i in range(Nmax):
X[i]=math.cos(Vt[i])
past=CubicHermiteSpline(n=3)
for time, datum in zip(Vt,X):
regular_past = [10.,0.]
past.append((
time-totT,
np.hstack((regular_past,datum)),
np.zeros(3)
))
noise= lambda t: y(2,t-totT)
# Integration of the DDE
g = [
y(1),
-y(0)/τ**2-2*y(1)/τ+0.008*noise(t)
]
g.append(0)
DDE = jitcdde(g)
DDE.add_past_points(past)
DDE.adjust_diff()
data = []
for time in np.arange(DDE.t, DDE.t+totT, 1):
data.append( DDE.integrate(time)[0] )
plt.plot(data)
plt.show()
Incidentally, I noticed that even without noise, the solution seems to be discontinuous at the point zero (y is set to be equal to zero for negative times), and I don't understand why.
As the comments unveiled, your problem eventually boiled down to this:
step_on_discontinuities assumes delays that are small with respect to the integration time and performs steps that are placed on those times where the delayed components points to the integration start (0 in your case). This way initial discontinuities are handled.
However, implementing an input with a delayed dummy variable introduces a large delay into the system, totT in your case.
The respective step for step_on_discontinuities would be at totT itself, i.e., after the desired integration time.
Thus when you reach for time in np.arange(DDE.t, DDE.t+totT, 1): in your code, DDE.t is totT.
Therefore you have made a big step before you actually start integrating and observing which may seem like a discontinuity and lead to weird results, in particular you do not see the effect of your input, because it has already “ended” at this point.
To avoid this, use adjust_diff or integrate_blindly instead of step_on_discontinuities.
I am trying to create a Fibonacci number with high n by using matrix exponential but it gives me negative result. I have tried to change the integer objects but failed.
import numpy as np
def matrixmul(a,n):
a=np.array([[1,1],[1,0]])
return ((np.array([1,1],[1,0], dtype=np.object))**n)
matrixMul(a,100)
my output is
array([[-1869596475, -980107325],
[ -980107325, -889489150]])
but it was wrong. there should not be any negative number.
It's hard to answer on your question. Your code have some bugs:
You haven't initialize a
Also name of defined function is different than used (python is case-sensitive)
Then in function you are not using a (because it is not in return)
And most important thing is that you can not use **n too get exponential of matrix. Instead you can try to find right function in scipy library. Probably expm() function can be right for this perpuse.
I'm trying to define a complex custom likelihood function using pymc3. The likelihood function involves a lot of iteration, and therefore I'm trying to use theano's scan method to define iteration directly within theano. Here's a greatly simplified example that illustrates the challenge that I'm facing. The (fake) likelihood function I'm trying to define is simply the sum of two pymc3 random variables, p and theta. Of course, I could simply return p+theta, but the actual likelihood function I'm trying to write is more complicated, and I believe I need to use theano.scan since it involves a lot of iteration.
import pymc3 as pm
from pymc3 import Model, Uniform, DensityDist
import theano.tensor as T
import theano
import numpy as np
### theano test
theano.config.compute_test_value = 'raise'
X = np.asarray([[1.0,2.0,3.0],[1.0,2.0,3.0]])
### pymc3 implementation
with Model() as bg_model:
p = pm.Uniform('p', lower = 0, upper = 1)
theta = pm.Uniform('theta', lower = 0, upper = .2)
def logp(X):
f = p+theta
print("f",f)
get_ll = theano.function(name='get_ll',inputs = [p, theta], outputs = f)
print("p keys ",p.__dict__.keys())
print("theta keys ",theta.__dict__.keys())
print("p name ",p.name,"p.type ",p.type,"type(p)",type(p),"p.tag",p.tag)
result=get_ll(p, theta)
print("result",result)
return result
y = pm.DensityDist('y', logp, observed = X) # Nx4 y = f(f,x,tx,n | p, theta)
When I run this, I get the error:
TypeError: ('Bad input argument to theano function with name "get_ll" at index 0(0-based)', 'Expected an array-like object, but found a Variable: maybe you are trying to call a function on a (possibly shared) variable instead of a numeric array?')
I understand that the issue occurs in line
result=get_ll(p, theta)
because p and theta are of type pymc3.TransformedRV, and that the input to a theano function needs to be a scalar number of a simple numpy array. However, a pymc3 TransformedRV does not seem to have any obvious way of obtaining the current value of the random variable itself.
Is it possible to define a log likelihood function that involves the use of a theano function that takes as input a pymc3 random variable?
The problem is that your th.function get_ll is a compiled theano function, which takes as input numerical arrays. Instead, pymc3 is sending it a symbolic variable (theano tensor). That's why you're getting the error.
As to your solution, you're right in saying that just returning p+theta is the way to go. If you have scans and whatnot in your logp, then you would return the scan variable of interest; there is no need to compile a theano function here. For example, if you wanted to add 1 to each element of a vector (as an impractical toy example), you would do:
def logp(X):
the_sum, the_sum_upd = th.scan(lambda x: x+1, sequences=[X])
return the_sum
That being said, if you need gradients, you would need to calculate your the_sum variable in a theano Op and provide a grad() method along with it (you can see a toy example of that on the answer here). If you do not need gradients, you might be better off doing everything in python (or C, numba, cython, for performance) and using the as_op decorator.