I need to solve a nonlinear system of equations in Python using Sympy.
For this, I wrote the code below. However, when I run this code the Python remains busy without returning any error message and, additionally, does not return the solution.
For comparison, I did the same work in Matlab and within a few seconds, the program returns two solutions for this system.
How, using Sympy, I can solve the system?
Regards.
import sympy as sym
import numpy as np
# Variables of the system
S, V, E, A, I, R = sym.symbols('S, V, E, A, I, R')
# Parameters of the system
N = sym.Symbol("N", positive = True)
mi = sym.Symbol("mi", positive = True)
v = sym.Symbol("v", positive = True)
epsilon = sym.Symbol("epsilon", positive = True)
alpha = sym.Symbol("alpha", positive = True)
gamma_as = sym.Symbol("gamma_as", positive = True)
gamma_s = sym.Symbol("gamma_s", positive = True)
gamma_a = sym.Symbol("gamma_a", positive = True)
lamb = sym.Symbol("lamb", positive = True)
tau = sym.Symbol("tau", positive = True)
beta = sym.Symbol("beta", positive = True)
x = sym.Symbol("x")
# Declaration of the system equations
system = [mi*N - v*S + R - (beta*(A+I)/N)*S - mi*S,\
v*S - (1-epsilon)*(beta*(A+I)/N)*V - mi*V,\
(beta*(A+I)/N)*S + (1-epsilon)*(beta*(A+I)/N)*V - sym.exp(-mi*tau)*(beta*(A+I)/N)*S - mi*E,\
alpha*sym.exp(-mi*tau)*(beta*(A+I)/N)*S - (gamma_as + gamma_a + mi)*A,\
(1-alpha)*sym.exp(-mi*tau)*(beta*(A+I)/N)*S + gamma_as*A - (gamma_s + mi)*I,\
gamma_a*A + gamma_s*I - (1+mi)*R]
# Solution
solution_set = sym.nonlinsolve(system, [S, V, E, A, I, R])
pyS, pyV, pyE, pyA, pyI, pyR = solution_set[0]
````
SymPy generally solves a system of polynomial equations like this using Groebner bases. To compute the Groebner basis SymPy needs to identify each of the equations as a polynomial in the given unknowns with coefficients in a computable field (a "domain"). Your coefficients involve both mi and exp(-mi*tau) which SymPy's construct_domain doesn't like so it gives up constructing a computable domain and uses the "EX" domain instead which is very slow.
The solution then is to replace exp(mi*tau) with another symbol (I'll just use tau) and then compute the Groebner basis explicitly yourself:
In [103]: rep = {exp(-mi*tau):tau}
In [104]: system2 = [eq.subs(rep) for eq in system]
In [105]: for eq in system2: pprint(eq)
S⋅β⋅(A + I)
N⋅mi + R - S⋅mi - S⋅v - ───────────
N
V⋅β⋅(1 - ε)⋅(A + I)
S⋅v - V⋅mi - ───────────────────
N
S⋅β⋅τ⋅(A + I) S⋅β⋅(A + I) V⋅β⋅(1 - ε)⋅(A + I)
-E⋅mi - ───────────── + ─────────── + ───────────────────
N N N
S⋅α⋅β⋅τ⋅(A + I)
-A⋅(γₐ + γₐₛ + mi) + ───────────────
N
S⋅β⋅τ⋅(1 - α)⋅(A + I)
A⋅γₐₛ - I⋅(γₛ + mi) + ─────────────────────
N
A⋅γₐ + I⋅γₛ - R⋅(mi + 1)
Now we could use solve or nonlinsolve but it's faster to compute and solve the Groebner basis ourselves:
In [106]: %time gb = groebner(system2, [S, V, E, A, I, R])
CPU times: user 3min 1s, sys: 100 ms, total: 3min 1s
Wall time: 3min 1s
The Groebner basis puts the system of equations into an almost solved form known as a rational univariate representation (RUR). In this case it looks like
S - a*R
V - b*R
E - c*R
A - d*R
I - e*R
R**2 + f*R + g
where the coefficients a, b, c, d, e, f, g are complicated rational functions of the symbolic parameters in the equations (alpha, beta etc). From here we can follow these steps to solve the Groebner basis:
Solve the first 5 linear equations for S, V, E, A and I in terms of R.
Solve the final quadratic equation for R giving two solutions R1 and R2.
Substitute the the solutions for R1 and R2 into the solutions for S, V, E, A and I.
Put it all together as two solution tuples.
That is:
In [115]: syms = [S, V, E, A, I, R]
In [116]: [lsol] = linsolve(gb[:-1], syms[:-1])
In [117]: R1, R2 = roots(gb[-1], R)
In [118]: sol1 = lsol.subs(R, R1) + (R1,)
In [119]: sol2 = lsol.subs(R, R2) + (R2,)
Now we have the two solution tuples in the form that would have been returned by nonlinsolve. Unfortunately the solutions are quite complicated so I won't show them in full. You can get some idea of the complexity by seeing the length of their string representations:
In [122]: print(len(str(sol1)))
794100
In [123]: print(len(str(sol2)))
27850
Now at this point it's worth considering what you actually wanted these solutions for. Maybe it's just that you wanted to substitute some explicit numeric values in for the symbolic parameters. It's worth noting here that potentially it would have been more efficient in the first place to substitute those values into the equations before attempting to solve them symbolically. If you want to see how your solutions depend on some particular parameters say just mi then you can substitute values for everything else and obtain a simpler form of the solution involving only that parameter more quickly:
In [136]: rep = {alpha:1, beta:2, epsilon:3, gamma_as:4, gamma_s:5, gamma_a:6, exp(-mi*tau):7, N:8, v
...: :9}
In [137]: system2 = [eq.subs(rep) for eq in system]
In [138]: %time solve(system2, syms)
CPU times: user 3.92 s, sys: 4 ms, total: 3.92 s
Wall time: 3.92 s
Out[138]:
⎡ ⎛ ⎛ 2
⎢⎛ 8⋅mi 72 ⎞ ⎜4⋅(mi + 5)⋅(mi + 10) 36⋅(mi + 5)⋅(mi + 10)⋅(mi + 12)⋅⎝mi + 4⋅mi
⎢⎜──────, ──────, 0, 0, 0, 0⎟, ⎜────────────────────, ─────────────────────────────────────────────
⎢⎝mi + 9 mi + 9 ⎠ ⎜ 7⋅(mi + 9) ⎛ 4 3 2
⎣ ⎝ 7⋅(mi + 9)⋅⎝3⋅mi + 38⋅mi + 161⋅mi + 718⋅mi
⎞ ⎛ 2 ⎞ ⎛ 3 2 ⎞
- 25⎠ 24⋅(mi + 1)⋅(mi + 5)⋅(mi + 10)⋅⎝mi + mi + 50⎠⋅⎝3⋅mi + 41⋅mi + 209⋅mi + 787⎠ -4⋅(mi + 1
───────, ──────────────────────────────────────────────────────────────────────────────, ──────────
⎞ ⎛ 2 ⎞ ⎛ 4 3 2 ⎞
+ 900⎠ 7⋅(mi + 12)⋅⎝mi + 4⋅mi - 25⎠⋅⎝3⋅mi + 38⋅mi + 161⋅mi + 718⋅mi + 900⎠ (mi +
⎛ 2 ⎞ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎞⎤
)⋅(mi + 5)⋅⎝mi + mi + 50⎠ -16⋅(mi + 1)⋅⎝mi + mi + 50⎠ -8⋅(3⋅mi + 25)⋅⎝mi + mi + 50⎠ ⎟⎥
───────────────────────────, ─────────────────────────────, ───────────────────────────────⎟⎥
⎛ 2 ⎞ ⎛ 2 ⎞ ⎛ 2 ⎞ ⎟⎥
12)⋅⎝mi + 4⋅mi - 25⎠ (mi + 12)⋅⎝mi + 4⋅mi - 25⎠ (mi + 12)⋅⎝mi + 4⋅mi - 25⎠ ⎠⎦
If you substitute values for all parameters then it's a lot faster:
In [139]: rep = {alpha:1, beta:2, epsilon:3, gamma_as:4, gamma_s:5, gamma_a:6, exp(-mi*tau):7, N:8, v
...: :9, mi:10}
In [140]: system2 = [eq.subs(rep) for eq in system]
In [141]: %time solve(system2, syms)
CPU times: user 112 ms, sys: 0 ns, total: 112 ms
Wall time: 111 ms
Out[141]:
⎡⎛1200 124200 5224320 -960 -256 -640 ⎞ ⎛80 72 ⎞⎤
⎢⎜────, ──────, ───────, ─────, ─────, ─────⎟, ⎜──, ──, 0, 0, 0, 0⎟⎥
⎣⎝133 55727 67459 23 23 23 ⎠ ⎝19 19 ⎠⎦
If you look at your system you will see that the 4th and 5th equations have two solutions since solving the 4th for A and substituting into the 5th gives an expression that factors as I*f(S) -- giving, for the value of A, I = 0 and S such that f(S) = 0. Judicious selection of which equation(s) to solve next and taking time to lump constants together so you don't bog down the solver gives both solutions in about 10 seconds with relatively small operation counts (relative to the results of nonlinsolve above -- 10 and 5192 operations). The process gives the same solutions for the representative values above:
def condense(eq, *x, reps=None):
"""collapse additive/multiplicative constants into single
variables, returning condensed expression and replacement
values.
Note: use of the replacement dictionary may require topological sorting
if values depend on the keys.
"""
from sympy.core.traversal import bottom_up
from sympy.simplify.radsimp import collect
from sympy.utilities.iterables import numbered_symbols
if reps is None:
reps = {}
else:
reps = {v:k for k,v in reps.items()}
con = numbered_symbols('c')
free = eq.free_symbols
def c():
while True:
rv = next(con)
if rv not in free:
return rv
def do(e):
if not e.args:
return e
e = e.func(*[do(i) for i in e.args])
isAdd=e.is_Add
if not (isAdd or e.is_Mul):
return e
if isAdd:
ee = collect(e, x, exact=None)
if ee != e:
e = do(ee)
co, id = e.as_coeff_Add() if isAdd else e.as_coeff_Mul()
i, d = id.as_independent(*x, as_Add=isAdd)
if not i.args:
return e
return e.func(co, reps.get(i, reps.setdefault(i, c())), d)
rv = do(bottom_up(eq, do))
return rv, {v: k for k, v in reps.items()}
def repsort(*replace):
"""Return sorted replacement tuples `(o, n)` such that `(o_i, n_i)`
will appear before `(o_j, n_j)` if `o_j` appears in `n_i`. An error
will be raised if `o_j` appears in `n_i` and `o_i` appears in `n_k`
if `k >= i`.
Examples
========
>>> from sympy.abc import x, y, z, a
>>> repsort((x, y + 1), (z, x + 2))
[(z, x + 2), (x, y + 1)]
>>> repsort((x, y + 1), (z, x**2))
[(z, x**2), (x, y + 1)]
>>> repsort(*Tuple((x,y+z),(y,a),(z,1/y)))
[(x, y + z), (z, 1/y), (y, a)]
Any two of the following 3 tuples will not raise an error,
but together they contain a cycle that raises an error:
>>> repsort((x, y), (y, z), (z, x))
Traceback (most recent call last):
...
raise ValueError("cycle detected")
"""
from itertools import permutations
from sympy import default_sort_key, topological_sort
free = {i for i,_ in replace}
defs, replace = sift(replace,
lambda x: x[1].is_number or not x[1].has_free(*free),
binary=True)
edges = [(i, j) for i, j in permutations(replace, 2) if
i[1].has(j[0]) and (not j[0].is_Symbol or j[0] in i[1].free_symbols)]
rv = topological_sort([replace, edges], default_sort_key)
rv.extend(ordered(defs))
return rv
def dupdate(d, s):
"""update values in d with values from s and return the combined dictionaries"""
rv = {k: v.xreplace(s) for k,v in d.items()}
rv.update(s)
return rv
# Variables of the system
syms=S, V, E, A, I, R = symbols('S, V, E, A, I, R')
# Parameters of the system
const = var('a:j k')
system = [
-A*S*c/a - I*S*c/a + R + S*(-h - j) + a*h,
A*(V*c*d/a - V*c/a) + I*(V*c*d/a - V*c/a) + S*j - V*h,
A*(-S*c*k/a + S*c/a - V*c*d/a + V*c/a) - E*h +
I*(-S*c*k/a + S*c/a - V*c*d/a + V*c/a),
A*(S*b*c*k/a - e - f - h) + I*S*b*c*k/a,
A*(-S*b*c*k/a + S*c*k/a + f) + I*(-S*b*c*k/a + S*c*k/a - g - h),
A*e + I*g + R*(-h - 1)
]
import sympy as sym
# Variables of the system
syms = S, V, E, A, I, R = sym.symbols('S, V, E, A, I, R')
# Parameters of the system
N = sym.Symbol("N", positive = True)
mi = sym.Symbol("mi", positive = True)
v = sym.Symbol("v", positive = True)
epsilon = sym.Symbol("epsilon", positive = True)
alpha = sym.Symbol("alpha", positive = True)
gamma_as = sym.Symbol("gamma_as", positive = True)
gamma_s = sym.Symbol("gamma_s", positive = True)
gamma_a = sym.Symbol("gamma_a", positive = True)
lamb = sym.Symbol("lamb", positive = True)
tau = sym.Symbol("tau", positive = True)
beta = sym.Symbol("beta", positive = True)
# Declaration of the system equations
system = [
mi*N - v*S + R - (beta*(A+I)/N)*S - mi*S,
v*S - (1-epsilon)*(beta*(A+I)/N)*V - mi*V,
(beta*(A+I)/N)*S + (1-epsilon)*(beta*(A+I)/N)*V -
sym.exp(-mi*tau)*(beta*(A+I)/N)*S - mi*E,
alpha*sym.exp(-mi*tau)*(beta*(A+I)/N)*S - (gamma_as + gamma_a + mi)*A,
(1-alpha)*sym.exp(-mi*tau)*(beta*(A+I)/N)*S + gamma_as*A - (gamma_s + mi)*I,
gamma_a*A + gamma_s*I - (1+mi)*R]
system, srep = condense(Tuple(*system), *syms)
asol = solve(system[3], A, dict=True)[0]
aeq=Tuple(*[i.xreplace(asol) for i in system])
si = solve(aeq[4], *syms, dict=True)
sol1 = dupdate(asol, si[0])
sol1 = dupdate(sol1, solve(Tuple(*system).xreplace(sol1),syms,dict=1)[0]); sol1
aeqs4 = Tuple(*[i.xreplace(si[1]) for i in aeq])
ceq, crep = condense(Tuple(*aeqs4),*syms,reps=srep)
ir = solve([ceq[0], ceq[-1]], I, R, dict=1)[0]
ve = solve([i.simplify() for i in Tuple(*ceq).xreplace(ir)], syms, dict=True)[0] # if we don't simplify to make first element 0 we get no solution -- bug?
sol2 = dupdate(asol, si[1])
sol2 = dupdate(sol2, ir)
sol2 = dupdate(sol2, ve)
crep = repsort(*crep.items())
sol1 = Dict({k:v.subs(crep) for k,v in sol1.items()}) # 10 ops
sol2 = Dict({k:v.subs(crep) for k,v in sol2.items()}) # 5192 ops
Test for specific values (as above):
>>> rep = {alpha: 1, beta: 2, epsilon: 3, gamma_as: 4, gamma_s: 5,
... gamma_a: 6, exp(-mi*tau): 7, N: 8, v: 9, mi: 10}
...
>>> sol1.xreplace(rep)
{A: 0, E: 0, I: 0, R: 0, S: 80/19, V: 72/19}
>>> sol2.xreplace(rep)
{A: -960/23, E: 89280/851, I: -256/23,
R: -640/23, S: 1200/133, V: -124200/4921}
Of course, it took time to find this path to the solution. But if the solver could make better selections of what to solve (rather than trying to get the Groebner basis of the whole system) the time for obtaining a solution from SymPy could be greatly reduced.
im trying to write a program that gives the integral approximation of e(x^2) between 0 and 1 based on this integral formula:
Formula
i've done this code so far but it keeps giving the wrong answer (Other methods gives 1.46 as an answer, this one gives 1.006).
I think that maybe there is a problem with the two for cycles that does the Riemman sum, or that there is a problem in the way i've wrote the formula. I also tried to re-write the formula in other ways but i had no success
Any kind of help is appreciated.
import math
import numpy as np
def f(x):
y = np.exp(x**2)
return y
a = float(input("¿Cual es el limite inferior? \n"))
b = float(input("¿Cual es el limite superior? \n"))
n = int(input("¿Cual es el numero de intervalos? "))
x = np.zeros([n+1])
y = np.zeros([n])
z = np.zeros([n])
h = (b-a)/n
print (h)
x[0] = a
x[n] = b
suma1 = 0
suma2 = 0
for i in np.arange(1,n):
x[i] = x[i-1] + h
suma1 = suma1 + f(x[i])
alfa = (x[i]-x[i-1])/3
for i in np.arange(0,n):
y[i] = (x[i-1]+ alfa)
suma2 = suma2 + f(y[i])
z[i] = y[i] + alfa
int3 = ((b-a)/(8*n)) * (f(x[0])+f(x[n]) + (3*(suma2+f(z[i]))) + (2*(suma1)))
print (int3)
I'm not a math major but I remember helping a friend with this rule for something about waterplane area for ships.
Here's an implementation based on Wikipedia's description of the Simpson's 3/8 rule:
# The input parameters
a, b, n = 0, 1, 10
# Divide the interval into 3*n sub-intervals
# and hence 3*n+1 endpoints
x = np.linspace(a,b,3*n+1)
y = f(x)
# The weight for each points
w = [1,3,3,1]
result = 0
for i in range(0, 3*n, 3):
# Calculate the area, 4 points at a time
result += (x[i+3] - x[i]) / 8 * (y[i:i+4] * w).sum()
# result = 1.4626525814387632
You can do it using numpy.vectorize (Based on this wikipedia post):
a, b, n = 0, 1, 10**6
h = (b-a) / n
x = np.linspace(0,n,n+1)*h + a
fv = np.vectorize(f)
(
3*h/8 * (
f(x[0]) +
3 * fv(x[np.mod(np.arange(len(x)), 3) != 0]).sum() + #skip every 3rd index
2 * fv(x[::3]).sum() + #get every 3rd index
f(x[-1])
)
)
#Output: 1.462654874404461
If you use numpy's built-in functions (which I think is always possible), performance will improve considerably:
a, b, n = 0, 1, 10**6
x = np.exp(np.square(np.linspace(0,n,n+1)*h + a))
(
3*h/8 * (
x[0] +
3 * x[np.mod(np.arange(len(x)), 3) != 0].sum()+
2 * x[::3].sum() +
x[-1]
)
)
#Output: 1.462654874404461
I have two functions that compute the same metric. One ends up using a list comprehension to cycle through a calculation, the other uses only numpy tensor operations. The functions take in a (N, 3) array, where N is the number of points in 3D space. When N <~ 3000 the tensor function is faster, when N >~ 3000 the list comprehension is faster. Both seem to have linear time complexity in terms of N i.e two time-N lines cross at N=~3000.
def approximate_area_loop(section, num_area_divisions):
n_a_d = num_area_divisions
interp_vectors = get_section_interp_(section)
a1 = section[:-1]
b1 = section[1:]
a2 = interp_vectors[:-1]
b2 = interp_vectors[1:]
c = lambda u: (1 - u) * a1 + u * a2
d = lambda u: (1 - u) * b1 + u * b2
x = lambda u, v: (1 - v) * c(u) + v * d(u)
area = np.sum([np.linalg.norm(np.cross((x((i + 1)/n_a_d, j/n_a_d) - x(i/n_a_d, j/n_a_d)),\
(x(i/n_a_d, (j +1)/n_a_d) - x(i/n_a_d, j/n_a_d))), axis = 1)\
for i in range(n_a_d) for j in range(n_a_d)])
Dt = section[-1, 0] - section[0, 0]
return area, Dt
def approximate_area_tensor(section, num_area_divisions):
divisors = np.linspace(0, 1, num_area_divisions + 1)
interp_vectors = get_section_interp_(section)
a1 = section[:-1]
b1 = section[1:]
a2 = interp_vectors[:-1]
b2 = interp_vectors[1:]
c = np.multiply.outer(a1, (1 - divisors)) + np.multiply.outer(a2, divisors) # c_areas_vecs_divs
d = np.multiply.outer(b1, (1 - divisors)) + np.multiply.outer(b2, divisors) # d_areas_vecs_divs
x = np.multiply.outer(c, (1 - divisors)) + np.multiply.outer(d, divisors) # x_areas_vecs_Divs_divs
u = x[:, :, 1:, :-1] - x[:, :, :-1, :-1] # u_areas_vecs_Divs_divs
v = x[:, :, :-1, 1:] - x[:, :, :-1, :-1] # v_areas_vecs_Divs_divs
sub_area_norm_vecs = np.cross(u, v, axis = 1) # areas_crosses_Divs_divs
sub_areas = np.linalg.norm(sub_area_norm_vecs, axis = 1) # areas_Divs_divs (values are now sub areas)
area = np.sum(sub_areas)
Dt = section[-1, 0] - section[0, 0]
return area, Dt
Why does the list comprehension version work faster at large N? Surely the tensor version should be faster? I'm wondering if it's something to do with the size of the calculations meaning it's too big to be done in cache? Please ask if I haven't included enough information, I'd really like to get to the bottom of this.
The bottleneck in the fully vectorized function was indeed in the np.linalg.norm as #hpauljs comment suggested.
Norm was used only to get the magnitude of all the vectors contained in axis 1. A much simpler and faster method was to just:
sub_areas = np.sqrt((sub_area_norm_vecs*sub_area_norm_vecs).sum(axis = 1))
This gives exactly the same results and sped up the code by up to 25 times faster than the loop implementation (even when the loop doesn't use linalg.norm either).