Comment #23 on issue 2015 by smi...@gmail.com: Hangs attempting to solve a
system of linear equations
http://code.google.com/p/sympy/issues/detail?id=2015
It seems that the method of finding determinants of minors is actually very
good for a sparse system like this since minors of elements in a given row
that are 0 need not be computed. Here is a very rudimentary implementation
that just uses the first row in the matrix to compute the determinant. The
total time to turn the equations into a matrix system and compute the
inverse is about 0.2 seconds:
def inv(self, factored=True, det=None):
"""Return the inverse of self using the determinant method `det`.
"""
n = self.rows
d = det(self)
if d is S.Zero:
raise ValueError("Matrix det == 0; not invertible.")
ret = zeros(n)
for i in range(n):
s = (-1)**i
for j in range(n):
di = det(self.minorMatrix(i, j))
ret[j, i] = s*di
s = -s
return (ret, d) if factored else ret/d
def det_minor(self):
"""Return the determinant of self computed from minors without
introducing new nesting in products.
"""
n = self.rows
if n == 2:
return self[0,0]*self[1,1] - self[1,0]*self[0,1]
else:
args = []
sgn = 1
for i in range(n):
mi = self[0, i]
if mi:
args.append(sgn*Add(*[mi*d for d in Add.make_args(det_minor(
self.minorMatrix(0, i)))]))
sgn = -sgn
return Add(*args)
from sympy.abc import *
from sympy import *
eqs=Matrix([[b - c/d + r/d], [c*(1/g + 1/e + 1/d) - f/g - r/d], [-c/g +
f*(1/j + 1/i + 1/g) - h/i], [-f/i + h*(1/m + 1/l + 1/i) - k/m], [-h/m
+ k*(1/p + 1/o + 1/m) - n/p], [-k/p + n*(1/q + 1/p)]])
v=Matrix([f,h,k,n,b,c])
from time import time
t=time(); J = eqs.jacobian(v); r = J*v-eqs; i = inv(J, factored=False,
det=det_minor); sol=i*r; time()-t
0.1979999542236328
# check the inverse
ii,d=inv(J, det=det_minor)
(ii*J/d).expand()
Matrix([
[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 1, 0, 0, 0],
[0, 0, 0, 1, 0, 0],
[0, 0, 0, 0, 1, 0],
[0, 0, 0, 0, 0, 1]])
The solution sizes are much smaller, too, than those from comment 19:
[s.count_ops() for s in sol]
[164, 148, 138, 135, 203, 196]
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