How to expand matrix expressions in SymPy?

Question

In SymPy, I am trying to perform a matrix multiplication and expand it afterwards. However, SymPy does not seem to support the expansion of matrix expressions. For example, here is the 4th order Runge-Kutta (RK4) for matrices:

from sympy import init_session
init_session()
from sympy import *

A = MatrixSymbol('A', 3, 3)
x = MatrixSymbol('x', 3, 1)
dt = symbols('dt')

k1 = A*x
k2 = A*(x + S(1)/2*k1*dt)
k3 = A*(x + S(1)/2*k2*dt)
k4 = A*(x + k3*dt)
final = dt*S(1)/6*(k1 + 2*k2 + 2*k3 + k4)
final.expand()

which produces the result

Traceback (most recent call last)
<ipython-input-38-b3ff67883c61> in <module>()
     12 final = dt*1/6*(k1+2*k2+2*k3+k4)
     13 
---> 14 final.expand()

AttributeError: 'MatMul' object has no attribute 'expand'

I hope the expression can be expanded just like the scalar variant:

A,x,dt = symbols('A x dt')
k1 = A*x
k2 = A*(x+k1*dt*S(1)/2)
k3 = A*(x+k2*dt*S(1)/2)
k4 = A*(x+k3*dt)
final = x+dt*(S(1)/6)*(k1+k2+k3+k4)
collect(expand((final)),x)

with the result:

x*(A**4*dt**4/24 + A**3*dt**3/8 + A**2*dt**2/3 + 2*A*dt/3 + 1)

Is it possible to alter a matrix expression likewise?

nicoguaro's answer takes the error away, but expands the the whole expression into one matrix. As illustrated with the scalar example, not what I'm looking for.

It ought to work. I opened https://github.com/sympy/sympy/issues/10360. — asmeurer, Jan 05 '16 at 21:53

smichr · Answer 1 · 2019-07-09T19:58:44.650

Matrix(final) creates and explicit Matrix with your individual equations. It may be convenient to leave them in the matrix so that whatever you do to one entry can be done to all. To do so, define the manipulation that you want to do as a function argument for applyfunc:

>>> ex = Matrix(final)
>>> ex = ex.applyfunc(expand)
>>> ex = ex.applyfunc(lambda i: collect(i, dt))
...

For printing economy I use Matrices with compact symbol entries to compute the same and then run cse over the simplified matrix to give:

>>> A = Matrix(3, 3, var('a:3:3'))
>>> x = Matrix(3, 1, var('x:3'))
>>> dt = symbols('dt')
>>> k1 = A*x
>>> k2 = A*(x + S(1)/2*k1*dt)
>>> k3 = A*(x + S(1)/2*k2*dt)
>>> k4 = A*(x + k3*dt)
>>> final = dt*S(1)/6*(k1 + 2*k2 + 2*k3 + k4)
>>> eqi = Matrix(final)
>>> print(cse(eqi.applyfunc(simplify)))
([(x3, 4*x0), (x4, a01*x1), (x5, a02*x2), (x6, dt*(a00*x0 + x4 + x5) +
2*x0), (x7, a00*x6), (x8, a11*x1), (x9, a12*x2), (x10, dt*(a10*x0 + x8
+ x9) + 2*x1), (x11, a01*x10), (x12, a21*x1), (x13, a22*x2), (x14,
dt*(a20*x0 + x12 + x13) + 2*x2), (x15, a02*x14), (x16, dt*(x11 + x15 +
x7) + x3), (x17, a00*x16), (x18, 4*x1), (x19, a10*x6), (x20, a11*x10),
(x21, a12*x14), (x22, dt*(x19 + x20 + x21) + x18), (x23, a01*x22),
(x24, 4*x2), (x25, a20*x6), (x26, a21*x10), (x27, a22*x14), (x28,
dt*(x25 + x26 + x27) + x24), (x29, a02*x28), (x30, dt*(x17 + x23 +
x29) + x3), (x31, a10*x16), (x32, a11*x22), (x33, a12*x28), (x34,
dt*(x31 + x32 + x33) + x18), (x35, a20*x16), (x36, a21*x22), (x37,
a22*x28), (x38, dt*(x35 + x36 + x37) + x24), (x39, dt/24)], [Matrix([
[   x39*(a00*x3 + a00*x30 + a01*x34 + a02*x38 + 4*x11 + 4*x15 + 2*x17
+ 2*x23 + 2*x29 + 4*x4 + 4*x5 + 4*x7)], [  x39*(a10*x3 + a10*x30 +
a11*x34 + a12*x38 + 4*x19 + 4*x20 + 4*x21 + 2*x31 + 2*x32 + 2*x33 +
4*x8 + 4*x9)], [x39*(a20*x3 + a20*x30 + a21*x34 + a22*x38 + 4*x12 +
4*x13 + 4*x25 + 4*x26 + 4*x27 + 2*x35 + 2*x36 + 2*x37)]])])

Limited support of noncommutative expressions is also available and can help in this situation:

>>> A, x = symbols("A x", commutative=False)
>>> dt = symbols('dt')
>>> k1 = A*x
>>> k2 = A*(x + S(1)/2*k1*dt)
>>> k3 = A*(x + S(1)/2*k2*dt)
>>> k4 = A*(x + k3*dt)
>>> final = dt*S(1)/6*(k1 + 2*k2 + 2*k3 + k4)
>>> final.expand()
dt**4*A**4*x/24 + dt**3*A**3*x/6 + dt**2*A**2*x/2 + dt*A*x
>>> factor(_)
dt*A*(dt**3*A**3/24 + dt**2*A**2/6 + dt*A/2 + 1)*x

But not all simplification routines are nc aware (and this is a known issue):

>>> collect(final,x)
Traceback (most recent call last):
...
AttributeError: Can not collect noncommutative symbol

This was not the desired result. The desired result is `(A**4*dt**4/24 + A**3*dt**3/6 + A**2*dt**2/2 + A*dt + I)*x`. — Lutz Lehmann, Jul 09 '19 at 14:28
I see - you want a MatExpr to allow for expansion and collection of scalars. — smichr, Jul 09 '19 at 15:51
Or, if it is possible, declare `A` and `x` as non-commuting scalars. — Lutz Lehmann, Jul 09 '19 at 16:18

score 0 · Answer 2 · answered Jan 05 '16 at 19:03

I think that you can expand Matrix expressions. But what you have is not a matrix but the multiplication of two symbolic matrices (Matsymbols). If you turn your expression to a matrix you can get the expansion you wanted. See the extra line below

from sympy import init_session
init_session()
from sympy import *

A = MatrixSymbol('A', 3, 3)
x = MatrixSymbol('x', 3, 1)
dt = symbols('dt')

k1 = A*x
k2 = A*(x + S(1)/2*k1*dt)
k3 = A*(x + S(1)/2*k2*dt)
k4 = A*(x + k3*dt)
final = dt*S(1)/6*(k1 + k2 + k3 + k4)
Matrix(final).expand()

Thanks for you answer!, This does not give an error indeed, but it is not what I'm looking for, I hope that the expression can be simplified, expanded, and collected just as the scalar variant (I have added the scalar variant to the question) — D.Thomas, Jan 06 '16 at 07:49

How to expand matrix expressions in SymPy?

2 Answers2