Convert from a system of linear equations to a matrix

[Aaron Meurer]

Is there a function in SymPy that makes it easy to convert from a
system of linear equations in symbolic form to a Matrix? I'm asking
because solve_linear_system takes a Matrix as input.

Aaron Meurer

[Cristóvão Duarte Sousa]

Hi Aaron,

I've found Expr.as_coefficients_dict() method useful for that, although I had to workaround constant and single variable expressions.

Check the function lin_expr_coeffs (https://github.com/cdsousa/PyLMI-SDP/blob/master/lmi_sdp/lm.py#L27) of my PyLMI-SDP package,

to see how I used it.

[Aaron Meurer]

That's useful, we should put that in SymPy, though I personally think
it would be more useful if it just took a system and returned a
matrix.

Aaron Meurer

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[Cristóvão Duarte Sousa]

Hi Aaron,

Sorry for the late reply.
Yes, the lin_expr_coeffs function can be extended to accept a linear
equation system.
Feel free, anyone, to take it and make a PR. If I find some time I can
do it, but that will not happen soon...

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[Chris Smith]

You can just pass the system to solve and let it call solve_linear_system, can't you?  Otherwise

>>> s

(x + 2*y == 4, y/2 == -2*c)

>>> sym = (x, y)

>>> m = []

>>> rhs = []

>>> for si in s:

...   if isinstance(si, Equality): si = si.lhs - si.rhs

...   i, d = si.as_independent(*sym)

...   m.append(d)

...   rhs.append(-i)

...

>>> Matrix(m).jacobian(sym)

Matrix([

[1,   2],

[0, 1/2]])

>>> Matrix(rhs)

Matrix([

[ 4],

[-2*c]])

(This doesn't check that the system is linear, but you could just check that it is symbol-free:

>>> Matrix(m).jacobian(sym).atoms(Symbol)

set([])

[There is no free_symbols attribute for it: AttributeError: MutableDenseMatrix has no attribute free_symbols.]

[Chris Smith]

I forgot that as_independent, without the as_Add=True flag will treat Muls differently. The following will be more robust:

def eqs2matrix(eqs, syms, augment=False):

    """

    >>> s

    [x + 2*y == 4, 2*c + y/2 == 0]

    >>> eqs2matrix(s, (x, c))

    (Matrix([

    [1, 0],

    [0, 2]]), Matrix([

    [-2*y + 4],

    [    -y/2]]))

    >>> eqs2matrix([2*c*(x+y)-4],(x, y))

    (Matrix([[2*c, 2*c]]), Matrix([[4]]))

    """

    s = Matrix([si.lhs - si.rhs if isinstance(si, Equality) else si for si in eqs])

    sym = syms

    j = s.jacobian(sym)

    rhs = -(s - j*Matrix(sym))

    rhs.simplify()

    if augment:

        j.col_insert(0, rhs)

    else:

        j = (j, rhs)

    return j

[Andrei Berceanu]

Was this implemented into sympy at any point? It could be the equivalent of Mathematica's CoefficientArrays function.

[James Crist]

I just answered this on gitter earlier today, but you can just take the jacobian of the system to get its matrix form. For example:

In [1]: from sympy import *

In [2]: a, b, c, d = symbols('a, b, c, d')

In [3]: x1, x2, x3, x4 = symbols('x1:5')

In [4]: x = Matrix([x1, x2, x3, x4])

In [5]: system = Matrix([a*x1 + b*x2 + c,
   ...: c*x1 + d*x3 + 2,
   ...: c*x3 + b*x4 + a])

In [6]: A = system.jacobian(x)

In [7]: B = A*x - system

In [8]: A
Out[8]:
Matrix([
[a, b, 0, 0],
[c, 0, d, 0],
[0, 0, c, b]])

In [9]: B
Out[9]:
Matrix([
[-c],
[-2],
[-a]])

In [10]: assert A*x - B == system

The functionality I'm adding for my GSoC for linearizing a system of equations will also be able to return these matrices in a convenient form. But it's not terribly difficult to solve for these arrangements using the current functionality.

[Aaron Meurer]

That's a clever trick. I should have thought of that.

Is there any reason you let system = A*x - B instead of A*x + B? The
latter seems more natural.

Aaron Meurer

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[James Crist]

It's just the convention I'm most used to. Systems that can be expressed as A*x = B I usually solve for x, or if A isn't square, the least squares solution x. In both cases you need A and B in this form. I suppose Ax + B could seem more natural though.

[Aaron Meurer]

Oh, of course. B is on the rhs. This is probably more natural to me too.

Should we make a convenience function that does this? I think this use
of jacobian would be lost on most people.

Aaron Meurer

> https://groups.google.com/d/msgid/sympy/a9c5f7ba-1c2d-4673-a8d4-b1253c150054%40googlegroups.com.

[James Crist]

We certainly could. The question would then be what the scope of the method should be. Should it only handle systems that can be expressed as Ax = b? Or should it behave like `CoefficientArrays` mentioned above, and handle Ax + Bx^2 + Cx^3 + D = 0? Either way, I think it should error if the form can't be matched exactly (i.e. don't linearize, just express a linear, or polynomial, system as matrices).

[Aaron Meurer]

A multidimensional version of collect() would probably be the best abstraction.

Aaron Meurer

> https://groups.google.com/d/msgid/sympy/0b486c56-8a2b-48a5-a210-f49b6d39899f%40googlegroups.com.

[Adam Leeper]

Hi all-

Interested party just wondering if there is any update on this.

Cheers,

Adam

[Jason Moore]

We've just introduced a function called linsolve from the GSoC work this summer. It has the capability to do this.

Jason
moorepants.info
+01 530-601-9791

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[AMiT Kumar]

Yeah, It's linear_eq_to_matrix in sympy.solvers.solveset.

Example:

    >>> eqns = [x + 2*y + 3*z - 1, 3*x + y + z + 6, 2*x + 4*y + 9*z - 2]
    >>> A, b = linear_eq_to_matrix(eqns, [x, y, z])
    >>> A
    Matrix([
    [1, 2, 3],
    [3, 1, 1],
    [2, 4, 9]])
    >>> b
    Matrix([
    [ 1],
    [-6],
    [ 2]])

AMiT Kumar

[Adam Leeper]

Yeah, It's linear_eq_to_matrix in sympy.solvers.solveset.

Looks like it's just sympy.solveset.linear_eq_to_matrix.

Thanks!

Adam 

[Riku Kivelä]

Stupid question: How do I import this thing?

I updated sympy using Anacondas command 

conda update sympy

and this installed me sympy

sympy.__version__

Out[14]: '0.7.6.1'

I cannot import or use linear_eq_to_matrix

sympy.solveset.linear_eq_to_matrix

AttributeError: 'module' object has no attribute 'solveset'

import sympy.solvers.solveset

ImportError: No module named 'sympy.solvers.solveset'

Then I went looking into Anacondas packages (Anaconda3\pkgs\sympy-0.7.6.1-py34_0\Lib\site-packages\sympy\solvers) and there is no such file as solveset.py. On the Github page this file exists under the same version (https://github.com/sympy/sympy/tree/master/sympy/solvers). I also tried downloading that solveset.py to the directory above and that did not help. Doing that is a horrible way of doing things?

[Jason Moore]

This is only available in the development version of SymPy. It will be in the next release.

You can install the development version if you want to run that. See: http://docs.sympy.org/0.7.6/install.html for info.

Jason
moorepants.info
+01 530-601-9791

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[RCU]

Hello.
I was very happy to find this thread.

I'm using the eqs2matrix(eqs, syms, augment=False) function provided in this thread
earlier (see below), in order to take some equations I parsed and build a matrix I can
then provide to the function solve_linear_system(resEqs2Matrix, xdst, ydst) in order to
determine the unknowns xdst and ydst. (If interested see the script and input equations at
https://github.com/alexsusu/video-diff/tree/master/TransformationSpecAndCodeGen ).
The function eqs2matrix provided by Chris Smith works greatly on quite a few input
equation examples. Unfortunately, it is not able to work greatly (it does not do good
"simplification") with the following "system" of equations:
xsrc = (M11*xdst + M12*ydst + M13)/(M31*xdst + M32*ydst + M33)
ysrc = (M21*xdst + M22*ydst + M23)/(M31*xdst + M32*ydst + M33)

More exactly, I need to compute the xdst and ydst from M11..M33, xsrc and ysrc. But
the final result I obtain with the script is big and redundant. More exactly, eqs2matrix
does not take into consideration it could keep linear the expressions above if we multiply
with the respective denominators the equations. (If I do this manually, it is OK, but I
would like eqs2matrix to automatically do this "preprocessing" intelligently, since there
can be probably many non-trivial cases with denominators, etc).

I'd like to ask you if there is a better way to implement eqs2matrix to obtain
simplified results.

Thank you,
Alex

On 10/7/2015 6:25 PM, Adam Leeper wrote:
> Hi all-
>
> Interested party just wondering if there is any update on this.
>
> Cheers,
> Adam
>
> On Tuesday, June 24, 2014 at 10:01:56 AM UTC-7, Aaron Meurer wrote:
>
> A multidimensional version of collect() would probably be the best abstraction.
>
> Aaron Meurer
>

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