On 09/09/2012 02:31 AM, akhil wrote:
> Hello,
>
>
> Given a set of m linear equations in n unknowns, how do I use SAGE to
> give me the coefficient matrix*A*(size m *n) and column vector *b*(size
> m*1) such that Ax = b; where *x*(size n*1) is the vector of unknowns ?
It depends on how those equations are given. You can get the `b` vector
easy enough. Let's declare some symbols:
sage: x1,x2 = var('x1,x2')
sage: a1,a2,a3,a4 = var('a1,a2,a3,a4')
sage: b1,b2 = var('b1,b2')
Now, define one equation:
sage: eqn1 = a1*x1 + a2*x2 == b1
We can use the `rhs` method to get the right-hand-side of an equation:
sage: eqn1.rhs()
b1
If we define another equation,
sage: eqn2 = a3*x1 + a4*x2 == b2
And put both of them in a list,
sage: eqns = [eqn1,eqn2]
We can write,
sage: b = [eqn.rhs() for eqn in eqns]
Now, `b` is the vector you seek. I don't think there's a general way to
get `A` or `x` unless you input one of them yourself. Sage doesn't know
that we're solving for the x1,x2 -- we could just as well be solving for
a1,a2,a3,a4!
If you're using constants and not symbols, though, it's possible. Here's
an equation with fixed a1,a2, and b1:
sage: eqn3 = 2*x1 + 3*x2 == 5
We can use the `variables` method to get the `x` vector:
sage: x = eqn3.variables()
sage: x
(x1, x2)
If we define another equation with fixed coefficients,
eqn4 = 4*x1 + 5*x2 == 6
We can then pick out the constants and stick them in a matrix `A`,
sage: eqns = [eqn3,eqn4]
sage: A = matrix(QQ,
[[row.lhs().coefficient(xi) for xi in x] for row in eqns])
Now,
sage: A
[2 3]
[4 5]
sage: x
(x1, x2)
sage: b
[b1, b2]
as desired.
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