Simple Equation Substitution Fails

[Matt]

I am trying to apply a lot of multivariate derivatives to a complex vector and I'm attempting to break apart my answer into sub-expressions to reduce the number of terms.

To support this in Sage 7.3,  I need to identify subexpressions that can be simplified back into my original set of variables.

In this toy example I have a radius variable r,  the norm of the vector (x,y,z) .  A simple direct substitution for r works, but it fails when even the simplest operations are performed on the variables.

An example follows:

var('x y z', domain='real')  

var('r', domain='positive')  

r2 = x^2 + y^2 + z^2

show(r2.subs(r2 == r^2))

q = (r2 - 3 * r^2)/ (r^3)

show(q)

qr = q.subs(r2 == r^2)

show(qr.simplify())

The result is:

r2

−3r2−x2−y2−z2r3

−3r2−x2−y2−z2r3

So the first substitution of x^2+y^2+z^2 directly for r^2 works fine,  but the second case,  which involves a simple rational combination of terms fails miserably.  

Is there any way to make this work?  Apparently subs is unable to identify the simplest subexpressions.

[slelievre]

2016-09-07 09:55:41 UTC+2, Matt:

> I am trying to apply a lot of multivariate derivatives to a complex vector

> and I'm attempting to break apart my answer into sub-expressions to reduce

> the number of terms.

>

> To support this in Sage 7.3, I need to identify subexpressions that can

> be simplified back into my original set of variables.

>

> In this toy example I have a radius variable r, the norm of the vector (x,y,z).

> A simple direct substitution for r works, but it fails when even the simplest

> operations are performed on the variables.

>

> An example follows:

>

> [...]

>

> So the first substitution of x^2+y^2+z^2 directly for r^2 works fine,

> but the second case, which involves a simple rational combination of

> terms fails miserably.

>

> Is there any way to make this work? Apparently subs is unable to identify

> the simplest subexpressions.

Let's look at this example more closely. I'm using the Sage REPL

and avoiding "show" to make things easier to copy and paste.

    $ sage -v

    SageMath version 7.3, Release Date: 2016-08-04

    $ sage -q

    sage: x, y, z = SR.var('x y z', domain='real')

    sage: r = SR.var('r', domain='positive')

    sage: r2 = x^2 + y^2 + z^2

    sage: r2.subs(r2 == r^2)

    r^2

    sage: q = (r2 - 3 * r^2)/ (r^3)

    sage: q

    -(3*r^2 - x^2 - y^2 - z^2)/r^3

    sage: qr = q.subs(r2 == r^2)

    sage: qr

    -(3*r^2 - x^2 - y^2 - z^2)/r^3

    sage: qr.simplify()

    -(3*r^2 - x^2 - y^2 - z^2)/r^3

The problem is that subtracting `r2 - 3 * r^2` yields an expression

whose expression tree no longer contains `(x^2 + y^2 + z^2)`. Indeed:

    sage: q.operands()

    [3*r^2 - x^2 - y^2 - z^2, r^(-3), -1]

    sage: a, b, c = q.operands()

    sage: a

    3*r^2 - x^2 - y^2 - z^2

    sage: a.operands()

    [3*r^2, -x^2, -y^2, -z^2]

One workaround in this case is as follows:

    sage: q1 = q.subs({r^2: r2})

    sage: q1

    -2*(x^2 + y^2 + z^2)/r^3

    sage: q2 = q1.subs({r2: r^2})

    sage: q2

    -2/r

[Ralf Stephan]

On Wednesday, September 7, 2016 at 11:10:48 AM UTC+2, slelievre wrote:

The problem is that subtracting `r2 - 3 * r^2` yields an expression

whose expression tree no longer contains `(x^2 + y^2 + z^2)`.

I consider it a bug and an equivalent case is substituting in the

denominator, the fix of which needs review:

https://trac.sagemath.org/ticket/21071

sage: ((1+x^2)/x^2).subs({x^2: 42})
43/x^2

Thanks for the report.