Using the Coefficient command with xCoba.

[JBates]

Hi, everyone! I'm a new user to xAct working my way up the learning
curve, but I've run into a problem I haven't been able to solve
involving trying to break up a large tensor expression using the
Coefficient command. The relevant portion of the code is this:

NKR[a_, b_, i_, j_] = NKX[a, b, i, j];
Do[
Print[ToValues[A[{n, -Zs}]]];
ComponentValue[B[{n, Zs}, a, b, i, j],
Coefficient[NKR[a, b, i, j], ToValues[A[{n, -Zs}]]]];
Tmp[a_, b_, i_, j_] =
ToValues[A[{n, -Zs}]]*ToValues[B[{n, Zs}, a, b, i, j]] // Expand;
NKR[a_, b_, i_, j_] = NKR[a, b, i, j] - Tmp[a, b, i, j];,
{n, 1, 2}];

The NKX tensor is the one I would like to break up into the form
NKX=A_n * B^n, where A is the list of coefficients and B is the list
of associated tensors. The problem is that NKX is a very large
expression, and as written I cannot get this to evaluate in a
reasonable timeframe.

However, if I don't define NKR as a tensor, and and write the
expression as

NKR = NKX[a, b, i, j];
Do[
Print[ToValues[A[{n, -Zs}]]];
ComponentValue[B[{n, Zs}, a, b, i, j],
Coefficient[NKR, ToValues[A[{n, -Zs}]]]];
Tmp[a_, b_, i_, j_] =
ToValues[A[{n, -Zs}]]*ToValues[B[{n, Zs}, a, b, i, j]] // Expand;
NKR = NKR - Tmp[a, b, i, j];,
{n, 1, 2}];

then it works fine. (This way takes about ten minutes to evaluate on
my system.) Unfortunately, when I do things this way, xCoba no longer
recognizes the B elements as tensors.

Is there a way around this?

[JMM]

Hi!

On Jun 4, 1:24 am, JBates <bat...@wfu.edu> wrote:
> Hi, everyone!  I'm a new user to xAct working my way up the learning
> curve, but I've run into a problem I haven't been able to solve
> involving trying to break up a large tensor expression using the
> Coefficient command.  The relevant portion of the code is this:
>
> NKR[a_, b_, i_, j_] = NKX[a, b, i, j];
> Do[
>   Print[ToValues[A[{n, -Zs}]]];
>   ComponentValue[B[{n, Zs}, a, b, i, j],
>    Coefficient[NKR[a, b, i, j], ToValues[A[{n, -Zs}]]]];
>   Tmp[a_, b_, i_, j_] =
>    ToValues[A[{n, -Zs}]]*ToValues[B[{n, Zs}, a, b, i, j]] // Expand;
>   NKR[a_, b_, i_, j_] = NKR[a, b, i, j] - Tmp[a, b, i, j];,
>   {n, 1, 2}];

I don't understand why you use ToValues in the previous code. Is the A
tensor explicitly present in your NKR expression or not? Is the result
of ToValues[A[...]] a complicated expression? Note that Coefficient[ a
x + b x, a + b ] does not return x.

Have you tried to play with Mathematica's Collect ? It is very useful
when trying to isolate the respective coefficients of a given list of
expressions.

I see that you use NKR as temporary storage. As you mention further
down, you do not need a tensor expression for that. A simple variable
is enough.

Does n go only to 2 in your real example, or is this only an example?
If you are looking for just 2 coefficients, perhaps it is simpler to
replace by hand the respective expressions by 0 in the original full
expression.

> The NKX tensor is the one I would like to break up into the form
> NKX=A_n * B^n, where A is the list of coefficients and B is the list
> of associated tensors.  The problem is that NKX is a very large
> expression, and as written I cannot get this to evaluate in a
> reasonable timeframe.
>
> However, if I don't define NKR as a tensor, and and write the
> expression as
>
> NKR = NKX[a, b, i, j];
> Do[
>   Print[ToValues[A[{n, -Zs}]]];
>   ComponentValue[B[{n, Zs}, a, b, i, j],
>    Coefficient[NKR, ToValues[A[{n, -Zs}]]]];
>   Tmp[a_, b_, i_, j_] =
>    ToValues[A[{n, -Zs}]]*ToValues[B[{n, Zs}, a, b, i, j]] // Expand;
>   NKR = NKR - Tmp[a, b, i, j];,
>   {n, 1, 2}];
>
> then it works fine.  (This way takes about ten minutes to evaluate on
> my system.)  Unfortunately, when I do things this way, xCoba no longer
> recognizes the B elements as tensors.

I'm confused again. ComponentValue expects all indices to be basis-
indices. Otherwise we do not really have a component. I mean, your
indices a,b,i,j must also be basis-indices, not abstract indices.

> Is there a way around this?

My impression is that you are not using ToValues and/or ComponentValue
correctly, but I cannot really tell with the information provided.

Cheers,
Jose.

[magma]

Hi JBates,
I would suggest you upload the WHOLE notebook and some cells
describing what you would like to achieve.
It is just easier for you and much clearer for everybody else.