I have a rank three n-dim tensor A. For two of the axes I want to perform v^t A v (a quadratic form with scalar output, where v is a vector). The final output should be a vector. I also need to compute the derivative of this with respect to v. This involves symmetrizing and matrix-vector multiplication (2 sym(A)v using two axes of A only, which gives a vector) with the final result being a matrix.
Is there something elegant I can do short of extending numpy with fortrash routines? Perhaps something like SciPy's 3-d tensor ops but for n-dim?
Thanks in advance for any tips, - d
PS One more dumb question: I just installed the ScientificPython-2.4.1 rpm on my reincarnated Mandrake linux machine running python2.2. Do I need to do something to configure it? My scripts aren't finding things (e.g. indexing.py).
I have a rank three n-dim tensor A. For two of the axes I want to perform v^t A v (a quadratic form with scalar output, where v is a vector). The final output should be a vector. I also need to compute the derivative of this with respect to v. This involves symmetrizing and matrix-vector multiplication (2 sym(A)v using two axes of A only, which gives a vector) with the final result being a matrix.
I'm not exactly sure what you mean by a rank three n-dim tensor (in Numeric the rank is the number of dimensions in the array).
But, I think you can accomplish what you desire in two different ways:
1) If A has N dimensions and you want to perform the reduction over axis I'll label a1 and a2 (that is a1 is the axis for the v^t*A sum while a2 is the axis for the A*v sum). Then I think this should work (if a2 > a1).
ex1 = [Numeric.NewAxis]*(N-1) ex1[a1] = slice(None) ex2 = [Numeric.NewAxis]*N ex2[a2] = slice(None)
Nar = Numeric.add.reduce result = Nar(v[ex1]*Nar(A*v[ex2],axis=a2),axis=a1)
# I think you need a recent version of Numeric for the axis keyword # to be defined here. Otherwise, just pass a2 and a1 as arguments # without the keyword.
2) Using dot (requires transposing the matrix) as dot only operates over certain dimensions --- I would not try this.
-Travis Oliphant
"Dr. Dmitry Gokhman" gokhman@sphere.math.utsa.edu writes:
I have a rank three n-dim tensor A. For two of the axes I want to perform v^t A v (a quadratic form with scalar output, where v is a vector). The final output should be a vector. I also need to compute the derivative of this with respect to v. This involves symmetrizing and matrix-vector multiplication (2 sym(A)v using two axes of A only, which gives a vector) with the final result being a matrix.
Whenever dealing with somewhat more complex operations of this time, I think it's best to go back to the basic NumPy functionality rather then figuring out if there happens to be a function that magically does it. In this case, assuming that the first axis of A is the one that is not summed over:
sum( sum(A*v[NewAxis, NewAxis, :], -1) * v[NewAxis, :], -1)
The idea is to align v with one of the dimensions of A, then multiply elementwise and sum over the common axis. Note that the first (inner) sum leaves a rank-2 array, so for the second multiplication v gets extended to rank-2 only.
PS One more dumb question: I just installed the ScientificPython-2.4.1 rpm on my reincarnated Mandrake linux machine running python2.2. Do I need to do something to configure it? My scripts aren't finding things (e.g. indexing.py).
If you took the binary RPM from my site, they might not work correctly with Mandrake, as they were made for RedHat. The source RPM should work with all RPM-based Linux distributions. There is nothing that needs configuring with an RPM.
Also note that Scientific is a package, so the correct way to import the indexing module is
import Scientific.indexing
Konrad.
participants (3)
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Dr. Dmitry Gokhman -
Konrad Hinsen -
Travis Oliphant