Dot product of tensors is not computed in parallel

• To: mathgroup at smc.vnet.net
• Subject: [mg83224] Dot product of tensors is not computed in parallel
• From: chmarkakis at gmail.com
• Date: Thu, 15 Nov 2007 05:42:10 -0500 (EST)

```For tensors of rank 2 or lower (matrices and vectors), the Dot command
is executed very efficiently in parallel. Here is an example:

Define two square matrices of real numbers:

In:    m1 = Table[Random[Real],{256},{256}];
m2 = Table[Random[Real],{256},{256}];

In a machine with 2 or more CPU cores, compute their dot product (this
machine has 8 cores). The sum of CPU time used in all cores is:

In:     Tsingle = Timing[m1.m2][[1]]
Out:  0.010999 Second

The actual time that passed is:

In:     Tparallel = AbsoluteTiming[m1.m2][[1]]
Out:   0.001546 Second

For large matrices, the ratio is close to the number of cores (8):
In:     Tsingle / Tparallel
Out:   7.114489004

But if we try the same when m1 is a rank 3 tensor, there is no
improvement:

In:    m1 = Table[Random[Real],{256},{256},{256}];
m2 = Table[Random[Real],{256},{256}];

In:     Tsingle=Timing[m1.m2][[1]]
Out:  8.272743 Second

In:     Tparallel=AbsoluteTiming[m1.m2][[1]]
Out:   8.264623 Second

Now the ratio is just 1, i.e. there is no parallelization for tensors!
In:     Tsingle / Tparallel
Out:   1.000982501

Is there an efficient way to do the computation of dot products of
tensors (of rank higher than two) in parallel?

```

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