<s>
Tensor	B-General_Concept
software	I-General_Concept
is	O
a	O
class	O
of	O
mathematical	B-Application
software	I-Application
designed	O
for	O
manipulation	O
and	O
calculation	O
with	O
tensors	B-Device
.	O
</s>
<s>
SPLATT	O
is	O
an	O
open	B-Application
source	I-Application
software	I-Application
package	O
for	O
high-performance	O
sparse	O
tensor	B-Device
factorization	O
.	O
</s>
<s>
SPLATT	O
ships	O
a	O
stand-alone	O
executable	O
,	O
C/C	O
++	O
library	O
,	O
and	O
Octave/MATLAB	O
API	B-Application
.	O
</s>
<s>
Cadabra	B-Language
is	O
a	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
(	O
CAS	O
)	O
designed	O
specifically	O
for	O
the	O
solution	O
of	O
problems	O
encountered	O
in	O
field	O
theory	O
.	O
</s>
<s>
It	O
has	O
extensive	O
functionality	O
for	O
tensor	B-Device
polynomial	O
simplification	O
including	O
multi-term	O
symmetries	O
,	O
fermions	O
and	O
anti-commuting	O
variables	O
,	O
Clifford	O
algebras	O
and	O
Fierz	O
transformations	O
,	O
implicit	O
coordinate	O
dependence	O
,	O
multiple	O
index	O
types	O
and	O
many	O
more	O
.	O
</s>
<s>
Tela	O
is	O
a	O
software	O
package	O
similar	O
to	O
MATLAB	B-Language
and	O
GNU	B-Language
Octave	I-Language
,	O
but	O
designed	O
specifically	O
for	O
tensors	B-Device
.	O
</s>
<s>
Tensor	B-Device
is	O
a	O
tensor	B-Device
package	O
written	O
for	O
the	O
Mathematica	O
system	O
.	O
</s>
<s>
Ricci	O
is	O
a	O
system	O
for	O
Mathematica	O
2.x	O
and	O
later	O
for	O
doing	O
basic	O
tensor	B-Device
analysis	O
,	O
available	O
for	O
free	O
.	O
</s>
<s>
TTC	O
Tools	O
of	O
Tensor	B-Device
Calculus	O
is	O
a	O
Mathematica	O
package	O
for	O
doing	O
tensor	B-Device
and	O
exterior	O
calculus	O
on	O
differentiable	O
manifolds	O
.	O
</s>
<s>
EDC	O
and	O
RGTC	O
,	O
"	O
Exterior	O
Differential	O
Calculus	O
"	O
and	O
"	O
Riemannian	O
Geometry	O
&	O
Tensor	B-Device
Calculus	O
,	O
"	O
are	O
free	O
Mathematica	O
packages	O
for	O
tensor	B-Device
calculus	O
especially	O
designed	O
but	O
not	O
only	O
for	O
general	O
relativity	O
.	O
</s>
<s>
Tensorial	O
"	O
Tensorial	O
4.0	O
"	O
is	O
a	O
general	O
purpose	O
tensor	B-Device
calculus	O
package	O
for	O
Mathematica	O
.	O
</s>
<s>
xAct	O
:	O
Efficient	O
Tensor	B-Device
Computer	O
Algebra	O
for	O
Mathematica	O
.	O
</s>
<s>
xAct	O
is	O
a	O
collection	O
of	O
packages	O
for	O
fast	O
manipulation	O
of	O
tensor	B-Device
expressions	O
.	O
</s>
<s>
GREAT	O
is	O
a	O
free	O
package	O
for	O
Mathematica	O
that	O
computes	O
the	O
Christoffel	O
connection	O
and	O
the	O
basic	O
tensors	B-Device
of	O
General	O
Relativity	O
from	O
a	O
given	O
metric	O
tensor	B-Device
.	O
</s>
<s>
GRTensorM	O
is	O
a	O
computer	B-General_Concept
algebra	I-General_Concept
package	I-General_Concept
for	O
performing	O
calculations	O
in	O
the	O
general	O
area	O
of	O
differential	O
geometry	O
.	O
</s>
<s>
MathGR	O
is	O
a	O
package	O
to	O
manipulate	O
tensor	B-Device
and	O
GR	O
calculations	O
with	O
either	O
abstract	O
or	O
explicit	O
indices	O
,	O
simplify	O
tensors	B-Device
with	O
permutational	O
symmetries	O
,	O
decompose	O
tensors	B-Device
from	O
abstract	O
indices	O
to	O
partially	O
or	O
completely	O
explicit	O
indices	O
and	O
convert	O
partial	O
derivatives	O
into	O
total	O
derivatives	O
.	O
</s>
<s>
TensoriaCalc	O
is	O
a	O
tensor	B-Device
calculus	O
package	O
written	O
for	O
Mathematica	O
9	O
and	O
higher	O
,	O
aimed	O
at	O
providing	O
user-friendly	O
functionality	O
and	O
a	O
smooth	O
consistency	O
with	O
the	O
Mathematica	O
language	O
itself	O
.	O
</s>
<s>
As	O
of	O
January	O
2015	O
,	O
given	O
a	O
metric	O
and	O
the	O
coordinates	O
used	O
,	O
TensoriaCalc	O
can	O
compute	O
Christoffel	O
symbols	O
,	O
the	O
Riemann	O
curvature	O
tensor	B-Device
,	O
and	O
Ricci	O
tensor/scalar	O
;	O
it	O
allows	O
for	O
user-defined	O
tensors	B-Device
and	O
is	O
able	O
to	O
perform	O
basic	O
operations	O
such	O
as	O
taking	O
the	O
covariant	O
derivatives	O
of	O
tensors	B-Device
.	O
</s>
<s>
OGRe	O
is	O
a	O
modern	O
free	B-License
and	I-License
open-source	I-License
Mathematica	O
package	O
for	O
tensor	B-Device
calculus	O
,	O
released	O
in	O
2021	O
for	O
Mathematica	O
12.0	O
and	O
later	O
.	O
</s>
<s>
OGRe	O
allows	O
performing	O
arbitrarily	O
complicated	O
tensor	B-Device
operations	O
,	O
and	O
automatically	O
transforms	O
between	O
index	O
configurations	O
and	O
coordinate	O
systems	O
behind	O
the	O
scenes	O
as	O
needed	O
for	O
each	O
operation	O
.	O
</s>
<s>
GRTensorII	O
is	O
a	O
computer	B-General_Concept
algebra	I-General_Concept
package	I-General_Concept
for	O
performing	O
calculations	O
in	O
the	O
general	O
area	O
of	O
differential	O
geometry	O
.	O
</s>
<s>
DifferentialGeometry	O
is	O
a	O
package	O
which	O
performs	O
fundamental	O
operations	O
of	O
calculus	O
on	O
manifolds	O
,	O
differential	O
geometry	O
,	O
tensor	B-Device
calculus	O
,	O
General	O
Relativity	O
,	O
Lie	O
algebras	O
,	O
Lie	O
groups	O
,	O
transformation	O
groups	O
,	O
jet	O
spaces	O
,	O
and	O
the	O
variational	O
calculus	O
.	O
</s>
<s>
It	O
includes	O
objects	O
from	O
general	O
relativity	O
(	O
tensors	B-Device
,	O
metrics	O
,	O
covariant	O
derivatives	O
,	O
tetrads	O
etc	O
.	O
</s>
<s>
Tensorlab	O
is	O
a	O
MATLAB	B-Language
toolbox	O
for	O
multilinear	O
algebra	O
and	O
structured	O
data	O
fusion	O
.	O
</s>
<s>
Tensor	B-Device
Toolbox	O
Multilinear	O
algebra	O
MATLAB	B-Language
software	I-Language
.	O
</s>
<s>
MPCA	O
and	O
MPCA+LDA	O
Multilinear	O
subspace	O
learning	O
software	O
:	O
Multilinear	B-Algorithm
principal	I-Algorithm
component	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
UMPCA	O
Multilinear	O
subspace	O
learning	O
software	O
:	O
Uncorrelated	O
multilinear	B-Algorithm
principal	I-Algorithm
component	I-Algorithm
analysis	I-Algorithm
.	O
</s>
<s>
Maxima	B-Language
is	O
a	O
free	O
open	B-Application
source	I-Application
general	O
purpose	O
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
which	O
includes	O
several	O
packages	O
for	O
tensor	B-Device
algebra	O
calculations	O
in	O
its	O
core	O
distribution	O
.	O
</s>
<s>
It	O
is	O
particularly	O
useful	O
for	O
calculations	O
with	O
abstract	O
tensors	B-Device
,	O
i.e.	O
,	O
when	O
one	O
wishes	O
to	O
do	O
calculations	O
without	O
defining	O
all	O
components	O
of	O
the	O
tensor	B-Device
explicitly	O
.	O
</s>
<s>
It	O
comes	O
with	O
three	O
tensor	B-Device
packages	O
:	O
</s>
<s>
itensor	O
for	O
abstract	O
(	O
indicial	O
)	O
tensor	B-Device
manipulation	O
,	O
</s>
<s>
atensor	O
for	O
algebraic	O
tensor	B-Device
manipulation	O
.	O
</s>
<s>
Tensor	B-Device
is	O
an	O
R	O
package	O
for	O
basic	O
tensor	B-Device
operations	O
.	O
</s>
<s>
rTensor	O
provides	O
several	O
tensor	B-Device
decomposition	O
approaches	O
.	O
</s>
<s>
nnTensor	O
provides	O
several	O
non-negative	O
tensor	B-Device
decomposition	O
approaches	O
.	O
</s>
<s>
ttTensor	O
provides	O
several	O
tensor-train	O
decomposition	O
approaches	O
.	O
</s>
<s>
tensorBF	O
is	O
an	O
R	O
package	O
for	O
Bayesian	O
Tensor	B-Device
decomposition	O
.	O
</s>
<s>
MTF	O
Bayesian	O
Multi-Tensor	O
Factorization	O
for	O
data	O
fusion	O
and	O
Bayesian	O
versions	O
of	O
Tensor	B-Device
PCA	O
and	O
Tensor	B-Device
CCA	O
.	O
</s>
<s>
TensorLy	O
provides	O
several	O
tensor	B-Device
decomposition	O
approaches	O
.	O
</s>
<s>
TensorDecompositions.jl	O
provides	O
several	O
tensor	B-Device
decomposition	O
approaches	O
.	O
</s>
<s>
TensorToolbox.jl	O
provides	O
several	O
tensor	B-Device
decomposition	O
approaches	O
.	O
</s>
<s>
This	O
follows	O
the	O
functionality	O
of	O
MATLAB	B-Language
Tensor	B-Device
toolbox	O
and	O
Hierarchical	O
Tucker	O
Toolbox	O
.	O
</s>
<s>
ITensors.jl	O
is	O
a	O
library	O
for	O
rapidly	O
creating	O
correct	O
and	O
efficient	O
tensor	B-Device
network	O
algorithms	O
.	O
</s>
<s>
TensorOperations.jl	O
provides	O
fast	O
tensor	B-Device
operations	O
using	O
a	O
convenient	O
Einstein	O
index	O
notation	O
.	O
</s>
<s>
TensorRules.jl	O
provides	O
a	O
macro	O
,	O
which	O
enable	O
us	O
to	O
use	O
automatic	O
differentiation	O
(	O
AD	O
)	O
libraries	O
(	O
e.g.	O
,	O
Zygote.jl	O
,	O
ForwardDiff.jl	O
)	O
with	O
@tensor	O
and	O
@tensoropt	O
macros	O
in	O
TensorOperations.jl	O
.	O
</s>
<s>
SageManifolds	B-Application
:	O
tensor	B-Device
calculus	O
on	O
smooth	O
manifolds	O
;	O
all	O
SageManifolds	B-Application
code	O
is	O
included	O
in	O
SageMath	B-Application
since	O
version	O
7.5	O
;	O
it	O
allows	O
for	O
computations	O
in	O
various	O
vector	O
frames	O
and	O
coordinate	O
charts	O
,	O
the	O
manifold	O
not	O
being	O
required	O
to	O
be	O
parallelizable	O
.	O
</s>
<s>
ND4J	O
:	O
N-dimensional	O
arrays	O
for	O
the	O
JVM	O
is	O
a	O
Java	B-Language
library	O
for	O
basic	O
tensor	B-Device
operations	O
and	O
scientific	O
computing	O
.	O
</s>
<s>
:	O
computation	O
for	O
regular	O
or	O
unstructured	O
multi-dimensional	O
tensors	B-Device
.	O
</s>
<s>
API	B-Application
inspired	O
by	O
Mathematica	O
.	O
</s>
<s>
Java	B-Language
8	O
library	O
in	O
with	O
no	O
external	O
dependencies	O
.	O
</s>
<s>
Redberry	O
is	O
an	O
open	B-Application
source	I-Application
computer	B-General_Concept
algebra	I-General_Concept
system	I-General_Concept
designed	O
for	O
symbolic	O
tensor	B-Device
manipulation	O
.	O
</s>
<s>
Redberry	O
provides	O
common	O
tools	O
for	O
expression	O
manipulation	O
,	O
generalized	O
on	O
tensorial	O
objects	O
,	O
as	O
well	O
as	O
tensor-specific	O
features	O
:	O
indices	O
symmetries	O
,	O
LaTeX-style	O
input	O
,	O
natural	O
dummy	O
indices	O
handling	O
,	O
multiple	O
index	O
types	O
etc	O
.	O
</s>
<s>
Redberry	O
is	O
written	O
in	O
Java	B-Language
and	O
provides	O
extensive	O
Groovy-based	O
programming	B-Language
language	I-Language
.	O
</s>
<s>
libxm	O
is	O
a	O
lightweight	O
distributed-parallel	O
tensor	B-Device
library	O
written	O
in	O
C	O
.	O
</s>
<s>
FTensor	O
is	O
a	O
high	O
performance	O
tensor	B-Device
library	O
written	O
in	O
C++	O
.	O
</s>
<s>
TL	O
is	O
a	O
multi-threaded	O
tensor	B-Device
library	O
implemented	O
in	O
C++	O
used	O
in	O
Dynare++	O
.	O
</s>
<s>
The	O
library	O
allows	O
for	O
folded/unfolded	O
,	O
dense/sparse	O
tensor	B-Device
representations	O
,	O
general	O
ranks	O
(	O
symmetries	O
)	O
.	O
</s>
<s>
vmmlib	O
is	O
a	O
C++	O
linear	O
algebra	O
library	O
that	O
supports	O
3-way	O
tensors	B-Device
,	O
emphasizing	O
computation	O
and	O
manipulation	O
of	O
several	O
tensor	B-Device
decompositions	O
.	O
</s>
<s>
Spartns	O
is	O
a	O
Sparse	O
Tensor	B-Device
framework	O
for	O
Common	B-Language
Lisp	I-Language
.	O
</s>
<s>
FAstMat	O
is	O
a	O
thread-safe	O
general	O
tensor	B-Device
algebra	O
library	O
written	O
in	O
C++	O
and	O
specially	O
designed	O
for	O
FEM/FVM/BEM/FDM	O
element/edge	O
wise	O
computations	O
.	O
</s>
<s>
Cyclops	O
Tensor	B-Device
Framework	O
is	O
a	O
distributed	O
memory	O
library	O
for	O
efficient	O
decomposition	O
of	O
tensors	B-Device
of	O
arbitrary	O
type	O
and	O
parallel	O
MPI+OpenMP	O
execution	O
of	O
tensor	B-Device
contractions/functions	O
.	O
</s>
<s>
TiledArray	O
is	O
a	O
scalable	O
,	O
block-sparse	O
tensor	B-Device
library	O
that	O
is	O
designed	O
to	O
aid	O
in	O
rapid	O
composition	O
of	O
high-performance	O
algebraic	O
tensor	B-Device
equation	O
.	O
</s>
<s>
libtensor	O
is	O
a	O
set	O
of	O
performance	O
linear	O
tensor	B-Device
algebra	O
routines	O
for	O
large	O
tensors	B-Device
found	O
in	O
post-Hartree	O
–	O
Fock	O
methods	O
in	O
quantum	O
chemistry	O
.	O
</s>
<s>
ITensor	O
features	O
automatic	O
contraction	O
of	O
matching	O
tensor	B-Device
indices	O
.	O
</s>
<s>
It	O
is	O
written	O
in	O
C++	O
and	O
has	O
higher-level	O
features	O
for	O
quantum	O
physics	O
algorithms	O
based	O
on	O
tensor	B-Device
networks	O
.	O
</s>
<s>
Fastor	O
is	O
a	O
high	O
performance	O
C++	O
tensor	B-Device
algebra	O
library	O
that	O
supports	O
tensors	B-Device
of	O
any	O
arbitrary	O
dimensions	O
and	O
all	O
their	O
possible	O
contraction	O
and	O
permutation	O
thereof	O
.	O
</s>
<s>
It	O
employs	O
compile-time	O
graph	O
search	O
optimisations	O
to	O
find	O
the	O
optimal	O
contraction	O
sequence	O
between	O
arbitrary	O
number	O
of	O
tensors	B-Device
in	O
a	O
network	O
.	O
</s>
<s>
Xerus	O
is	O
a	O
C++	O
tensor	B-Device
algebra	O
library	O
for	O
tensors	B-Device
of	O
arbitrary	O
dimensions	O
and	O
tensor	B-Device
decomposition	O
into	O
general	O
tensor	B-Device
networks	O
(	O
focusing	O
on	O
matrix	O
product	O
states	O
)	O
.	O
</s>
<s>
It	O
offers	O
Einstein	O
notation	O
like	O
syntax	O
and	O
optimizes	O
the	O
contraction	O
order	O
of	O
any	O
network	O
of	O
tensors	B-Device
at	O
runtime	O
so	O
that	O
dimensions	O
need	O
not	O
be	O
fixed	O
at	O
compile-time	O
.	O
</s>
