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This project is no longer maintained, please consider to use its GPU counterpart:

Tensor Contraction Library (TCL) for C++ (and Python)

TCL is a C++ library for high-performance tensor contractions; TCL also includes a wrapper for python and can be easily integrated into native python code.

From a computational perspective, tensors can be interpreted as a generalization of matrices to multiple dimensions or simply as multidimensional arrays; likewise, tensor contractions are a generalization of the matrix-matrix multiplication to higher dimensions. For instance, A[i,k], B[k,j] and C[i,j] denote two-dimensional tensors (i.e., matrices) and C[i,j] = A[i,k] * B[k,j] represents a tensor contraction where the sum over 'k' as well as the loops over 'i' and 'j' are implicit. Further examples of tensor contractions are:

  • C[i0,j0,j1] = A[i0,k0] * B[j1,k0,j0]
  • C[i0,j0,j1,i1] = A[i0,k0,i1] * B[j1,k0,j0]
  • C[i0,j0,j1,i1] = A[k0,i0,k1,i1] * B[k1,j1,k0,j0]
  • C[i1,j1,j0,i0] = A[k0,i0,k1,k2,i1] * B[k1,j1,k0,j0,k2]
  • ...

You can find additional information on tensor contractions in the paper listed below.


  • A C++ compiler with c++11 support (I've tested it with: g++ 5.1.0, icpc 17.0.2).
  • Some BLAS library (e.g., BLIS, ATLAS, MKL, OpenBlas)
  • HPTT for high-performance tensor transpositions


C/C++ library

Install TCL's dependencies (see above). Then clone the repository into a desired directory and change to that location:

git clone
cd tcl 

You might have to update the Makefile and specify the location of your BLAS and HPTT library, then continue with:


This should be it and you should see a in the ./lib/ directory.

Python API

To install the python API you have to:

cd pythonAPI
python install

At this point you can import the tcl module in your python scripts and call the tcl.tensorMult() function (see ./benchmark/python/ for further examples).

Keep in mind that TCL is a multi-threaded and performance critical library. Thus, it is of great importance that you follow the following steps before you run your python script:

  • Specify the thread affinity (e.g., via environment variable KMP_AFFINITY, via taskset, ...)
  • Specify the amount of threads to be used via the OMP_NUM_THREADS environment variable.
  • Ensure that your python environment links against a multi-threaded BLAS (see

Getting started

First off, TCL supports any kind of tensor contractions (i.e., it is not limited to tensor contractions that can be mapped to GEMM). The idea behind TCL is that you only have to call a single function for any contraction: tcl::tensorMult(). Once you have specified the tensor contraction, TCL will automatically map this tensor contraction to the most efficient kernel.

TCL supports both column-major (default) and row-major data layouts. Column-major: indices are stored from left to right with the leftmost and rightmost index respectively being the fastest-varying (stride-1) index and the slowest-varying index; row-major: indices are stored from right to left.

You can find an self-explanatory example under ./examples/contraction.cpp

#include <tcl.h>

tcl::sizeType m = 5;
tcl::sizeType n = 4;
tcl::sizeType k1 = 2;
tcl::sizeType k2 = 3;

// Allocation of the Tensors (data is owned by the tensors)
tcl::Tensor<float> A({k2,m,k1});
tcl::Tensor<float> B({n,k2,k1});
tcl::Tensor<float> C({m,n});

// Data initialization (omitted) ...

// tensor contarction: C_{m,n} = alpha * A_{k2,m,k1} * B_{n,k2,k1} + beta * C_{m,n}
auto ret = tcl::tensorMult<float>( alpha, A["k2,m,k1"], B["n,k2,k1"], beta, C["m,n"], 0 );

You just have to include the header (which can be found in ./include/) and link against tcl; an exemplary Makefile can be found in ./examples/Makefile.


TCL also provides a C interface:

void sTensorMult(const float alpha, const float *A, const long *sizeA, const long *outerSizeA, const char* indA,
                                    const float *B, const long *sizeB, const long *outerSizeB, const char* indB,
                 const float beta ,       float *C, const long *sizeC, const long *outerSizeC, const char* indC, int useRowMajor = 0);

void dTensorMult(const double alpha, const double *A, const long *sizeA, const long *outerSizeA, const char* indA,
                                     const double *B, const long *sizeB, const long *outerSizeB, const char* indB,
                 const double beta ,       double *C, const long *sizeC, const long *outerSizeC, const char* indC, int useRowMajor = 0);

The outerSizes enable the user to operate on subtensors; the outerSize may be NULL, in that case a dense tensor with size=outerSize is assumed.


TCL now also offers a python-interface. The functionality offered by TCL is comparable to that of numpy.einsum:

tensorMult( alpha, A, indicesA, B, indicesB, beta,  C, indicesC)

See docstring for additional information.

Several examples can be found under ./benchmark/python/

Key Features

  • Multi-threading support
  • TCL's current implementation is based on the Transpose-Transpose-GEMM-Transpose (TTGT) approach (see paper).
  • Support for single- and double-precision as well as complex data types.

Performance Results


The above Figure presents the speedup of TCL over the best reference among multiple state-of-the-art implementations (i.e., Eigen, Tensor Toolbox, NumPy, ITensor) for 1000 random tensor contractions running on a two socket Intel Haswell-EP E5-2680 v3 utilizing 24 threads. The cases are sorted with respect to the arithmetic intensity of an equally-sized matrix-matrix multiplication.

We make the following observations:

  • All speedups are well above 1.0x; phrased differently, TCL exhibits positive speedups across all 1000 random tensor contractions.
  • The speedups are especially high for tensor contractions with a low arithmetic intensity (left side of the plot), reaching up to 18x.
  • The speedups decrease with increasing arithmetic intensity, this is due to the fact that the runtime of those contractions is dominated by a large GEMM, thus attaining close to the theoretical peak floating-point performance of the CPU.

You can run your own benchmarks via:

./benchmark/python/ <numThreads>

Notice that the full benchmark may take hours to complete.

Current limitations

TCL currently requires additional auxiliary memory for the transposed tensors. This library should eventually also support the GEMM-like Tensor-Tensor (GETT) contraction approach (see paper), which yields better performance and does not require any auxiliary memory.


This project is under LGPLv3 for now. If this license is too restrictive for you, please feel free to contact me via email (


In case you want to refer to TCL as part of a research paper, please cite the following article (pdf):

   author      = {Paul Springer and Paolo Bientinesi},
   title       = {{Design of a {H}igh-{P}erformance {GEMM}-like {T}ensor-{T}ensor {M}ultiplication}},
   archivePrefix = "arXiv",
   eprint = {1607.00145},
   primaryClass = {cs.MS, cs.PF},
   journal     = {CoRR},
   year        = {2016},
   issue_date  = {July 2016},
   url         = {}