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Linnea is an experimental tool for the automatic generation of optimized code for linear algebra problems. It is developed at the High-Performance and Automatic Computing group at RWTH Aachen University. An online demo of Linnea can be found here.


Linnea requires Python 3.6 and can be installed with pip install git+git:// (Depending on your setup, you may have to use pip3 instead of pip.) To uninstall Linnea, use pip uninstall linnea. This also removes the commandline tool.

Development Installation

If you intend to contribute to Linnea, you can install it from local sources by running pip install -e . in your local development directory.


Linnea is a prototype of a compiler/program synthesis tool that automates the translation of the mathematical description of a linear algebra problem to an efficient sequence of calls to BLAS and LAPACK kernels. The main idea of Linnea is to construct a search graph that represents a large number of programs, taking into account knowledge about linear algebra, numerical linear algebra and high-performance computing. The algebraic nature of the domain is used to reduce the size of the search graph, without reducing the size of the search space that is explored.

The input to Linnea are linear algebra expressions. As operands, matrices, vectors and scalars are supported. Operands can be annotated with properties, such as 'lower triangular' or 'symmetric'. Supported operations are addition, multiplication, transposition and inversion. At the moment, Linnea generates Julia code (see, using BLAS and LAPACK wrappers whenever possible.


Linnea can be used in two different ways.

Python Module

At the moment, Linnea is primarily a Python module. An example script for how to use Linnea within Python can found in examples/ The input expressions are represented as Python objects. As an example, consider the description of a lower triangular linear system (omitting imports):

n = 1000

L = Matrix("L", (n, n))
x = Vector("x", (n, 1))
y = Vector("y", (n, 1))

input = Equations(Equal(y, Times(Inverse(L), x)))

Further examples of input problems are provided in the examples/ files.

Options can be set with a number of linnea.config.set_X() functions.

Commandline Tool

When installing Linnea via pip, the commandline tool linnea is installed. As input, it takes a description of the input problem in a simple custom language. With this language, the same lower triangular system is described as:

n = 1000

Matrix L(n, n) <LowerTriangular, FullRank>
ColumnVector x(n) <>
ColumnVector y(n) <>

y = inv(L)*x

Further examples are provided in examples/ Notice that the primary purpose of this input format is to make it slightly easier to try out Linnea. There are no plans to establish this as an actual language. New features will probably not be immediately available in this language, and the language may change in the future without being backward compatible.

The list of commandline options is available via linnea -h.


As output, Linnea generates a directory structure that contains code files, as well a file containing a description of the search graph, the primary datastructure used by Linnea. Which files are generated can be set as options. Likewise, the location of the output can be specified. By default, it is the current directory.

For the linear system from the previous examples, the following code will be generated:

using LinearAlgebra.BLAS
using LinearAlgebra

    algorithm0(ml0::Array{Float64,2}, ml1::Array{Float64,1})

# Arguments
- `ml0::Array{Float64,2}`: Operand L of size 1000 x 1000 with properties LowerTriangular, Non-singular.
- `ml1::Array{Float64,1}`: Operand x of size 1000.
function algorithm0(ml0::Array{Float64,2}, ml1::Array{Float64,1})
    # cost: 1e+06 FLOPs
    # L: ml0, full, x: ml1, full
    # tmp1 = (L^-1 x)
    trsv!('L', 'N', 'N', ml0, ml1)

    # tmp1: ml1, full
    # y = tmp1
    return (ml1)


Linnea offers a number of options which can be set through linnea.config in Python or as commandline options for the commandline tool. Alternatively, all options can also be specified in a linnea_config.json file (see examples) which has to be located in the same directory where Linnea is run, or at the user's $HOME folder. Both commandline options and linnea.config options override what is specified in linnea_config.json. As a fallback, reasonable default values are used.

There are the following options (those are the names used in Python, the commandline options have slightly different names. See linnea -h):

  • output_code_path The output of Linnea will be stored in this directory. The default is the current directory.

  • output_name Linnea creates a new directory that contains all output files. This is the name of this directory. The default is tmp.

  • time_limit The maximum time spent to find algorithms, in seconds. A higher limit allows Linnea to find better solutions. Linnea usually finds a (potentially suboptimal) solution in less than one second.

  • julia_data_type The data type used in the generated code. Either Float32 or Float64. The default is Float64.

  • merging_branches Whether or not to merge branches in the search graph. The default is true.

  • dead_ends Whether or not to eliminate dead ends in the search graph early. The default is true.

  • algorithms_limit The upper limit for the number of algorithms that are written to files. The default is 100.

  • generate_graph Whether or not to generate a .gv file of the search graph. The default is false.

  • graph_style Style of the search graph. Either full, simple, or minimal. The default is full. Only applies if generate_graph is set to True.

  • generate_steps Whether or not to generate a description of how the algorithms were generated. The default is false.

  • generate_code Whether or not to generate the actual code of the algorithms. The default is true.

  • generate_experiments Whether or not to generate code that can be used to run the algorithms. The default is false.

  • verbosity Level of verbosity. The default is 1.


A number of publications that discuss different aspects of Linnea can be found here. If you want to cite Linnea, please reference this paper:

    author = {Barthels, Henrik and Psarras, Christos and Bientinesi, Paolo},
    title = {{L}innea: {A}utomatic {G}eneration of {E}fficient {L}inear {A}lgebra {P}rograms},
    year = {2021},
    issue_date = {June 2021},
    publisher = {Association for Computing Machinery},
    address = {New York, NY, USA},
    volume = {47},
    number = {3},
    issn = {0098-3500},
    url = {},
    doi = {10.1145/3446632},
    journal = {ACM Trans. Math. Softw.},
    month = jun,
    articleno = {22},
    numpages = {26},


  • Henrik Barthels
  • Marcin Copik
  • Diego Fabregat Traver
  • Julius Hohnerlein
  • Manuel Krebber


Linnea is an experimental tool for the automatic generation of optimized code for linear algebra problems.








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