Linnea

Input a linear algebra expression to generate Julia code that computes it. The following operations are allowed: Multiplication (*), addition (+), subtraction (-), transposition (trans(X)), and inversion (inv(X)).

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Least Squares: $b := (X^{T} X)^{- 1} X^{T} y$
Generalized Least Squares: $b := (X^{T} M^{- 1} X)^{- 1} X^{T} M^{- 1} y$
Triangular Matrix Inversion: $\begin{aligned} X_{10} & := L_{10} L_{00}^{- 1} \\ X_{20} & := L_{20} + L_{22}^{- 1} L_{21} L_{11}^{- 1} L_{10} \\ X_{11} & := L_{11}^{- 1} \\ X_{21} & := - L_{22}^{- 1} L_{21} \end{aligned}$
Image Restoration: $\begin{aligned} H^{†} & := H^{T} (H H^{T})^{- 1} \\ y_{k} & := H^{†} y + (I_{n} - H^{†} H) x_{k} \end{aligned}$
Stochastic Newton: $\begin{aligned} B_{k} := & \frac{k}{k - 1} B_{k - 1} (I_{n} - A^{T} W_{k} ((k - 1) I_{l} \\ + W_{k}^{T} A B_{k - 1} A^{T} W_{k})^{- 1} W_{k}^{T} A B_{k - 1}) \end{aligned}$

Linnea minimizes the number of floating-point operations. The generated algorithms are a selection of the best ones that are found within 10 seconds. Additional features of Linnea, such as a longer generation time or the option to generate a larger number of algorithms can be accessed by installing Linnea.

Algorithm Generation Steps

                    # The generated algorithm will appear here.

The generated code is provided "as is", without warranty of any kind. In order to improve Linnea, all submitted input expressions are stored. If you have questions, comments, or encounter bugs, please post an issue on GitHub.

All Properties:
FullRank $A$ has full rank.	SPD $m = n$ and $x^{T} A x > 0$ for all $x \in R^{n}$ .	Symmetric $m = n$ and $a_{i j} = a_{j i}$
Orthogonal $m = n$ and $A^{T} A = A A^{T} = I$	OrthogonalRows $n ⩾ m$ and $A A^{T} = I$	OrthogonalColumns $m ⩾ n$ and $A^{T} A = I$
SPSD $m = n$ and $x^{T} A x \geq 0$ for all $x \in R^{n}$ .	LowerTriangular $a_{i j} = 0$ if $i < j$	UpperTriangular $a_{i j} = 0$ if $i > j$
Diagonal $a_{i j} = 0$ if $i \neq j$	UnitDiagonal $a_{i j} = 1$ if $i = j$ requires UpperTriangular or LowerTriangular.	Permutation $m = n$ . There is exactly one $1$ in each row and column, all other entries are $0$ .