faer
faer is a collection of crates that implement low level linear algebra routines in pure Rust.
The aim is to eventually provide a fully featured library for linear algebra with focus on portability, correctness, and performance.
See the official website and the docs.rs documentation for code examples and usage instructions.
Questions about using the library, contributing, and future directions can be discussed in the Discord server.
faer-core
The core module implements matrix structures, as well as BLAS-like matrix operations such as matrix multiplication and solving triangular linear systems.
faer-cholesky
The Cholesky module implements the LLT and LDLT matrix decompositions. These allow for solving symmetric/hermitian (+positive definite for LLT) linear systems.
faer-lu
The LU module implements the LU factorization with row pivoting, as well as the version with full pivoting.
faer-qr
The QR module implements the QR decomposition with no pivoting, as well as the version with column pivoting.
faer-svd
The SVD module implements the singular value decomposition for real matrices (complex support will be following soon).
Coming soon
faer-eigen
Contributing
If you'd like to contribute to faer, check out the list of "good first issue"
issues. These are all (or should be) issues that are suitable for getting
started, and they generally include a detailed set of instructions for what to
do. Please ask questions on the Discord server or the issue itself if anything
is unclear!
Benchmarks
The benchmarks were run on an 11th Gen Intel(R) Core(TM) i5-11400 @ 2.60GHz with 12 threads.
nalgebrais used with thematrixmultiplybackendndarrayis used with theopenblasbackendeigenis compiled with-march=native -O3 -fopenmp
All computations are done on column major matrices containing f64 values.
Matrix multiplication
Multiplication of two square matrices of dimension n.
n faer faer(par) ndarray nalgebra eigen
32 1.2µs 1.1µs 1.1µs 1.9µs 1.2µs
64 8µs 8µs 7.8µs 10.8µs 5µs
96 27.5µs 10.9µs 26.1µs 34µs 10.1µs
128 65.2µs 17µs 35.2µs 79.2µs 32.6µs
192 216.1µs 54.1µs 53.7µs 258µs 51.8µs
256 509.8µs 118.1µs 154.9µs 603.9µs 155µs
384 1.7ms 368.5µs 346.3µs 2ms 327.4µs
512 4.1ms 846.8µs 1.2ms 4.7ms 1.2ms
640 7.9ms 1.7ms 2.3ms 9.3ms 2ms
768 14ms 3ms 3.6ms 16.1ms 3.2ms
896 22.3ms 4.9ms 6.5ms 25.9ms 5.9ms
1024 34.3ms 7.1ms 9ms 39.3ms 8.3ms
Triangular solve
Solving AX = B in place where A and B are two square matrices of dimension n, and A is a triangular matrix.
n faer faer(par) ndarray nalgebra eigen
32 2.4µs 2.4µs 8.1µs 7µs 2.8µs
64 10µs 10µs 25.7µs 34.5µs 13.7µs
96 29.1µs 27µs 54.3µs 100.9µs 36.7µs
128 58µs 42.9µs 144.7µs 232.3µs 81.1µs
192 172.5µs 93.2µs 262.6µs 809.7µs 213.8µs
256 374.3µs 165.2µs 648.9µs 1.9ms 487.2µs
384 1.1ms 333.8µs 1.4ms 6.7ms 1.4ms
512 2.6ms 683.4µs 3.5ms 15ms 3.2ms
640 4.8ms 1.6ms 5.6ms 29.1ms 5.4ms
768 8.1ms 2.4ms 9.2ms 51.8ms 9.1ms
896 12.4ms 3.6ms 13.4ms 81.9ms 13.8ms
1024 18.9ms 5.3ms 25.2ms 124.9ms 22.8ms
Triangular inverse
Computing A^-1 where A is a square triangular matrix with dimension n.
n faer faer(par) ndarray nalgebra eigen
32 3.3µs 10.9µs 8.2µs 7µs 2.9µs
64 10.7µs 17.9µs 25.6µs 34.4µs 13.7µs
96 25.8µs 36.8µs 54.1µs 101µs 36.8µs
128 39.2µs 51.1µs 144.8µs 234.7µs 81.2µs
192 99.9µs 91.4µs 262.3µs 807.4µs 213.8µs
256 184.6µs 140.3µs 647.7µs 1.9ms 487.2µs
384 511.3µs 265.1µs 1.4ms 6.7ms 1.4ms
512 1.1ms 442.3µs 3.5ms 15ms 3.2ms
640 2ms 633.1µs 5.7ms 29.1ms 5.4ms
768 3.2ms 985.7µs 9.2ms 51.8ms 9.1ms
896 4.7ms 2.3ms 13.4ms 81.9ms 13.8ms
1024 7.1ms 2.4ms 25.1ms 123ms 22.8ms
Cholesky decomposition
Factorizing a square matrix with dimension n as L×L.T, where L is lower triangular.
n faer faer(par) ndarray nalgebra eigen
32 4.3µs 4.3µs 3.4µs 2.6µs 2.3µs
64 12.1µs 12.1µs 36.8µs 12.8µs 8.7µs
96 28µs 28µs 73.5µs 37.5µs 20.3µs
128 40.3µs 40.3µs 135.7µs 89.1µs 38.4µs
192 109.3µs 117.9µs 411.3µs 291.4µs 99.7µs
256 185µs 175.8µs 700.2µs 680.1µs 207.3µs
384 500.4µs 472.1µs 1.2ms 2.3ms 564.2µs
512 1.1ms 675.8µs 3.7ms 6.3ms 1.2ms
640 2ms 1.3ms 3.3ms 11.9ms 2.1ms
768 3.4ms 1.8ms 5.3ms 20.7ms 3.6ms
896 5.2ms 2.7ms 6.9ms 32.1ms 5.5ms
1024 8ms 3.4ms 14.8ms 47.4ms 8.3ms
LU decomposition with partial pivoting
Factorizing a square matrix with dimension n as P×L×U, where P is a permutation matrix, L is unit lower triangular and U is upper triangular.
n faer faer(par) ndarray nalgebra eigen
32 5.6µs 5.6µs 5.6µs 5µs 3.8µs
64 23.2µs 23.2µs 17.6µs 22.2µs 15.1µs
96 50.3µs 52.8µs 34.4µs 69.2µs 36.2µs
128 98.2µs 102.4µs 96.8µs 172.7µs 130.5µs
192 258.1µs 285µs 187.6µs 550.5µs 429.9µs
256 510.8µs 538.2µs 318µs 1.4ms 838µs
384 1.3ms 1.3ms 747.1µs 4.7ms 1.9ms
512 2.9ms 2.4ms 2.1ms 11.7ms 4.3ms
640 4.9ms 3.8ms 2.3ms 21.5ms 5.7ms
768 7.8ms 5.7ms 3.3ms 37.2ms 8.7ms
896 11.5ms 8.5ms 4.7ms 58.8ms 11.3ms
1024 17.5ms 11.5ms 6.7ms 91.5ms 17.2ms
LU decomposition with full pivoting
Factorizing a square matrix with dimension n as P×L×U×Q.T, where P and Q are permutation matrices, L is unit lower triangular and U is upper triangular.
n faer faer(par) ndarray nalgebra eigen
32 12.7µs 12.6µs - 14.8µs 10.8µs
64 52.2µs 52.2µs - 105.3µs 68.8µs
96 127.2µs 127.2µs - 349.9µs 196.6µs
128 257.9µs 259.2µs - 844.8µs 455.1µs
192 654µs 652.4µs - 2.8ms 1.4ms
256 1.5ms 1.5ms - 6.6ms 3.3ms
384 4.6ms 4.7ms - 22ms 10.7ms
512 11.7ms 9ms - 52.7ms 26.5ms
640 19.8ms 13.1ms - 101.5ms 49.7ms
768 33.1ms 18.6ms - 175.5ms 85.8ms
896 50.8ms 26ms - 278.2ms 135.1ms
1024 79.9ms 35.8ms - 426.1ms 204.5ms
QR decomposition with no pivoting
Factorizing a square matrix with dimension n as QR, where Q is unitary and R is upper triangular.
n faer faer(par) ndarray nalgebra eigen
32 12.6µs 12.6µs 15.5µs 7.7µs 6.9µs
64 37µs 37µs 60.4µs 43.2µs 44.3µs
96 74.9µs 75µs 324.6µs 141.1µs 79.3µs
128 132.1µs 132.2µs 825.1µs 321.1µs 156.7µs
192 338.5µs 338.5µs 2ms 1.1ms 384.4µs
256 678.7µs 733.7µs 7.3ms 2.5ms 800.8µs
384 1.9ms 1.8ms 8.2ms 8.1ms 2.1ms
512 4.2ms 3.2ms 16.2ms 19.4ms 4.5ms
640 7.6ms 4.7ms 22.8ms 37ms 8.2ms
768 12.5ms 7ms 41.7ms 62.6ms 13.4ms
896 19ms 9.8ms 56.4ms 100ms 20.7ms
1024 28.3ms 13.8ms 80.5ms 150.7ms 30.8ms
QR decomposition with column pivoting
Factorizing a square matrix with dimension n as QRP, where P is a permutation matrix, Q is unitary and R is upper triangular.
n faer faer(par) ndarray nalgebra eigen
32 26.4µs 39.3µs - 17.9µs 9.1µs
64 107.5µs 136.6µs - 127.1µs 36.3µs
96 241.1µs 272.7µs - 420.2µs 99µs
128 447.5µs 490.7µs - 981µs 222.4µs
192 1.2ms 1.2ms - 3.3ms 623.3µs
256 2.3ms 2.2ms - 7.7ms 1.6ms
384 6.9ms 4.6ms - 25.5ms 5.8ms
512 15.7ms 8.3ms - 60.8ms 16.2ms
640 28.8ms 12.5ms - 117.3ms 28.7ms
768 47.5ms 17.4ms - 201.2ms 49.3ms
896 73.5ms 23.5ms - 321.2ms 78.3ms
1024 107.2ms 42ms - 500.3ms 121.9ms
Matrix inverse
Computing the inverse of a square matrix with dimension n.
n faer faer(par) ndarray nalgebra eigen
32 14.7µs 33.2µs 10.4µs 21.1µs 10.5µs
64 53.5µs 79µs 38µs 99µs 45.6µs
96 135.3µs 141.8µs 156.2µs 285.8µs 119µs
128 221.6µs 209.7µs 333.6µs 658.1µs 343µs
192 595.3µs 486µs 653.5µs 2.2ms 974.8µs
256 1.2ms 813µs 1.1ms 5.6ms 2ms
384 3.2ms 1.9ms 2.3ms 20.4ms 5.1ms
512 6.9ms 3.7ms 4.6ms 47.8ms 12ms
640 12.6ms 6.6ms 7.3ms 92.5ms 19.4ms
768 21.1ms 10.5ms 11ms 156.5ms 31.2ms
896 31.7ms 14.8ms 16.7ms 242.9ms 44.4ms
1024 46.8ms 21.3ms 23.8ms 377.4ms 69.7ms
Square matrix singular value decomposition
Computing the SVD of a square matrix with dimension n.
n faer faer(par) ndarray nalgebra eigen
32 146.8µs 166.1µs 91.5µs 102.5µs 239.4µs
64 484.5µs 444.5µs 613.1µs 565.9µs 984.2µs
96 1.1ms 1.1ms 1.7ms 1.7ms 2.6ms
128 1.9ms 1.9ms 2.9ms 4.6ms 4.3ms
192 4.6ms 4.4ms 6.6ms 14.9ms 9.7ms
256 8.8ms 7.6ms 11.3ms 45.3ms 17.4ms
384 23.1ms 16.2ms 25.1ms 121.7ms 42.4ms
512 49.6ms 29.7ms 50.6ms 450.1ms 80.7ms
640 85.8ms 47.5ms 78.1ms 655.2ms 127.2ms
768 139.8ms 70.9ms 121.6ms 1.42s 196.7ms
896 210.5ms 99.2ms 170.8ms 2.09s 277.1ms
1024 313ms 136ms 270.8ms 3.8s 412.6ms
Thin matrix singular value decomposition
Computing the SVD of a rectangular matrix with shape (4096, n).
n faer faer(par) ndarray nalgebra eigen
32 1.3ms 1.4ms 5.2ms 5.1ms 3ms
64 3.7ms 3.5ms 15.3ms 20.2ms 8.1ms
96 7.3ms 6.2ms 30.3ms 44.7ms 17.1ms
128 12ms 9.4ms 47.5ms 79.3ms 28.3ms
192 24.9ms 17.9ms 62.7ms 181.6ms 54.8ms
256 42.9ms 28.1ms 83.6ms 367.8ms 92.3ms
384 93.8ms 51.8ms 132.5ms 912.4ms 206ms
512 167.8ms 84ms 299.9ms 2.01s 385.4ms
640 263.8ms 125ms 289.9ms 3.25s 625.2ms
768 388.5ms 190.6ms 439.3ms 5.18s 912ms
896 540.6ms 260.4ms 551.6ms 7.26s 1.3s
1024 733.6ms 335.6ms 849.9ms 10.68s 1.71s