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 8.1µs 8.1µs 7.8µs 11µs 5.1µs
96 27.7µs 10.9µs 26.1µs 33.9µs 10µs
128 65.6µs 17.2µs 35.3µs 78.6µs 32.7µs
192 218µs 54.2µs 54µs 258.3µs 51.7µs
256 512.8µs 118.5µs 156.8µs 606.9µs 142.9µs
384 1.7ms 369.1µs 407.6µs 2ms 330.4µs
512 4.1ms 853.1µs 1.3ms 4.7ms 1.2ms
640 8ms 1.7ms 2.3ms 9.3ms 2ms
768 13.9ms 3ms 3.6ms 16.1ms 3.2ms
896 22.2ms 4.9ms 6.5ms 25.8ms 5.9ms
1024 34ms 7.2ms 8.9ms 39.2ms 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 2.5µs 2.4µs 8.2µs 7.2µs 3µs
64 10.2µs 10.3µs 25.8µs 34.7µs 13.6µs
96 29.5µs 26.6µs 54.4µs 102.4µs 36.8µs
128 59.1µs 40.8µs 145.2µs 239.2µs 83µs
192 174µs 93.3µs 263.4µs 832.8µs 214.3µs
256 382.1µs 164.3µs 625.9µs 1.9ms 485.9µs
384 1.2ms 328.7µs 1.4ms 6.9ms 1.4ms
512 2.7ms 683.1µs 3.6ms 15.6ms 3.2ms
640 4.8ms 1.7ms 5.7ms 29.9ms 5.4ms
768 8.2ms 2.5ms 9.3ms 53.1ms 9.1ms
896 12.5ms 3.6ms 13.5ms 83.8ms 13.8ms
1024 19.1ms 5.3ms 25.2ms 127.7ms 22.9ms
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 10.8µs 8.2µs 7.2µs 3µs
64 10.7µs 17.7µs 25.8µs 34.7µs 13.6µs
96 25.5µs 37.3µs 54.1µs 102.6µs 36.7µs
128 39.5µs 51.1µs 145.1µs 241.7µs 83µs
192 100.3µs 86.9µs 263.4µs 832.7µs 214.1µs
256 188.6µs 136.6µs 633µs 1.9ms 485.4µs
384 519.1µs 268.1µs 1.4ms 7ms 1.4ms
512 1.1ms 450.2µs 3.6ms 15.6ms 3.2ms
640 2ms 654.2µs 5.7ms 29.9ms 5.4ms
768 3.2ms 923.6µs 9.3ms 53.2ms 9.1ms
896 4.8ms 1.5ms 13.6ms 83.8ms 13.8ms
1024 7.3ms 2.5ms 25.3ms 127.5ms 22.9ms
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 4.6µs 4.6µs 3.2µs 2.3µs 2.2µs
64 12.7µs 12.7µs 38.4µs 11.1µs 8.7µs
96 28.9µs 28.9µs 100.6µs 32.1µs 20.2µs
128 41.7µs 41.8µs 282.3µs 77.6µs 38.2µs
192 110.9µs 116.6µs 303.7µs 252.6µs 99µs
256 189.8µs 178.5µs 696.7µs 617.8µs 204.4µs
384 509.6µs 483.2µs 1.2ms 2.1ms 562.1µs
512 1.2ms 693.1µs 3.7ms 6ms 1.2ms
640 2ms 1.3ms 3.3ms 11.3ms 2.1ms
768 3.4ms 1.9ms 5.4ms 19.8ms 3.6ms
896 5.2ms 2.7ms 6.9ms 30.9ms 5.5ms
1024 8.1ms 3.5ms 14.7ms 46.4ms 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 5.3µs 5.4µs 5.6µs 5µs 3.8µs
64 24.1µs 24.2µs 17.4µs 22.4µs 15.2µs
96 53.2µs 53.2µs 34.3µs 67.8µs 36.2µs
128 109.6µs 108.6µs 97.1µs 160.2µs 128.6µs
192 270.7µs 287.5µs 186.7µs 496.6µs 423.8µs
256 524.7µs 588.1µs 315.9µs 1.3ms 825.8µs
384 1.4ms 1.3ms 672.2µs 4.6ms 1.8ms
512 3ms 2.6ms 1.9ms 11.3ms 4.3ms
640 5.1ms 4.1ms 2.8ms 21ms 5.6ms
768 8.3ms 6ms 3.3ms 36.1ms 8.7ms
896 12ms 9.2ms 4.7ms 56.9ms 11.4ms
1024 17.7ms 12.3ms 6.8ms 90.1ms 17.3ms
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 10.8µs 10.8µs - 14.4µs 10.4µs
64 44.5µs 44.5µs - 104.9µs 67.5µs
96 109.7µs 110.5µs - 344.8µs 196.4µs
128 228.9µs 228.2µs - 823µs 452.6µs
192 589.7µs 589.2µs - 2.7ms 1.4ms
256 1.3ms 1.3ms - 6.6ms 3.3ms
384 4.6ms 4.4ms - 22ms 10.7ms
512 11.8ms 8.7ms - 52.6ms 26.6ms
640 20.1ms 12.9ms - 101.4ms 49.3ms
768 33.4ms 18.1ms - 175.3ms 85.1ms
896 52.3ms 25.9ms - 277.6ms 133.8ms
1024 80.1ms 34.2ms - 438.1ms 203.3ms
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 10.3µs 10.3µs 15.5µs 8µs 7.2µs
64 32.8µs 32.8µs 60.3µs 43.7µs 44.9µs
96 68.7µs 68.6µs 319.7µs 141.4µs 78.9µs
128 123.5µs 123.3µs 806.9µs 326µs 157.1µs
192 323.2µs 322.9µs 1.7ms 1ms 383.4µs
256 661.5µs 713.2µs 4.9ms 2.5ms 797.5µs
384 1.9ms 1.7ms 8.1ms 8.2ms 2.1ms
512 4.2ms 3.2ms 16.4ms 19.5ms 4.5ms
640 7.6ms 4.7ms 22.9ms 37.7ms 8.2ms
768 12.5ms 7ms 41.8ms 63.3ms 13.5ms
896 19.1ms 9.8ms 56.2ms 100.8ms 20.8ms
1024 28.4ms 13.9ms 80.5ms 153.1ms 30.9ms
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 12.4µs 25.7µs - 18µs 9.3µs
64 50.8µs 71.4µs - 128.5µs 39.1µs
96 120.4µs 147.2µs - 423.3µs 102.1µs
128 241.2µs 272.9µs - 996.2µs 218.8µs
192 675µs 729.1µs - 3.3ms 626µs
256 1.5ms 1.5ms - 7.7ms 1.7ms
384 5.1ms 3.7ms - 25.5ms 5.5ms
512 12.1ms 7ms - 60ms 14.2ms
640 23.4ms 10.8ms - 116.6ms 25.4ms
768 39.5ms 15.1ms - 199.9ms 44ms
896 63.2ms 24.2ms - 318.4ms 68.6ms
1024 93.2ms 37.6ms - 494.3ms 108.2ms
Matrix inverse
Computing the inverse of a square matrix with dimension n.
n faer faer(par) ndarray nalgebra eigen
32 14.4µs 32.9µs 10.6µs 21.3µs 10.8µs
64 55.3µs 79.6µs 37.8µs 98.6µs 46.7µs
96 138.2µs 144.7µs 162.8µs 284.4µs 118.8µs
128 227.1µs 216µs 487.6µs 646.1µs 342.4µs
192 606.2µs 509.8µs 658.8µs 2.2ms 969.2µs
256 1.2ms 843µs 1.2ms 5.5ms 2ms
384 3.2ms 2ms 2.3ms 20ms 5.1ms
512 6.9ms 3.8ms 4.5ms 47.8ms 12ms
640 12.7ms 6.5ms 7.3ms 90.9ms 19.3ms
768 21.4ms 10.4ms 11ms 154.1ms 31.2ms
896 32ms 16ms 16.7ms 241ms 44.2ms
1024 47.7ms 21.7ms 23.8ms 375.3ms 69.5ms
Matrix singular value decomposition
Computing the SVD of a square matrix with dimension n.
n faer faer(par) ndarray nalgebra eigen
32 120.2µs 137.1µs 92.6µs 103.6µs 235.7µs
64 403.8µs 362.6µs 559.1µs 551.1µs 983.6µs
96 918µs 899.1µs 1.6ms 1.7ms 2.6ms
128 1.7ms 1.6ms 2.9ms 4.7ms 4.3ms
192 4ms 4ms 6.5ms 15.2ms 9.7ms
256 7.9ms 7ms 11.2ms 46.5ms 17.4ms
384 21.1ms 15.4ms 25.1ms 120.2ms 42.4ms
512 45.9ms 29.3ms 51.6ms 454.2ms 82.1ms
640 80.7ms 45.8ms 78.4ms 651.1ms 129.5ms
768 132ms 68.7ms 121.9ms 1.44s 201.2ms
896 199.9ms 94.8ms 171.4ms 2.1s 282.4ms
1024 298.2ms 131.7ms 270.7ms 3.83s 422.4ms