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matflat's Introduction

MatFlat

This library aims to provide a pure C# implementation of reasonably fast low-level routines for linear algebra operations.

This library was created as a matrix decomposition implementation for NumFlat, a general purpose numerical computing library written in pure C#. If you're interested in numerical computing with C#, please check out NumFlat.

This library is based on the following great projects:

Currently the following routines are implemented:

  • LU decomposition
  • Cholesky decomposition
  • QR decomposition
  • Singular value decomposition
  • Eigenvalue decomposition
  • Generalized eigenvalue decomposition
  • Forward and backward substitution
  • Inverse matrix
  • Matrix-vector multiplication
  • Matrix-matrix multiplication
  • Dot and outer product
  • Vector norm

Features

  • Supports float, double, and Complex matrices.
  • Faster than the managed matrix decompositions in Math.NET in many cases.
  • Small code size, with no dependencies other than .NET 8.
  • No internal multi-threaded optimization, making it safe to use in any multi-threaded code.
  • BLAS and LAPACK-like interface that allows arbitrary leading dimension.

Limitations

  • Unsafe pointers are required, similar to the original BLAS and LAPACK routines.
  • Only column-major order is supported.
  • EVD and GEVD support only symmetric (Hermitian) matrices.

Installation

.NET 8 is required.

The NuGet package is available.

Install-Package MatFlat

All the classes are in the MatFlat namespace.

using MatFlat;

Performance

The performance of MatFlat is compared with that of Math.NET Numerics and OpenBLAS below. The execution times of various double matrix decompositions for square matrices were measured. The matrix sizes range from 10x10 to 100x100. The plots show the execution times, with the execution time of Math.NET Numerics considered as 1.

Measurement condition

The benchmarks were run under the following condition:

BenchmarkDotNet v0.13.12, Windows 11 (10.0.22631.3447/23H2/2023Update/SunValley3)
12th Gen Intel Core i7-12700K, 1 CPU, 20 logical and 12 physical cores
.NET SDK 8.0.202
  [Host]     : .NET 8.0.3 (8.0.324.11423), X64 RyuJIT AVX2
  DefaultJob : .NET 8.0.3 (8.0.324.11423), X64 RyuJIT AVX2

LU decomposition

Method Order Mean Error StdDev Allocated
MathNet 10 356.3 ns 1.03 ns 0.91 ns 104 B
MatFlat 10 318.4 ns 1.55 ns 1.45 ns -
OpenBlas 10 632.7 ns 3.47 ns 3.25 ns -
MathNet 20 1,884.5 ns 9.26 ns 8.66 ns 184 B
MatFlat 20 1,511.7 ns 9.80 ns 9.17 ns -
OpenBlas 20 1,731.4 ns 8.02 ns 7.50 ns -
MathNet 30 6,141.2 ns 22.08 ns 20.65 ns 264 B
MatFlat 30 4,062.0 ns 24.81 ns 23.21 ns -
OpenBlas 30 4,109.6 ns 9.17 ns 8.13 ns -
MathNet 40 12,300.8 ns 53.74 ns 47.64 ns 344 B
MatFlat 40 8,590.4 ns 41.08 ns 38.43 ns -
OpenBlas 40 5,613.0 ns 11.54 ns 10.80 ns -
MathNet 50 25,438.9 ns 112.91 ns 105.61 ns 424 B
MatFlat 50 15,788.6 ns 69.35 ns 64.87 ns -
OpenBlas 50 9,751.5 ns 21.30 ns 19.92 ns -
MathNet 60 39,288.8 ns 151.06 ns 141.30 ns 504 B
MatFlat 60 25,824.7 ns 81.10 ns 75.86 ns -
OpenBlas 60 14,091.9 ns 28.01 ns 26.20 ns -
MathNet 70 69,376.4 ns 315.96 ns 280.09 ns 584 B
MatFlat 70 39,621.0 ns 173.00 ns 161.82 ns -
OpenBlas 70 18,460.6 ns 34.77 ns 32.52 ns -
MathNet 80 101,273.9 ns 329.08 ns 307.82 ns 664 B
MatFlat 80 54,074.5 ns 273.94 ns 256.24 ns -
OpenBlas 80 22,488.7 ns 64.39 ns 60.23 ns -
MathNet 90 131,540.5 ns 671.69 ns 628.30 ns 744 B
MatFlat 90 82,170.1 ns 422.07 ns 394.81 ns -
OpenBlas 90 30,344.1 ns 46.13 ns 40.89 ns -
MathNet 100 180,358.0 ns 359.72 ns 336.48 ns 824 B
MatFlat 100 111,801.6 ns 136.84 ns 128.00 ns -
OpenBlas 100 396,249.3 ns 7,715.05 ns 7,922.79 ns -

A graphical plot of the LU performance comparison above.

Cholesky decomposition

Method Order Mean Error StdDev Allocated
MathNet 10 200.9 ns 0.60 ns 0.54 ns 664 B
MatFlat 10 163.4 ns 0.85 ns 0.79 ns -
OpenBlas 10 592.0 ns 2.33 ns 2.18 ns 24 B
MathNet 20 1,074.6 ns 5.24 ns 4.91 ns 1304 B
MatFlat 20 907.6 ns 4.37 ns 4.08 ns -
OpenBlas 20 704.4 ns 2.83 ns 2.65 ns 24 B
MathNet 30 3,289.6 ns 15.63 ns 14.62 ns 1944 B
MatFlat 30 2,469.6 ns 12.36 ns 11.56 ns -
OpenBlas 30 1,486.7 ns 2.35 ns 2.20 ns 24 B
MathNet 40 7,381.9 ns 29.87 ns 27.94 ns 2584 B
MatFlat 40 5,024.0 ns 25.70 ns 24.04 ns -
OpenBlas 40 3,076.4 ns 5.84 ns 5.47 ns 24 B
MathNet 50 14,076.3 ns 70.74 ns 66.17 ns 3224 B
MatFlat 50 8,803.8 ns 46.68 ns 43.66 ns -
OpenBlas 50 4,668.1 ns 9.87 ns 9.23 ns 24 B
MathNet 60 25,819.8 ns 81.21 ns 75.97 ns 3864 B
MatFlat 60 14,346.9 ns 36.56 ns 34.20 ns -
OpenBlas 60 7,256.3 ns 12.05 ns 11.27 ns 24 B
MathNet 70 38,900.2 ns 145.13 ns 135.76 ns 4504 B
MatFlat 70 21,853.4 ns 119.00 ns 111.31 ns -
OpenBlas 70 52,870.4 ns 1,056.20 ns 2,707.44 ns 24 B
MathNet 80 62,603.4 ns 358.44 ns 335.29 ns 5144 B
MatFlat 80 31,830.3 ns 149.25 ns 139.61 ns -
OpenBlas 80 73,786.8 ns 609.29 ns 508.78 ns 24 B
MathNet 90 83,681.4 ns 286.78 ns 268.26 ns 5784 B
MatFlat 90 44,244.4 ns 223.02 ns 208.62 ns -
OpenBlas 90 84,149.8 ns 1,003.21 ns 938.41 ns 24 B
MathNet 100 115,551.9 ns 672.86 ns 596.47 ns 6424 B
MatFlat 100 59,350.4 ns 334.89 ns 313.25 ns -
OpenBlas 100 89,611.6 ns 1,711.85 ns 1,681.26 ns 24 B

A graphical plot of the Cholesky performance comparison above.

QR decomposition

Method Order Mean Error StdDev Allocated
MathNet 10 16,395.9 ns 60.26 ns 53.41 ns 29206 B
MatFlat 10 680.4 ns 1.53 ns 1.43 ns -
OpenBlas 10 1,655.8 ns 4.50 ns 4.21 ns -
MathNet 20 54,226.6 ns 549.03 ns 513.57 ns 68867 B
MatFlat 20 4,645.9 ns 22.98 ns 21.49 ns -
OpenBlas 20 5,148.5 ns 20.61 ns 19.28 ns -
MathNet 30 118,586.5 ns 668.11 ns 592.27 ns 114887 B
MatFlat 30 12,199.9 ns 56.84 ns 53.17 ns -
OpenBlas 30 10,844.1 ns 37.50 ns 35.08 ns -
MathNet 40 200,877.5 ns 430.67 ns 381.78 ns 156810 B
MatFlat 40 26,532.9 ns 163.90 ns 153.31 ns -
OpenBlas 40 19,615.5 ns 42.48 ns 37.66 ns -
MathNet 50 302,011.2 ns 1,808.26 ns 1,691.45 ns 201012 B
MatFlat 50 47,273.9 ns 308.30 ns 288.38 ns -
OpenBlas 50 30,619.7 ns 61.78 ns 54.77 ns -
MathNet 60 426,465.4 ns 7,785.35 ns 7,282.42 ns 242501 B
MatFlat 60 80,619.4 ns 307.43 ns 272.53 ns -
OpenBlas 60 45,598.1 ns 74.76 ns 69.93 ns -
MathNet 70 584,892.0 ns 5,543.57 ns 4,914.23 ns 286236 B
MatFlat 70 135,624.1 ns 526.02 ns 492.04 ns -
OpenBlas 70 66,024.7 ns 143.80 ns 134.51 ns -
MathNet 80 764,410.9 ns 14,198.79 ns 14,581.12 ns 324848 B
MatFlat 80 187,239.2 ns 1,068.00 ns 999.00 ns -
OpenBlas 80 93,071.6 ns 261.52 ns 244.62 ns -
MathNet 90 1,025,887.2 ns 8,722.75 ns 8,159.26 ns 366159 B
MatFlat 90 264,657.7 ns 509.46 ns 451.62 ns -
OpenBlas 90 130,194.9 ns 392.21 ns 366.88 ns -
MathNet 100 1,270,333.6 ns 17,520.01 ns 16,388.22 ns 406464 B
MatFlat 100 345,101.4 ns 2,451.92 ns 2,293.52 ns -
OpenBlas 100 650,895.2 ns 10,389.01 ns 8,675.29 ns -

A graphical plot of the QR performance comparison above.

Singular value decomposition

Method Order Mean Error StdDev Allocated
MathNet 10 10.369 μs 0.0390 μs 0.0365 μs 1136 B
MatFlat 10 6.261 μs 0.0190 μs 0.0168 μs -
OpenBlas 10 8.218 μs 0.0441 μs 0.0413 μs 48 B
MathNet 20 60.701 μs 0.2754 μs 0.2576 μs 3776 B
MatFlat 20 34.161 μs 0.1679 μs 0.1570 μs -
OpenBlas 20 32.076 μs 0.1371 μs 0.1282 μs 48 B
MathNet 30 165.329 μs 0.6463 μs 0.5729 μs 8016 B
MatFlat 30 97.288 μs 0.2930 μs 0.2741 μs -
OpenBlas 30 80.800 μs 0.2287 μs 0.2028 μs 48 B
MathNet 40 359.838 μs 1.3219 μs 1.2365 μs 13856 B
MatFlat 40 212.872 μs 0.6934 μs 0.6486 μs -
OpenBlas 40 159.768 μs 0.3706 μs 0.3285 μs 48 B
MathNet 50 660.509 μs 2.0123 μs 1.8823 μs 21296 B
MatFlat 50 392.559 μs 1.3545 μs 1.2670 μs -
OpenBlas 50 261.269 μs 1.3967 μs 1.3064 μs 48 B
MathNet 60 1,144.149 μs 4.0893 μs 3.8251 μs 30337 B
MatFlat 60 664.029 μs 3.4705 μs 3.2463 μs 1 B
OpenBlas 60 431.300 μs 0.8319 μs 0.7782 μs 48 B
MathNet 70 1,832.909 μs 9.0039 μs 8.4223 μs 40976 B
MatFlat 70 1,073.801 μs 3.6979 μs 3.4590 μs 1 B
OpenBlas 70 642.370 μs 1.4747 μs 1.3794 μs 48 B
MathNet 80 2,600.145 μs 16.8514 μs 15.7628 μs 53218 B
MatFlat 80 1,542.437 μs 8.6110 μs 8.0548 μs 1 B
OpenBlas 80 918.765 μs 2.7510 μs 2.5733 μs 48 B
MathNet 90 3,615.089 μs 16.6810 μs 15.6034 μs 67058 B
MatFlat 90 2,136.354 μs 12.5674 μs 11.7555 μs 3 B
OpenBlas 90 1,319.471 μs 2.5227 μs 2.3597 μs 49 B
MathNet 100 4,905.273 μs 28.3718 μs 26.5390 μs 82499 B
MatFlat 100 3,036.419 μs 13.4627 μs 11.9343 μs 3 B
OpenBlas 100 2,625.432 μs 33.5104 μs 31.3456 μs 50 B

A graphical plot of the SVD performance comparison above.

Todo

  • ✅ LU decomposition
  • ✅ Cholesky decomposition
  • ✅ QR decomposition
  • ✅ Singular value decomposition
  • ✅ Eigenvalue decomposition
  • ✅ Generalized eigenvalue decomposition
  • ✅ Forward and backward substitution
  • ✅ Inverse matrix
  • ✅ Matrix-vector multiplication
  • ✅ Matrix-matrix multiplication
  • ✅ Dot and outer product
  • ✅ Vector norm

License

MatFlat is available under the MIT license.

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