About

This is an implementation of the Zohar-Trench algorithm for solving a set of linear Toeplitz equations, based on the original article by S. Zohar "The Solution of a Toeplitz Set of Linear Equations" (https://doi.org/10.1145/321812.321822). There is also an option for an iterative solution that gives a valid result for increasing sub-matrix size.

For more info on the algorithm and its limitations see the referenced paper. There is also many papers about algorithm numerical stability.

Performance

The solution includes a small benchmark to compare with MathNet.Numerics+MKL implementation. Although MathNet does not implement Toeplitz matrices and algorithms optimized for them, it is still worth the comparison. Here are my results:

BenchmarkDotNet=v0.12.0, OS=Windows 10.0.18363
Intel Core i5-6600 CPU 3.30GHz (Skylake), 1 CPU, 4 logical and 4 physical cores
.NET Core SDK=3.1.302
  [Host]     : .NET Core 3.1.6 (CoreCLR 4.700.20.26901, CoreFX 4.700.20.31603), X64 RyuJIT
  DefaultJob : .NET Core 3.1.6 (CoreCLR 4.700.20.26901, CoreFX 4.700.20.31603), X64 RyuJIT

Method	N	Mean	Error	StdDev	Median	Ratio	Gen 0	Gen 1	Gen 2	Allocated
MainSolver	8	621.0 ns	3.36 ns	2.98 ns	621.7 ns	0.11	0.1707	-	-	536 B
MathNetNativeMkl	8	5,594.6 ns	20.61 ns	18.27 ns	5,589.4 ns	1.00	0.3052	-	-	968 B
MainSolver	32	11,690.5 ns	126.34 ns	118.17 ns	11,641.2 ns	0.33	0.2899	-	-	920 B
MathNetNativeMkl	32	35,077.0 ns	256.05 ns	239.51 ns	35,094.7 ns	1.00	1.5259	-	-	4904 B
MainSolver	128	72,167.5 ns	378.92 ns	316.42 ns	72,159.9 ns	0.39	0.7324	-	-	2457 B
MathNetNativeMkl	128	185,198.8 ns	1,108.50 ns	1,036.89 ns	184,818.0 ns	1.00	20.5078	-	-	66728 B
MainSolver	512	219,324.7 ns	1,027.25 ns	910.63 ns	219,068.5 ns	0.17	2.6855	-	-	8603 B
MathNetNativeMkl	512	1,274,336.5 ns	25,380.10 ns	47,043.64 ns	1,249,485.4 ns	1.00	248.0469	248.0469	248.0469	1053296 B