Sparse Matrices in C# QuickStart Sample

Illustrates using sparse vectors and matrices using the classes in the Numerics.NET.LinearAlgebra.Sparse namespace in C#.

This sample is also available in: Visual Basic, F#, IronPython.

Overview

This QuickStart sample demonstrates how to work with sparse vectors and matrices in Numerics.NET using the classes in the LinearAlgebra.Sparse namespace.

Sparse vectors and matrices are data structures optimized for situations where most elements are zero. Instead of storing all elements, they only store the non-zero values and their positions. This makes them memory efficient for large but sparse data.

The sample shows:

Different ways to create sparse vectors and matrices
How to specify initial capacity and fill factors
Methods to insert individual elements and groups of elements
Accessing elements and iterating over non-zero entries
Working with rows, columns and submatrices (cliques)
Mathematical operations optimized for sparse structures
Performance considerations when working with large sparse matrices

The code includes examples of both successful operations and potential pitfalls, such as operations that may revert to dense algorithms and fail on very large matrices.

The code

using System;
// The sparse vector and matrix classes reside in the
// Numerics.NET.LinearAlgebra namespace.
using Numerics.NET;

// Illustrates using sparse vectors and matrices using the classes
// in the Numerics.NET.LinearAlgebra.Sparse namespace
// of Numerics.NET.

// The license is verified at runtime. We're using
// a 30 day trial key here. For more information, see
//     https://numerics.net/trial-key
Numerics.NET.License.Verify("your-trial-key-here");
//
// Sparse vectors
//

// The SparseVector class has three constructors. The
// first simply takes the length of the vector. All elements
// are initially zero.
var v1 = Vector.CreateSparse<double>(1000000);

// The second constructor lets you specify how many elements
// are expected to be nonzero. This 'fill factor' is a number
// between 0 and 1.
var v2 = Vector.CreateSparse<double>(1000000, 1e-4);

// The second constructor lets you specify how many elements
// are expected to be nonzero. This 'fill factor' is a number
// between 0 and 1.
var v3 = Vector.CreateSparse<double>(1000000, 100);

// The fourth constructor lets you specify the indexes of the nonzero
// elements and their values as arrays:
int[] indexes = { 1, 10, 100, 1000, 10000 };
double[] values = { 1.0, 10.0, 100.0, 1000.0, 10000.0 };
var v4 = Vector.CreateSparse(1000000, indexes, values);

// Elements can be accessed individually:
v1[1000] = 2.0;
Console.WriteLine($"v1[1000] = {v1[1000]}");

// The NonzeroCount returns how many elements are non zero:
Console.WriteLine($"v1 has {v1.NonzeroCount} nonzeros");
Console.WriteLine($"v4 has {v4.NonzeroCount} nonzeros");

// The NonzeroElements property returns a collection of
// IndexValuePair structures that you can use to iterate
// over the elements of the vector:
Console.WriteLine("Nonzero elements of v4:");
foreach(var pair in v4.NonzeroElements)
    Console.WriteLine("Element {0} = {1}", pair.Index, pair.Value);

// All other vector methods and properties are also available,
// Their implementations take advantage of sparsity.
Console.WriteLine($"Norm(v4) = {v4.Norm()}");
Console.WriteLine($"Sum(v4) = {v4.Sum()}");

// Note that some operations convert a sparse vector to a
// DenseVector, causing memory to be allocated for all
// elements.

//
// Sparse Matrices
//

// All sparse matrix classes inherit from SparseMatrix. This is an abstract class.
// There currently is only one implementation class:
// SparseCompressedColumnMatrix.

// Sparse matrices are created by calling the CreateSparse factory
// method on the Matrix class. It has 4 overloads:

// The first overload takes the number of rows and columns as arguments:
var m1 = Matrix.CreateSparse<double>(100000, 100000);

// The second overload adds a fill factor:
var m2 = Matrix.CreateSparse<double>(100000, 100000, 1e-5);

// The third overload uses the actual number of nonzero elements rather than
// the fraction:
var m3 = Matrix.CreateSparse<double>(10000, 10000, 20000);

// The fourth overload lets you specify the locations and values of the
// nonzero elements:
int[] rows = { 1, 11, 111, 1111 };
int[] columns = { 2, 22, 222, 2222 };
double[] matrixValues = { 3.0, 33.0, 333.0, 3333.0 };
var m4 = Matrix.CreateSparse(10000, 10000, rows, columns, matrixValues);

// You can access elements as before...
Console.WriteLine("m4[111, 222] = {0}", m4[111, 222]);
m4[99, 22] = 99.0;

// A series of Insert methods lets you build a sparse matrix from scratch:
// A single value:
m1.InsertEntry(25.0, 200, 500);
// Multiple values:
m1.InsertEntries(matrixValues, rows, columns);
// Multiple values all in the same column:
m1.InsertColumn(33, values, indexes);
// Multiple values all in the same row:
m1.InsertRow(55, values, indexes);

// A clique is a 2-dimensional submatrix with indexed rows and columns.
var clique = Matrix.CreateFromArray(2, 2, new double[] {11, 12, 21, 22},
    MatrixElementOrder.ColumnMajor);
int[] cliqueIndexes = new int[] { 5, 8 };
m1.InsertClique(clique, cliqueIndexes, cliqueIndexes);

// You can use the NonzeroElements collection to iterate
// over the nonzero elements of the matrix. The items
// are of type RowColumnValueTriplet:
Console.WriteLine("Nonzero elements of m1:");
foreach(var triplet in m1.NonzeroElements)
    Console.WriteLine("m1[{0},{1}] = {2}", triplet.Row, triplet.Column, triplet.Value);

// ... including rows and columns.
var column = m4.GetColumn(22);
Console.WriteLine("Nonzero elements in column 22 of m4:");
foreach (var pair in column.NonzeroElements)
    Console.WriteLine("Element {0} = {1}", pair.Index, pair.Value);

// Many matrix methods have been optimized to take advantage of sparsity:
Console.WriteLine($"F-norm(m1) = {m1.FrobeniusNorm()}");

// But beware: some revert to a dense algorithm and will fail on huge matrices:
try
{
    var inverse = m1.GetInverse();
}
catch (OutOfMemoryException e)
{
    Console.WriteLine(e.Message);
}

Console.Write("Press Enter key to exit...");
Console.ReadLine();