Sparse Matrices in F# QuickStart Sample
Illustrates using sparse vectors and matrices using the classes in the Numerics.NET.LinearAlgebra.Sparse namespace in F#.
This sample is also available in: C#, Visual Basic, 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
//=====================================================================
//
// File: sparse-matrices.fs
//
//---------------------------------------------------------------------
//
// This file is part of the Numerics.NET Code Samples.
//
// Copyright (c) 2004-2025 ExoAnalytics Inc. All rights reserved.
//
//=====================================================================
module SparseMatrices
// Illustrates using sparse vectors and matrices using the classes
// in the Numerics.NET.LinearAlgebra namespace
// of Numerics.NET.
#light
open System
// The sparse vector and matrix classes reside in the
// Numerics.NET.LinearAlgebra namespace.
open 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
let licensed = 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.
let v1 = Vector.CreateSparse<float>(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.
let v2 = Vector.CreateSparse<float>(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.
let v3 = Vector.CreateSparse<float>(1000000, 100)
// The fourth constructor lets you specify the indexes of the nonzero
// elements and their values as arrays:
let indexes = [| 1; 10; 100; 1000; 10000 |]
let values = [| 1.0; 10.0; 100.0; 1000.0; 10000.0 |]
let v4 = Vector.CreateSparse(1000000, indexes, values)
// Elements can be accessed individually:
v1[1000] <- 2.0
printfn "v1[1000] = %A" v1[1000]
// The NonzeroCount returns how many elements are non zero:
printfn "v1 has %d nonzeros" v1.NonzeroCount
printfn "v4 has %d nonzeros" v4.NonzeroCount
// The NonzeroElements property returns a collection of
// IndexValuePair structures that you can use to iterate
// over the elements of the vector:
printfn "Nonzero elements of v4:"
for pair in v4.NonzeroElements do
printfn "Element %d = %A" pair.Index pair.Value
// All other vector methods and properties are also available,
// Their implementations take advantage of sparsity.
printfn "Norm(v4) = %A" (v4.Norm())
printfn "Sum(v4) = %A" (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:
let m1 = Matrix.CreateSparse<float>(100000, 100000)
// The second overload adds a fill factor:
let m2 = Matrix.CreateSparse<float>(100000, 100000, 1e-5)
// The third overload uses the actual number of nonzero elements rather than
// the fraction:
let m3 = Matrix.CreateSparse<float>(10000, 10000, 20000)
// The fourth overload lets you specify the locations and values of the
// nonzero elements:
let rows = [| 1; 11; 111; 1111 |]
let columns = [| 2; 22; 222; 2222 |]
let matrixValues = [| 3.0; 33.0; 333.0; 3333.0 |]
let m4 = Matrix.CreateSparse(10000, 10000, rows, columns, matrixValues)
// You can access elements as before...
printfn "m4[111, 222] = %A" 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.
let clique = Matrix.CreateFromArray(2, 2, [| 11.0; 12.0; 21.0; 22.0|],
MatrixElementOrder.ColumnMajor)
let cliqueIndexes = [| 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:
printfn "Nonzero elements of m1:"
for triplet in m1.NonzeroElements do
printfn "m1[%d,%d] = %A" triplet.Row triplet.Column triplet.Value
// ... including rows and columns.
let column = m4.GetColumn(22)
printfn "Nonzero elements in column 22 of m4:"
for pair in column.NonzeroElements do
printfn "Element %d = %A" pair.Index pair.Value
// Many matrix methods have been optimized to take advantage of sparsity:
printfn "F-norm(m1) = %A" (m1.FrobeniusNorm())
// But beware: some revert to a dense algorithm and will fail on huge matrices:
try
let inverse = m1.GetInverse()
printfn "%O" inverse
with
| :? OutOfMemoryException as e -> printfn "%s" e.Message