Multiplication matricielle en Java

1. Vue d'ensemble

Dans ce tutoriel, nous verrons comment nous pouvons multiplier deux matrices en Java.

Comme le concept de matrice n'existe pas nativement dans le langage, nous allons l'implémenter nous-mêmes, et nous travaillerons également avec quelques bibliothèques pour voir comment elles gèrent la multiplication des matrices.

Au final, nous ferons un petit benchmarking des différentes solutions que nous avons explorées afin de déterminer la plus rapide.

2. L'exemple

Commençons par mettre en place un exemple auquel nous pourrons nous référer tout au long de ce tutoriel.

Tout d'abord, nous allons imaginer une matrice 3 × 2:

Imaginons maintenant une deuxième matrice, deux lignes par quatre colonnes cette fois:

Ensuite, la multiplication de la première matrice par la deuxième matrice, qui se traduira par une matrice 3 × 4:

Pour rappel, ce résultat est obtenu en calculant chaque cellule de la matrice résultante avec cette formule :

r est le nombre de rangées de la matrice A , c est le nombre de colonnes de la matrice B et n est le nombre de colonnes de la matrice A , qui doit correspondre au nombre de rangées de la matrice B .

3. Multiplication matricielle

3.1. Propre mise en œuvre

Commençons par notre propre implémentation de matrices.

Nous allons garder les choses simples et utiliser simplement des tableaux doubles à deux dimensions :

double[][] firstMatrix = { new double[]{1d, 5d}, new double[]{2d, 3d}, new double[]{1d, 7d} }; double[][] secondMatrix = { new double[]{1d, 2d, 3d, 7d}, new double[]{5d, 2d, 8d, 1d} };

Ce sont les deux matrices de notre exemple. Créons celui attendu comme résultat de leur multiplication:

double[][] expected = { new double[]{26d, 12d, 43d, 12d}, new double[]{17d, 10d, 30d, 17d}, new double[]{36d, 16d, 59d, 14d} };

Maintenant que tout est configuré, implémentons l'algorithme de multiplication. Nous allons d'abord créer un tableau de résultats vide et parcourir ses cellules pour stocker la valeur attendue dans chacune d'elles:

double[][] multiplyMatrices(double[][] firstMatrix, double[][] secondMatrix) { double[][] result = new double[firstMatrix.length][secondMatrix[0].length]; for (int row = 0; row < result.length; row++) { for (int col = 0; col < result[row].length; col++) { result[row][col] = multiplyMatricesCell(firstMatrix, secondMatrix, row, col); } } return result; }

Enfin, implémentons le calcul d'une seule cellule. Pour y parvenir, nous utiliserons la formule indiquée précédemment dans la présentation de l'exemple :

double multiplyMatricesCell(double[][] firstMatrix, double[][] secondMatrix, int row, int col) { double cell = 0; for (int i = 0; i < secondMatrix.length; i++) { cell += firstMatrix[row][i] * secondMatrix[i][col]; } return cell; }

Enfin, vérifions que le résultat de l'algorithme correspond à notre résultat attendu:

double[][] actual = multiplyMatrices(firstMatrix, secondMatrix); assertThat(actual).isEqualTo(expected);

3.2. EJML

La première bibliothèque que nous examinerons est EJML, qui signifie Efficient Java Matrix Library. Au moment de la rédaction de ce tutoriel, il s'agit de l' une des bibliothèques de matrices Java les plus récemment mises à jour . Son objectif est d'être le plus efficace possible en matière de calcul et d'utilisation de la mémoire.

Nous devrons ajouter la dépendance à la bibliothèque dans notre pom.xml :

 org.ejml ejml-all 0.38 

Nous allons utiliser à peu près le même schéma que précédemment: créer deux matrices selon notre exemple et vérifier que le résultat de leur multiplication est celui que nous avons calculé précédemment.

Alors, créons nos matrices en utilisant EJML. Pour ce faire, nous utiliserons la classe SimpleMatrix proposée par la bibliothèque .

Il peut prendre un double tableau à deux dimensions comme entrée pour son constructeur:

SimpleMatrix firstMatrix = new SimpleMatrix( new double[][] { new double[] {1d, 5d}, new double[] {2d, 3d}, new double[] {1d ,7d} } ); SimpleMatrix secondMatrix = new SimpleMatrix( new double[][] { new double[] {1d, 2d, 3d, 7d}, new double[] {5d, 2d, 8d, 1d} } );

Et maintenant, définissons notre matrice attendue pour la multiplication:

SimpleMatrix expected = new SimpleMatrix( new double[][] { new double[] {26d, 12d, 43d, 12d}, new double[] {17d, 10d, 30d, 17d}, new double[] {36d, 16d, 59d, 14d} } );

Now that we're all set up, let's see how to multiply the two matrices together. The SimpleMatrix class offers a mult() method taking another SimpleMatrix as a parameter and returning the multiplication of the two matrices:

SimpleMatrix actual = firstMatrix.mult(secondMatrix);

Let's check if the obtained result matches the expected one.

As SimpleMatrix doesn't override the equals() method, we can't rely on it to do the verification. But, it offers an alternative: the isIdentical() method which takes not only another matrix parameter but also a double fault tolerance one to ignore small differences due to double precision:

assertThat(actual).matches(m -> m.isIdentical(expected, 0d));

That concludes matrices multiplication with the EJML library. Let's see what the other ones are offering.

3.3. ND4J

Let's now try the ND4J Library. ND4J is a computation library and is part of the deeplearning4j project. Among other things, ND4J offers matrix computation features.

First of all, we've to get the library dependency:

 org.nd4j nd4j-native 1.0.0-beta4 

Note that we're using the beta version here because there seems to have some bugs with GA release.

For the sake of brevity, we won't rewrite the two dimensions double arrays and just focus on how they are used with each library. Thus, with ND4J, we must create an INDArray. In order to do that, we'll call the Nd4j.create() factory method and pass it a double array representing our matrix:

INDArray matrix = Nd4j.create(/* a two dimensions double array */);

As in the previous section, we'll create three matrices: the two we're going to multiply together and the one being the expected result.

After that, we want to actually do the multiplication between the first two matrices using the INDArray.mmul() method:

INDArray actual = firstMatrix.mmul(secondMatrix);

Then, we check again that the actual result matches the expected one. This time we can rely on an equality check:

assertThat(actual).isEqualTo(expected);

This demonstrates how the ND4J library can be used to do matrix calculations.

3.4. Apache Commons

Let's now talk about the Apache Commons Math3 module, which provides us with mathematic computations including matrices manipulations.

Again, we'll have to specify the dependency in our pom.xml:

 org.apache.commons commons-math3 3.6.1 

Once set up, we can use the RealMatrix interface and its Array2DRowRealMatrix implementation to create our usual matrices. The constructor of the implementation class takes a two-dimensional double array as its parameter:

RealMatrix matrix = new Array2DRowRealMatrix(/* a two dimensions double array */);

As for matrices multiplication, the RealMatrix interface offers a multiply() method taking another RealMatrix parameter:

RealMatrix actual = firstMatrix.multiply(secondMatrix);

We can finally verify that the result is equal to what we're expecting:

assertThat(actual).isEqualTo(expected);

Let's see the next library!

3.5. LA4J

This one's named LA4J, which stands for Linear Algebra for Java.

Let's add the dependency for this one as well:

 org.la4j la4j 0.6.0 

Now, LA4J works pretty much like the other libraries. It offers a Matrix interface with a Basic2DMatrix implementation that takes a two-dimensional double array as input:

Matrix matrix = new Basic2DMatrix(/* a two dimensions double array */);

As in the Apache Commons Math3 module, the multiplication method is multiply() and takes another Matrix as its parameter:

Matrix actual = firstMatrix.multiply(secondMatrix);

Once again, we can check that the result matches our expectations:

assertThat(actual).isEqualTo(expected);

Let's now have a look at our last library: Colt.

3.6. Colt

Colt is a library developed by CERN. It provides features enabling high performance scientific and technical computing.

As with the previous libraries, we must get the right dependency:

 colt colt 1.2.0 

In order to create matrices with Colt, we must make use of the DoubleFactory2D class. It comes with three factory instances: dense, sparse and rowCompressed. Each is optimized to create the matching kind of matrix.

For our purpose, we'll use the dense instance. This time, the method to call is make() and it takes a two-dimensional double array again, producing a DoubleMatrix2D object:

DoubleMatrix2D matrix = doubleFactory2D.make(/* a two dimensions double array */);

Once our matrices are instantiated, we'll want to multiply them. This time, there's no method on the matrix object to do that. We've got to create an instance of the Algebra class which has a mult() method taking two matrices for parameters:

Algebra algebra = new Algebra(); DoubleMatrix2D actual = algebra.mult(firstMatrix, secondMatrix);

Then, we can compare the actual result to the expected one:

assertThat(actual).isEqualTo(expected);

4. Benchmarking

Now that we're done with exploring the different possibilities of matrix multiplication, let's check which are the most performant.

4.1. Small Matrices

Let's begin with small matrices. Here, a 3×2 and a 2×4 matrices.

In order to implement the performance test, we'll use the JMH benchmarking library. Let's configure a benchmarking class with the following options:

public static void main(String[] args) throws Exception { Options opt = new OptionsBuilder() .include(MatrixMultiplicationBenchmarking.class.getSimpleName()) .mode(Mode.AverageTime) .forks(2) .warmupIterations(5) .measurementIterations(10) .timeUnit(TimeUnit.MICROSECONDS) .build(); new Runner(opt).run(); }

This way, JMH will make two full runs for each method annotated with @Benchmark, each with five warmup iterations (not taken into the average computation) and ten measurement ones. As for the measurements, it'll gather the average time of execution of the different libraries, in microseconds.

We then have to create a state object containing our arrays:

@State(Scope.Benchmark) public class MatrixProvider { private double[][] firstMatrix; private double[][] secondMatrix; public MatrixProvider() { firstMatrix = new double[][] { new double[] {1d, 5d}, new double[] {2d, 3d}, new double[] {1d ,7d} }; secondMatrix = new double[][] { new double[] {1d, 2d, 3d, 7d}, new double[] {5d, 2d, 8d, 1d} }; } }

That way, we make sure arrays initialization is not part of the benchmarking. After that, we still have to create methods that do the matrices multiplication, using the MatrixProvider object as the data source. We won't repeat the code here as we saw each library earlier.

Finally, we'll run the benchmarking process using our main method. This gives us the following result:

Benchmark Mode Cnt Score Error Units MatrixMultiplicationBenchmarking.apacheCommonsMatrixMultiplication avgt 20 1,008 ± 0,032 us/op MatrixMultiplicationBenchmarking.coltMatrixMultiplication avgt 20 0,219 ± 0,014 us/op MatrixMultiplicationBenchmarking.ejmlMatrixMultiplication avgt 20 0,226 ± 0,013 us/op MatrixMultiplicationBenchmarking.homemadeMatrixMultiplication avgt 20 0,389 ± 0,045 us/op MatrixMultiplicationBenchmarking.la4jMatrixMultiplication avgt 20 0,427 ± 0,016 us/op MatrixMultiplicationBenchmarking.nd4jMatrixMultiplication avgt 20 12,670 ± 2,582 us/op

As we can see, EJML and Colt are performing really well with about a fifth of a microsecond per operation, where ND4j is less performant with a bit more than ten microseconds per operation. The other libraries have performances situated in between.

Also, it's worth noting that when increasing the number of warmup iterations from 5 to 10, performance is increasing for all the libraries.

4.2. Large Matrices

Now, what happens if we take larger matrices, like 3000×3000? To check what happens, let's first create another state class providing generated matrices of that size:

@State(Scope.Benchmark) public class BigMatrixProvider { private double[][] firstMatrix; private double[][] secondMatrix; public BigMatrixProvider() {} @Setup public void setup(BenchmarkParams parameters) { firstMatrix = createMatrix(); secondMatrix = createMatrix(); } private double[][] createMatrix() { Random random = new Random(); double[][] result = new double[3000][3000]; for (int row = 0; row < result.length; row++) { for (int col = 0; col < result[row].length; col++) { result[row][col] = random.nextDouble(); } } return result; } }

As we can see, we'll create 3000×3000 two-dimensions double arrays filled with random real numbers.

Let's now create the benchmarking class:

public class BigMatrixMultiplicationBenchmarking { public static void main(String[] args) throws Exception { Map parameters = parseParameters(args); ChainedOptionsBuilder builder = new OptionsBuilder() .include(BigMatrixMultiplicationBenchmarking.class.getSimpleName()) .mode(Mode.AverageTime) .forks(2) .warmupIterations(10) .measurementIterations(10) .timeUnit(TimeUnit.SECONDS); new Runner(builder.build()).run(); } @Benchmark public Object homemadeMatrixMultiplication(BigMatrixProvider matrixProvider) { return HomemadeMatrix .multiplyMatrices(matrixProvider.getFirstMatrix(), matrixProvider.getSecondMatrix()); } @Benchmark public Object ejmlMatrixMultiplication(BigMatrixProvider matrixProvider) { SimpleMatrix firstMatrix = new SimpleMatrix(matrixProvider.getFirstMatrix()); SimpleMatrix secondMatrix = new SimpleMatrix(matrixProvider.getSecondMatrix()); return firstMatrix.mult(secondMatrix); } @Benchmark public Object apacheCommonsMatrixMultiplication(BigMatrixProvider matrixProvider) { RealMatrix firstMatrix = new Array2DRowRealMatrix(matrixProvider.getFirstMatrix()); RealMatrix secondMatrix = new Array2DRowRealMatrix(matrixProvider.getSecondMatrix()); return firstMatrix.multiply(secondMatrix); } @Benchmark public Object la4jMatrixMultiplication(BigMatrixProvider matrixProvider) { Matrix firstMatrix = new Basic2DMatrix(matrixProvider.getFirstMatrix()); Matrix secondMatrix = new Basic2DMatrix(matrixProvider.getSecondMatrix()); return firstMatrix.multiply(secondMatrix); } @Benchmark public Object nd4jMatrixMultiplication(BigMatrixProvider matrixProvider) { INDArray firstMatrix = Nd4j.create(matrixProvider.getFirstMatrix()); INDArray secondMatrix = Nd4j.create(matrixProvider.getSecondMatrix()); return firstMatrix.mmul(secondMatrix); } @Benchmark public Object coltMatrixMultiplication(BigMatrixProvider matrixProvider) { DoubleFactory2D doubleFactory2D = DoubleFactory2D.dense; DoubleMatrix2D firstMatrix = doubleFactory2D.make(matrixProvider.getFirstMatrix()); DoubleMatrix2D secondMatrix = doubleFactory2D.make(matrixProvider.getSecondMatrix()); Algebra algebra = new Algebra(); return algebra.mult(firstMatrix, secondMatrix); } }

When we run this benchmarking, we obtain completely different results:

Benchmark Mode Cnt Score Error Units BigMatrixMultiplicationBenchmarking.apacheCommonsMatrixMultiplication avgt 20 511.140 ± 13.535 s/op BigMatrixMultiplicationBenchmarking.coltMatrixMultiplication avgt 20 197.914 ± 2.453 s/op BigMatrixMultiplicationBenchmarking.ejmlMatrixMultiplication avgt 20 25.830 ± 0.059 s/op BigMatrixMultiplicationBenchmarking.homemadeMatrixMultiplication avgt 20 497.493 ± 2.121 s/op BigMatrixMultiplicationBenchmarking.la4jMatrixMultiplication avgt 20 35.523 ± 0.102 s/op BigMatrixMultiplicationBenchmarking.nd4jMatrixMultiplication avgt 20 0.548 ± 0.006 s/op

As we can see, the homemade implementations and the Apache library are now way worse than before, taking nearly 10 minutes to perform the multiplication of the two matrices.

Colt is taking a bit more than 3 minutes, which is better but still very long. EJML and LA4J are performing pretty well as they run in nearly 30 seconds. But, it's ND4J which wins this benchmarking performing in under a second on a CPU backend.

4.3. Analysis

That shows us that the benchmarking results really depend on the matrices' characteristics and therefore it's tricky to point out a single winner.

5. Conclusion

In this article, we've learned how to multiply matrices in Java, either by ourselves or with external libraries. After exploring all solutions, we did a benchmark of all of them and saw that, except for ND4J, they all performed pretty well on small matrices. On the other hand, on larger matrices, ND4J is taking the lead.

As usual, the full code for this article can be found over on GitHub.