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SIMD-based linear algebra and statistics, efficient manipulation with numeric data

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SIMD-based linear algebra and statistics for data science with Dart

Table of contents

Linear algebra #

In a few words, linear algebra is a branch of mathematics that works with vectors and matrices.

Vectors and matrices are extremely powerful tools which can be used in real-life applications, such as machine learning algorithms. There are many implementations of these great mathematical entities in a plenty of programming languages, and as Dart offers developers good instrumentarium, e.g. highly optimized virtual machine and rich out-of-the-box library, Dart-based implementation of vectors and matrices has to be quite performant.

Among myriad of standard Dart tools there are SIMD data types. Namely support of SIMD computational architecture served as inspiration for creating this library.

What is SIMD? #

SIMD stands for Single instruction, multiple data - it's a computer architecture that allows to perform uniform mathematical operations in parallel on a list-like data structure. For instance, one has two arrays:

final a = [10, 20, 30, 40];
final b = [50, 60, 70, 80];

and one needs to add these arrays element-wise. Using the regular architecture this operation could be done in the following manner:

final c = List(4);

c[0] = a[0] + b[0]; // operation 1
c[1] = a[1] + b[1]; // operation 2
c[2] = a[2] + b[2]; // operation 3
c[3] = a[3] + b[3]; // operation 4

As you may have noticed, we need to do 4 operations one by one in a row using regular computational approach. But with help of SIMD architecture we may do one arithmetic operation on several operands in parallel, thus element-wise sum of two arrays can be done for just one step:

Vectors #

A couple of words about the underlying architecture #

The library contains two high performant vector classes based on Float32x4 and Float64x2 data types - Float32x4Vector and Float64x2Vector (the second one is generated from the source code of the first vector's implementation)

Most of element-wise operations in the first one are performed in four "threads" and in the second one - in two "threads".

Implementation of both classes is hidden from the library's users. You can create a Float32x4Vector or a Float64x2Vector instance via Vector factory (see examples below).

The vectors are immutable: once created, the vector cannot be changed. All the vector operations lead to creation of a new vector instance (of course, if the operation is supposed to return a Vector).

Both classes implement Iterable<double> interface, so it's possible to use them as regular iterable collections.

It's possible to use vector instances as keys for HashMap and similar data structures and to look up a value by the vector-key, since the hash code for equal vectors is the same:

import 'package:ml_linalg/vector.dart';

final map = HashMap<Vector, bool>();

map[Vector.fromList([1, 2, 3, 4, 5])] = true;

print(map[Vector.fromList([1, 2, 3, 4, 5])]); // true
print(Vector.fromList([1, 2, 3, 4, 5]).hashCode == Vector.fromList([1, 2, 3, 4, 5]).hashCode); // true

Vector benchmarks #

To see the performance benefits provided by the library's vector classes, one may visit benchmark directory: one may find there a baseline benchmark - element-wise summation of two regular List instances and a benchmark of a similar operation, but performed on two Float32x4Vector instances on the same amount of elements and compare the timings:

  • Baseline benchmark (executed on Macbook Air mid 2017), 2 regular lists each with 10,000,000 elements:

  • Actual benchmark (executed on Macbook Air mid 2017), 2 vectors each with 10,000,000 elements:

It took 15 seconds to create a new regular list by summing the elements of two lists, and 0.7 second to sum two vectors - the difference is significant.

Vector operations examples #

Vectors sum

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1 + vector2;

  print(result.toList()); // [3.0, 5.0, 7.0, 9.0, 11.0]

Vectors subtraction

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([4.0, 5.0, 6.0, 7.0, 8.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 2.0, 3.0, 2.0]);
  final result = vector1 - vector2;

  print(result.toList()); // [2.0, 2.0, 4.0, 4.0, 6.0]

Element wise vector by vector multiplication

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1 * vector2;

  print(result.toList()); // [2.0, 6.0, 12.0, 20.0, 30.0]

Element wise vector by vector division

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([6.0, 12.0, 24.0, 48.0, 96.0]);
  final vector2 = Vector.fromList([3.0, 4.0, 6.0, 8.0, 12.0]);
  final result = vector1 / vector2;

  print(result.toList()); // [2.0, 3.0, 4.0, 6.0, 8.0]

Euclidean norm

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.norm();

  print(result); // sqrt(2^2 + 3^2 + 4^2 + 5^2 + 6^2) = sqrt(90) ~~ 9.48

Manhattan norm

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.norm(Norm.manhattan);

  print(result); // 2 + 3 + 4 + 5 + 6 = 20.0

Mean value

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.mean();

  print(result); // (2 + 3 + 4 + 5 + 6) / 5 = 4.0

Sum of all vector elements

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector.sum();

  print(result); // 2 + 3 + 4 + 5 + 6 = 20.0

Product of all vector elements

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector.prod();

  print(result); // 2 * 3 * 4 * 5 * 6 = 720

Element-wise power

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector.pow(3);
  
  print(result); // [2 ^ 3 = 8.0, 3 ^ 3 = 27.0, 4 ^ 3 = 64.0, 5 ^3 = 125.0, 6 ^ 3 = 216.0]

Element-wise exp

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector.exp();
  
  print(result); // [e ^ 2, e ^ 3, e ^ 4, e ^ 5, e ^ 6]

Dot product of two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.dot(vector2);

  print(result); // 1.0 * 2.0 + 2.0 * 3.0 + 3.0 * 4.0 + 4.0 * 5.0 + 5.0 * 6.0 = 70.0

Sum of a vector and a scalar

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector1 + scalar;

  print(result.toList()); // [6.0, 7.0, 8.0, 9.0, 10.0]

Subtraction of a scalar from a vector

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector - scalar;

  print(result.toList()); // [-4.0, -3.0, -2.0, -1.0, 0.0]

Multiplication of a vector by a scalar

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final scalar = 5.0;
  final result = vector * scalar;

  print(result.toList()); // [5.0, 10.0, 15.0, 20.0, 25.0]

Division of a vector by a scalar

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([25.0, 50.0, 75.0, 100.0, 125.0]);
  final scalar = 5.0;
  final result = vector.scalarDiv(scalar);

  print(result.toList()); // [5.0, 10.0, 15.0, 20.0, 25.0]

Euclidean distance between two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2, distance: Distance.euclidean);

  print(result); // ~~2.23

Manhattan distance between two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2, distance: Distance.manhattan);

  print(result); // 5.0

Cosine distance between two vectors

  import 'package:ml_linalg/linalg.dart';

  final vector1 = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final vector2 = Vector.fromList([2.0, 3.0, 4.0, 5.0, 6.0]);
  final result = vector1.distanceTo(vector2, distance: Distance.cosine);

  print(result); // 0.00506

Vector normalization using Euclidean norm

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, 2.0, 3.0, 4.0, 5.0]);
  final result = vector.normalize(Norm.euclidean);

  print(result); // [0.134, 0.269, 0.404, 0.539, 0.674]

Vector normalization using Manhattan norm

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, -2.0, 3.0, -4.0, 5.0]);
  final result = vector.normalize(Norm.manhattan);

  print(result); // [0.066, -0.133, 0.200, -0.266, 0.333]

Vector rescaling (min-max normalization)

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, -2.0, 3.0, -4.0, 5.0, 0.0]);
  final result = vector.rescale();

  print(result); // [0.555, 0.222, 0.777, 0.0, 1.0, 0.444]

Vector serialization

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, -2.0, 3.0, -4.0, 5.0, 0.0]);
  final serialized = vector.toJson();
  print(serialized); // it yields a serializable representation of the vector

  final restoredVector = Vector.fromJson(serialized);
  print(restoredVector); // [1.0, -2.0, 3.0, -4.0, 5.0, 0.0]

Vector mapping

  import 'package:ml_linalg/linalg.dart';

  final vector = Vector.fromList([1.0, -2.0, 3.0, -4.0, 5.0, 0.0]);
  final mapped = vector.mapToVector((el) => el * 2);
  
  print(mapped); // [2.0, -4.0, 6.0, -8.0, 10.0, 0.0]
  print(mapped is Vector); // true
  print(identical(vector, mapped)); // false

Matrices #

Along with SIMD vectors, the library contains SIMD-based Matrices. One can use the matrices via Matrix factory. The matrices are immutable as well as vectors and also they implement Iterable interface (to be more precise, Iterable<Iterable<double>>), thus it's possible to use them as a regular iterable collection.

Matrices are serializable, and that means that one can easily convert a Matrix instance to a json-serializable map via toJson method, see the examples below.

Matrix operations examples #

Creation of diagonal matrix

import 'package:ml_linalg/matrix.dart';

final matrix = Matrix.diagonal([1, 2, 3, 4, 5]);

print(matrix);

The output:

Matrix 5 x 5:
(1.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 2.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 3.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 4.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 5.0)

Creation of scalar matrix

import 'package:ml_linalg/matrix.dart';

final matrix = Matrix.scalar(3, 5);

print(matrix);

The output:

Matrix 5 x 5:
(3.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 3.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 3.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 3.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 3.0)

Creation of identity matrix

import 'package:ml_linalg/matrix.dart';

final matrix = Matrix.identity(5);

print(matrix);

The output:

Matrix 5 x 5:
(1.0, 0.0, 0.0, 0.0, 0.0)
(0.0, 1.0, 0.0, 0.0, 0.0)
(0.0, 0.0, 1.0, 0.0, 0.0)
(0.0, 0.0, 0.0, 1.0, 0.0)
(0.0, 0.0, 0.0, 0.0, 1.0)

Creation of column matrix

final matrix = Matrix.column([1, 2, 3, 4, 5]);

print(matrix);

The output:

Matrix 5 x 1:
(1.0)
(2.0)
(3.0)
(4.0)
(5.0)

Creation of row matrix

final matrix = Matrix.row([1, 2, 3, 4, 5]);

print(matrix);

The output:

Matrix 1 x 5:
(1.0, 2.0, 3.0, 4.0, 5.0)

Sum of a matrix and another matrix

import 'package:ml_linalg/linalg.dart';

final matrix1 = Matrix.fromList([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
final matrix2 = Matrix.fromList([
  [10.0, 20.0, 30.0, 40.0],
  [-5.0, 16.0, 2.0, 18.0],
  [2.0, -1.0, -2.0, -7.0],
]);
print(matrix1 + matrix2);
// [
//  [11.0, 22.0, 33.0, 44.0],
//  [0.0, 22.0, 9.0, 26.0],
//  [11.0, -1.0, -4.0, -10.0],
// ];

Sum of a matrix and a scalar

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
print(matrix + 7);
//  [
//    [8.0, 9.0, 10.0, 11.0],
//    [12.0, 13.0, 14.0, 15.0],
//    [16.0, 7.0, 5.0, 4.0],
//  ];

Multiplication of a matrix and a vector

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final vector = Vector.fromList([2.0, 3.0, 4.0, 5.0]);
  final result = matrix * vector;
  print(result); 
  // a vector-column [
  //  [40],
  //  [96],
  //  [-5],
  //]

Multiplication of a matrix and another matrix

  import 'package:ml_linalg/linalg.dart';

  final matrix1 = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final matrix2 = Matrix.fromList([
    [1.0, 2.0],
    [5.0, 6.0],
    [9.0, .0],
    [-9.0, 1.0],
  ]);
  final result = matrix1 * matrix2;
  print(result);
  //[
  // [2.0, 18.0],
  // [26.0, 54.0],
  // [18.0, 15.0],
  //]

Multiplication of a matrix and a scalar

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
print(matrix * 3);
// [
//   [3.0, 6.0, 9.0, 12.0],
//   [15.0, 18.0, 21.0, 24.0],
//   [27.0, .0, -6.0, -9.0],
// ];

Hadamard product (element-wise matrices multiplication)

import 'package:ml_linalg/linalg.dart';

final matrix1 = Matrix.fromList([
  [1.0, 2.0,  3.0,  4.0],
  [5.0, 6.0,  7.0,  8.0],
  [9.0, 0.0, -2.0, -3.0],
]);
final matrix2 = Matrix.fromList([
  [7.0,   1.0,  9.0,  2.0],
  [2.0,   4.0,  3.0, -8.0],
  [0.0, -10.0, -2.0, -3.0],
]);
print(matrix1.multiply(matrix2));
// [
//   [ 7.0,  2.0, 27.0,   8.0],
//   [10.0, 24.0, 21.0, -64.0],
//   [ 0.0,  0.0,  4.0,   9.0],
// ];

Element wise matrices subtraction

import 'package:ml_linalg/linalg.dart';

final matrix1 = Matrix.fromList([
  [1.0, 2.0, 3.0, 4.0],
  [5.0, 6.0, 7.0, 8.0],
  [9.0, .0, -2.0, -3.0],
]);
final matrix2 = Matrix.fromList([
  [10.0, 20.0, 30.0, 40.0],
  [-5.0, 16.0, 2.0, 18.0],
  [2.0, -1.0, -2.0, -7.0],
]);
print(matrix1 - matrix2);
// [
//   [-9.0, -18.0, -27.0, -36.0],
//   [10.0, -10.0, 5.0, -10.0],
//   [7.0, 1.0, .0, 4.0],
// ];

Matrix transposition

  import 'package:ml_linalg/linalg.dart';
  
  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
    [9.0, .0, -2.0, -3.0],
  ]);
  final result = matrix.transpose();
  print(result);
  //[
  // [1.0, 5.0, 9.0],
  // [2.0, 6.0, .0],
  // [3.0, 7.0, -2.0],
  // [4.0, 8.0, -3.0],
  //]

Matrix LU decomposition

  final matrix = Matrix.fromList([
    [4, 12, -16],
    [12, 37, -43],
    [-16, -43, 98],
  ], dtype: dtype);
  final decomposed = matrix.decompose(Decomposition.LU);
  
  // yields approximately the same matrix as the original one:
  print(decomposed.first * decomposed.last);

Matrix Cholesky decomposition

  final matrix = Matrix.fromList([
    [4, 12, -16],
    [12, 37, -43],
    [-16, -43, 98],
  ], dtype: dtype);
  final decomposed = matrix.decompose(Decomposition.cholesky);
  
  // yields approximately the same matrix as the original one:
  print(decomposed.first * decomposed.last);

Keep in mind that Cholesky decomposition is applicable only for positive definite and symmetric matrices

Matrix LU inversion

  final matrix = Matrix.fromList([
    [-16, -43, 98],
    [33, 12.4, 37],
    [12, -88.3, 4],
  ], dtype: dtype);
  final inverted = matrix.inverse(Inverse.LU);

  print(inverted * matrix);
  // The output (there can be some round-off errors):
  // [1, 0, 0],
  // [0, 1, 0],
  // [0, 0, 1],

Matrix Cholesky inversion

  final matrix = Matrix.fromList([
    [4, 12, -16],
    [12, 37, -43],
    [-16, -43, 98],
  ], dtype: dtype);
  final inverted = matrix.inverse(Inverse.cholesky);

  print(inverted * matrix);
  // The output (there can be some round-off errors):
  // [1, 0, 0],
  // [0, 1, 0],
  // [0, 0, 1],

Keep in mind that since this kind of inversion is based on Cholesky decomposition, the inversion is applicable only for positive definite and symmetric matrices

Lower triangular matrix inversion

  final matrix = Matrix.fromList([
    [  4,   0,  0],
    [ 12,  37,  0],
    [-16, -43, 98],
  ], dtype: dtype);
  final inverted = matrix.inverse(Inverse.forwardSubstitution);

  print(inverted * matrix);
  // The output (there can be some round-off errors):
  // [1, 0, 0],
  // [0, 1, 0],
  // [0, 0, 1],

Upper triangular matrix inversion

  final matrix = Matrix.fromList([
    [4, 12, -16],
    [0, 37, -43],
    [0,  0, -98],
  ], dtype: dtype);
  final inverted = matrix.inverse(Inverse.backwardSubstitution);

  print(inverted * matrix);
  // The output (there can be some round-off errors):
  // [1, 0, 0],
  // [0, 1, 0],
  // [0, 0, 1],

Matrix row-wise reduce

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
  ]); 
  final reduced = matrix.reduceRows((combine, row) => combine + row);
  print(reduced); // [6.0, 8.0, 10.0, 12.0]

Matrix column-wise reduce

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 17.0, 18.0],
    [21.0, 22.0, 23.0, 24.0],
  ]);
  final result = matrix.reduceColumns((combine, vector) => combine + vector);
  print(result); // [50, 66, 90]

Matrix row-wise mapping

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
  ]); 
  final modifier = Vector.filled(4, 2.0);
  final newMatrix = matrix.rowsMap((row) => row + modifier);
  print(newMatrix); 
  // [
  //  [3.0, 4.0, 5.0, 6.0],
  //  [7.0, 8.0, 9.0, 10.0],
  // ]

Matrix column-wise mapping

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0, 4.0],
    [5.0, 6.0, 7.0, 8.0],
  ]); 
  final modifier = Vector.filled(2, 2.0);
  final newMatrix = matrix.columnsMap((column) => column + modifier);
  print(newMatrix); 
  // [
  //  [3.0, 4.0, 5.0, 6.0],
  //  [7.0, 8.0, 9.0, 10.0],
  // ]

Matrix element-wise mapping

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [11.0, 12.0, 13.0, 14.0],
  [15.0, 16.0, 0.0, 18.0],
  [21.0, 22.0, -23.0, 24.0],
], dtype: DType.float32);
final result = matrix.mapElements((element) => element * 2);

print(result);
// [
//  [22.0, 24.0,  26.0, 28.0],
//  [30.0, 32.0,   0.0, 36.0],
//  [42.0, 44.0, -46.0, 48.0],
// ]

Matrix' columns filtering (by column index)

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [11.0, 12.0, 13.0, 14.0],
  [15.0, 16.0, 17.0, 18.0],
  [21.0, 22.0, 23.0, 24.0],
], dtype: dtype);

final indicesToExclude = [0, 3];
final result = matrix.filterColumns((column, idx) => !indicesToExclude.contains(idx));

print(result);
// [
//   [12.0, 13.0],
//   [16.0, 17.0],
//   [22.0, 23.0],
// ]

Matrix' columns filtering (by column)

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [11.0, 33.0, 13.0, 14.0],
  [15.0, 92.0, 17.0, 18.0],
  [21.0, 22.0, 23.0, 24.0],
], dtype: dtype);

final result = matrix.filterColumns((column, _) => column.sum() > 100);

print(result);
// [
//   [33.0],
//   [92.0],
//   [22.0],
// ];

Getting max value of the matrix

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 17.0, 18.0],
    [21.0, 22.0, 23.0, 24.0],
    [24.0, 32.0, 53.0, 74.0],
  ]);
  final maxValue = matrix.max();
  print(maxValue);
  // 74.0

Getting min value of the matrix

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 0.0, 18.0],
    [21.0, 22.0, -23.0, 24.0],
    [24.0, 32.0, 53.0, 74.0],
  ]);
  final minValue = matrix.min();
  print(minValue);
  // -23.0

Matrix element-wise power

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0],
    [4.0, 5.0, 6.0],
    [7.0, 8.0, 9.0],
  ]);
  final result = matrix.pow(3.0);
  
  print(result);
  // [1 ^ 3 = 1,   2 ^ 3 = 8,   3 ^ 3 = 27 ]
  // [4 ^ 3 = 64,  5 ^ 3 = 125, 6 ^ 3 = 216]
  // [7 ^ 3 = 343, 8 ^ 3 = 512, 9 ^ 3 = 729]

Matrix element-wise exp

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0],
    [4.0, 5.0, 6.0],
    [7.0, 8.0, 9.0],
  ]);
  final result = matrix.exp();
  
  print(result);
  // [e ^ 1, e ^ 2, e ^ 3]
  // [e ^ 4, e ^ 5, e ^ 6]
  // [e ^ 7, e ^ 8, e ^ 9]

Sum of all matrix elements

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0],
    [4.0, 5.0, 6.0],
    [7.0, 8.0, 9.0],
  ]);
  final result = matrix.sum();
  
  print(result); // 1.0 + 2.0 + 3.0 + 4.0 + 5.0 + 6.0 + 7.0 + 8.0 + 9.0

Product of all matrix elements

  import 'package:ml_linalg/linalg.dart';

  final matrix = Matrix.fromList([
    [1.0, 2.0, 3.0],
    [4.0, 5.0, 6.0],
    [7.0, 8.0, 9.0],
  ]);
  final result = matrix.product();
  
  print(result); // 1.0 * 2.0 * 3.0 * 4.0 * 5.0 * 6.0 * 7.0 * 8.0 * 9.0

Matrix indexing and sampling

To access a certain row vector of the matrix one may use [] operator:

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 0.0, 18.0],
    [21.0, 22.0, -23.0, 24.0],
    [24.0, 32.0, 53.0, 74.0],
  ]);

final row = matrix[2];

print(row); // [21.0, 22.0, -23.0, 24.0]

The library's matrix interface offers sample method that is supposed to return a new matrix, consisting of different segments of a source matrix. It's possible to build a new matrix from certain columns and vectors and they should not be necessarily subsequent.

For example, one needs to create a matrix from rows 1, 3, 5 and columns 1 and 3. To do so, it's needed to perform the following:

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
//| 1 |         | 3 |                
  [4.0,   8.0,   12.0,   16.0,  34.0], // 1 Range(0, 1)
  [20.0,  24.0,  28.0,   32.0,  23.0],
  [36.0,  .0,   -8.0,   -12.0,  12.0], // 3 Range(2, 3)
  [16.0,  1.0,  -18.0,   3.0,   11.0],
  [112.0, 10.0,  34.0,   2.0,   10.0], // 5 Range(4, 5)
]);
final result = matrix.sample(
  rowIndices: [0, 2, 4],
  columnIndices: [0, 2],
);
print(result);
/*
  [4.0,   12.0],
  [36.0,  -8.0],
  [112.0, 34.0]
*/

Add new columns to a matrix

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
  [11.0, 12.0, 13.0, 14.0],
  [15.0, 16.0, 0.0, 18.0],
  [21.0, 22.0, -23.0, 24.0],
  [24.0, 32.0, 53.0, 74.0],
], dtype: DType.float32);

final updatedMatrix = matrix.insertColumns(0, [
  Vector.fromList([1.0, 2.0, 3.0, 4.0]),
  Vector.fromList([-1.0, -2.0, -3.0, -4.0]),
]);

print(updatedMatrix);
// [
//  [1.0, -1.0, 11.0, 12.0, 13.0, 14.0],
//  [2.0, -2.0, 15.0, 16.0, 0.0, 18.0],
//  [3.0, -3.0, 21.0, 22.0, -23.0, 24.0],
//  [4.0, -4.0, 24.0, 32.0, 53.0, 74.0],
// ]

print(updatedMatrix == matrix); // false

Matrix serialization/deserialization

To convert a matrix to a json-serializable map one may use toJson method:

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromList([
    [11.0, 12.0, 13.0, 14.0],
    [15.0, 16.0, 0.0, 18.0],
    [21.0, 22.0, -23.0, 24.0],
    [24.0, 32.0, 53.0, 74.0],
  ]);

final serialized = matrix.toJson();

To restore a serialized matrix one may use Matrix.fromJson constructor:

import 'package:ml_linalg/linalg.dart';

final matrix = Matrix.fromJson(serialized);

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SIMD-based linear algebra and statistics, efficient manipulation with numeric data

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