ch.alpine.tensor

Library for tensor computations in Java.

The tensor library was developed with the following objectives in mind

support for exact precision using integer fractions
support for calculation with physical units
suitable for use in safety-critical real-time systems
API and string expressions inspired by Mathematica

Diverse projects rely on the tensor library:

Mobility-on-Demand

SwissTrolley+

Motion Planning

Autonomous Gokart

Features

multi-dimensional arrays: scalars, vectors, matrices, n-linear forms, Lie-algebra ad-tensor, ...
unstructured, nested tensors, for instance {{1+2*I[A], -3/4}, {{5.678}, 9[kg*s^-1], 2[m^3]}}
scalars are real-, or complex numbers, from finite fields, or quantities with physical units
values are encoded as exact integer fractions, in double precision, and as java.math.BigDecimal
probability distributions for random variate generation: Binomial-, Poisson-, Exponential-distribution, etc.
linear solvers GaussianElimination, CholeskyDecomposition, QRDecomposition, SingularValueDecomposition
matrix functions MatrixExp, MatrixLog, MatrixSqrt, ...
tensor functions TensorProduct, TensorWedge, Trace, HodgeDual, ...
Lie theory: BakerCampbellHausdorff
parametric functions LinearInterpolation, BSplineFunction
window functions: Gaussian, Hamming, Hann, Blackman, ...
spectral analysis: Fourier, SpectrogramArray
import from and export to Mathematica, CSV, and image files

Gallery

Gamma function

Trigonometry

Nylander's formula

Newton's method

Examples

Exact Precision

Solving systems of linear equations

Tensor matrix = Tensors.matrixInt(new int[][] { { 2, -3, 2 }, { 4, 9, -3 }, { -1, 3, 2 } });
System.out.println(Pretty.of(Inverse.of(matrix)));

[
 [   9/37    4/37   -3/37 ]
 [ -5/111    2/37  14/111 ]
 [   7/37   -1/37   10/37 ]
]

Linear programming

Tensor x = LinearOptimization.maxLessEquals( //
    Tensors.vector(1, 1), // rewards
    Tensors.fromString("{{4, -1}, {2, 1}, {-5, 2}}"), // matrix
    Tensors.vector(8, 7, 2)); // rhs
System.out.println(x);

{4/3, 13/3}

Pseudoinverse, Moore-Penrose inverse

Tensor matrix = Tensors.fromString("{{-1 + I, 0}, {-I, 2}, {2 - I, 2 * I}}");
System.out.println(Pretty.of(PseudoInverse.of(matrix)));

[
 [   -1/3-I/3     1/6-I/6     1/6+I/6 ]
 [    1/6-I/3   5/12+I/12  -1/12-I/12 ]
]

Nullspace

Tensor matrix = Tensors.fromString("{{-1/3, 0, I}}");
System.out.println(Pretty.of(NullSpace.of(matrix)));

[
 [    1     0  -I/3 ]
 [    0     1     0 ]
]

Statistics

Distribution distribution = HypergeometricDistribution.of(10, 50, 100);
System.out.println(RandomVariate.of(distribution, 20));
PDF pdf = PDF.of(distribution);
System.out.println("P(X=3)=" + pdf.at(RealScalar.of(3)));

{6, 5, 1, 4, 3, 4, 7, 5, 7, 4, 6, 3, 5, 4, 5, 4, 6, 2, 6, 7}
P(X=3)=84000/742729

Physical Quantities

The tensor library implements Quantity, i.e. numbers with physical units. Several algorithms are verified to work with scalars of type Quantity.

Tensor matrix = Tensors.fromString( //
  "{{60[m^2], 30[m*rad], 20[kg*m]}, {30[m*rad], 20[rad^2], 15[kg*rad]}, {20[kg*m], 15[kg*rad], 12[kg^2]}}");
CholeskyDecomposition cd = CholeskyDecomposition.of(matrix);
System.out.println(cd.diagonal());
System.out.println(Pretty.of(cd.getL()));
System.out.println(cd.det().divide(Quantity.of(20, "m^2*rad")));

{60[m^2], 5[rad^2], 1/3[kg^2]}
[
 [             1              0              0 ]
 [ 1/2[m^-1*rad]              1              0 ]
 [  1/3[kg*m^-1]   1[kg*rad^-1]              1 ]
]
5[kg^2*rad]

The units of a quantity are chosen by the application layer. For instance, Quantity.of(3, "Apples") is valid syntax.

The tensor library contains the resource /unit/si.properties that encodes the SI unit system in the familiar strings such as m, kg, s, but the use of this convention is optional. The example below makes use of these provided definitions

Scalar mass = Quantity.of(300, "g"); // in gram
Scalar a = Quantity.of(981, "cm*s^-2"); // in centi-meters per seconds square
Scalar force = mass.multiply(a);
System.out.println(force);
Scalar force_N = UnitConvert.SI().to(Unit.of("N")).apply(force);
System.out.println(force_N);

294300[cm*g*s^-2]
2943/1000[N]

The scalar type Quantity was developed in collaboration with SwissTrolley+.

Date and Time

The tensor library implements DateTime for calendar arithmetic and data sets with calendar entries. The arithmetic and string expressions are identical to those of the java class LocalDateTime.

Scalar mean = DateTime.of(2022, Month.FEBRUARY, 28, 12, 00);
Scalar sigma = Quantity.of(30, "h");
Distribution distribution = NormalDistribution.of(mean, sigma);
Scalar guess = RandomVariate.of(distribution);
System.out.println(mean.add(sigma));
System.out.println(guess);

2022-03-01T18:00
2022-03-02T10:12:06.641540174

The scalar type DateTime was developed in collaboration with GRZ Technologies.

Miscellaneous

Tensors of rank 3

Tensor ad = LeviCivitaTensor.of(3).negate();
Tensor x = Tensors.vector(7, 2, -4);
Tensor y = Tensors.vector(-3, 5, 2);
System.out.println(ad);
System.out.println(ad.dot(x).dot(y)); // coincides with cross product of x and y

{{{0, 0, 0}, {0, 0, -1}, {0, 1, 0}}, {{0, 0, 1}, {0, 0, 0}, {-1, 0, 0}}, {{0, -1, 0}, {1, 0, 0}, {0, 0, 0}}}
{24, -2, 41}

Functions for complex numbers

System.out.println(Sqrt.of(RationalScalar.of(-9, 16)));

3/4*I

Several functions support evaluation to higher than machine precision for type DecimalScalar.

System.out.println(Exp.of(DecimalScalar.of(10)));
System.out.println(Sqrt.of(DecimalScalar.of(2)));

220255.6579480671651695790064528423`34
1.414213562373095048801688724209698`34

The number after the prime indicates the precision of the decimal. The string representation is compatible with Mathematica.

Indices for the set and get functions start from zero like in C/Java:

Tensor matrix = Array.zeros(3, 4);
matrix.set(Tensors.vector(9, 8, 4, 5), 2);
matrix.set(Tensors.vector(6, 7, 8), Tensor.ALL, 1);
System.out.println(Pretty.of(matrix));
System.out.println(matrix.get(Tensor.ALL, 3)); // extraction of the 4th column

[
 [ 0  6  0  0 ]
 [ 0  7  0  0 ]
 [ 9  8  4  5 ]
]
{0, 0, 5}

Optimization

Distance-based queries for point sets in Euclidean space

k-nearest neighbors

radius search

Visualization

Predefined color gradients

Predefined color lists

Integration

From time to time, a version is deployed and made available for maven integration. Specify repository and dependency of the library tensor in the pom.xml file of your maven project:

<dependencies>
  <!-- other dependencies -->
  <dependency>
    <groupId>ch.alpine</groupId>
    <artifactId>tensor</artifactId>
    <version>1.0.6</version>
  </dependency>
</dependencies>

<repositories>
  <!-- other repositories -->
  <repository>
    <id>tensor-mvn-repo</id>
    <url>https://raw.github.com/datahaki/tensor/mvn-repo/</url>
    <snapshots>
      <enabled>true</enabled>
      <updatePolicy>always</updatePolicy>
    </snapshots>
  </repository>
</repositories>

For Java 17, for version use 1.0.6 .

For Java 11, for version use 1.1.1-jdk-11.

The source code is attached to every release.

The branch master always contains the latest features for Java 17, and does not correspond to the most recent deployed version generally.

datahaki / tensor