# DMelt:Numeric/1 Linear Algebra

## Linear Algebra

Linear algebra is the branch of mathematics concerning linear equations. Read about linear algebra. The DataMelt contains many high-performance Java packages for linear algebra and matrix operations, such as

- jhplot.math.LinearAlgebra (matrix libraries)
- Jama.package-summary (matrix package)
- Parallel Colt (high-performance calculations on multiple cores)
- EJML (efficient Java Matrix Library)
- La4J (La4J Matrix Library)
- ND4J (N-dimensional arrays)
- VectorZ (fast double-precision vector and matrix maths library based on N-dimensional arrays)

## Vectors

For manipulations with vectors, use the following "core" classes with useful static methods:

- ArrayMath - manipulate with 1D arrays
- IntegerArray - to construct integer 1D arrays
- P0I - the standard jhplot 1D integer arrays with many methods
- P0D- the standard jhplot 1D double arrays with many methods

You can also use the Python list as a container to hold and manipulate with 1D data structures, such as P0I and P0D arrays. In addition, DataMelt supports 3rd-party vectors and their methods:

- Colt lists
- La4j vectors
- jSCi vectors
- Jas vectors
- Rit vectors
- jScience vectors
- VectorZ (mikera) package

Below we show how to use static methods by mixing Python lists with the static methods of the ArrayMath Java class:

from jhplot.math.ArrayMath import * a=[-1,-2,3,4,5,-6,7,10] # make a Python list print a b=invert(a) # invert it print b.tolist() c=scalarMultiply(10, b) # scalar multiply by 10 print c.tolist() print mean(a) print sumSquares(a) # sums the squares

This code generates the following output:

[-1, -2, 3, 4, 5, -6, 7, 10] [10, 7, -6, 5, 4, 3, -2, -1] [100.0, 70.0, -60.0, 50.0, 40.0, 30.0, -20.0, -10.0] 2.5 240

## Matrices

A large choice matrix manipulation provided by DataMelt is shown below. The core DataMelt packages include the following implementation:

Please look at DataMelt API to see all Java implementations of matricies from different packages. Here are examples:

- Colt matrix package
- jsci.maths.matrices
- Jama.Matrix
- org.la4j.matrix
- org.apache.commons.math3.linear.RealMatrix
- cern.colt.matrix.tfloat
- SparseRealMatrix
- Universal Java Matrix Package (UJMP)
- VectorZ (mikera) package

For matrix calculations, consider the package LinearAlgebra. A simple example below can illustrate how to get started:

from jhplot.math.LinearAlgebra import * array = [[1.,2.,3],[4.,5.,6.],[7.,8.,10.]] inverse=inverse(array) # calculate inverse matrix print inverse print trace(array) # calculate trace

While working with NxM matrices, consider another important library DoubleArray which helps to manipulate with double arrays. For example, this class has toString() method to print double arrays in a convenient format. Consider this example:

from jhplot.math.LinearAlgebra import * from jhplot.math.DoubleArray import * print dir() # list all imported methods array = [[1.,2.,3],[4.,5.,6.],[7.,8.,10.]] inverse=inverse(array) print toString("%7.3f", inverse.tolist()) # print the matrix

The above script prints all the methods for matrix manipulation and the inverse matrix itself:

-0.667 -1.333 1.000 -0.667 3.667 -2.000 1.000 -2.000 1.000

## Scripting using Jama

The Java Jama package allows allows matrix creation and manipulation. Below is a simple example of how to call Jama package to create a matrix to perform some manipulations.

from Jama import * array = [[1.,2.,3],[4.,5.,6.],[7.,8.,10.]] a = Matrix(array) b = Matrix.random(3,1) x = a.solve(b) Residual = a.times(x).minus(b); rnorm = Residual.normInf();

To print a matrix, one can make a simple function that converts a matrix to a string:

from Jama import * def toString(a): s="" for i in range(a.getRowDimension()): for j in range(a.getColumnDimension()): s=s+str(a.get(i,j))+" " s=s+ "\n" return s print toString(a) # print "a" (must be Matrix object)

Here is a summary of Jama capability. Please read Jama API for detailed description.

<html>

Object Manipulation |
constructors
clone |

Elementary Operations |
addition
norm |

Decompositions |
Cholesky
nonsymmetric eigenvalue |

Equation Solution |
nonsingular systems
least squares |

Derived Quantities |
condition number
pseudoinverse |

</html>

The following simple example solves a 3x3 linear system Ax=b and computes the norm of the residual. Using the Java syntax, this can be written as:

double[][] array = {{1.,2.,3},{8.,9.,10.},{29.,23.,34.}}; Matrix A = new Matrix(array); Matrix b = Matrix.random(3,1); Matrix x = A.solve(b); Matrix Residual = A.times(x).minus(b); double rnorm = Residual.normInf();

## Linear Algebra with Apache Math

For matrix manipulation, one can also use Apache Math Common Linear Algebra package: Look at the Apache API for linear algebra. Below we show a simple example of how to create and manipulate with matrices:

from org.apache.commons.math3.linear import * # Create a real matrix with two rows and three columns matrixData = [[1,2,3], [2,5,3]] m=Array2DRowRealMatrix(matrixData) # One more with three rows, two columns matrixData2 = [[1,2], [2,5], [1, 7]] n=Array2DRowRealMatrix(matrixData2) # Now multiply m by n p = m.multiply(n); print p.getRowDimension() # print 2 print p.getColumnDimension() # print 2 # Invert p, using LU decomposition inverse =LUDecompositionImpl(p).getSolver().getInverse();

## Dense and sparse matrices

la4j package provides a simple API to handle sparse and dense matrices. According to the La4j authors, the package has Linear systems solving (Gaussian, Jacobi, Zeidel, Square Root, Sweep and other), Matrices decomposition (Eigenvalues/Eigenvectors, SVD, QR, LU, Cholesky and other), and useful I/O (CSV and MatrixMarket format).

Let us consider how we define such matrices in this package:

""" <h1>Lightweight Java sparse/dense matrix library</h1> See <a href="http://la4j.org/">La4j</a> for detail <p> The <strong>la4j</strong> is open source and 100% Java library that provides <a href="http://mathworld.wolfram.com/LinearAlgebra.html">Linear Algebra</a> primitives (matrices and vectors) and algorithms. The la4j was initially designed to be lightweight and simple tool for passionate Java developers. It has been started as student project and turned into one of the most popular Java packages for matrices and vectors. </p> <p> The key features of the la4j are listed bellow: </p> <ul> <li>Great performance allied with beautiful design</li> <li>No dependencies and tiny size (~150kb jar)</li> <li>Fluent object-oriented/functional API</li> <li>Sparse (<a href="http://netlib.org/linalg/html_templates/node91.html#SECTION00931100000000000000">CRS</a>, <a href="http://netlib.org/linalg/html_templates/node92.html#SECTION00931200000000000000">CCS</a>) and dense (1D/2D arrays) matrices</li> <li>Linear systems solving (Gaussian, Jacobi, Zeidel, Square Root, Sweep and other)</li> <li>Matrices decomposition (Eigenvalues/Eigenvectors, SVD, QR, LU, Cholesky and other)</li> <li><a href="http://math.nist.gov/MatrixMarket/formats.html">MatrixMarket</a>/<a href="http://en.wikipedia.org/wiki/Comma-separated_values">CSV</a> IO formats support for matrices and vectors</li> </ul> <br /> <hr /> """ from org.la4j.matrix.dense import * from org.la4j.matrix.sparse import * from org.la4j.matrix import * print "Dense matrix" m=Basic2DMatrix.from2DArray([[1.0, 2.0, 3.0 ], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) m.multiplyByItsTranspose() print m

Let us show how to perform manipulations with such matrices. In the example shown below, we multiply matrices and then perform a transformation of matrices using an arbitrary function:

""" <h1>Lightweight Java sparse/dense matrix library</h1> The <a href="http://la4j.org/">La4j</a> is open source and 100% Java library that provides <a href="http://mathworld.wolfram.com/LinearAlgebra.html">Linear Algebra</a> primitives (matrices and vectors) and algorithms. The la4j was initially designed to be lightweight and simple tool for passionate Java developers. It has been started as student project and turned into one of the most popular Java packages for matrices and vectors. """ from org.la4j.matrix.dense import * from org.la4j.matrix.sparse import * from org.la4j.matrix import * from org.la4j.matrix.functor import * from org.la4j.vector.sparse import * from org.la4j.vector.dense import * from org.la4j import * print "Create Dense matrix" a=Basic2DMatrix.from2DArray([[1.0, 2.0, 3.0 ], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) print "Create Sparse matrix" b=CRSMatrix.from2DArray( [[1.0, 2.0, 3.0 ], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) # define a function for this matrix class myFunction(MatrixFunction): def evaluate(self, i,j,value): if (i==j): return value else: return 0 print "Matrix before transform=\n", a.toString() print "We keep only diagonal elements of matrix 'a'" c= a.transform(myFunction()) print c.toString()

Next, let us make transformation using a user-defined function which keeps only diagonal elements:

""" <h1>Lightweight Java sparse/dense matrix library</h1> The <a href="http://la4j.org/">La4j</a> is open source and 100% Java library that provides <a href="http://mathworld.wolfram.com/LinearAlgebra.html">Linear Algebra</a> primitives (matrices and vectors) and algorithms. The la4j was initially designed to be lightweight and simple tool for passionate Java developers. It has been started as student project and turned into one of the most popular Java packages for matrices and vectors. """ from org.la4j.matrix.dense import * from org.la4j.matrix.sparse import * from org.la4j.matrix import * from org.la4j.matrix.functor import * from org.la4j.vector.sparse import * from org.la4j.vector.dense import * from org.la4j import * print "Create Dense matrix" a=Basic2DMatrix.from2DArray([[1.0, 2.0, 3.0 ], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) print "Create Sparse matrix" b=CRSMatrix.from2DArray( [[1.0, 2.0, 3.0 ], [4.0, 5.0, 6.0], [7.0, 8.0, 9.0]]) # define a function for this matrix class myFunction(MatrixFunction): def evaluate(self, i,j,value): if (i==j): return value else: return 0 print "Matrix before transform=\n", a.toString() print "We keep only diagonal elements of matrix 'a'" c= a.transform(myFunction()) print c.toString()

## Dense matrices in EJML

The EJML package provides 2 types of matrices:

- DenseMatrix64F - a dense matrix with elements that are 64-bit floats (doubles)
- SimpleMatrix - a wrapper around DenseMatrix64F that provides an easy to use object oriented interface for performing matrix operations.

EJML library provides the following operations:

- Basic Matrix Operators (addition, multiplication ... )
- Matrix Manipulation (extract, insert, combine... )
- Linear Solvers (linear, least squares, incremental... )
- Matrix Decompositions (LU, QR, Cholesky, SVD, Eigenvalue ...)
- Matrix Features (rank, symmetric, definitiveness ... )
- Creating random Matrices (covariance, orthogonal, symmetric ... )
- Different Internal Formats (row-major, block)
- Unit Testing
- Saving matrices into CSV files

Let us give a simple example using Jython: We create a few matrices and perform some algebra (multiplication, inverse etc). We also computes the eigen value decomposition and will print the answer:

from org.ejml.simple import SimpleMatrix # create 3 matricies S=SimpleMatrix([[1,2,3,4],[3,4,4,5],[3,3,3,1],[3,3,1,1]]) H=SimpleMatrix([[1,0,0,4],[1,4,0,5],[2,3,1,1],[2,3,1,1]]) P=SimpleMatrix([[1,0,0,4],[1,4,0,5],[2,1,1,2],[1,2,2,1]]) SIN=S.invert() # invert # complex operation: invert, multiply,transporse K = P.mult(H.transpose().mult(S.invert())); K.eig() # Computes the eigen value decomposition of 'this' matrix K.print()

You can test various features of a matrix using this MatrixFeatures API. For example, let's check "SkewSymmetric" feature of a given matrix:

from org.ejml.data import DenseMatrix64F from org.ejml.ops import MatrixFeatures A=DenseMatrix64F(2,2) A.set(0,1,2) A.set(1,0,-2.0000000001) A.print() if MatrixFeatures.isSkewSymmetric(A,1e-8): print "Is skew symmetric!" else: print "Should be skew symmetric!"

You can save matrices in CVS files or binary formats. The example below shows how to do this:

from org.ejml.simple import SimpleMatrix S=SimpleMatrix([[1,2,3,4],[3,4,4,5],[3,3,3,1],[3,3,1,1]]) S.print() S.print("%e") S.print("%10.2f"); # save and read S.saveToFileCSV("matrix_file.csv"); B = SimpleMatrix.loadCSV("matrix_file.csv"); # save in binary form S.saveToFileBinary("matrix_file.data"); B=SimpleMatrix.loadBinary("matrix_file.data");

Finally, you can visualize the matrices. The example below creates a matrix and then shows it's state in a window. Block means an element is zero. Red positive and blue negative. More intense the color larger the element's absolute value is.

from org.ejml.ops import RandomMatrices from java.util import Random from org.ejml.ops import MatrixVisualization from org.ejml.data import DenseMatrix64F A=DenseMatrix64F(4,4,True,[0,2,3,4,-2,0,2,3,-3,-2,0,2,-4,-3,-2,0]) # Creates a window visually showing the matrix's state. # Block means an element is zero. Red positive and blue negative. # More intense the color larger the element's absolute value is. MatrixVisualization.show(A,"Small Matrix"); B = DenseMatrix64F(25,50) for i in range(25): B.set(i,i,i+1) MatrixVisualization.show(B,"Larger Diagonal Matrix")

The above example shows a graphic representation of the matrix defined as:

A=DenseMatrix64F(4,4,True,[0,2,3,4,-2,0,2,3,-3,-2,0,2,-4,-3,-2,0])

## Dense and sparse matrices in UJMP

The UJMP package provides severals types of matrices, such as dense, sparse and multidimentional. Look at org.ujmp.core. In addition, the package provides manipulation and visualization environment for such matrices. This image shows how to visualize a random matrix:

with the code shown below:

from org.ujmp.core import Matrix # create matrix with random values between 0 and 1 rand=Matrix.Factory.rand(100, 10) # create matrix with random values between -1 and - 1 randn = Matrix.Factory.randn(100, 10) rand.showGUI() randn.showGUI()

Here is another example to create and manipulate with dense and parse matricies:

from org.ujmp.core import Matrix,DenseMatrix,SparseMatrix # dense empty matrix with 4 rows and 4 columns dense = DenseMatrix.Factory.zeros(4, 4) # set entry at row 2 and column 3 to the value 5.0 dense.setAsDouble(5.0, 2, 3) # set some other values dense.setAsDouble(1.0, 0, 0) dense.setAsDouble(3.0, 1, 1) dense.setAsDouble(4.0, 2, 2) dense.setAsDouble(-2.0, 3, 3) dense.setAsDouble(-2.0, 1, 3) print dense # create a sparse empty matrix with 4 rows and 4 columns sparse = SparseMatrix.Factory.zeros(4, 4) sparse.setAsDouble(2.0, 0, 0) # basic calculations transpose = dense.transpose() sum = dense.plus(sparse) difference = dense.minus(sparse) matrixProduct = dense.mtimes(sparse) scaled = dense.times(2.0) inverse = dense.inv() pseudoInverse = dense.pinv() determinant = dense.det() singularValueDecomposition = dense.svd() eigenValueDecomposition = dense.eig() luDecomposition = dense.lu() qrDecomposition = dense.qr() choleskyDecomposition = dense.chol() print choleskyDecomposition choleskyDecomposition.showGUI()

# Multidimensional matrices

DataMelt supports multidimensional matrices and operations similar to Numpy. The difference, however, you can use native Java and plus other scripting languages, such as Python or Groovy. Let us build a matrix in 4 dimensions in Python using org.nd4j.linalg.factory.Nd4j factory:

from org.nd4j.linalg.api.ndarray import INDArray from org.nd4j.linalg.factory import Nd4j; n = Nd4j.create(Nd4j.ones(81).data(), [3,3,3,3]) print n

Here we build a matrix 3x3x3x3 and filled it with 1. The last arguments specifies the dimension of the matrix (3x3x3x3), while the first argument its values. In a more general approach, you can assign any values at the initialization step:

nd = Nd4j.create([1., 2., 3., 4., 5., 6., 7., 8., 9., 10., 11., 12.], [2, 6]) # 2x6 matrix with values nd = Nd4j.create([2, 2]) # empty 2x2

Now let us show how to manipulate and transform with multidimensional matrices.

from org.nd4j.linalg.api.ndarray import INDArray from org.nd4j.linalg.factory import Nd4j; from org.nd4j.linalg.ops.transforms import Transforms # make 4D matrix with 3 rows and value 1 n = Nd4j.create(Nd4j.ones(81).data(), [3,3,3,3]) # transform to exp ndv = Transforms.exp(n) print n print ndv # very large matrox n = Nd4j.linspace(1,10000000,10000000) # multiply it on transpose print "MMUL=", n.mmul(n.transpose())

Here is the list with useful API for arbitrary matrices in N-dimensions

Read more on ND4J web page on how to use more methods and how to program in Java.

Another package that can be used for multidimensional matrices is called VectorZ. See VectorZ (mikera) package</javadoc>

## Input and output

You can save arrays and matrices in a compressed serialized form, as well as using XML form. Look at the Section DMelt:IO/1_File_Input_and_Output. Generally, each linear-algebra packages inside DataMelt has its own methods for reading/writing linear algebra objects from/to files.

## Matrix operations using multiple cores

Matrix manipulation can be performed on multiple cores taking the advantage of parallel processing supported by Java virtual machine. In this approach, all processing cores of your computer will be used for calculations (or only a certain number of core as you have specified). Below we give a simple example. In the example below we create a large matrix 2000x2000 and calculate various characteristics of such matrix (cardinality, vectorize). We compare single threaded calculations with multithreaded ones (in this case, we set the number of cores to 2, but feel free to set to a large value).

To build random 2D matrices use cern.colt.matrix.DoubleFactory2D. Here is a short example to create 1000x1000 matrix and fill it with random numbers:

from cern.colt.matrix import * from edu.emory.mathcs.utils import ConcurrencyUtils ConcurrencyUtils.setNumberOfThreads(4) # set 4 numbers of threads M=tdouble.DoubleFactory2D.dense.random(1000, 1000) # random matrix

Below is a simple example which shows matrix operations on several cores. We set 2 cores, but you should remove the method "ConcurrencyUtils.setNumberOfThreads()" if you what to let the machine to determine the core automatically.

from cern.colt.matrix import tdouble from edu.emory.utils import ConcurrencyUtils import time # do some calculations on DenseDoubleMatrix2D def process(M): M.cardinality() M.dctColumns(0) M.dctRows(0) M.dct2(0) M.dht2() M.dhtColumns() M.dhtRows() M.dst2(0) M.dstColumns(0) M.dstRows(0) M.vectorize() M.zSum() M.idct2(0) Ncores=2 print " benchmarks of DenseDoubleMatrix2D for "+str(Ncores)+" CPU core. Wait!" ConcurrencyUtils.setNumberOfThreads(Ncores) start = time.clock() M=tdouble.DoubleFactory2D.dense.random(1000, 1000) # random matrix process(M) print ' Multiple CPU time (s)=',time.clock()-start

# Vectorz. A fast double-precision vector and matrix library

Vectorz is a fast double-precision vector and matrix maths library for Java (vectorz) developed by Mike Anderson. There are a lot of helpful static factory methods in the following classes:

from mikera.matrixx import Matrix m = Matrix.createSquareMatrix(10) # 10x10 square matrix, initially filled with zeros

Joined vectors are a powerful capability in Vectorz, that allow you to connect vectors together to make larger vectors. See mikera.vectorz.Vectorz.

Vectorz includes support for large sparse vectors and matrices. A sparse array is an array where storage of the array data is designed to be much more efficient in the presence of large numbers of identical (usually zero) elements. If stored in a traditional dense data format (one double value for each element of the array) then such arrays could easily become too large for available memory and operations would be likely to fail with an OutOfMemoryException. Constructing sparse arrays is complicated by the fact that you usually want to construct the array incrementally. Typically you will want to use some combination of the following:

SparseRowMatrix.create(rowCount, columnCount) # to create a large, empty sparse matrix set(i,j,value) #to set individual sparse values. replaceRow(i,vector) # to set an entire row of the sparse matrix (ideally using a sparse vector)

Vectorz supports views over all of its vector / matrix types. A view is an array structure that references the data in one or more underlying arrays. This has several important implications:

- If the underlying data changes, the elements you observe in the view will change
- If you mutate elements in the the view, you will actually mutate the underlying data. This change may be visible to other views accessing the same data.
- Views are lightweight in the sense that they do not take a copy of the underlying data. This means that they are very memory efficient when used appropriately.

Vectorz has special support for 1D, 2D, 3D and 4D vectors. These have the following properties:

- The elements are stored in public fields (x,y,z and t)
- They have specialised, efficient functions that operate on and return vectors of the same size / type
- They require less memory than regular Vector values of the same size, improving efficiency
- They still extend AVector, so you get all the standard Vectorz functionality as usual

These small fixed-size vectors are very useful in many circumstances, e.g.

- 2D and 3D graphics
- Representing RGB and ARGB colours (as 3D and 4D vectors)
- Physical modelling
- Complex numbers (as 2D vectors)
- Quaternions (as 4D vectors)

v=Vector3.of(1,2,3) # create a new 3D vector val = v.x; # directly access the x-coordinate of a 3D vector v.add(Vector3.of(2,3,4)) # add to a 3D vector (uses optimised 3D addition)

Vectorz supports a wide range of specialised matrix classes which provide much more efficient operations in certain situations. For example, multiplying a large vector by a diagonal matrix can easily be 100x faster using the specialised DiagonalMatrix implementation.