# Announcement

MDArray version 0.5.5 has Just been released. MDArray is a multi

dimensional array implemented

for JRuby inspired by NumPy (www.numpy.org) and Masahiro TanakaÂ´s

Narray (narray.rubyforge.org).

MDArray stands on the shoulders of Java-NetCDF and Parallel Colt. At

this point MDArray has

libraries for linear algebra, mathematical, trigonometric and

descriptive statistics methods.

NetCDF-Java Library is a Java interface to NetCDF files, as well as to

many other types of

scientific data formats. It is developed and distributed by Unidata

(http://www.unidata.ucar.edu).

Parallel Colt

(https://sites.google.com/site/piotrwendykier/software/parallelcolt is a

multithreaded version of Colt (redirect...).

Colt provides a set of

Open Source Libraries for High Performance Scientific and Technical

Computing in Java.

Scientific and technical computing is characterized by demanding problem

sizes and a need for

high performance at reasonably small memory footprint.

# What´s new:

## Class MDMatrix and Linear Algebra Methods

This version of MDArray introduces class MDMatrix. MDMatrix is a matrix

class wrapping many

linear algebra methods from Parallel Colt (see below). MDMatrix support

only the following

types: i) int; ii) long; iii) float and iv) double.

Differently from other libraries, in which matrix is a subclass of

array, MDMatrix is a

twin class of MDArray. MDMatrix is a twin class of MDArray as every

time an MDMatrix is

instantiated, an MDArray class is also instantiated. In reality, there

is only one backing

store that can be viewed by either MDMatrix or MDArray.

Creation of MDMatrix follows the same API as MDArray API. For instance,

creating a double

square matrix of size 5 x 5 can be done by:

```
matrix = MDMatrix.double([5, 5], [2.0, 0.0, 8.0, 6.0, 0.0,\
1.0, 6.0, 0.0, 1.0, 7.0,\
5.0, 0.0, 7.0, 4.0, 0.0,\
7.0, 0.0, 8.0, 5.0, 0.0,\
0.0, 10.0, 0.0, 0.0, 7.0])
```

Creating an int matrix filled with zero can be done by:

```
matrix = MDMatrix.int([4, 3])
```

MDMatrix also supports creation based on methods such as fromfunction,

linspace, init_with,

arange, typed_arange and ones:

```
array = MDArray.typed_arange("double", 0, 15)
array = MDMatrix.fromfunction("double", [4, 4]) { |x, y| x + y }
```

An MDMatrix can also be created from an MDArray as follows:

```
d2 = MDArray.typed_arange("double", 0, 15)
double_matrix = MDMatrix.from_mdarray(d2)
```

An MDMatrix can only be created from MDArrays of one, two or three

dimensions. However,

one can take a view from an MDArray to create an MDMatrix, as long as

the view is at most

three dimensional:

```
# Instantiate an MDArray and shape it with 4 dimensions
> d1 = MDArray.typed_arange("double", 0, 420)
> d1.reshape!([5, 4, 3, 7])
# slice the array, getting a three dimensional array and from that,
```

make a matrix

> matrix = MDMatrix.from_mdarray(d1.slice(0, 0))

# get a region from the array, with the first two dimensions of size

0, reduce it to a

# size two array and then build a two dimensional matrix

> matrix = MDMatrix.from_mdarray(d1.region(:spec => “0:0, 0:0, 0:2,

0:6”).reduce)

printing the above two dimensional matrix gives us:

```
> matrix.print
3 x 7 matrix
0,00000 1,00000 2,00000 3,00000 4,00000 5,00000 6,00000
7,00000 8,00000 9,00000 10,0000 11,0000 12,0000 13,0000
14,0000 15,0000 16,0000 17,0000 18,0000 19,0000 20,0000
```

Every MDMatrix instance has a twin MDArray instance that uses the same

backing store. This

allows the user to treat the data as either a matrix or an array and use

methods either from

matrix or array. The above matrix can be printed as an array:

```
> matrix.mdarray.print
[[0.00 1.00 2.00 3.00 4.00 5.00 6.00]
[7.00 8.00 9.00 10.00 11.00 12.00 13.00]
[14.00 15.00 16.00 17.00 18.00 19.00 20.00]]
```

With an MDMatrix, many linear algebra methods can be easily calculated:

```
# basic operations on matrix can be done, such as, â€˜+â€™, â€˜-â€˜,
```

Â´*â€™, â€˜/â€™

# make a 4 x 4 matrix and fill it with Â´doubleÂ´ 2.5

> a = MDMatrix.double([4, 4])

> a.fill(2.5)

> make a 4 x 4 matrix Â´bÂ´ from a given function (block)

> b = MDMatrix.fromfunction(“double”, [4, 4]) { |x, y| x + y }

# add both matrices

> c = a + b

# multiply by scalar

> c = a * 2

# divide two matrices. Matrix Â´bÂ´ has to be non-singular,

otherwise an exception is

# raised.

# generate a non-singular matrix from a given matrix

> b.generate_non_singular!

> c = a / b

Linear algebra methods:

```
# create a matrix with the given data
> pos = MDArray.double([3, 3], [4, 12, -16, 12, 37, -43, -16, -43,
```

98])

> matrix = MDMatrix.from_mdarray(pos)

# Cholesky decomposition from wikipedia example

> chol = matrix.chol

> assert_equal(2, chol[0, 0])

> assert_equal(6, chol[1, 0])

> assert_equal(-8, chol[2, 0])

> assert_equal(5, chol[2, 1])

> assert_equal(3, chol[2, 2])

All other linear algebra methods are called the same way.

# MDArray and SciRuby:

MDArray subscribes fully to the SciRuby Manifesto (http://sciruby.com/).

â€œRuby has for some time had no equivalent to the beautifully

constructed NumPy, SciPy, and

matplotlib libraries for Python.

We believe that the time for a Ruby science and visualization package

has come. Sometimes

when a solution of sugar and water becomes super-saturated, from it

precipitates a pure,

delicious, and diabetes-inducing crystal of sweetness, induced by no

more than the tap of a

finger. So is occurring now, we believe, with numeric and visualization

libraries for Ruby.â€

# MDArray main properties are:

- Homogeneous multidimensional array, a table of elements (usually

numbers), all of the

same type, indexed by a tuple of positive integers; - Support for many linear algebra methods (see bellow);
- Easy calculation for large numerical multi dimensional arrays;
- Basic types are: boolean, byte, short, int, long, float, double,

string, structure; - Based on JRuby, which allows importing Java libraries;
- Operator: +,-,*,/,%,**, >, >=, etc.;
- Functions: abs, ceil, floor, truncate, is_zero, square, cube, fourth;
- Binary Operators: &, |, ^, ~ (binary_ones_complement), <<, >>;
- Ruby Math functions: acos, acosh, asin, asinh, atan, atan2, atanh,

cbrt, cos, erf, exp,

gamma, hypot, ldexp, log, log10, log2, sin, sinh, sqrt, tan, tanh,

neg; - Boolean operations on boolean arrays: and, or, not;
- Fast descriptive statistics from Parallel Colt (complete list found

bellow); - Easy manipulation of arrays: reshape, reduce dimension, permute,

section, slice, etc.; - Support for reading and writing NetCDF-3 files;
- Reading of two dimensional arrays from CSV files (mainly for

debugging and simple testing

purposes); - StatList: a list that can grow/shrink and that can compute Parallel

Colt descriptive

statistics; - Experimental lazy evaluation (still slower than eager evaluation).

# Supported linear algebra methods:

- backwardSolve: Solves the upper triangular system U*x=b;
- chol: Constructs and returns the cholesky-decomposition of the given

matrix. - cond: Returns the condition of matrix A, which is the ratio of

largest to smallest singular value. - det: Returns the determinant of matrix A.
- eig: Constructs and returns the Eigenvalue-decomposition of the

given matrix. - forwardSolve: Solves the lower triangular system L*x=b;
- inverse: Returns the inverse or pseudo-inverse of matrix A.
- kron: Computes the Kronecker product of two real matrices.
- lu: Constructs and returns the LU-decomposition of the given matrix.
- mult: Inner product of two vectors; Sum(x[i] * y[i]).
- mult: Linear algebraic matrix-vector multiplication; z = A * y.
- mult: Linear algebraic matrix-matrix multiplication; C = A x B.
- multOuter: Outer product of two vectors; Sets A[i,j] = x[i] * y[j].
- norm1: Returns the one-norm of vector x, which is Sum(abs(x[i])).
- norm1: Returns the one-norm of matrix A, which is the maximum

absolute column sum. - norm2: Returns the two-norm (aka euclidean norm) of vector x;

equivalent to Sqrt(mult(x,x)). - norm2: Returns the two-norm of matrix A, which is the maximum

singular value; obtained from SVD. - normF: Returns the Frobenius norm of matrix A, which is

Sqrt(Sum(A[i]2)). - normF: Returns the Frobenius norm of matrix A, which is

Sqrt(Sum(A[i,j]2)). - normInfinity: Returns the infinity norm of vector x, which is

Max(abs(x[i])). - normInfinity: Returns the infinity norm of matrix A, which is the

maximum absolute row sum. - pow: Linear algebraic matrix power; B = Ak <==> B = A
*A*…*A. - qr: Constructs and returns the QR-decomposition of the given matrix.
- rank: Returns the effective numerical rank of matrix A, obtained

from Singular Value Decomposition. - solve: Solves A*x = b.
- solve: Solves A*X = B.
- solveTranspose: Solves X*A = B, which is also A’*X’ = B’.
- svd: Constructs and returns the SingularValue-decomposition of the

given matrix. - trace: Returns the sum of the diagonal elements of matrix A;

Sum(A[i,i]). - trapezoidalLower: Modifies the matrix to be a lower trapezoidal

matrix. - vectorNorm2: Returns the two-norm (aka euclidean norm) of vector

X.vectorize(); - xmultOuter: Outer product of two vectors; Returns a matrix with

A[i,j] = x[i] * y[j]. - xpowSlow: Linear algebraic matrix power; B = Ak <==> B = A
*A*…*A.

# PropertiesÂ´ methods tested on matrices:

- density: Returns the matrix’s fraction of non-zero cells;

A.cardinality() / A.size(). - generate_non_singular!: Modifies the given square matrix A such that

it is diagonally dominant by row and column, hence non-singular, hence

invertible. - diagonal?: A matrix A is diagonal if A[i,j] == 0 whenever i != j.
- diagonally_dominant_by_column?: A matrix A is diagonally dominant by

column if the absolute value of each diagonal element is larger than the

sum of the absolute values of the off-diagonal elements in the

corresponding column. - diagonally_dominant_by_row?: A matrix A is diagonally dominant by

row if the absolute value of each diagonal element is larger than the

sum of the absolute values of the off-diagonal elements in the

corresponding row. - identity?: A matrix A is an identity matrix if A[i,i] == 1 and all

other cells are zero. - lower_bidiagonal?: A matrix A is lower bidiagonal if A[i,j]==0

unless i==j || i==j+1. - lower_triangular?: A matrix A is lower triangular if A[i,j]==0

whenever i < j. - nonnegative?: A matrix A is non-negative if A[i,j] >= 0 holds for

all cells. - orthogonal?: A square matrix A is orthogonal if A*transpose(A) = I.
- positive?: A matrix A is positive if A[i,j] > 0 holds for all cells.
- singular?: A matrix A is singular if it has no inverse, that is, iff

det(A)==0. - skew_symmetric?: A square matrix A is skew-symmetric if A =

-transpose(A), that is A[i,j] == -A[j,i]. - square?: A matrix A is square if it has the same number of rows and

columns. - strictly_lower_triangular?: A matrix A is strictly lower triangular

if A[i,j]==0 whenever i <= j. - strictly_triangular?: A matrix A is strictly triangular if it is

triangular and its diagonal elements all equal 0. - strictly_upper_triangular?: A matrix A is strictly upper triangular

if A[i,j]==0 whenever i >= j. - symmetric?: A matrix A is symmetric if A = tranpose(A), that is

A[i,j] == A[j,i]. - triangular?: A matrix A is triangular iff it is either upper or

lower triangular. - tridiagonal?: A matrix A is tridiagonal if A[i,j]==0 whenever

Math.abs(i-j) > 1. - unit_triangular?: A matrix A is unit triangular if it is triangular

and its diagonal elements all equal 1. - upper_bidiagonal?: A matrix A is upper bidiagonal if A[i,j]==0

unless i==j || i==j-1. - upper_triangular?: A matrix A is upper triangular if A[i,j]==0

whenever i > j. - zero?: A matrix A is zero if all its cells are zero.
- lower_bandwidth: The lower bandwidth of a square matrix A is the

maximum i-j for which A[i,j] is nonzero and i > j. - semi_bandwidth: Returns the semi-bandwidth of the given square

matrix A. - upper_bandwidth: The upper bandwidth of a square matrix A is the

maximum j-i for which A[i,j] is nonzero and j > i.

# Descriptive statistics methods imported from Parallel Colt:

- auto_correlation, correlation, covariance, durbin_watson,

frequencies, geometric_mean, - harmonic_mean, kurtosis, lag1, max, mean, mean_deviation, median,

min, moment, moment3, - moment4, pooled_mean, pooled_variance, product, quantile,

quantile_inverse, - rank_interpolated, rms, sample_covariance, sample_kurtosis,

sample_kurtosis_standard_error, - sample_skew, sample_skew_standard_error, sample_standard_deviation,

sample_variance, - sample_weighted_variance, skew, split, standard_deviation,

standard_error, sum, - sum_of_inversions, sum_of_logarithms, sum_of_powers,

sum_of_power_deviations, - sum_of_squares, sum_of_squared_deviations, trimmed_mean, variance,

weighted_mean, - weighted_rms, weighted_sums, winsorized_mean.

# Double and Float methods from Parallel Colt:

- acos, asin, atan, atan2, ceil, cos, exp, floor, greater,

IEEEremainder, inv, less, lg, - log, log2, rint, sin, sqrt, tan.

# Double, Float, Long and Int methods from Parallel Colt:

- abs, compare, div, divNeg, equals, isEqual (is_equal), isGreater

(is_greater), - isles (is_less), max, min, minus, mod, mult, multNeg (mult_neg),

multSquare (mult_square), - neg, plus (add), plusAbs (plus_abs), pow (power), sign, square.

# Long and Int methods from Parallel Colt

- and, dec, factorial, inc, not, or, shiftLeft (shift_left),

shiftRightSigned

(shift_right_signed), shiftRightUnsigned (shift_right_unsigned),

xor.

# MDArray installation and download:

- Install Jruby
- jruby â€“S gem install mdarray

# MDArray Homepages:

# Contributors:

Contributors are welcome.

# MDArray History:

- 14/11/2013: Version 0.5.5 - Support for linear algebra methods
- 07/08/2013: Version 0.5.4 - Support for reading and writing NetCDF-3

files - 24/06/2013: Version 0.5.3 - Over 90% Performance improvements for

methods imported

from Parallel Colt and over 40% performance improvements for all

other methods

(implemented in Ruby); - 16/05/2013: Version 0.5.0 - All loops transferred to Java with over

50% performance

improvements. Descriptive statistics from Parallel Colt; - 19/04/2013: Version 0.4.3 - Fixes a simple, but fatal bug in 0.4.2.

No new features; - 17/04/2013: Version 0.4.2 - Adds simple statistics and boolean

operators; - 05/04/2013: Version 0.4.0 - Initial release.