Neo - A Matrix library

This library is meant to provide basic linear algebra operations for Nim applications. The ambition would be to become a stable basis on which to develop a scientific ecosystem for Nim, much like Numpy does for Python.

The library has been tested on Ubuntu Linux 16.04 64-bit using either ATLAS, OpenBlas or Intel MKL. It was also tested on OSX Yosemite. The GPU support has been tested using NVIDIA CUDA 8.0.

The library is currently aligned with latest Nim devel.

API documentation is here

A lot of examples are available in the tests.

Working on the CPU
- Dense linear algebra
- Sparse linear algebra
Working on the GPU
TODO
Contributing

Working on the CPU

Dense linear algebra

Introduction

The library revolves around operations on vectors and matrices of floating point numbers. It allows to compute operations either on the CPU or on the GPU offering identical APIs.

The library defines types Matrix[A] and Vector[A], where A is sometimes restricted to be float32 or float64 (usually to use BLAS and LAPACK routines). Actually, Vector[A] is just a small wrapper around seq[A], which allows to perform linear algebra operations on standard Nim sequences without copying.

Initialization

Here we show a few ways to create matrices and vectors. All matrices methods accept a parameter to define whether to store the matrix in row-major (that is, data are laid out in memory row by row) or column-major order (that is, data are laid out in memory column by column). The default is in each case column-major.

Whenever possible, we try to deduce whether to use 32 or 64 bits by appropriate parameters. When this is not possible, there is an optional parameter float32 that can be passed to specify the precision (the default is 64 bit).

Static matrices and vectors can be created like this:

import neo

let
  v1 = makeVector(5, proc(i: int): float64 = (i * i).float64)
  v2 = randomVector(7, max = 3.0) # max is optional, default 1
  v3 = constantVector(5, 3.5)
  v4 = zeros(8)
  v5 = ones(9)
  v6 = vector(1.0, 2.0, 3.0, 4.0, 5.0) # `vector` also accepts a seq
  m1 = makeMatrix(6, 3, proc(i, j: int): float64 = (i + j).float64)
  m2 = randomMatrix(2, 8, max = 1.6) # max is optional, default 1
  m3 = constantMatrix(3, 5, 1.8, order = rowMajor) # order is optional, default colMajor
  m4 = ones(3, 6)
  m5 = zeros(5, 2)
  m6 = eye(7)
  m7 = matrix(@[
    @[1.2, 3.5, 4.3],
    @[1.1, 4.2, 1.7]
  ])

All constructors that take as input an existing array or seq perform a copy of the data for memory safety.

Working with 32-bit

Some constructors (such as zeros) allow a type specifier if one wants to create a 32-bit vector or matrix. The following example all return 32-bit vectors and matrices

import neo

let
  v1 = makeVector(5, proc(i: int): float32 = (i * i).float32)
  v2 = randomVector(7, max = 3'f32) # max is no longer optional, to distinguish 32/64 bit
  v3 = constantVector(5, 3.5'f32)
  v4 = zeros(8, float32)
  v5 = ones(9, float32)
  v6 = vector(@[1'f32, 2'f32, 3'f32, 4'f32, 5'f32]) # this `seq` shares data with the vector
  m1 = makeMatrix(6, 3, proc(i, j: int): float32 = (i + j).float32)
  m2 = randomMatrix(2, 8, max = 1.6'f32)
  m3 = constantMatrix(3, 5, 1.8'f32, order = rowMajor) # order is optional, default colMajor
  m4 = ones(3, 6, float32)
  m5 = zeros(5, 2, float32)
  m6 = eye(7, float32)
  m7: Matrix32[2, 3] = matrix(@[
    @[1.2'f32, 3.5'f32, 4.3'f32],
    @[1.1'f32, 4.2'f32, 1.7'f32]
  ])

One can convert precision with to32 or to64:

let
  v64 = randomVector(10)
  v32 = v64.to32()
  m32 = randomMatrix(3, 8, max = 1'f32)
  m64 = m32.to64()

Once vectors and matrices are created, everything is inferred, so there are no differences in working with 32-bit or 64-bit. All examples that follow are for 64-bit, but they would work as well for 32-bit.

Accessors

Vectors can be accessed as expected:

var v = randomVector(6)
v[4] = 1.2
echo v[3]

Same for matrices, where m[i, j] denotes the item on row i and column j, regardless of the matrix order:

var m = randomMatrix(3, 7)
m[1, 3] = 0.8
echo m[2, 2]

One can also map vectors and matrices via a proc:

let
  v1 = v.map(proc(x: float64): float64 = 2 - 3 * x)
  m1 = m.map(proc(x: float64): float64 = 1 / x)

Slicing

The row and column procs will return vectors that share memory with their parent matrix:

let
  m = randomMatrix(10, 10)
  r2 = m.row(2)
  c5 = m.column(5)

Similarly, one can slice a matrix with the familiar notation:

let
  m = randomMatrix(10, 10)
  m1 = m[2 .. 4, 3 .. 8]
  m2 = m[All, 1 .. 5]

where All is a placeholder that denotes that no slicing occurs on that dimension.

In general it is convenient to have slicing, rows and columns that do not copy data but share the underlying data sequence. This can have two possible drawbacks:

the result may need to be modified while the original matrix stays unchanged, or viceversa;
a small matrix or vector may hold a reference to a large data sequence, preventing it to be garbage collected.

In this case, it is enough to call the .clone() proc to obtain a copy of the matrix or vector with its own storage.

Iterators

One can iterate over vector or matrix elements, as well as over rows and columns

let
  v = randomVector(6)
  m = randomMatrix(3, 5)
for x in v: echo x
for i, x in v: echo i, x
for x in m: echo x
for t, x in m:
  let (i, j) = t
  echo i, j, x
for row in m.rows:
  echo row[0]
for column in m.columns:
  echo column[1]

One important point about performance. When iterating over rows or columns, the same ref is reused throughout - this entails that the loop is allocation-free, resulting in orders of magnitude faster loops. One should pay attention not to hold to these references, because they will be mutated.

This means that - for instance - the following is correct:

let m = randomMatrix(1000, 1000)
var columnSum = zeros(1000)
for c in m.columns =
  columnSum += c

but the following will give wrong results (all elements of cols will be identical at the end):

let m = randomMatrix(1000, 1000)
var cols = newSeq[Vector[float64]]()
for c in m.columns =
  cols.add(c)

If one needs a fresh reference for each element of the iteration, the rowsSlow and columnSlow iterators are available, hence the following modification is ok:

let m = randomMatrix(1000, 1000)
var cols = newSeq[Vector[float64]]()
for c in m.columnsSlow =
  cols.add(c)

Equality

There are two kinds of equality. The usual == operator will compare the contents of vector and matrices exactly

let
  u = vector(1.0, 2.0, 3.0, 4.0)
  v = vector(1.0, 2.0, 3.0, 4.0)
  w = vector(1.0, 3.0, 3.0, 4.0)
u == v # true
u == w # false

Usually, though, one wants to take into account the errors introduced by floating point operations. To do this, use the =~ operator, or its negation !=~:

let
  u = vector(1.0, 2.0, 3.0, 4.0)
  v = vector(1.0, 2.000000001, 2.99999999, 4.0)
u == v # false
u =~ v # true

Pretty-print

Both vectors and matrix have a pretty-print operation, so one can do

let m = randomMatrix(3, 7)
echo m8

and get something like

[ [ 0.5024584865674662  0.0798945419892334  0.7512423051567048  0.9119041361916302  0.5868388894943912  0.3600554448403415  0.4419034543022882 ]
  [ 0.8225964245706265  0.01608615513584155 0.1442007939324697  0.7623388321096165  0.8419745686508193  0.08792951865247645 0.2902529012579151 ]
  [ 0.8488187232786935  0.422866666087792 0.1057975175658363  0.07968277822379832 0.7526946339452074  0.7698915909784674  0.02831893268471575 ] ]

Reshape operations

The following operations do not change the underlying memory layout of matrices and vectors. This means they run in very little time even on big matrices, but you have to pay attention when mutating matrices and vectors produced in this way, since the underlying data is shared.

let
  m1 = randomMatrix(6, 9)
  m2 = randomMatrix(9, 6)
  v1 = randomVector(9)
echo m1.t # transpose, done in constant time without copying
echo m1 + m2.t
let m3 = m1.reshape(9, 6)
let m4 = v1.asMatrix(3, 3)
let v2 = m2.asVector

In case you need to allocate a copy of the original data, say in order to transpose a matrix and then mutate the transpose without altering the original matrix, a clone operation is available:

let m5 = m1.clone

Notice that clone() will be called internally anyway when using one of the reshape operations with a matrix that is not contiguous (that is, a matrix obtained by slicing).

BLAS Operations

A few linear algebra operations are available, wrapping BLAS libraries:

var v1 = randomVector(7)
let
  v2 = randomVector(7)
  m1 = randomMatrix(6, 9)
  m2 = randomMatrix(9, 7)
echo 3.5 * v1
v1 *= 2.3
echo v1 + v2
echo v1 - v2
echo v1 * v2 # dot product
echo v1 |*| v2 # Hadamard (component-wise) product
echo l_1(v1) # l_1 norm
echo l_2(v1) # l_2 norm
echo m2 * v1 # matrix-vector product
echo m1 * m2 # matrix-matrix product
echo m1 |*| m2 # Hadamard (component-wise) product
echo max(m1)
echo min(v2)

Universal functions

Universal functions are real-valued functions that are extended to vectors and matrices by working element-wise. There are many common functions that are implemented as universal functions:

sqrt
cbrt
log10
log2
log
exp
arccos
arcsin
arctan
cos
cosh
sin
sinh
tan
tanh
erf
erfc
lgamma
tgamma
trunc
floor
ceil
degToRad
radToDeg

This means that, for instance, the following check passes:

  let
    v1 = vector(1.0, 2.3, 4.5, 3.2, 5.4)
    v2 = log(v1)
    v3 = v1.map(log)

  assert v2 == v3

Universal functions work both on 32 and 64 bit precision, on vectors and matrices.

If you have a function f of type proc(x: float64): float64 you can use

makeUniversal(f)

to turn f into a (public) universal function. If you do not want to export f, there is the equivalent template makeUniversalLocal.

Rewrite rules

A few rewrite rules allow to optimize a chain of linear algebra operations into a single BLAS call. For instance, if you try

echo v1 + 5.3 * v2

this is not implemented as a scalar multiplication followed by a sum, but it is turned into a single function call.

Solving linear systems

Some linear algebraic functions are included, currently for solving systems of linear equations of the form Ax = b, for square matrices A. Functions to invert square invertible matrices are also provided. These throw floating-point errors in the case of non-invertible matrices.

These functions require a LAPACK implementation.

let
  a = randomMatrix(5, 5)
  b = randomVector(5)

echo solve(a, b)
echo a \ b # equivalent
echo a.inv()

Computing eigenvalues and eigenvectors

These functions require a LAPACK implementation.

To be documented.

Linking BLAS and LAPACK implementations

The library requires to link some BLAS implementation to perform the actual linear algebra operations. By default, it tries to link whatever is the default system-wide BLAS implementation.

A few compile flags are available to link specific BLAS implementations

-d:atlas
-d:openblas
-d:mkl
-d:mkl -d:threaded

Packages for various BLAS implementations are available from the package managers of many Linux distributions. On OSX one can add the brew formulas from Homebrew Science, such as brew install homebrew/science/openblas.

You may also need to add suitable paths for the includes and library dirs. On OSX, this should do the trick

switch("clibdir", "/usr/local/opt/openblas/lib")
switch("cincludes", "/usr/local/opt/openblas/include")

If you have problems with MKL, you may want to link it statically. Just pass the options

--dynlibOverride:mkl_intel_lp64
--passL:${PATH_TO_MKL}/libmkl_intel_lp64.a

to enable static linking.

Sparse linear algebra

To be documented.

Working on the GPU

Linking CUDA

It is possible to delegate work to the GPU using CUDA. The library has been tested to work with NVIDIA CUDA 8.0, but it is possible that earlier versions will work as well. In order to compile and link against CUDA, you should make the appropriate headers and libraries available. If they are not globally set, you can pass suitable options to the Nim compiler, such as

--cincludes:"/usr/local/cuda/include"
--clibdir:"/usr/local/cuda/lib64"

Support for CUDA is under the package neo/cuda, that needs to be imported explicitly.

Dense linear algebra

If you have a matrix or vector, you can move it on the GPU, and back like this:

import neo, neo/cuda
let
  v = randomVector(12, max=1'f32)
  vOnTheGpu = v.gpu()
  vBackOnTheCpu = vOnTheGpu.cpu()

Vectors and matrices on the GPU support linear-algebraic operations via cuBLAS, exactly like their CPU counterparts. A few operation - such as reading a single element - are not supported, as it does not make much sense to copy a single value back and forth from the GPU. Usually it is advisable to move vectors and matrices to the GPU, make as many computations as possible there, and finally move the result back to the CPU.

The following are all valid operations, assuming v and w are vectors on the GPU, m and n are matrices on the GPU and the dimensions are compatible:

v * 3'f32
v + w
v -= w
m * v
m - n
m * n

For more information, look at the tests in tests/cudadense.

Sparse linear algebra

To be documented.

TODO

Add support for matrices and vector on the stack
Use rewrite rules to optimize complex operations into a single BLAS call
More specialized BLAS operations
Try on more platforms/configurations
Make a proper benchmark
Improve documentation
Better pretty-print

Contributing

Every contribution is very much appreciated! This can range from:

using the library and reporting any issues and any configuration on which it works fine
building other parts of the scientific environment on top of it
writing blog posts and tutorials
contributing actual code (see the TODO section)

singularperturbation/neo

Neo - A Matrix library

Table of contents

Working on the CPU

Dense linear algebra

Introduction

Initialization

Working with 32-bit

Accessors

Slicing

Iterators

Equality

Pretty-print

Reshape operations

BLAS Operations

Universal functions

Rewrite rules

Solving linear systems

Computing eigenvalues and eigenvectors

Linking BLAS and LAPACK implementations

Sparse linear algebra

Working on the GPU

Linking CUDA

Dense linear algebra

Sparse linear algebra

TODO

Contributing