1. 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 and 20.04 64-bit using either ATLAS, OpenBlas or Intel MKL. It was also tested on OSX Yosemite to Monterey. The GPU support has been tested using NVIDIA CUDA 8.0 up to 10.x.

The library is currently aligned with latest Nim devel.

API documentation is here

A lot of examples are available in the tests.

1.1. 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.

Similar types exist on the GPU side, and there are facilities to move them back and forth from the CPU.

Neo makes use of many standard libraries such as BLAS, LAPACK and CUDA. See this section to learn how to link the correct implementation for your platform.

1.2. Working on the CPU

1.2.1. Dense linear algebra

1.2.1.1. 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)
  v7 = vector([1.2, 3.4, 5.6])
  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.

1.2.1.2. 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)
  v7 = vector([1.2'f32, 3.4'f32, 5.6'f32])
  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 = 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.

1.2.1.3. 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)

1.2.1.4. 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.

1.2.1.5. 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)

1.2.1.6. 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

1.2.1.7. 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 ] ]

1.2.1.8. 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).

There is also a hard transpose operation which, unlike t() will not try to share storage but will always create a new matrix instead and copy the data to the new matrix (this way, it will also preserve the row-major or colum-major order). The hard transpose is denoted T(), so that

m.t == m.T

always holds, although the internal representations differ.

1.2.1.9. 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)

1.2.1.10. 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.

1.2.1.11. 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.

1.2.1.12. Stacking vectors and matrices

Vectors can be stacked both horizontally (which gives a new vector)

let
  v1 = vector([1.0, 2.0])
  v2 = vector([5.0, 7.0, 9.0])
  v3 = vector([9.9, 8.8, 7.7, 6.6])

echo hstack(v1, v2, v3) #  vector([1.0, 2.0, 5.0, 7.0, 9.0, 9.9, 8.8, 7.7, 6.6])

or vertically (which gives a matrix having the vectors as rows)

let
  v1 = vector([1.0, 2.0, 3.0])
  v2 = vector([5.0, 7.0, 9.0])
  v3 = vector([9.9, 8.8, 7.7])

echo vstack(v1, v2, v3)
# matrix(@[
#   @[1.0, 2.0, 3.0],
#   @[5.0, 7.0, 9.0],
#   @[9.9, 8.8, 7.7]
# ])

Also, concat is an alias for hstack.

Matrices can be stacked similarly, for instance

let
  m1 = matrix(@[
    @[1.0, 2.0],
    @[3.0, 4.0]
  ])
  m2 = matrix(@[
    @[5.0, 7.0, 9.0],
    @[6.0, 2.0, 1.0]
  ])
  m3 = matrix(@[
    @[2.0, 2.0],
    @[1.0, 3.0]
  ])
echo hstack(m1, m2, m3)
# m = matrix(@[
#   @[1.0, 2.0, 5.0, 7.0, 9.0, 2.0, 2.0],
#   @[3.0, 4.0, 6.0, 2.0, 1.0, 1.0, 3.0]
# ])

TODO: stack matrices

1.2.1.13. 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()

1.2.1.14. Computing eigenvalues and eigenvectors

These functions require a LAPACK implementation.

To be documented.

1.2.2. Sparse linear algebra

To be documented.

1.3. Working on the GPU

1.3.1. 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.

1.3.2. Sparse linear algebra

To be documented.

1.4. Static typing for dimensions

Under neo/statics there exist types that encode vectors and matrices whose dimensions are known at compile time. They are defined as aliases of their dynamic counterparts:

type
  StaticVector*[N: static[int]; A] = distinct Vector[A]
  StaticMatrix*[M, N: static[int]; A] = distinct Matrix[A]

In this way, these types are fully interoperable with the dynamic ones. One can freely convert between the two representations:

import neo, neo/statics

let
  u = randomVector(5) # static, of known dimension 5
  v = u.asDynamic
  w = v.asStatic(5)

assert(u == w)

All operations implemented by neo are also avaiable for static vectors and matrices. The difference are that:

operations on static vectors and matrices will not compile if the dimensions do not match
operations on static vectors and matrices will return other static vectors and matrices, thereby automatically tracking dimensions.

An example of an operation that will not compile is

import neo, neo/statics

let
  m = statics.randomMatrix(5, 7) # static, of known dimension 5x7
  n = statics.randomMatrix(4, 6) # static, of known dimension 4x6
  p = statics.randomMatrix(7, 3) # static, of known dimension 7x3

discard m * n # this will not compile
let x = m * p # this will infer dimension 5x3

By converting back and forth between static and dynamic vectors and matrices - which can be done at no cost - one can incorporate data whose dimension is only known at runtime, while at the same time having guaranteed dimension compatibility whenever enough information is known at compile time.

For now, statics are only available on the CPU. It would be a nice contribution to extend this to GPU types.

1.5. Design

1.5.1. On the CPU

On the CPU, dense vectors and matrices are stored using this structure:

type
  MatrixShape* = enum
    Diagonal, UpperTriangular, LowerTriangular, UpperHessenberg, LowerHessenberg, Symmetric
  Vector*[A] = ref object
    data*: seq[A]
    fp*: ptr A # float pointer
    len*, step*: int
  Matrix*[A] = ref object
    order*: OrderType
    M*, N*, ld*: int # ld = leading dimension
    fp*: ptr A # float pointer
    data*: seq[A]
    shape*: set[MatrixShape]

Each store some information on dimensions (len for vectors, M and N for matrices) and a pointer to the beginning of the actual data fp.

The order of a matrix can be colMajor or rowMajor: the first one means that the matrix is stored column by column, the second row by row.

To make it easier to share data without copying, but still keep the data garbage collected, the actual data is usually allocated in a seq, here called data, which can be shared between matrices and their slices, or row and column vectors. The pointer fp is usually a pointer somewhere inside this sequence, although this is not required.

All operations are expressed in terms of fp, so data is not really required. When the last reference to data goes away, though, the sequence is garbage collected and there will be no more pointers inside it. If there is a small vector or matrix holding the last reference to a big chunk of data, it may be more convenient to copy it to a fresh location and free the data itself: this can be done by using the clone() operation.

Vectors are not required to be contiguous, and they have a step parameter that determines how far apart are their elements. This is useful when taking a row of a column major matrix or the column of a row major one.

Matrices can also not be contiguous. When taking a minor of a column major matrix, one gets a submatrix whose elements are contiguous in a column, but whose column are not contiguously placed. Rather, the distance (in elements) between the start of two successive columns is the same as the parent matrix, and is called the leading dimension of the matrix (here stored as ld). A similar remark holds for row major matrices, where ld is the number of elements between the beginning of rows.

This design allows to have matrices or vectors that are not managed by the garbage collector. In this case, it is enough to set fp manually, and leave data nil. This allows to support

matrices and vectors with data on the stack, which can be constructed using the stackVector and stackMatrix constructors (and which are only valid as long as the relevant data lives on the stack), and
matrices and vectors allocated manually on the shared heap, which can be constructed using the sharedVector and sharedMatrix constructors, and destructed with dealloc.

1.5.2. Why fields are public

Notice that all members of the types are public, but in general it is not safe to change them if you don't know what you are doing. These fields are not generally meant to be changed, and a previous version of the library had them private. In general, though, a user may need access to some of these fields for performance reasons, so they are exposed.

An example of this case is the rows (or columns) iterator. Neo keeps vector and matrix types on the heap (they are ref types). This prevents accidental copying, but has the downside that creating a new one requires allocation. When iterating over rows in a loop, one wants to avoid to trigger many costly allocations, since the underlying data is always the same, and the only thing that changes is the position of the vectors inside this data array. For this reason, the iterator is implemented as follows:

iterator rows*[A](m: Matrix[A]): auto {. inline .} =
  let
    mp = cast[CPointer[A]](m.fp)
    step = if m.order == rowMajor: m.ld else: 1
  var v = m.row(0)
  yield v
  for i in 1 ..< m.M:
    v.fp = addr(mp[i * step])
    yield v

There is a single vector which is reused at each step and the iterator always yields this vector, where fp is changed. A user that wants - say - to implement a similar iteration over some minors of a matrix may need to perform a similar trick, and preventing to change fp would impede this optimization.

1.5.3. On the GPU

On the GPU side, the definitions are similar:

type
  CudaVector*[A] = object
    data*: ref[ptr A]
    fp*: ptr A
    len, step*: int32
  CudaMatrix*[A] = object
    M*, N*, ld*: int32
    data*: ref[ptr A]
    fp*: ptr A
    shape*: set[MatrixShape]

The main difference here is that one cannot store the underlying data in a sequence, because data is allocated on a device, and the CUDA api returns the relevant pointers, over which we have no control.

To have a similar approach to the former case, the CUDA pointer to the data is wrapped inside a ref. The actual pointer used in computation is, again, fp, while data is shared for slices, or rows and vectors of a matrix.

When the last reference to data goes away, a finalizer calls the CUDA routines to clean up the allocated memory.

Also, CUDA matrices are only column major for now, although this is going to change in the future.

1.6. Linking external libraries

1.6.1. Linking BLAS and LAPACK implementations

Neo requires to link some BLAS and LAPACK implementation to perform the actual linear algebra operations. By default, it tries to link whatever are the default system-wide implementations.

You can link against different implementations by a combination of:

changing the path for linked libraries (use --clibdir for this)
using the --define:blas flag. By default, the system tries to load a BLAS library called blas, which translates into something called blas.dll or libblas.so according to the underling operating system. To link, say, the library libopenblas.so.3 on Linux, you should pass to Nim the option --define:blas=openblas.
using the --define:lapack flag. By default, the system tries to load a LAPACK library called lapack, which translates into something called lapack.dll or liblapack.so according to the underling operating system. To link, say, the library libopenblas.so.3 on Linux, you should pass to Nim the option --define:lapack=openblas.

See the tasks inside neo.nimble for a few examples.

Packages for various BLAS or LAPACK 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.

On Windows, it is recommended to use MSYS2 to install the mingw compiler toolchain and compatible OpenBLAS library. For 64-bit builds, this would be:

pacman -S mingw-w64-x86_64-gcc mingw-w64-x86_64-openblas

You should then add MSYS2_ROOT\mingw64\bin to your PATH. Programs using nimblas can then be compiled using the -d:blas=libopenblas switch. At runtime, libopenblas,dll should be loaded from the mingw64 bin directory you added to your PATH, though it is suggested to distribute this DLL file alongside your executable if your are publishing binary packages.

1.6.2. 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.

1.7. TODO

See the issue list

1.8. 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
helping with the documentation
contributing actual code (see the issue list section)

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LICENSE		LICENSE
README.md		README.md
_config.yml		_config.yml
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1. Neo - A Matrix library

Table of contents

1.1. Introduction

1.2. Working on the CPU

1.2.1. Dense linear algebra

1.2.1.1. Initialization

1.2.1.2. Working with 32-bit

1.2.1.3. Accessors

1.2.1.4. Slicing

1.2.1.5. Iterators

1.2.1.6. Equality

1.2.1.7. Pretty-print

1.2.1.8. Reshape operations

1.2.1.9. BLAS Operations

1.2.1.10. Universal functions

1.2.1.11. Rewrite rules

1.2.1.12. Stacking vectors and matrices

1.2.1.13. Solving linear systems

1.2.1.14. Computing eigenvalues and eigenvectors

1.2.2. Sparse linear algebra

1.3. Working on the GPU

1.3.1. Dense linear algebra

1.3.2. Sparse linear algebra

1.4. Static typing for dimensions

1.5. Design

1.5.1. On the CPU

1.5.2. Why fields are public

1.5.3. On the GPU

1.6. Linking external libraries

1.6.1. Linking BLAS and LAPACK implementations

1.6.2. Linking CUDA

1.7. TODO

1.8. Contributing

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