#Linear Algebra for Nim

This library is not mantained anymore. It still works fine, but new development happens on Neo. The main difference between the two libraries is that Neo only focuses in what we call here dynamic vectors and matrices. Using static types to encode dimensions was a nice experiment, but it turned out to be one more dimension to support (along with 32 vs 64 bit, CPU vs GPU, dense vs sparse...).

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 14.10 through 15.10 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 7.0 and 7.5.

The library is currently aligned with latest Nim devel. For versions of Nim up to 0.13 use version 0.4.2 of linalg.

API documentation is here

A lot of examples are available in the tests.

Table of contents

• Introduction
• Initialization
• Working with 32-bit
• Accessors
• Iterators
• Equality
• Pretty-print
• Operations
• Trivial operations
• Universal functions
• Linear Algebra functions
• Rewrite rules
• Type safety guarantees
• Linking BLAS implementations
• GPU support
• TODO
• Contributing

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. Also, whenever possible, the dimension of vectors and matrices are encoded into the types in the form of static[int] type parameters. This allow to check dimensions at compile time and refuse to compile invalid operations, such as summing two vectors of different sizes, or multiplying two matrices of incompatible dimensions.

The library defines types Matrix64[M, N] and Vector64[N] for 64-bit matrices and vectors of static dimension, as well as their 32-bit counterparts Matrix32[M, N] and Vector32[N].

For the case where the dimension is not known at compile time, one can use so-called dynamic vectors and matrices, whose types are DVector64 and DMatrix64 (resp. DVector32 and DMatrix32). Note that DVector64 is just and alias for seq[float64] (and similarly for 32-bit), which allows to perform linear algebra operations on standard Nim sequences.

In all examples, types are inferred, and are shown just for explanatory purposes.

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 linalg

let
  v1: Vector64[5] = makeVector(5, proc(i: int): float64 = (i * i).float64)
  v2: Vector64[7] = randomVector(7, max = 3.0) # max is optional, default 1
  v3: Vector64[5] = constantVector(5, 3.5)
  v4: Vector64[8] = zeros(8)
  v5: Vector64[9] = ones(9)
  v6: Vector64[5] = vector([1.0, 2.0, 3.0, 4.0, 5.0]) # initialize from an array...
  m1: Matrix64[6, 3] = makeMatrix(6, 3, proc(i, j: int): float64 = (i + j).float64)
  m2: Matrix64[2, 8] = randomMatrix(2, 8, max = 1.6) # max is optional, default 1
  m3: Matrix64[3, 5] = constantMatrix(3, 5, 1.8, order = rowMajor) # order is optional, default colMajor
  m4: Matrix64[3, 6] = ones(3, 6)
  m5: Matrix64[5, 2] = zeros(5, 2)
  m6: Matrix64[7, 7] = eye(7)
  m7: Matrix64[2, 3] = matrix([
    [1.2, 3.5, 4.3],
    [1.1, 4.2, 1.7]
  ])
  m8: Matrix64[2, 3] = matrix(@[
    @[1.2, 3.5, 4.3],
    @[1.1, 4.2, 1.7]
  ], 2, 3)

The last matrix constructor takes a seq of seqs, but also requires statically passing the dimensions to be used. The following are equivalent when xs is a seq[seq[float64]] and M, N are integers known at compile time:

let
  m1 = matrix(xs).toStatic(M, N)
  m2 = matrix(xs, M, N)

but the latter form avoids the construction of an intermediate matrix.

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

Dynamic matrices and vectors have similar constructors - the difference is that the dimension parameter are not known at compile time:

import linalg

let
  M = 5
  N = 7
  v1: DVector64 = makeVector(M, proc(i: int): float64 = (i * i).float64)
  v2: DVector64 = randomVector(N, max = 3.0) # max is optional, default 1
  v3: DVector64 = constantVector(M, 3.5)
  v4: DVector64 = zeros(M)
  v5: DVector64 = ones(N)
  v6: DVector64 = @[1.0, 2.0, 3.0, 4.0, 5.0] # DVectors are just seqs...
  m1: DMatrix64 = makeMatrix(M, N, proc(i, j: int): float64 = (i + j).float64)
  m2: DMatrix64 = randomMatrix(M, N, max = 1.6) # max is optional, default 1
  m3: DMatrix64 = constantMatrix(M, N, 1.8, order = rowMajor) # order is optional, default colMajor
  m4: DMatrix64 = ones(M, N)
  m5: DMatrix64 = zeros(M, N)
  m6: DMatrix64 = eye(M)
  m7: DMatrix64 = matrix(@[
    @[1.2, 3.5, 4.3],
    @[1.1, 4.2, 1.7]
  ])

If for some reason you want to create a dynamic vector of matrix, but you want to write literal dimensions, you can either assign these numbers to variables or use the toDynamic proc to convert the static instances to dynamic ones:

import linalg

let
  M = 5
  v1 = makeVector(M, proc(i: int): float64 = (i * i).float64)
  v2 = makeVector(5, proc(i: int): float64 = (i * i).float64)

assert v1.toStatic(5) == v2
assert v2.toDynamic == v1

Working with 32-bit

One can also instantiate 32-bit matrices and vectors. Whenever possible, the API is identical. In cases that would be ambiguous (such as zeros), one can explicitly specify the float32 parameter.

import linalg

let
  v1: Vector32[5] = makeVector(5, proc(i: int): float32 = (i * i).float32)
  v2: Vector32[7] = randomVector(7, max = 3'f32) # max is no longer optional, to distinguish 32/64 bit
  v3: Vector32[5] = constantVector(5, 3.5'f32)
  v4: Vector32[8] = zeros(8, float32)
  v5: Vector32[9] = ones(9, float32)
  v6: Vector32[5] = vector([1'f32, 2'f32, 3'f32, 4'f32, 5'f32])
  m1: Matrix32[6, 3] = makeMatrix(6, 3, proc(i, j: int): float32 = (i + j).float32)
  m2: Matrix32[2, 8] = randomMatrix(2, 8, max = 1.6'f32)
  m3: Matrix32[3, 5] = constantMatrix(3, 5, 1.8'f32, order = rowMajor) # order is optional, default colMajor
  m4: Matrix32[3, 6] = ones(3, 6, float32)
  m5: Matrix32[5, 2] = zeros(5, 2, float32)
  m6: Matrix32[7, 7] = 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]
  ])
  m8: Matrix32[2, 3] = matrix(@[
    @[1.2'f32, 3.5'f32, 4.3'f32],
    @[1.1'f32, 4.2'f32, 1.7'f32]
  ], 2, 3)

Similarly,

import linalg

let
  M = 5
  N = 7
  v1: DVector32 = makeVector(M, proc(i: int): float32 = (i * i).float32)
  v2: DVector32 = randomVector(N, max = 3'f32) # max is not optional
  v3: DVector32 = constantVector(M, 3.5'f32)
  v4: DVector32 = zeros(M, float32)
  v5: DVector32 = ones(N, float32)
  v6: DVector32 = @[1'f32, 2'f32, 3'f32, 4'f32, 5'f32] # DVectors are just seqs...
  m1: DMatrix32 = makeMatrix(M, N, proc(i, j: int): float32 = (i + j).float32)
  m2: DMatrix32 = randomMatrix(M, N, max = 1.6'f32) # max is not optional
  m3: DMatrix32 = constantMatrix(M, N, 1.8'f32, order = rowMajor) # order is optional, default colMajor
  m4: DMatrix32 = ones(M, N, float32)
  m5: DMatrix32 = zeros(M, N, float32)
  m6: DMatrix32 = eye(M, float32)
  m7: DMatrix32 = 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: Vector64[10] = randomVector(10)
  v32: Vector32[10] = v64.to32()
  m32: Matrix32[3, 8] = randomMatrix(3, 8, max = 1'f32)
  m64: Matrix64[3, 8] = 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]

Also one can see rows and columns as vectors

let
  r2: Vector64[7] = m.row(2)
  c5: Vector64[3] = m.column(5)

For memory safety, this performs a copy of the row or column values, at least for now. 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)

Similar operations are available for dynamic matrices and vectors as well.

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]

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 ] ]

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)

Trivial 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: Matrix64[9, 6] = m1.reshape(9, 6)
let m4: Matrix64[3, 3] = v1.asMatrix(3, 3)
let v2: Vector64[54] = 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

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, both static and dynamic.

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.

Linear Algebra functions

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.

At the moment, only static matrices are supported for system solution and matrix inversion.

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

import linalg/rewrites

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.

Type safety guarantees

The library is designed with the use case of having dimensions known at compile time, and leverages the compiles to ensure that dimensions match when performing the appropriate operations - for instance in matrix multiplication.

To see some examples where the compiler avoids malformed operations, look inside tests/compilation (yes, in Nim one can actually test that some operations do not compile!).

Also, operations that mutate a vector of matrix in place are only available if that vector of matrix is defined via var instead of let.

Linking BLAS 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.

GPU support

It is possible to delegate work to the GPU using CUDA. The library has been tested to work with NVIDIA CUDA 7.0 and 7.5, 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/targets/x86_64-linux/include" \
--clibdir:"/usr/local/cuda/targets/x86_64-linux/lib"

You will also need to explicitly add linalg support for CUDA with the flag

-d:cublas

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

let
  v: Vector32[12] = randomVector(12, max=1'f32)
  vOnTheGpu: CudaVector[12] = v.gpu()
  vBackOnTheCpu: Vector32[12] = 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/cuda.

TODO

• Add support for matrices and vector on the stack
• Add more functional interfaces (foldl, scanl)
• Use rewrite rules to optimize complex operations into a single BLAS call
• More specialized BLAS operations
• Add operations from LAPACK
• Support slicing/nonconstant steps
• Make row and column operations non-copying
• Better types to avoid out of bounds exceptions when statically checkable
• Add a fallback Nim implementation, that is valid over other rings
• Move CUBLAS and LAPACK to separate libraries required via nimble
• 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)

The library has to cater many different use cases, hence the vector and matrix types differ in various axes:

• 32/64 bit
• CPU/GPU
• static/dynamic
• (on the stack? non-contiguous? non GC pointers?)

In order to avoid a combinatorial explosion of operations, a judicious use of templates and union types helps to limit the actual implementations that have to be written.