linalg
linalg
Linear algebra library in Crystal, uses LAPACK.
 direct access to LAPACK methods
 convenient Matrix(T) class, supports T=Float32, Float64 and Complex.
 highlevel interface similar to scipy.linalg or MATLAB.
Killing SciPy, one module at a time.
Installation

Install LAPACK and BLAS.
sudo apt install libopenblasbase liblapack3
for Ubuntu,sudo pacman S lapack
(for better performance useopenblaslapack
package from AUR) for Arch. For Windows you need libopenblas.dll (and libopenblas.lib) from https://github.com/xianyi/OpenBLAS/releases 
(for Ubuntu 18) it seems package doesn't create symlink, so use
sudo ln s /usr/lib/lapack/liblapack.so.3 /usr/lib/liblapack.so
sudo ln s /usr/lib/openblasbase/libblas.so.3 /usr/lib/libcblas.so
 Add this to your application's
shard.yml
:
dependencies:
linalg:
github: konovod/linalg
 Run
shards install
Usage
require "linalg"
Basic type aliases are
 Mat = Matrix(Float64)
 Mat32 = Matrix(Float32)
 MatComplex = Matrix(Complex)
Complex consisting of two Float32 isn't supported for now (it is easy, but I'm not sure if it's useful).
Types with prefix G (GMat, GMat32, GMatComplex) are for actually allocated matrices, others are automatically converted to them when needed.
#suggested to don't prefix LA:: everywhere
include LA
# create matrix from array of arrays (or tuple... everything Indexable)
m = GMat[
[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
[10, 11, 12],
]
# or using block
m = GMat32.new(3, 4) { i, j i*3 + j + 1 }
# or using one of other ways, check "spec" directory
# do basic arithmetics
a = Mat.eye(3)
pp 2 * a  Mat.diag([2, 2, 2]) == Mat.zeros(3, 3) # => true
# basic algebra
a = Mat.rand(5, 5) + 2 * Mat.identity(5)
pp (a.inv * a  Mat.identity(5)).norm < 1e6
b = Mat.rand(5, 1)
x = LA.solve(a, b) # or a.solve(b)
pp (a*x  b).norm < 1e6
m = GMat[[2, 4, 1], [2, 4, 1], [1, 1, 1]]
pp m.eigvals # => [6.0, 1.0, 2.0]
# extract submatrices (memory isn't copied as they reference to basic matrix)
m = GMat[
[1, 2, 3],
[4, 5, 6],
[7, 8, 9],
]
pp m.columns[2] # LA::SubMatrix(Float64) (3x1, None):
# [3.0]
# [6.0]
# [9.0]
x = m[1..1, 1..2]
pp x # => [5.0, 6.0]
x[0, 0] = 0 # m[1,1] is now 0 (questionable feature? maybe should be ##[]! for modifiable submatrices and ##[] for CoW?)
y = x.clone # now y is a separate matrix
y[0, 0] = 1 # m[1,1] is still 0
pp m[1, 1]
other present features:
 svd (
Mat#svd
orMat#svdvals
for just values)  lu decomposition (
Mat#lu
)
# to just get P L U matrices
p, l, u = a.lu
# to get them in compact form and use for solving linear equations:
a = GMat32[
[2, 4],
[2, 8]
]
lu = a.lu_factor # lu is LUMatrix(T)  immutable object that can return it's content and solve systems
puts lu.solve(GMat32[[2], [4]])
 matrix rank determination (using SVD or QRP)
 linear least squares problem (
LA.solvels
to just get decision orLA.lstsq
to also get rank and singular values (TODO  and residues))  cholesky decomposition (
#cholesky
,#cholesky!
,#cho_solve
) hessenberg
formqr
,rq
,lq
,ql
decompositionsschur
andqz
(generalized schur) decomposition generalized eigenproblem (
eigs(a, b, ...)
)  creating special matrices like
pascal
ortoeplitz
(check scipy.md for a full list)  matrix exponent and trigonomertic functions
 matrix exponentiation (to integer powers only atm (TODO  fractional))
There is also concept of Mat#flags
that represent properties of matrix (symmetric, positive definite etc), they are used to automatically select faster algorithms from LAPACK. Flags are partially enforced by runtime checks, with the possibility of user override. For example, if we say that a.assume!(MatrixFlags::Symmetric)
then a.transpose
or a + Mat.diag(*a.size)
will also have this flag, so the LAPACK routines for symmetrical matrices will be used. In fact, a.transpose
will return matrix clone as for symmetric matrices A=A'.
Supported flags:
enum MatrixFlags
Symmetric
Hermitian
PositiveDefinite
Orthogonal
UpperTriangular
LowerTriangular
Triangular = UpperTriangular  LowerTriangular
NOTE for complex matrices Orthogonal
flag means Unitary
.
Main functions for flags are:
a.assume!(flag) # sets matrix flag without check, can lead to incorrect results if matrix do not have corresponding property.
a.detect?(flag) # checks if matrix has property, if yes sets the flag. Returns true if check positive
a.detect(flag) # same as `detect?`, but returns matrix a
a.detect # detect all possible flags
a.flags # returns matrix flags
Most operations  matrix addition, multiplication, inversion, transposition and decompositions correctly update flags, but any direct access like a[i,j] = 0
or a.map!{v v+1}
resets flags to None
, so use a.detect
after them if you need to preserve flags (or a.assume!(f)
if detection is too slow).
Development
Roadmap:
Important
 [x] saving/loading from files
 [x] ways to evade allocations during calculations
 [x] Matrix exponent and trigonometric
 [ ] other matrix functions
 [ ] Banded matrices
 [x] Columnmajor storage (optional?)
 [ ] Other missing features from LAPACK (mostly selectable and orderable eigenvalues)
 [ ] Sparse matrices (perhaps out of scope/deserves separate shard)
 [ ] Other missing features from scipy.linalg (lyapunov/ricatti/sylvester equations, other things i don't know algorithms for)
 [x] Support multithreading\reenterability
 [x] Allow matrices of unsupported types (use
check_type
only when actually needed) [ ] type still has to support
T.new(value)
, maybe this requirement can be relaxed  [ ] also
map
signature has to be changed?
 [ ] type still has to support
Not so important
 [x] saving/loading to matlablike string
 [ ] better prettyprint, with alignment and various precision
 [x] use blas for multiplication
 [ ] more flags support (inversion of diagonal matrix and other trivial cases)
Contributing
 Fork it ( https://github.com/konovod/linalg/fork )
 Create your feature branch (git checkout b mynewfeature)
 Commit your changes (git commit am 'Add some feature')
 Push to the branch (git push origin mynewfeature)
 Create a new Pull Request
Contributors
 konovod  creator and maintainer
 RainbowZephyr  pseudoinverse implementation
 jtanderson 
arange
, openended ranges in submatrices  MatthiasWinkelmann 
GMat.max
fix