V Linear Algebra System

This package implements BLAS and LAPACKE functions. It provides different backends:

Therefore, its routines are a little more lower level than the ones in the package vsl.la.

OpenBLAS Backend

Use the flag -d vsl_vlas_cblas to use the OpenBLAS backend.

Install dependencies

Debian/Ubuntu GNU Linux

libopenblas-dev is not needed when using the pure V backend.

sudo apt-get install -y --no-install-recommends

Arch Linux/Manjaro GNU Linux

The best way of installing OpenBlas is using lapack-openblas.

yay -S lapack-openblas

or

git clone https://aur.archlinux.org/lapack-openblas.git /tmp/lapack-openblas
cd /tmp/lapack-openblas
makepkg -si

macOS

brew install openblas

LAPACKE Backend

Use the flag -d vsl_vlas_lapacke to use the LAPACKE backend (enabled by default for now).

Install dependencies

Debian/Ubuntu GNU Linux

sudo apt-get install -y --no-install-recommends

Arch Linux/Manjaro GNU Linux

The best way of installing LAPACKE is using lapack-openblas.

yay -S lapack-openblas

or

git clone https://aur.archlinux.org/lapack-openblas.git /tmp/lapack-openblas
cd /tmp/lapack-openblas
makepkg -si

fn c_trans(trans bool) blas.Transpose

fn c_uplo(up bool) blas.Uplo

fn col_major_complex_to_slice(m int, n int, data []complex.Complex) [][]complex.Complex

col_major_complex_to_slice converts col-major matrix to nested slice

fn col_major_to_slice(m int, n int, data []f64) [][]f64

col_major_to_slice converts col-major matrix to nested slice

fn dasum(n int, x []f64, incx int) f64

fn daxpy(n int, alpha f64, x []f64, incx int, mut y []f64, incy int)

fn dcopy(n int, x []f64, incx int, mut y []f64, incy int)

fn ddot(n int, x []f64, incx int, y []f64, incy int) f64

fn dgeev(calc_vl bool, calc_vr bool, n int, mut a []f64, lda int, wr []f64, wi []f64, vl []f64, ldvl_ int, vr []f64, ldvr_ int)

dgeev computes for an N-by-N real nonsymmetric matrix A, the eigenvalues and, optionally, the left and/or right eigenvectors.

See: http://www.netlib.org/lapack/explore-html/d9/d28/dgeev_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-geev

See: https://www.nag.co.uk/numeric/fl/nagdoc_fl26/html/f08/f08naf.html

The right eigenvector v(j) of A satisfies

A * v(j) = lambda(j) * v(j)

where lambda(j) is its eigenvalue.

The left eigenvector u(j) of A satisfies

u(j)**H * A = lambda(j) * u(j)**H

where u(j)**H denotes the conjugate-transpose of u(j).

The computed eigenvectors are normalized to have Euclidean norm equal to 1 and largest component real.

fn dgemm(trans_a bool, trans_b bool, m int, n int, k int, alpha f64, a []f64, lda int, b []f64, ldb int, beta f64, mut cc []f64, ldc int)

fn dgemv(trans bool, m int, n int, alpha f64, a []f64, lda int, x []f64, incx int, beta f64, mut y []f64, incy int)

fn dger(m int, n int, alpha f64, x []f64, incx int, y []f64, incy int, mut a []f64, lda int)

fn dgesv(n int, nrhs int, mut a []f64, lda int, ipiv []int, mut b []f64, ldb int)

dgesv computes the solution to a real system of linear equations.

See: http://www.netlib.org/lapack/explore-html/d8/d72/dgesv_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-gesv

The system is:

A * X = B,

where A is an N-by-N matrix and X and B are N-by-NRHS matrices.

The LU decomposition with partial pivoting and row interchanges is used to factor A as

A = P * L * U,

where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then used to solve the system of equations A * X = B.

Note: matrix 'a' will be modified

fn dgesvd(jobu &char, jobvt &char, m int, n int, a []f64, lda int, s []f64, u []f64, ldu int, vt []f64, ldvt int, superb []f64)

dgesvd computes the singular value decomposition (SVD) of a real M-by-N matrix A, optionally computing the left and/or right singular vectors.

See: http://www.netlib.org/lapack/explore-html/d8/d2d/dgesvd_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-gesvd

The SVD is written

A = U * SIGMA * transpose(V)

where SIGMA is an M-by-N matrix which is zero except for its min(m,n) diagonal elements, U is an M-by-M orthogonal matrix, and V is an N-by-N orthogonal matrix. The diagonal elements of SIGMA are the singular values of A; they are real and non-negative, and are returned in descending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.

Note that the routine returns V**T, not V.

Note: matrix 'a' will be modified

fn dgetrf(m int, n int, mut a []f64, lda int, ipiv []int)

dgetrf computes an LU factorization of a general M-by-N matrix A using partial pivoting with row interchanges.

See: http://www.netlib.org/lapack/explore-html/d3/d6a/dgetrf_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-getrf

The factorization has the form A = P * L * U where P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m > n), and U is upper triangular (upper trapezoidal if m < n).

This is the right-looking Level 3 BLAS version of the algorithm.

Note: (1) matrix 'a' will be modified(2) ipiv indices are 1-based (i.e. Fortran)

fn dgetri(n int, mut a []f64, lda int, ipiv []int)

dgetri computes the inverse of a matrix using the LU factorization computed by DGETRF.

See: http://www.netlib.org/lapack/explore-html/df/da4/dgetri_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-getri

This method inverts U and then computes inv(A) by solving the system inv(A)*L = inv(U) for inv(A).

fn dlange(norm rune, m int, n int, a []f64, lda int, work []f64) f64

fn dnrm2(n int, x []f64, incx int) f64

fn dpotrf(up bool, n int, mut a []f64, lda int)

dpotrf computes the Cholesky factorization of a real symmetric positive definite matrix A.

See: http://www.netlib.org/lapack/explore-html/d0/d8a/dpotrf_8f.html

See: https://software.intel.com/en-us/mkl-developer-reference-c-potrf

The factorization has the form

A = U**T * U, if UPLO = 'U'

or

A = L * L**T, if UPLO = 'L'

where U is an upper triangular matrix and L is lower triangular.

This is the block version of the algorithm, calling Level 3 BLAS.

fn drot(n int, mut x []f64, incx int, mut y []f64, incy int, c f64, s f64)

fn dscal(n int, alpha f64, mut x []f64, incx int)

fn dswap(n int, mut x []f64, incx int, mut y []f64, incy int)

fn dsyr(uplo bool, n int, alpha f64, x []f64, incx int, mut a []f64, lda int)

fn dsyr2(uplo bool, n int, alpha f64, x []f64, incx int, y []f64, incy int, mut a []f64, lda int)

fn dtrmv(uplo bool, trans_a bool, diag blas.Diagonal, n int, a []f64, lda int, mut x []f64, incx int)

fn dtrsv(uplo bool, trans_a bool, diag blas.Diagonal, n int, a []f64, lda int, mut x []f64, incx int)

fn eigenvecs_build(mut vv []complex.Complex, wr []f64, wi []f64, v []f64)

eigenvecs_build builds complex eigenvectros created by Dgeev function

input: wr, wi: real and imag parts of eigenvalues. v: left or right eigenvectors from Dgeev.

output: vv: complex version of left or right eigenvector [pre-allocated].

NOTE: (no checks made).

n = wr.len = wi.len = v.len 2 * n = vv.len

fn eigenvecs_build_both(mut vvl []complex.Complex, mut vvr []complex.Complex, wr []f64, wi []f64, vl []f64, vr []f64)

eigenvecs_build_both builds complex left and right eigenvectros created by Dgeev function

input: wr, wi:real and imag parts of eigenvalues. vl, vr:left and right eigenvectors from Dgeev.

output: vvl, vvr:complex version of left and right eigenvectors [pre-allocated].

NOTE: (no checks made).

n = wr.len = wi.len = vl.len = vr.len 2 * n = vvl.len = vvr.len

fn extract_col(j int, m int, n int, a []f64) []f64

extract_col extracts j column from (m,n) col-major matrix

fn extract_col_complex(j int, m int, n int, a []complex.Complex) []complex.Complex

extract_col_complex extracts j column from (m,n) col-major matrix (complex version)

fn extract_row(i int, m int, n int, a []f64) []f64

extract_row extracts i row from (m,n) col-major matrix

fn extract_row_complex(i int, m int, n int, a []complex.Complex) []complex.Complex

extract_row_complex extracts i row from (m,n) col-major matrix (complex version)

fn get_join_complex(v_real []f64, v_imag []f64) []complex.Complex

get_join_complex joins real and imag parts of array

fn get_split_complex(v []complex.Complex) ([]f64, []f64)

get_split_complex splits real and imag parts of array

fn join_complex(v_real []f64, v_imag []f64) []complex.Complex

join_complex joins real and imag parts of array

fn print_col_major(m int, n int, data []f64, nfmt_ string) string

print_col_major prints matrix (without commas or brackets)

fn print_col_major_complex(m int, n int, data []complex.Complex, nfmt_r_ string, nfmt_i_ string) string

print_col_major_complex prints matrix (without commas or brackets). NOTE: if non-empty, nfmt_i must have '+' e.g. %+g

fn print_col_major_complex_v(m int, n int, data []complex.Complex, nfmt_r_ string, nfmt_i_ string) string

print_col_major_complex_v prints matrix in v format NOTE: if non-empty, nfmt_i must have '+' e.g. %+g

fn print_col_major_omplex_py(m int, n int, data []complex.Complex, nfmt_r_ string, nfmt_i_ string) string

print_col_major_omplex_py prints matrix in Python format NOTE: if non-empty, nfmt_i must have '+' e.g. %+g

fn print_col_major_py(m int, n int, data []f64, nfmt_ string) string

print_col_major_py prints matrix in Python format

fn print_col_major_v(m int, n int, data []f64, nfmt_ string) string

print_col_major_v prints matrix in v format

fn set_num_threads(n int)

set_num_threads sets the number of threads in VLAS

fn slice_to_col_major(a [][]f64) []f64

slice_to_col_major converts nested slice into an array representing a col-major matrix

NOTE: make sure to have at least 1x1 item

fn slice_to_col_major_complex(a [][]complex.Complex) []complex.Complex

slice_to_col_major_complex converts nested slice into an array representing a col-major matrix of complex numbers.

data[i+j*m] = a[i][j]

NOTE: make sure to have at least 1x1 item

fn split_complex(v []complex.Complex) ([]f64, []f64)

split_complex splits real and imag parts of array