Russell Lab - Scientific laboratory for linear algebra and numerical mathematics

This crate is part of Russell - Rust Scientific Library

Contents

• Introduction
  • Documentation
• Installation
  • TL;DR (Debian/Ubuntu/Linux)
  • Details
  • Setting Cargo.toml up
• Complex numbers
• Examples
  • Running an example with Intel MKL
  • Sorting small tuples
  • Check first and second derivatives
  • Bessel functions
  • Linear fitting
  • Lagrange interpolation
  • Solution of a 1D PDE using spectral collocation
  • Numerical integration: perimeter of ellipse
  • Finding a local minimum and a root
  • Computing the pseudo-inverse matrix
  • Matrix visualization
  • Computing eigenvalues and eigenvectors
  • Cholesky factorization
• About the column major representation
• Benchmarks
  • Jacobi Rotation versus LAPACK DSYEV
• Notes for developers

Introduction

This library implements specialized mathematical functions (e.g., Bessel, Erf, Gamma) and functions to perform linear algebra computations (e.g., Matrix, Vector, Matrix-Vector, Eigen-decomposition, SVD). This library also implements a set of helpful function for comparing floating-point numbers, measuring computer time, reading table-formatted data, and more.

The code shall be implemented in native Rust code as much as possible. However, light interfaces ("wrappers") are implemented for some of the best tools available in numerical mathematics, including OpenBLAS and Intel MKL.

The code is organized in modules:

• algo — algorithms that depend on the other modules (e.g, Lagrange interpolation)
• base — "base" functionality to help other modules
• check — functions to assist in unit and integration testing
• math — mathematical (specialized) functions and constants
• matrix — [NumMatrix] struct and associated functions
• matvec — functions operating on matrices and vectors
• vector — [NumVector] struct and associated functions

For linear algebra, the main structures are NumVector and NumMatrix, that are generic Vector and Matrix structures. The Matrix data is stored as column-major. The Vector and Matrix are f64 and Complex64 aliases of NumVector and NumMatrix, respectively.

The linear algebra functions currently handle only (f64, i32) pairs, i.e., accessing the (double, int) C functions. We also consider (Complex64, i32) pairs.

There are many functions for linear algebra, such as (for Real and Complex types):

• Vector addition, copy, inner and outer products, norms, and more
• Matrix addition, multiplication, copy, singular-value decomposition, eigenvalues, pseudo-inverse, inverse, norms, and more
• Matrix-vector multiplication, and more
• Solution of dense linear systems with symmetric or non-symmetric coefficient matrices, and more
• Reading writing files, linspace, grid generators, Stopwatch, linear fitting, and more
• Checking results, comparing float point numbers, and verifying the correctness of derivatives; see russell_lab::check

Documentation

Installation

At this moment, Russell works on Linux (Debian/Ubuntu; and maybe Arch). It has some limited functionality on macOS too. In the future, we plan to enable Russell on Windows; however, this will take time because some essential libraries are not easily available on Windows.

TL;DR (Debian/Ubuntu/Linux)

First:

sudo apt-get install -y --no-install-recommends \
    g++ \
    gdb \
    gfortran \
    liblapacke-dev \
    libmumps-seq-dev \
    libopenblas-dev \
    libsuitesparse-dev

Then:

cargo add russell_lab

Details

This crate depends on an efficient BLAS library such as OpenBLAS and Intel MKL.

The root README file presents the steps to install the required dependencies.

Setting Cargo.toml up

👆 Check the crate version and update your Cargo.toml accordingly:

[dependencies]
russell_lab = "*"

Or, considering the optional features (see more about these here):

[dependencies]
russell_lab = { version = "*", features = ["intel_mkl"] }

Complex numbers

Note: For the functions dealing with complex numbers, the following line must be added to all derived code:

use num_complex::Complex64;

This line will bring Complex64 to the scope. For convenience the (russell_lab) macro cpx! may be used to allocate complex numbers.

Examples

See also:

• russell_lab/examples

Running an example with Intel MKL

Consider the following code:

use russell_lab::*;

fn main() -> Result<(), StrError> {
    println!("Using Intel MKL  = {}", using_intel_mkl());
    println!("BLAS num threads = {}", get_num_threads());
    set_num_threads(2);
    println!("BLAS num threads = {}", get_num_threads());
    Ok(())
}

First, run the example without Intel MKL (default):

cargo run --example base_auxiliary_blas

The output looks like this:

Using Intel MKL  = false
BLAS num threads = 24
BLAS num threads = 2

Second, run the code with the intel_mkl feature:

cargo run --example base_auxiliary_blas --features intel_mk

Then, the output looks like this:

Using Intel MKL  = true
BLAS num threads = 24
BLAS num threads = 2

Sorting small tuples

See the code

use russell_lab::base::{sort2, sort3, sort4};
use russell_lab::StrError;

fn main() -> Result<(), StrError> {
    // sorting slices with the standard function
    let mut u2 = vec![2.0, 1.0];
    let mut u3 = vec![3.0, 1.0, 2.0];
    let mut u4 = vec![3.0, 1.0, 4.0, 2.0];
    u2.sort_by(|a, b| a.partial_cmp(b).unwrap());
    u3.sort_by(|a, b| a.partial_cmp(b).unwrap());
    u4.sort_by(|a, b| a.partial_cmp(b).unwrap());
    println!("u2 = {:?}", u2);
    println!("u3 = {:?}", u3);
    println!("u4 = {:?}", u4);
    assert_eq!(&u2, &[1.0, 2.0]);
    assert_eq!(&u3, &[1.0, 2.0, 3.0]);
    assert_eq!(&u4, &[1.0, 2.0, 3.0, 4.0]);

    // sorting small tuples
    let mut v2 = (2.0, 1.0);
    let mut v3 = (3.0, 1.0, 2.0);
    let mut v4 = (3.0, 1.0, 4.0, 2.0);
    sort2(&mut v2);
    sort3(&mut v3);
    sort4(&mut v4);
    println!("v2 = {:?}", v2);
    println!("v3 = {:?}", v3);
    println!("v4 = {:?}", v4);
    assert_eq!(v2, (1.0, 2.0));
    assert_eq!(v3, (1.0, 2.0, 3.0));
    assert_eq!(v4, (1.0, 2.0, 3.0, 4.0));
    Ok(())
}

Check first and second derivatives

Check the implementation of the first and second derivatives of f(x) (illustrated below).

See the code

use russell_lab::algo::NoArgs;
use russell_lab::check::{deriv1_approx_eq, deriv2_approx_eq};
use russell_lab::{StrError, Vector};

fn main() -> Result<(), StrError> {
    // f(x)
    let f = |x: f64, _: &mut NoArgs| Ok(1.0 / 2.0 - 1.0 / (1.0 + 16.0 * x * x));

    // g(x) = df/dx(x)
    let g = |x: f64, _: &mut NoArgs| Ok((32.0 * x) / f64::powi(1.0 + 16.0 * f64::powi(x, 2), 2));

    // h(x) = d²f/dx²(x)
    let h = |x: f64, _: &mut NoArgs| {
        Ok((-2048.0 * f64::powi(x, 2)) / f64::powi(1.0 + 16.0 * f64::powi(x, 2), 3)
            + 32.0 / f64::powi(1.0 + 16.0 * f64::powi(x, 2), 2))
    };

    let xx = Vector::linspace(-2.0, 2.0, 9)?;
    let args = &mut 0;
    println!("{:>4}{:>23}{:>23}", "x", "df/dx", "d²f/dx²");
    for x in xx {
        let dfdx = g(x, args)?;
        let d2dfx2 = h(x, args)?;
        println!("{:>4}{:>23}{:>23}", x, dfdx, d2dfx2);
        deriv1_approx_eq(dfdx, x, args, 1e-10, f);
        deriv2_approx_eq(d2dfx2, x, args, 1e-9, f);
    }
    Ok(())
}

Output:

   x                  df/dx                d²f/dx²
  -2   -0.01514792899408284  -0.022255803368229403
-1.5   -0.03506208911614317   -0.06759718081851025
  -1   -0.11072664359861592   -0.30612660289029103
-0.5                  -0.64                 -2.816
   0                      0                     32
 0.5                   0.64                 -2.816
   1    0.11072664359861592   -0.30612660289029103
 1.5    0.03506208911614317   -0.06759718081851025
   2    0.01514792899408284  -0.022255803368229403

Bessel functions

Plotting the Bessel J0, J1, and J2 functions:

use plotpy::{Curve, Plot};
use russell_lab::math::{bessel_j0, bessel_j1, bessel_jn, GOLDEN_RATIO};
use russell_lab::{StrError, Vector};

const OUT_DIR: &str = "/tmp/russell_lab/";

fn main() -> Result<(), StrError> {
    // values
    let xx = Vector::linspace(0.0, 15.0, 101)?;
    let j0 = xx.get_mapped(|x| bessel_j0(x));
    let j1 = xx.get_mapped(|x| bessel_j1(x));
    let j2 = xx.get_mapped(|x| bessel_jn(2, x));
    // plot
    if false { // <<< remove this condition
    let mut curve_j0 = Curve::new();
    let mut curve_j1 = Curve::new();
    let mut curve_j2 = Curve::new();
    curve_j0.set_label("J0").draw(xx.as_data(), j0.as_data());
    curve_j1.set_label("J1").draw(xx.as_data(), j1.as_data());
    curve_j2.set_label("J2").draw(xx.as_data(), j2.as_data());
    let mut plot = Plot::new();
    let path = format!("{}/math_bessel_functions_1.svg", OUT_DIR);
    plot.add(&curve_j0)
        .add(&curve_j1)
        .add(&curve_j2)
        .grid_labels_legend("$x$", "$J_0(x),\\,J_1(x),\\,J_2(x)$")
        .set_figure_size_points(GOLDEN_RATIO * 280.0, 280.0)
        .save(path.as_str())?;
    }
    Ok(())
}

Output:

Linear fitting

Fit a line through a set of points. The line has slope m and intercepts the y axis at x=0 with y(x=0) = c.

use russell_lab::algo::linear_fitting;
use russell_lab::{approx_eq, StrError, Vector};

fn main() -> Result<(), StrError> {
    // model: c is the y value @ x = 0; m is the slope
    let x = Vector::from(&[0.0, 1.0, 3.0, 5.0]);
    let y = Vector::from(&[1.0, 0.0, 2.0, 4.0]);
    let (c, m) = linear_fitting(&x, &y, false)?;
    println!("c = {}, m = {}", c, m);
    approx_eq(c, 0.1864406779661015, 1e-15);
    approx_eq(m, 0.6949152542372882, 1e-15);
    Ok(())
}

Results:

Lagrange interpolation

This example illustrates the use of InterpLagrange with at Chebyshev-Gauss-Lobatto grid to interpolate Runge's equation.

See the code

Results:

Solution of a 1D PDE using spectral collocation

This example illustrates the solution of a 1D PDE using the spectral collocation method. It employs the InterpLagrange struct.

d²u     du          x
——— - 4 —— + 4 u = e  + C
dx²     dx

    -4 e
C = ——————
    1 + e²

x ∈ [-1, 1]

Boundary conditions:

u(-1) = 0  and  u(1) = 0

Reference solution:

        x   sinh(1)  2x   C
u(x) = e  - ——————— e   + —
            sinh(2)       4

See the code

Results:

Numerical integration: perimeter of ellipse

use russell_lab::algo::Quadrature;
use russell_lab::math::{elliptic_e, PI};
use russell_lab::{approx_eq, StrError};

fn main() -> Result<(), StrError> {
    //  Determine the perimeter P of an ellipse of length 2 and width 1
    //
    //      2π
    //     ⌠   ____________________
    // P = │ \╱ ¼ sin²(θ) + cos²(θ)  dθ
    //     ⌡
    //    0

    let mut quad = Quadrature::new();
    let args = &mut 0;
    let (perimeter, _) = quad.integrate(0.0, 2.0 * PI, args, |theta, _| {
        Ok(f64::sqrt(
            0.25 * f64::powi(f64::sin(theta), 2) + f64::powi(f64::cos(theta), 2),
        ))
    })?;
    println!("\nperimeter = {}", perimeter);

    // complete elliptic integral of the second kind E(0.75)
    let ee = elliptic_e(PI / 2.0, 0.75)?;

    // reference solution
    let ref_perimeter = 4.0 * ee;
    approx_eq(perimeter, ref_perimeter, 1e-14);
    Ok(())
}

Finding a local minimum and a root

This example finds the local minimum between 0.1 and 0.3 and the root between 0.3 and 0.4 for the function illustrated below

See the code

The output looks like:

x_optimal = 0.20000000003467466

Number of function evaluations   = 18
Number of Jacobian evaluations   = 0
Number of iterations             = 18
Error estimate                   = unavailable
Total computation time           = 6.11µs

x_root = 0.3397874957748173

Number of function evaluations   = 10
Number of Jacobian evaluations   = 0
Number of iterations             = 9
Error estimate                   = unavailable
Total computation time           = 907ns

Computing the pseudo-inverse matrix

use russell_lab::{mat_pseudo_inverse, Matrix, StrError};

fn main() -> Result<(), StrError> {
    // set matrix
    let mut a = Matrix::from(&[
      [1.0, 0.0], //
      [0.0, 1.0], //
      [0.0, 1.0], //
    ]);
    let a_copy = a.clone();

    // compute pseudo-inverse matrix
    let mut ai = Matrix::new(2, 3);
    mat_pseudo_inverse(&mut ai, &mut a)?;

    // compare with solution
    let ai_correct = "┌                ┐\n\
                      │ 1.00 0.00 0.00 │\n\
                      │ 0.00 0.50 0.50 │\n\
                      └                ┘";
    assert_eq!(format!("{:.2}", ai), ai_correct);

    // compute a ⋅ ai
    let (m, n) = a.dims();
    let mut a_ai = Matrix::new(m, m);
    for i in 0..m {
        for j in 0..m {
            for k in 0..n {
                a_ai.add(i, j, a_copy.get(i, k) * ai.get(k, j));
            }
        }
    }

    // check: a ⋅ ai ⋅ a = a
    let mut a_ai_a = Matrix::new(m, n);
    for i in 0..m {
        for j in 0..n {
            for k in 0..m {
                a_ai_a.add(i, j, a_ai.get(i, k) * a_copy.get(k, j));
            }
        }
    }
    let a_ai_a_correct = "┌           ┐\n\
                          │ 1.00 0.00 │\n\
                          │ 0.00 1.00 │\n\
                          │ 0.00 1.00 │\n\
                          └           ┘";
    assert_eq!(format!("{:.2}", a_ai_a), a_ai_a_correct);
    Ok(())
}

Matrix visualization

We can use the fantastic tool named vismatrix to visualize the pattern of non-zero values of a matrix. With vismatrix, we can click on each circle and investigate the numeric values as well.

The function mat_write_vismatrix writes the input data file for vismatrix.

See the code

After generating the "dot-smat" file, run the following command:

vismatrix /tmp/russell_lab/matrix_visualization.smat

Output:

Computing eigenvalues and eigenvectors

use russell_lab::*;

fn main() -> Result<(), StrError> {
    // set matrix
    let data = [[2.0, 0.0, 0.0], [0.0, 3.0, 4.0], [0.0, 4.0, 9.0]];
    let mut a = Matrix::from(&data);

    // allocate output arrays
    let m = a.nrow();
    let mut l_real = Vector::new(m);
    let mut l_imag = Vector::new(m);
    let mut v_real = Matrix::new(m, m);
    let mut v_imag = Matrix::new(m, m);

    // perform the eigen-decomposition
    mat_eigen(&mut l_real, &mut l_imag, &mut v_real, &mut v_imag, &mut a)?;

    // check results
    assert_eq!(
        format!("{:.1}", l_real),
        "┌      ┐\n\
         │ 11.0 │\n\
         │  1.0 │\n\
         │  2.0 │\n\
         └      ┘"
    );
    assert_eq!(
        format!("{}", l_imag),
        "┌   ┐\n\
         │ 0 │\n\
         │ 0 │\n\
         │ 0 │\n\
         └   ┘"
    );

    // check eigen-decomposition (similarity transformation) of a
    // symmetric matrix with real-only eigenvalues and eigenvectors
    let a_copy = Matrix::from(&data);
    let lam = Matrix::diagonal(l_real.as_data());
    let mut a_v = Matrix::new(m, m);
    let mut v_l = Matrix::new(m, m);
    let mut err = Matrix::filled(m, m, f64::MAX);
    mat_mat_mul(&mut a_v, 1.0, &a_copy, &v_real, 0.0)?;
    mat_mat_mul(&mut v_l, 1.0, &v_real, &lam, 0.0)?;
    mat_add(&mut err, 1.0, &a_v, -1.0, &v_l)?;
    approx_eq(mat_norm(&err, Norm::Max), 0.0, 1e-15);
    Ok(())
}

Cholesky factorization

use russell_lab::*;

fn main() -> Result<(), StrError> {
    // set matrix
    let sym = 0.0;
    #[rustfmt::skip]
    let mut a = Matrix::from(&[
        [  4.0,   sym,   sym],
        [ 12.0,  37.0,   sym],
        [-16.0, -43.0,  98.0],
    ]);

    // perform factorization
    mat_cholesky(&mut a, false)?;

    // define alias (for convenience)
    let l = &a;

    // compare with solution
    let l_correct = "┌          ┐\n\
                     │  2  0  0 │\n\
                     │  6  1  0 │\n\
                     │ -8  5  3 │\n\
                     └          ┘";
    assert_eq!(format!("{}", l), l_correct);

    // check:  l ⋅ lᵀ = a
    let m = a.nrow();
    let mut l_lt = Matrix::new(m, m);
    for i in 0..m {
        for j in 0..m {
            for k in 0..m {
                l_lt.add(i, j, l.get(i, k) * l.get(j, k));
            }
        }
    }
    let l_lt_correct = "┌             ┐\n\
                        │   4  12 -16 │\n\
                        │  12  37 -43 │\n\
                        │ -16 -43  98 │\n\
                        └             ┘";
    assert_eq!(format!("{}", l_lt), l_lt_correct);
    Ok(())
}

About the column major representation

Only the COL-MAJOR representation is considered here.

    ┌     ┐  row_major = {0, 3,
    │ 0 3 │               1, 4,
A = │ 1 4 │               2, 5};
    │ 2 5 │
    └     ┘  col_major = {0, 1, 2,
    (m × n)               3, 4, 5}

Aᵢⱼ = col_major[i + j·m] = row_major[i·n + j]
        ↑
COL-MAJOR IS ADOPTED HERE

The main reason to use the col-major representation is to make the code work better with BLAS/LAPACK written in Fortran. Although those libraries have functions to handle row-major data, they usually add an overhead due to temporary memory allocation and copies, including transposing matrices. Moreover, the row-major versions of some BLAS/LAPACK libraries produce incorrect results (notably the DSYEV).

Benchmarks

Need to install:

cargo install cargo-criterion

Run the benchmarks with:

bash ./zscripts/benchmark.bash

Jacobi Rotation versus LAPACK DSYEV

Comparison of the performances of mat_eigen_sym_jacobi (Jacobi rotation) versus mat_eigen_sym (calling LAPACK DSYEV).

Notes for developers

• The c_code directory contains a thin wrapper to the BLAS libraries (OpenBLAS or Intel MKL)
• The c_code directory also contains a wrapper to the C math functions
• The build.rs file uses the crate cc to build the C-wrappers