use ndarray::Array2;

/// Initializes the electric and magnetic field grids as 2D arrays.
/// 
/// # Arguments
/// 
/// * `nx` - Number of grid points along the x-axis.
/// * `ny` - Number of grid points along the y-axis.
/// 
/// # Returns
/// 
/// A tuple containing two 2D arrays (electric field and magnetic field) initialized to zero.
fn initialize_fields(nx: usize, ny: usize) -> (Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny)); // Electric field grid
    let b_field = Array2::<f64>::zeros((nx, ny)); // Magnetic field grid
    (e_field, b_field)
}

/// Updates the electric and magnetic fields using a finite-difference scheme.
/// 
/// This function approximates spatial derivatives by calculating differences between adjacent
/// grid points and updates the field values over one time step. The update for the electric field
/// is based on the difference in the magnetic field along the x-direction, while the magnetic field
/// update uses the difference in the electric field along the y-direction.
/// 
/// # Arguments
/// 
/// * `e_field` - Mutable reference to the electric field array.
/// * `b_field` - Mutable reference to the magnetic field array.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step increment.
fn update_fields(e_field: &mut Array2<f64>, b_field: &mut Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.shape()[0];
    let ny = e_field.shape()[1];

    // Update the electric field using finite-difference approximation in the x-direction.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            e_field[[i, j]] += (b_field[[i + 1, j]] - b_field[[i, j]]) * dt / dx;
        }
    }

    // Update the magnetic field using finite-difference approximation in the y-direction.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            b_field[[i, j]] -= (e_field[[i, j + 1]] - e_field[[i, j]]) * dt / dx;
        }
    }
}

fn main() {
    // Define grid dimensions and simulation parameters.
    let nx = 100;  // Number of grid points along x-axis
    let ny = 100;  // Number of grid points along y-axis
    let dx = 0.1;  // Grid spacing (spatial resolution)
    let dt = 0.01; // Time step for simulation

    // Initialize the electric and magnetic fields.
    let (mut e_field, mut b_field) = initialize_fields(nx, ny);

    // Run the simulation for 100 time steps.
    for _ in 0..100 {
        update_fields(&mut e_field, &mut b_field, dx, dt);
    }

    // Indicate that the simulation has completed.
    println!("Simulation complete!");
}

// Import the ndarray crate to work with multi-dimensional arrays.
use ndarray::Array2;

/// Initializes the electric and magnetic field grids along with the material property grids.
/// 
/// This function creates 2D arrays for the electric field, magnetic field, permittivity, and permeability.
/// Each grid is defined with dimensions (nx, ny). The material property grids are filled with constant values,
/// representing uniform material properties across the domain.
/// 
/// # Arguments
/// 
/// * `nx` - Number of grid points along the x-axis.
/// * `ny` - Number of grid points along the y-axis.
/// * `eps` - Permittivity value to be assigned across the grid.
/// * `mu` - Permeability value to be assigned across the grid.
/// 
/// # Returns
/// 
/// A tuple containing four 2D arrays: electric field, magnetic field, permittivity grid, and permeability grid.
fn initialize_fields(
    nx: usize, 
    ny: usize, 
    eps: f64, 
    mu: f64
) -> (Array2<f64>, Array2<f64>, Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny));      // Electric field grid initialized to zero.
    let b_field = Array2::<f64>::zeros((nx, ny));      // Magnetic field grid initialized to zero.
    let eps_grid = Array2::<f64>::from_elem((nx, ny), eps); // Permittivity grid with constant value.
    let mu_grid = Array2::<f64>::from_elem((nx, ny), mu);   // Permeability grid with constant value.
    (e_field, b_field, eps_grid, mu_grid)
}

/// Updates the electric and magnetic fields using a finite-difference scheme.
/// 
/// The function computes spatial derivatives using the differences between adjacent grid points.
/// The electric field is updated based on the gradient of the magnetic field along the x-axis,
/// and the magnetic field is updated based on the gradient of the electric field along the y-axis.
/// The local permittivity and permeability values are incorporated in the update formulas.
/// 
/// # Arguments
/// 
/// * `e_field` - Mutable reference to the 2D array representing the electric field.
/// * `b_field` - Mutable reference to the 2D array representing the magnetic field.
/// * `eps` - Reference to the 2D array representing the permittivity grid.
/// * `mu` - Reference to the 2D array representing the permeability grid.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step for the simulation.
fn update_fields(
    e_field: &mut Array2<f64>, 
    b_field: &mut Array2<f64>, 
    eps: &Array2<f64>, 
    mu: &Array2<f64>, 
    dx: f64, 
    dt: f64
) {
    let nx = e_field.shape()[0];
    let ny = e_field.shape()[1];

    // Finite-difference update for the electric field.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            // Calculate the spatial derivative of the magnetic field along the x-axis.
            let d_b = (b_field[[i + 1, j]] - b_field[[i, j]]) / dx;
            // Update the electric field using the local permittivity.
            e_field[[i, j]] += dt / eps[[i, j]] * d_b;
        }
    }

    // Finite-difference update for the magnetic field.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            // Calculate the spatial derivative of the electric field along the y-axis.
            let d_e = (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
            // Update the magnetic field using the local permeability.
            b_field[[i, j]] -= dt / mu[[i, j]] * d_e;
        }
    }
}

fn main() {
    // Define simulation parameters: grid dimensions, spatial resolution, time step, and material properties.
    let nx = 100;       // Number of grid points in the x-direction.
    let ny = 100;       // Number of grid points in the y-direction.
    let dx = 0.1;       // Spatial grid spacing.
    let dt = 0.01;      // Time step for the simulation.
    let eps = 8.85e-12; // Permittivity of free space (F/m).
    let mu = 1.26e-6;   // Permeability of free space (H/m).

    // Initialize the fields and material property grids.
    let (mut e_field, mut b_field, eps_grid, mu_grid) = initialize_fields(nx, ny, eps, mu);

    // Run the simulation for 100 time steps.
    for _ in 0..100 {
        update_fields(&mut e_field, &mut b_field, &eps_grid, &mu_grid, dx, dt);
    }

    // Output a message indicating that the simulation has completed.
    println!("Simulation complete!");
}

use ndarray::Array2;

/// Initializes the electric and magnetic field arrays to zero.
/// 
/// # Arguments
/// 
/// * `nx` - Number of grid points along the x-axis.
/// * `ny` - Number of grid points along the y-axis.
/// 
/// # Returns
/// 
/// A tuple containing two 2D arrays representing the electric field and the magnetic field.
fn initialize_fields(nx: usize, ny: usize) -> (Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny));  // Electric field initialized to zeros.
    let b_field = Array2::<f64>::zeros((nx, ny));  // Magnetic field initialized to zeros.
    (e_field, b_field)
}

/// Updates the electric and magnetic fields using a finite-difference time-domain scheme.
/// 
/// This function approximates spatial derivatives with finite differences. The electric field is
/// updated based on the gradient of the magnetic field along the x-direction, and the magnetic field
/// is updated based on the gradient of the electric field along the y-direction. Only interior points
/// are updated while boundary points remain unchanged.
/// 
/// # Arguments
/// 
/// * `e_field` - Mutable reference to the electric field array.
/// * `b_field` - Mutable reference to the magnetic field array.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step increment.
fn update_fields(e_field: &mut Array2<f64>, b_field: &mut Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();
    
    // Update electric field: compute derivative along the x-direction.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            let d_b_dx = (b_field[[i + 1, j]] - b_field[[i, j]]) / dx;
            e_field[[i, j]] += dt * d_b_dx;
        }
    }

    // Update magnetic field: compute derivative along the y-direction.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            let d_e_dy = (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
            b_field[[i, j]] -= dt * d_e_dy;
        }
    }
}

fn main() {
    // Simulation parameters: grid dimensions, spatial resolution, and time step.
    let nx = 100;
    let ny = 100;
    let dx = 0.1;   // Grid spacing.
    let dt = 0.01;  // Time step.

    // Initialize the fields.
    let (mut e_field, mut b_field) = initialize_fields(nx, ny);

    // Run the simulation for 100 time steps.
    for _ in 0..100 {
        update_fields(&mut e_field, &mut b_field, dx, dt);
    }

    println!("FDTD simulation complete!");
}

use nalgebra::{DMatrix, DVector};

/// Assembles a simplified stiffness matrix for a 2D FEM mesh.
/// 
/// The stiffness matrix represents a discretized system of equations. In this simplified example
/// the diagonal entries are set to 1.0 and adjacent off-diagonals are assigned -0.5 to mimic connectivity.
/// 
/// # Arguments
/// 
/// * `nx` - Number of elements along the x-axis.
/// * `ny` - Number of elements along the y-axis.
/// * `dx` - Grid spacing (included for context in a more detailed model).
/// 
/// # Returns
/// 
/// A DMatrix representing the stiffness matrix of size (nx*ny) x (nx*ny).
fn assemble_stiffness_matrix(nx: usize, ny: usize, dx: f64) -> DMatrix<f64> {
    let size = nx * ny;
    let mut stiffness_matrix = DMatrix::<f64>::zeros(size, size);

    for i in 0..size {
        // Set the diagonal value.
        stiffness_matrix[(i, i)] = 1.0;
        // Set off-diagonal values for a simplified connectivity.
        if i + 1 < size {
            stiffness_matrix[(i, i + 1)] = -0.5;
            stiffness_matrix[(i + 1, i)] = -0.5;
        }
    }
    stiffness_matrix
}

/// Solves the FEM system defined by the stiffness matrix and the right-hand side vector.
/// 
/// This function uses LU decomposition to solve the linear system. If the stiffness matrix is singular
/// or the solution cannot be computed, an error is returned.
/// 
/// # Arguments
/// 
/// * `stiffness_matrix` - The assembled stiffness matrix.
/// * `rhs` - The right-hand side vector representing the source terms.
/// 
/// # Returns
/// 
/// A Result containing the solution vector or an error message.
fn solve_fem_system(
    stiffness_matrix: DMatrix<f64>, 
    rhs: DVector<f64>
) -> Result<DVector<f64>, String> {
    match stiffness_matrix.lu().solve(&rhs) {
        Some(solution) => Ok(solution),
        None => Err(String::from("Failed to solve the FEM system: Matrix may be singular.")),
    }
}

fn main() {
    // Define FEM mesh parameters.
    let nx = 10;
    let ny = 10;
    let dx = 0.1;

    // Assemble the stiffness matrix for the 2D FEM mesh.
    let stiffness_matrix = assemble_stiffness_matrix(nx, ny, dx);

    // Define a right-hand side vector (for example, representing a source term).
    let rhs = DVector::<f64>::from_element(nx * ny, 1.0);

    // Solve the FEM system and handle any potential errors.
    match solve_fem_system(stiffness_matrix, rhs) {
        Ok(solution) => println!("FEM solution: {:?}", solution),
        Err(e) => println!("Error solving FEM system: {}", e),
    }
}

use ndarray::Array1;

/// Initializes the 1D electric and magnetic field arrays.
/// 
/// This function creates two 1D arrays representing the electric field (E) and the magnetic field (H)
/// over a spatial domain defined by the number of grid points. Both fields are initialized to zero.
/// 
/// # Arguments
/// 
/// * `nx` - The number of grid points in the spatial domain.
/// 
/// # Returns
/// 
/// A tuple containing the electric field array and the magnetic field array.
fn initialize_fields(nx: usize) -> (Array1<f64>, Array1<f64>) {
    let e_field = Array1::<f64>::zeros(nx); // Electric field initialized to zero.
    let h_field = Array1::<f64>::zeros(nx); // Magnetic field initialized to zero.
    (e_field, h_field)
}

/// Updates the 1D electric and magnetic fields using the FDTD method.
/// 
/// In this function the electric field is updated based on the difference between adjacent magnetic field
/// values while the magnetic field is updated based on differences in the electric field values. Updates are
/// performed on interior grid points while the boundary values remain unchanged.
/// 
/// # Arguments
/// 
/// * `e_field` - A mutable reference to the electric field array.
/// * `h_field` - A mutable reference to the magnetic field array.
/// * `dx` - The spatial grid spacing.
/// * `dt` - The time step for the simulation.
fn update_fields(e_field: &mut Array1<f64>, h_field: &mut Array1<f64>, dx: f64, dt: f64) {
    let nx = e_field.len();

    // Update the electric field using a finite-difference approximation.
    for i in 1..nx - 1 {
        // Compute the difference in magnetic field values and update the electric field.
        e_field[i] += dt / dx * (h_field[i] - h_field[i - 1]);
    }

    // Update the magnetic field using a finite-difference approximation.
    for i in 0..nx - 1 {
        // Compute the difference in electric field values and update the magnetic field.
        h_field[i] -= dt / dx * (e_field[i + 1] - e_field[i]);
    }
}

fn main() {
    // Define simulation parameters.
    let nx = 200;           // Number of spatial grid points.
    let dx = 0.01;          // Spatial grid spacing.
    let dt = 0.005;         // Time step for the simulation.

    // Initialize the fields.
    let (mut e_field, mut h_field) = initialize_fields(nx);

    // Run the FDTD simulation for 1000 time steps.
    for _ in 0..1000 {
        update_fields(&mut e_field, &mut h_field, dx, dt);
    }

    // Indicate that the time-domain simulation is complete.
    println!("Time-domain simulation complete!");
}

use rustfft::{FftPlanner, FftDirection};
use num_complex::Complex;

/// Demonstrates frequency-domain analysis using FFT in Rust.
/// 
/// This example generates a time-domain sine wave signal and converts it into its frequency-domain
/// representation using the FFT. The resulting frequency components are then printed to the console.
fn main() {
    // Define the length of the signal.
    let signal_len = 1024;
    
    // Generate a time-domain signal: a sine wave represented as a vector of complex numbers.
    // The imaginary parts are set to zero.
    let mut time_domain_signal: Vec<Complex<f64>> = (0..signal_len)
        .map(|i| {
            // Calculate the phase angle for the current sample.
            let theta = 2.0 * std::f64::consts::PI * i as f64 / signal_len as f64;
            Complex::new(theta.sin(), 0.0)
        })
        .collect();

    // Prepare a vector to store the FFT output.
    let mut fft_output = vec![Complex::new(0.0, 0.0); signal_len];

    // Create an FFT planner instance and plan the FFT for the given signal length.
    let mut planner = FftPlanner::new();
    let fft = planner.plan_fft(signal_len, FftDirection::Forward);
    
    // Execute the FFT to convert the time-domain signal into the frequency domain.
    fft.process(&mut time_domain_signal);
    
    // Copy the FFT result into fft_output.
    fft_output.copy_from_slice(&time_domain_signal);

    // Output the frequency-domain data.
    println!("Frequency-domain output: {:?}", fft_output);
}

use ndarray::Array2;

/// Initializes an adaptive grid with varying spacing near a material interface.
/// 
/// This function creates a 2D grid of dimensions (nx, ny) in which the grid spacing is refined
/// near a specified material boundary. For indices less than `boundary` a finer spacing is used,
/// while indices greater than or equal to `boundary` are assigned a coarser spacing. This strategy
/// enables high resolution where the electromagnetic fields are expected to vary rapidly without
/// incurring excessive computational cost across the entire domain.
/// 
/// # Arguments
/// 
/// * `nx` - Number of grid points in the x-direction.
/// * `ny` - Number of grid points in the y-direction.
/// * `boundary` - The index along the y-axis representing the material interface.
/// 
/// # Returns
/// 
/// A 2D array where each element represents the local grid spacing.
fn initialize_adaptive_grid(nx: usize, ny: usize, boundary: usize) -> Array2<f64> {
    let mut grid = Array2::<f64>::zeros((nx, ny));

    // Assign finer grid spacing near the material interface and coarser spacing elsewhere.
    for i in 0..nx {
        for j in 0..ny {
            if j < boundary {
                grid[[i, j]] = 0.1;  // Fine grid spacing near the interface.
            } else {
                grid[[i, j]] = 0.5;  // Coarser grid spacing away from the interface.
            }
        }
    }
    grid
}

fn main() {
    let nx = 100;
    let ny = 100;
    let boundary = 50;

    // Create an adaptive grid with refined spacing near the material boundary.
    let adaptive_grid = initialize_adaptive_grid(nx, ny, boundary);

    println!("Adaptive grid initialized with finer spacing near the boundary.");
}

use ndarray::Array2;

/// Updates the electric field across a material interface using interface tracking.
/// 
/// This function applies distinct finite-difference update rules for the electric field depending on
/// whether a point lies in material 1 (above the boundary) or material 2 (below the boundary). The method
/// adjusts the update based on the local material properties by using different coefficients in the update
/// formula. To avoid index underflow the update is applied starting from the second grid point along the y-axis.
/// 
/// # Arguments
/// 
/// * `e_field` - Mutable reference to the 2D electric field array.
/// * `h_field` - Reference to the 2D magnetic field array.
/// * `boundary` - The index representing the material interface along the y-axis.
/// * `dx` - The grid spacing.
/// * `dt` - The time step.
fn update_fields_interface(e_field: &mut Array2<f64>, h_field: &Array2<f64>, boundary: usize, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();

    // Update the electric field in material 1 (above the boundary).
    for i in 0..nx {
        // Start from j = 1 to ensure valid indexing for j - 1.
        for j in 1..boundary.min(ny) {
            e_field[[i, j]] += dt / dx * (h_field[[i, j]] - h_field[[i, j - 1]]);
        }
    }

    // Update the electric field in material 2 (below the boundary) using a modified update rule.
    for i in 0..nx {
        for j in boundary..ny {
            e_field[[i, j]] += dt / (2.0 * dx) * (h_field[[i, j]] - h_field[[i, j - 1]]);
        }
    }
}

fn main() {
    let nx = 100;
    let ny = 100;
    let dx = 0.01;   // Spatial grid spacing.
    let dt = 0.005;  // Time step.
    let boundary = 50; // Index of the material interface.

    // Initialize the electric and magnetic fields as 2D arrays filled with zeros.
    let mut e_field = Array2::<f64>::zeros((nx, ny));
    let h_field = Array2::<f64>::zeros((nx, ny));

    // Run the interface tracking update for 1000 time steps.
    for _ in 0..1000 {
        update_fields_interface(&mut e_field, &h_field, boundary, dx, dt);
    }

    println!("Simulation complete with interface tracking.");
}

use ndarray::Array2;
use rayon::prelude::*;

/// Performs a parallel update of the electric and magnetic fields using a finite-difference time-domain scheme.
///
/// The function parallelizes the update process by iterating over each row of the field arrays concurrently.
/// - For the electric field the update is based on the difference between adjacent values in the corresponding row,
///   and for the magnetic field the update uses the difference between successive values from the electric field.
/// Interior grid points are updated while boundary points are left unchanged to avoid indexing errors.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the 2D electric field array.
/// * `h_field` - Mutable reference to the 2D magnetic field array.
/// * `dx` - The spatial grid spacing.
/// * `dt` - The time step.
fn parallel_update_fields(e_field: &mut Array2<f64>, h_field: &mut Array2<f64>, dx: f64, dt: f64) {
    // Get the number of columns (each row length). Array2 is stored in row-major order.
    let ny = e_field.shape()[1];

    // Update the electric field in parallel by processing each row as a mutable chunk.
    e_field
        .as_slice_mut()
        .expect("e_field is not contiguous in memory")
        .par_chunks_mut(ny)
        .zip(
            h_field
                .as_slice()
                .expect("h_field is not contiguous in memory")
                .par_chunks(ny)
        )
        .for_each(|(e_row, h_row)| {
            let len = e_row.len();
            // Update interior points to avoid boundary issues.
            for j in 1..len - 1 {
                e_row[j] += dt / dx * (h_row[j] - h_row[j - 1]);
            }
        });

    // Update the magnetic field in parallel by processing each row as a mutable chunk.
    h_field
        .as_slice_mut()
        .expect("h_field is not contiguous in memory")
        .par_chunks_mut(ny)
        .zip(
            e_field
                .as_slice()
                .expect("e_field is not contiguous in memory")
                .par_chunks(ny)
        )
        .for_each(|(h_row, e_row)| {
            let len = h_row.len();
            for j in 1..len - 1 {
                h_row[j] -= dt / dx * (e_row[j + 1] - e_row[j]);
            }
        });
}

fn main() {
    let nx = 100;
    let ny = 100;
    let dx = 0.01;  // Spatial grid spacing.
    let dt = 0.005; // Time step.

    // Initialize the electric and magnetic fields as 2D arrays of zeros.
    let mut e_field = Array2::<f64>::zeros((nx, ny));
    let mut h_field = Array2::<f64>::zeros((nx, ny));

    // Execute the parallel update for 1000 time steps.
    for _ in 0..1000 {
        parallel_update_fields(&mut e_field, &mut h_field, dx, dt);
    }

    println!("Parallel simulation complete!");
}

extern crate ndarray;
use ndarray::Array2;

/// Initializes the electric and magnetic fields over a 2D grid.
///
/// # Arguments
///
/// * `nx` - The number of grid points in the x-direction.
/// * `ny` - The number of grid points in the y-direction.
///
/// # Returns
///
/// A tuple containing two 2D arrays representing the electric field and the magnetic field,
/// both initialized to zero.
fn initialize_wave_fields(nx: usize, ny: usize) -> (Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny)); // Electric field grid initialized to zero.
    let h_field = Array2::<f64>::zeros((nx, ny)); // Magnetic field grid initialized to zero.
    (e_field, h_field)
}

/// Updates the wave fields using a finite-difference scheme based on Maxwell's equations.
///
/// The electric field is updated using the difference between adjacent magnetic field values
/// along the x-direction while the magnetic field is updated using the difference between adjacent
/// electric field values along the y-direction. The updates are performed for interior grid points,
/// leaving the boundaries unchanged.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the 2D electric field array.
/// * `h_field` - Mutable reference to the 2D magnetic field array.
/// * `dx` - The spatial grid spacing.
/// * `dt` - The time step for the simulation.
fn update_wave_fields(e_field: &mut Array2<f64>, h_field: &mut Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();

    // Update the electric field based on the spatial derivative of the magnetic field.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            e_field[[i, j]] += dt * (h_field[[i + 1, j]] - h_field[[i, j]]) / dx;
        }
    }

    // Update the magnetic field based on the spatial derivative of the electric field.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            h_field[[i, j]] -= dt * (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
        }
    }
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;  // Grid spacing.
    let dt = 0.005; // Time step.

    // Initialize the wave fields.
    let (mut e_field, mut h_field) = initialize_wave_fields(nx, ny);

    // Simulate wave propagation for 1000 time steps.
    for _ in 0..1000 {
        update_wave_fields(&mut e_field, &mut h_field, dx, dt);
    }

    println!("Wave propagation simulation complete!");
}

/// Applies a basic perfectly matched layer (PML) to attenuate fields near the boundaries.
///
/// The PML is applied over a region of thickness `pml_thickness` by multiplying the field values
/// near the edges by an attenuation factor that increases towards the boundary.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the 2D electric field array.
/// * `h_field` - Mutable reference to the 2D magnetic field array.
/// * `pml_thickness` - The thickness of the PML region (in grid points).
/// * `pml_factor` - The base attenuation factor used to compute the attenuation.
fn apply_pml(e_field: &mut Array2<f64>, h_field: &mut Array2<f64>, pml_thickness: usize, pml_factor: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();

    // Attenuate the fields along the boundaries.
    for i in 0..pml_thickness {
        let attenuation = pml_factor * (pml_thickness - i) as f64 / pml_thickness as f64;

        // Attenuate the top and bottom edges for the electric field.
        for j in 0..ny {
            e_field[[i, j]] *= attenuation;
            e_field[[nx - 1 - i, j]] *= attenuation;
        }

        // Attenuate the left and right edges for the magnetic field.
        for j in 0..nx {
            h_field[[j, i]] *= attenuation;
            h_field[[j, ny - 1 - i]] *= attenuation;
        }
    }
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;
    let dt = 0.005;
    let pml_thickness = 20; // Number of grid points used for the PML region.
    let pml_factor = 0.9;   // Base attenuation factor.

    // Initialize the wave fields.
    let (mut e_field, mut h_field) = initialize_wave_fields(nx, ny);

    // Run the simulation for 1000 time steps with the PML applied at each step.
    for _ in 0..1000 {
        update_wave_fields(&mut e_field, &mut h_field, dx, dt);
        apply_pml(&mut e_field, &mut h_field, pml_thickness, pml_factor);
    }

    println!("Wave propagation with PML complete!");
}

extern crate rayon;
extern crate ndarray;
use rayon::prelude::*;
use ndarray::{Array2, Axis};

/// Performs a parallel update of the wave fields using a finite-difference scheme.
/// 
/// The electric field is updated based on the spatial derivative of the magnetic field along the x-direction
/// and the magnetic field is updated based on the spatial derivative of the electric field along the y-direction.
/// This function parallelizes the update process by iterating over rows concurrently.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the 2D electric field array.
/// * `h_field` - Mutable reference to the 2D magnetic field array.
/// * `dx` - The spatial grid spacing.
/// * `dt` - The time step.
fn parallel_update_wave_fields(e_field: &mut Array2<f64>, h_field: &mut Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();

    // Parallel update for the electric field.
    e_field.axis_iter_mut(Axis(0))
        .into_par_iter()
        .enumerate()
        .for_each(|(i, mut row)| {
            if i >= 1 && i < nx - 1 {
                for j in 1..ny - 1 {
                    row[j] += dt * (h_field[[i + 1, j]] - h_field[[i, j]]) / dx;
                }
            }
        });

    // Parallel update for the magnetic field.
    h_field.axis_iter_mut(Axis(0))
        .into_par_iter()
        .enumerate()
        .for_each(|(i, mut row)| {
            if i >= 1 && i < nx - 1 {
                for j in 1..ny - 1 {
                    row[j] -= dt * (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
                }
            }
        });
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;
    let dt = 0.005;

    // Initialize the wave fields.
    let (mut e_field, mut h_field) = initialize_wave_fields(nx, ny);

    // Execute the parallel wave propagation simulation for 1000 time steps.
    for _ in 0..1000 {
        parallel_update_wave_fields(&mut e_field, &mut h_field, dx, dt);
    }

    println!("Parallel wave propagation simulation complete!");
}

extern crate nalgebra as na;
use na::{DMatrix, DVector};

/// Computes a simple Green's function for the interaction between two points on a circular surface.
///
/// This function calculates the interaction based on the absolute difference between two radial positions.
/// A small epsilon is added to the denominator to avoid division by zero in the case of coincident points.
/// 
/// # Arguments
///
/// * `r1` - The radial coordinate of the first point.
/// * `r2` - The radial coordinate of the second point.
/// 
/// # Returns
///
/// A scalar value representing the Green's function evaluation.
fn greens_function(r1: f64, r2: f64) -> f64 {
    let epsilon = 1e-8;
    let distance = (r1 - r2).abs().max(epsilon);
    0.25 * (-distance).exp() / distance
}

/// Sets up the Method of Moments system for a 2D scattering problem on a circular object.
///
/// The object's surface is discretized into `n` points. The matrix is constructed such that each entry
/// represents the interaction between points on the surface as computed by the Green's function, and the
/// right-hand side vector represents a simplified incident field.
/// 
/// # Arguments
///
/// * `n` - The number of discrete points along the object's circumference.
/// * `radius` - The radius of the circular object.
/// 
/// # Returns
///
/// A tuple containing the interaction matrix and the incident field vector.
fn setup_mom_system(n: usize, radius: f64) -> (DMatrix<f64>, DVector<f64>) {
    let mut matrix = DMatrix::<f64>::zeros(n, n);
    let mut rhs = DVector::<f64>::zeros(n);

    // The angular separation between discrete points.
    let delta_theta = 2.0 * std::f64::consts::PI / n as f64;
    for i in 0..n {
        let theta_i = i as f64 * delta_theta;
        // Using cosine of the angle multiplied by the radius as a simple representation of the radial coordinate.
        let r1 = radius * theta_i.cos();
        for j in 0..n {
            let theta_j = j as f64 * delta_theta;
            let r2 = radius * theta_j.cos();
            // If the points coincide, use a self-term that avoids the singularity of the Green's function.
            if i == j {
                // Approximate the self-term using the average separation between points.
                matrix[(i, j)] = 0.25 * (-radius * delta_theta).exp() / (radius * delta_theta);
            } else {
                matrix[(i, j)] = greens_function(r1, r2);
            }
        }
        // A simplified representation of an incident wave; here the cosine function is used.
        rhs[i] = theta_i.cos();
    }

    (matrix, rhs)
}

fn main() {
    let n = 100;       // Number of discretization points on the object's surface.
    let radius = 1.0;  // Radius of the circular object.

    // Set up the MoM system.
    let (matrix, rhs) = setup_mom_system(n, radius);

    // Solve for the surface currents using LU decomposition.
    let currents = matrix.lu().solve(&rhs).expect("Failed to solve the MoM system");

    println!("Surface currents: {:?}", currents);
}

extern crate nalgebra as na;
use na::{DMatrix, DVector};

/// Constructs a simple stiffness matrix for a one-dimensional FEM scattering problem.
///
/// The stiffness matrix is assembled using a finite element discretization where each element is assumed
/// to contribute a value of 2.0/dx on the diagonal and -1.0/dx on the off-diagonals. This simplified matrix
/// represents the discretization of the wave equation in one dimension.
/// 
/// # Arguments
///
/// * `n` - The number of finite elements.
/// * `dx` - The element size or grid spacing.
///
/// # Returns
///
/// A square matrix of size `n` representing the stiffness matrix.
fn setup_fem_stiffness_matrix(n: usize, dx: f64) -> DMatrix<f64> {
    let mut stiffness_matrix = DMatrix::<f64>::zeros(n, n);

    for i in 0..n {
        stiffness_matrix[(i, i)] = 2.0 / dx;
        if i > 0 {
            stiffness_matrix[(i, i - 1)] = -1.0 / dx;
            stiffness_matrix[(i - 1, i)] = -1.0 / dx;
        }
    }

    stiffness_matrix
}

fn main() {
    let n = 100;      // Number of finite elements.
    let dx = 0.01;    // Element size.

    // Set up the FEM stiffness matrix.
    let stiffness_matrix = setup_fem_stiffness_matrix(n, dx);

    // Define the right-hand side vector representing a simplified incident wave.
    let rhs = DVector::<f64>::from_element(n, 1.0);

    // Solve the FEM system using LU decomposition.
    let solution = stiffness_matrix.lu().solve(&rhs).expect("Failed to solve the FEM system");

    println!("FEM solution: {:?}", solution);
}

extern crate rayon;
extern crate nalgebra as na;
use rayon::prelude::*;
use na::{DMatrix, DVector};

/// Computes a simple Green's function for the interaction between two points on a circular surface.
///
/// This function calculates the interaction based on the absolute difference between two radial positions.
/// A small epsilon is added to the denominator to avoid division by zero in the case of coincident points.
/// 
/// # Arguments
///
/// * `r1` - The radial coordinate of the first point.
/// * `r2` - The radial coordinate of the second point.
/// 
/// # Returns
///
/// A scalar value representing the Green's function evaluation.
fn greens_function(r1: f64, r2: f64) -> f64 {
    let epsilon = 1e-8;
    let distance = (r1 - r2).abs().max(epsilon);
    0.25 * (-distance).exp() / distance
}

/// Sets up the Method of Moments system for a 2D scattering problem on a circular object.
///
/// The object's surface is discretized into `n` points. The matrix is constructed such that each entry
/// represents the interaction between points on the surface as computed by the Green's function, and the
/// right-hand side vector represents a simplified incident field.
/// 
/// # Arguments
///
/// * `n` - The number of discrete points along the object's circumference.
/// * `radius` - The radius of the circular object.
/// 
/// # Returns
///
/// A tuple containing the interaction matrix and the incident field vector.
fn setup_mom_system(n: usize, radius: f64) -> (DMatrix<f64>, DVector<f64>) {
    let mut matrix = DMatrix::<f64>::zeros(n, n);
    let mut rhs = DVector::<f64>::zeros(n);

    // The angular separation between discrete points.
    let delta_theta = 2.0 * std::f64::consts::PI / n as f64;
    for i in 0..n {
        let theta_i = i as f64 * delta_theta;
        // Using cosine of the angle multiplied by the radius as a simple representation of the radial coordinate.
        let r1 = radius * theta_i.cos();
        for j in 0..n {
            let theta_j = j as f64 * delta_theta;
            let r2 = radius * theta_j.cos();
            // If the points coincide, use a self-term that avoids the singularity of the Green's function.
            if i == j {
                // Approximate the self-term using the average separation between points.
                matrix[(i, j)] = 0.25 * (-radius * delta_theta).exp() / (radius * delta_theta);
            } else {
                matrix[(i, j)] = greens_function(r1, r2);
            }
        }
        // A simplified representation of an incident wave; here the cosine function is used.
        rhs[i] = theta_i.cos();
    }

    (matrix, rhs)
}

/// Sets up the MoM system in parallel for a 2D scattering problem on a circular object.
///
/// This function parallelizes the assembly of the interaction matrix using Rayon. The matrix is
/// constructed by iterating over each index in parallel, computing the interaction between surface points.
/// 
/// # Arguments
///
/// * `n` - The number of discretization points on the object's surface.
/// * `radius` - The radius of the circular object.
/// 
/// # Returns
///
/// A tuple containing the assembled interaction matrix and the right-hand side vector.
fn parallel_setup_mom_system(n: usize, radius: f64) -> (DMatrix<f64>, DVector<f64>) {
    let delta_theta = 2.0 * std::f64::consts::PI / n as f64;
    
    // Assemble the matrix in parallel by computing a vector of all matrix entries.
    let values: Vec<f64> = (0..(n * n))
        .into_par_iter()
        .map(|idx| {
            let i = idx / n;
            let j = idx % n;
            let theta_i = i as f64 * delta_theta;
            let theta_j = j as f64 * delta_theta;
            let r1 = radius * theta_i.cos();
            let r2 = radius * theta_j.cos();
            // If the points coincide, use a self-term that avoids the singularity of the Green's function.
            if i == j {
                0.25 * (-radius * delta_theta).exp() / (radius * delta_theta)
            } else {
                greens_function(r1, r2)
            }
        })
        .collect();
    let matrix = DMatrix::<f64>::from_vec(n, n, values);

    // Assemble the right-hand side vector in a sequential loop.
    let mut rhs = DVector::<f64>::zeros(n);
    for i in 0..n {
        let theta = i as f64 * delta_theta;
        rhs[i] = theta.cos();
    }

    (matrix, rhs)
}

fn main() {
    let n = 100;
    let radius = 1.0;

    // Set up the MoM system in parallel.
    let (matrix, rhs) = parallel_setup_mom_system(n, radius);

    // Solve for the surface currents using LU decomposition.
    let currents = matrix.lu().solve(&rhs).expect("Failed to solve the parallel MoM system");

    println!("Parallelized surface currents: {:?}", currents);
}

extern crate ndarray;
use ndarray::{Array2, s, Axis};
use std::thread;

/// Initializes the electric and magnetic fields as 2D arrays filled with zeros.
///
/// # Arguments
///
/// * `nx` - Number of grid points along the x-axis.
/// * `ny` - Number of grid points along the y-axis.
///
/// # Returns
///
/// A tuple containing the electric field and the magnetic field.
fn initialize_fields(nx: usize, ny: usize) -> (Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny));
    let h_field = Array2::<f64>::zeros((nx, ny));
    (e_field, h_field)
}

/// Updates the electric field using finite-difference approximations in parallel.
///
/// The update is applied to interior grid points only. The 2D domain is divided into chunks of rows,
/// and each chunk is processed concurrently using Rust’s scoped threads. A read-only reference to the
/// magnetic field is used to update the corresponding rows in the electric field.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the electric field array.
/// * `h_field` - Reference to the magnetic field array.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step.
fn update_fields_parallel(e_field: &mut Array2<f64>, h_field: &Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();
    // We update interior rows (rows 1 through nx-2) to avoid boundary issues.
    let interior_rows = nx - 2; // these are the rows to update (global rows 1..nx-1, where row nx-1 is not updated)
    let num_threads = 4;
    // Compute the chunk size using ceiling division.
    let rows_per_chunk = (interior_rows + num_threads - 1) / num_threads;
    
    // Obtain a mutable view of the interior of e_field (rows 1 through nx-1).
    // (We update rows 1..nx-1; note that the last row is a boundary and is not updated.)
    let mut interior = e_field.slice_mut(s![1..nx-1, ..]); // shape: (nx-2, ny)
    
    // Use scoped threads and split the interior view into disjoint chunks along the row axis.
    thread::scope(|s| {
        for (chunk_idx, mut chunk) in interior.axis_chunks_iter_mut(Axis(0), rows_per_chunk).enumerate() {
            // The global starting row index for this chunk is 1 + (chunk_idx * rows_per_chunk)
            let global_offset = 1 + chunk_idx * rows_per_chunk;
            s.spawn(move || {
                // Process each row in the current chunk.
                for (i, mut row) in chunk.outer_iter_mut().enumerate() {
                    let global_row = global_offset + i;
                    // Update interior columns only.
                    for j in 1..ny-1 {
                        // Update the electric field using finite differences along the x-direction.
                        // (global_row+1 is valid because global_row ranges from 1 to nx-2.)
                        row[j] += dt * (h_field[[global_row+1, j]] - h_field[[global_row, j]]) / dx;
                    }
                }
            });
        }
    });
}

/// A simple update for the magnetic field using finite-difference approximations.
///
/// This function processes the interior grid points sequentially for illustration.
/// In a full HPC implementation a similar parallel strategy can be applied.
///
/// # Arguments
///
/// * `h_field` - Mutable reference to the magnetic field array.
/// * `e_field` - Reference to the electric field array.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step.
fn update_magnetic_field(h_field: &mut Array2<f64>, e_field: &Array2<f64>, dx: f64, dt: f64) {
    let nx = h_field.nrows();
    let ny = h_field.ncols();
    for i in 1..nx-1 {
        for j in 1..ny-1 {
            h_field[[i, j]] -= dt * (e_field[[i, j+1]] - e_field[[i, j]]) / dx;
        }
    }
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;
    let dt = 0.005;
    
    // Initialize the electromagnetic fields.
    let (mut e_field, mut h_field) = initialize_fields(nx, ny);
    
    // Run the simulation for 1000 time steps with parallel updates.
    for _ in 0..1000 {
        update_fields_parallel(&mut e_field, &h_field, dx, dt);
        update_magnetic_field(&mut h_field, &e_field, dx, dt);
    }
    
    println!("Parallel simulation complete!");
}

extern crate cuda_sys as cuda;
use std::ptr;

fn main() {
    unsafe {
        // Set the CUDA device to device 0.
        let device_status = cuda::cudaSetDevice(0);
        assert!(device_status == cuda::cudaSuccess, "Failed to set CUDA device");

        // Define the size for a 100x100 matrix.
        let n = 100;
        let matrix_size = (n * n * std::mem::size_of::<f64>()) as usize;

        // Allocate memory on the GPU for the matrix.
        let mut d_a: *mut f64 = ptr::null_mut();
        let malloc_status = cuda::cudaMalloc(&mut d_a as *mut *mut f64, matrix_size);
        assert!(malloc_status == cuda::cudaSuccess, "Failed to allocate GPU memory");

        // Placeholder for GPU operations:
        // In a complete implementation, data would be copied to d_a, CUDA kernels launched to perform
        // field updates or matrix operations, and results copied back to the host.

        // Free the allocated GPU memory.
        let free_status = cuda::cudaFree(d_a as *mut std::ffi::c_void);
        assert!(free_status == cuda::cudaSuccess, "Failed to free GPU memory");
    }

    println!("GPU matrix operation complete!");
}

extern crate ndarray;
use ndarray::Array2;

/// Adjusts the time step based on the maximum values of the electric and magnetic fields.
///
/// If the maximum field value exceeds a threshold the time step is reduced to maintain stability.
/// Conversely if the fields are small the time step is increased up to a specified maximum.
///
/// # Arguments
///
/// * `e_field` - Reference to the electric field array.
/// * `h_field` - Reference to the magnetic field array.
/// * `current_dt` - The current time step.
/// * `max_dt` - The maximum allowable time step.
///
/// # Returns
///
/// The adjusted time step.
fn adaptive_time_step(e_field: &Array2<f64>, h_field: &Array2<f64>, current_dt: f64, max_dt: f64) -> f64 {
    // Compute the maximum absolute values of the fields.
    let max_e = e_field.iter().map(|v| v.abs()).fold(0.0, f64::max);
    let max_h = h_field.iter().map(|v| v.abs()).fold(0.0, f64::max);

    // Reduce time step if either field exceeds a threshold.
    if max_e > 1.0 || max_h > 1.0 {
        return current_dt / 2.0;
    }
    // Increase the time step gradually if it is safe to do so.
    if current_dt < max_dt {
        return current_dt * 1.1;
    }
    current_dt
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;
    let mut dt = 0.005;
    let max_dt = 0.01;

    let (mut e_field, mut h_field) = {
        // Reuse the field initialization function.
        let e = Array2::<f64>::zeros((nx, ny));
        let h = Array2::<f64>::zeros((nx, ny));
        (e, h)
    };

    // Run the simulation for 1000 time steps with adaptive time stepping.
    for _ in 0..1000 {
        dt = adaptive_time_step(&e_field, &h_field, dt, max_dt);
        // Update fields using a chosen update method; here we call a placeholder sequential update.
        // In a full simulation, you would combine this with parallel updates as shown earlier.
        // For demonstration, we perform simple sequential updates.
        for i in 1..nx - 1 {
            for j in 1..ny - 1 {
                e_field[[i, j]] += dt * (h_field[[i + 1, j]] - h_field[[i, j]]) / dx;
                h_field[[i, j]] -= dt * (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
            }
        }
    }

    println!("Simulation with adaptive time stepping complete!");
}

extern crate ndarray;
use ndarray::Array2;

/// Initializes the electric and magnetic field arrays for the antenna simulation.
///
/// # Arguments
///
/// * `nx` - Number of grid points in the x-direction.
/// * `ny` - Number of grid points in the y-direction.
///
/// # Returns
///
/// A tuple containing two 2D arrays: the electric field and the magnetic field, both initialized to zero.
fn initialize_antenna_fields(nx: usize, ny: usize) -> (Array2<f64>, Array2<f64>) {
    let e_field = Array2::<f64>::zeros((nx, ny)); // Initialize electric field grid.
    let h_field = Array2::<f64>::zeros((nx, ny)); // Initialize magnetic field grid.
    (e_field, h_field)
}

/// Updates the antenna fields using a simple FDTD scheme.
///
/// The function updates the electric field based on the difference between adjacent magnetic field values
/// along the x-direction and updates the magnetic field using differences in the electric field along the y-direction.
/// The updates are applied to the interior of the grid to avoid boundary issues.
///
/// # Arguments
///
/// * `e_field` - Mutable reference to the electric field array.
/// * `h_field` - Mutable reference to the magnetic field array.
/// * `dx` - Spatial grid spacing.
/// * `dt` - Time step for the simulation.
fn update_antenna_fields(e_field: &mut Array2<f64>, h_field: &mut Array2<f64>, dx: f64, dt: f64) {
    let nx = e_field.nrows();
    let ny = e_field.ncols();

    // Update electric field using finite differences.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            e_field[[i, j]] += dt * (h_field[[i + 1, j]] - h_field[[i, j]]) / dx;
        }
    }

    // Update magnetic field using finite differences.
    for i in 1..nx - 1 {
        for j in 1..ny - 1 {
            h_field[[i, j]] -= dt * (e_field[[i, j + 1]] - e_field[[i, j]]) / dx;
        }
    }
}

fn main() {
    let nx = 200;
    let ny = 200;
    let dx = 0.01;
    let dt = 0.005;

    // Initialize fields for the antenna simulation.
    let (mut e_field, mut h_field) = initialize_antenna_fields(nx, ny);

    // Run the simulation for 1000 time steps.
    for _ in 0..1000 {
        update_antenna_fields(&mut e_field, &mut h_field, dx, dt);
    }

    println!("Antenna radiation simulation complete!");
}

extern crate nalgebra as na;
use na::{DMatrix, DVector};

/// Defines a simple Green's function for RCS analysis.
///
/// This function calculates the interaction between two points based on their radial positions.
/// A simplified exponential decay model is used.
///
/// # Arguments
///
/// * `r1` - Radial coordinate of the first point.
/// * `r2` - Radial coordinate of the second point.
///
/// # Returns
///
/// A scalar representing the Green's function value.
fn greens_function(r1: f64, r2: f64) -> f64 {
    let distance = (r1 - r2).abs().max(1e-8); // Prevent division by zero.
    0.25 * (-distance).exp() / distance
}

/// Sets up the MoM system for RCS analysis on a circular object.
///
/// The object’s surface is discretized into `n` points. Each matrix element represents the interaction
/// between two points on the surface, and the right-hand side vector models a simplified incident radar signal.
///
/// # Arguments
///
/// * `n` - Number of discretization points along the object's circumference.
/// * `radius` - Radius of the circular object.
///
/// # Returns
///
/// A tuple containing the interaction matrix and the incident field vector.
fn setup_rcs_system(n: usize, radius: f64) -> (DMatrix<f64>, DVector<f64>) {
    let mut matrix = DMatrix::<f64>::zeros(n, n);
    let mut rhs = DVector::<f64>::zeros(n);
    let delta_theta = 2.0 * std::f64::consts::PI / n as f64;

    for i in 0..n {
        let theta_i = i as f64 * delta_theta;
        let r1 = radius * theta_i.cos();
        for j in 0..n {
            let theta_j = j as f64 * delta_theta;
            let r2 = radius * theta_j.cos();
            matrix[(i, j)] = greens_function(r1, r2);
        }
        // Define the incident radar signal (simplified as a cosine function).
        rhs[i] = theta_i.cos();
    }

    (matrix, rhs)
}

fn main() {
    let n = 100;      // Number of surface discretization points.
    let radius = 1.0; // Radius of the scatterer.

    // Set up the MoM system for RCS analysis.
    let (matrix, rhs) = setup_rcs_system(n, radius);

    // Solve for the surface currents using LU decomposition.
    let currents = matrix.lu().solve(&rhs).expect("Failed to solve MoM system");

    println!("Radar cross-section simulation complete. Surface currents: {:?}", currents);
}

extern crate nalgebra as na;
use na::{DMatrix, DVector};

/// Constructs a stiffness matrix for a one-dimensional FEM simulation of an optical waveguide.
///
/// The matrix is assembled by assigning 2.0/dx to diagonal entries and -1.0/dx to off-diagonals,
/// representing a simple discretization of the wave equation.
///
/// # Arguments
///
/// * `n` - Number of finite elements.
/// * `dx` - Element size or grid spacing.
///
/// # Returns
///
/// A stiffness matrix of size `n x n`.
fn setup_waveguide_stiffness_matrix(n: usize, dx: f64) -> DMatrix<f64> {
    let mut stiffness_matrix = DMatrix::<f64>::zeros(n, n);

    for i in 0..n {
        stiffness_matrix[(i, i)] = 2.0 / dx;
        if i > 0 {
            stiffness_matrix[(i, i - 1)] = -1.0 / dx;
            stiffness_matrix[(i - 1, i)] = -1.0 / dx;
        }
    }

    stiffness_matrix
}

fn main() {
    let n = 100;      // Number of finite elements.
    let dx = 0.01;    // Element size.

    // Assemble the stiffness matrix for the optical waveguide.
    let stiffness_matrix = setup_waveguide_stiffness_matrix(n, dx);

    // Define the right-hand side vector representing the wave source.
    let rhs = DVector::<f64>::from_element(n, 1.0);

    // Solve the FEM system using LU decomposition.
    let solution = stiffness_matrix.lu().solve(&rhs).expect("Failed to solve FEM system");

    println!("Optical waveguide simulation complete. Solution: {:?}", solution);
}

extern crate nalgebra as na;
use na::{DVector, DMatrix};

/// Nonlinear function f(x) = x^3 - x - 1.
/// 
/// # Arguments
/// 
/// * `x` - Input variable.
/// 
/// # Returns
/// 
/// The computed function value.
fn nonlinear_function(x: f64) -> f64 {
    x.powi(3) - x - 1.0
}

/// Derivative of the nonlinear function f'(x) = 3x^2 - 1.
/// 
/// # Arguments
/// 
/// * `x` - Input variable.
/// 
/// # Returns
/// 
/// The computed derivative value.
fn derivative_nonlinear_function(x: f64) -> f64 {
    3.0 * x.powi(2) - 1.0
}

/// Newton's method for solving a nonlinear equation.
/// 
/// This function iteratively refines the initial guess until the update is smaller than the specified tolerance
/// or the maximum number of iterations is reached.
/// 
/// # Arguments
/// 
/// * `initial_guess` - The starting value for the iteration.
/// * `tol` - The tolerance for convergence.
/// * `max_iter` - The maximum number of iterations.
/// 
/// # Returns
/// 
/// The approximated solution.
fn newtons_method(initial_guess: f64, tol: f64, max_iter: usize) -> f64 {
    let mut x = initial_guess;
    for _ in 0..max_iter {
        let f_x = nonlinear_function(x);
        let f_prime_x = derivative_nonlinear_function(x);
        let delta_x = f_x / f_prime_x;  // Compute the update.
        x -= delta_x;  // Update the guess.
        if delta_x.abs() < tol {  // Check for convergence.
            break;
        }
    }
    x
}

fn main() {
    let initial_guess = 1.0;
    let tol = 1e-6;
    let max_iter = 1000;
    let solution = newtons_method(initial_guess, tol, max_iter);
    println!("Nonlinear solution using Newton's method: {}", solution);
}

extern crate nalgebra as na;
use na::{DMatrix, DVector};

/// Constructs a FEM matrix for the near-field region using a simple finite element discretization.
/// 
/// The matrix is assembled with diagonal entries set to 2.0/dx and off-diagonals set to -1.0/dx.
/// 
/// # Arguments
/// 
/// * `n` - Number of finite elements.
/// * `dx` - Element size or grid spacing.
/// 
/// # Returns
/// 
/// A stiffness matrix for the near-field region.
fn setup_fem_matrix(n: usize, dx: f64) -> DMatrix<f64> {
    let mut fem_matrix = DMatrix::<f64>::zeros(n, n);
    for i in 0..n {
        fem_matrix[(i, i)] = 2.0 / dx;
        if i > 0 {
            fem_matrix[(i, i - 1)] = -1.0 / dx;
            fem_matrix[(i - 1, i)] = -1.0 / dx;
        }
    }
    fem_matrix
}

/// Constructs a BEM matrix for the far-field region.
/// 
/// Here the diagonal entries are set to 1.0/dx while the off-diagonals are set to 0.5/dx to represent
/// the boundary interactions in the far-field.
/// 
/// # Arguments
/// 
/// * `n` - Number of discretization points.
/// * `dx` - Spatial discretization size.
/// 
/// # Returns
/// 
/// A matrix representing the far-field interactions.
fn setup_bem_matrix(n: usize, dx: f64) -> DMatrix<f64> {
    let mut bem_matrix = DMatrix::<f64>::zeros(n, n);
    for i in 0..n {
        bem_matrix[(i, i)] = 1.0 / dx;
        if i > 0 {
            bem_matrix[(i, i - 1)] = 0.5 / dx;
            bem_matrix[(i - 1, i)] = 0.5 / dx;
        }
    }
    bem_matrix
}

fn main() {
    let n = 100;  // Number of elements or discretization points.
    let dx = 0.01;

    // Set up FEM and BEM matrices.
    let fem_matrix = setup_fem_matrix(n, dx);
    let bem_matrix = setup_bem_matrix(n, dx);

    // Combine the FEM and BEM matrices to form a hybrid system.
    let combined_matrix = &fem_matrix + &bem_matrix;

    // Define a right-hand side vector representing the source term.
    let rhs = DVector::<f64>::from_element(n, 1.0);

    // Solve the hybrid system using LU decomposition.
    let solution = combined_matrix.lu().solve(&rhs).expect("Failed to solve hybrid FEM-BEM system");

    println!("Hybrid FEM-BEM solution: {:?}", solution);
}

extern crate tch;
use tch::{nn, nn::Module, Tensor};

/// Builds a simple neural network model for predicting electromagnetic fields.
/// 
/// The network consists of three linear layers with ReLU activations between them. It takes an input of
/// dimension 2 and outputs a single scalar value.
/// 
/// # Arguments
/// 
/// * `vs` - A reference to the variable store path for model parameters.
/// 
/// # Returns
/// 
/// An instance of a sequential neural network model.
fn build_model(vs: &nn::Path) -> impl Module {
    nn::seq()
        .add(nn::linear(vs, 2, 64, Default::default()))
        .add_fn(|xs| xs.relu())
        .add(nn::linear(vs, 64, 64, Default::default()))
        .add_fn(|xs| xs.relu())
        .add(nn::linear(vs, 64, 1, Default::default()))
}

fn main() {
    // Generate synthetic input and target data for demonstration purposes.
    let xs = Tensor::randn(&[100, 2], tch::kind::FLOAT_CPU);
    let ys = Tensor::randn(&[100, 1], tch::kind::FLOAT_CPU);

    // Initialize the variable store and build the model.
    let vs = nn::VarStore::new(tch::Device::Cpu);
    let model = build_model(&vs.root());

    // Set up the Adam optimizer.
    let mut opt = nn::Adam::default().build(&vs, 1e-3).unwrap();

    // Train the model over 200 epochs.
    for epoch in 1..200 {
        let loss = model.forward(&xs).mse_loss(&ys);
        opt.backward_step(&loss);
        if epoch % 10 == 0 {
            println!("Epoch: {}, Loss: {}", epoch, f64::from(loss));
        }
    }
}