linreg_core/

linalg.rs

//! Minimal Linear Algebra module to replace nalgebra dependency.
//!
//! Implements matrix operations, QR decomposition, and solvers needed for OLS.
//! Uses row-major storage for compatibility with statistical computing conventions.

#![allow(clippy::needless_range_loop)]
//!
//! # Numerical Stability Considerations
//!
//! This implementation uses Householder QR decomposition with careful attention to
//! numerical stability:
//!
//! - **Sign convention**: Uses the numerically stable Householder sign choice
//!   (v = x + sgn(x₀)||x||e₁) to avoid cancellation
//! - **Tolerance checking**: Uses predefined tolerances to detect near-singular matrices
//! - **Zero-skipping**: Skips transformations when columns are already zero-aligned
//!
//! # Scaling Recommendations
//!
//! For optimal numerical stability when predictor variables have vastly different
//! scales (e.g., one variable in millions, another in thousandths), consider
//! standardizing predictors before regression. Z-score standardization
//! (`x_scaled = (x - mean) / std`) is already done in VIF calculation.
//!
//! However, the current implementation handles typical OLS cases without explicit
//! scaling, as QR decomposition is generally stable for well-conditioned matrices.

// ============================================================================
// Numerical Constants
// ============================================================================

/// Machine epsilon threshold for detecting zero values in QR decomposition.
/// Values below this are treated as zero to avoid numerical instability.
const QR_ZERO_TOLERANCE: f64 = 1e-12;

/// Threshold for detecting singular matrices during inversion.
/// Diagonal elements below this value indicate a near-singular matrix.
const SINGULAR_TOLERANCE: f64 = 1e-10;

/// A dense matrix stored in row-major order.
///
/// # Storage
///
/// Elements are stored in a single flat vector in row-major order:
/// `data[row * cols + col]`
///
/// # Example
///
/// ```
/// # use linreg_core::linalg::Matrix;
/// let m = Matrix::new(2, 3, vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
/// assert_eq!(m.rows, 2);
/// assert_eq!(m.cols, 3);
/// assert_eq!(m.get(0, 0), 1.0);
/// assert_eq!(m.get(1, 2), 6.0);
/// ```
#[derive(Clone, Debug)]
pub struct Matrix {
    /// Number of rows in the matrix
    pub rows: usize,
    /// Number of columns in the matrix
    pub cols: usize,
    /// Flat vector storing matrix elements in row-major order
    pub data: Vec<f64>,
}

impl Matrix {
    /// Creates a new matrix from the given dimensions and data.
    ///
    /// # Panics
    ///
    /// Panics if `data.len() != rows * cols`.
    ///
    /// # Arguments
    ///
    /// * `rows` - Number of rows
    /// * `cols` - Number of columns
    /// * `data` - Flat vector of elements in row-major order
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let m = Matrix::new(2, 2, vec![1.0, 2.0, 3.0, 4.0]);
    /// assert_eq!(m.get(0, 0), 1.0);
    /// assert_eq!(m.get(0, 1), 2.0);
    /// assert_eq!(m.get(1, 0), 3.0);
    /// assert_eq!(m.get(1, 1), 4.0);
    /// ```
    pub fn new(rows: usize, cols: usize, data: Vec<f64>) -> Self {
        assert_eq!(data.len(), rows * cols, "Data length must match dimensions");
        Matrix { rows, cols, data }
    }

    /// Creates a matrix filled with zeros.
    ///
    /// # Arguments
    ///
    /// * `rows` - Number of rows
    /// * `cols` - Number of columns
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let m = Matrix::zeros(3, 2);
    /// assert_eq!(m.rows, 3);
    /// assert_eq!(m.cols, 2);
    /// assert_eq!(m.get(1, 1), 0.0);
    /// ```
    pub fn zeros(rows: usize, cols: usize) -> Self {
        Matrix {
            rows,
            cols,
            data: vec![0.0; rows * cols],
        }
    }

    // NOTE: Currently unused but kept as reference implementation.
    // Uncomment if needed for convenience constructor.
    /*
    /// Creates a matrix from a row-major slice.
    ///
    /// # Arguments
    ///
    /// * `rows` - Number of rows
    /// * `cols` - Number of columns
    /// * `slice` - Slice containing matrix elements in row-major order
    pub fn from_row_slice(rows: usize, cols: usize, slice: &[f64]) -> Self {
        Matrix::new(rows, cols, slice.to_vec())
    }
    */

    /// Gets the element at the specified row and column.
    ///
    /// # Arguments
    ///
    /// * `row` - Row index (0-based)
    /// * `col` - Column index (0-based)
    pub fn get(&self, row: usize, col: usize) -> f64 {
        self.data[row * self.cols + col]
    }

    /// Sets the element at the specified row and column.
    ///
    /// # Arguments
    ///
    /// * `row` - Row index (0-based)
    /// * `col` - Column index (0-based)
    /// * `val` - Value to set
    pub fn set(&mut self, row: usize, col: usize, val: f64) {
        self.data[row * self.cols + col] = val;
    }

    /// Returns the transpose of this matrix.
    ///
    /// Swaps rows with columns: `result[col][row] = self[row][col]`.
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let m = Matrix::new(2, 3, vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
    /// let t = m.transpose();
    /// assert_eq!(t.rows, 3);
    /// assert_eq!(t.cols, 2);
    /// assert_eq!(t.get(0, 1), 4.0);
    /// ```
    pub fn transpose(&self) -> Matrix {
        let mut t_data = vec![0.0; self.rows * self.cols];
        for r in 0..self.rows {
            for c in 0..self.cols {
                t_data[c * self.rows + r] = self.get(r, c);
            }
        }
        Matrix::new(self.cols, self.rows, t_data)
    }

    /// Performs matrix multiplication: `self * other`.
    ///
    /// # Panics
    ///
    /// Panics if `self.cols != other.rows`.
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let a = Matrix::new(2, 3, vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
    /// let b = Matrix::new(3, 2, vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
    /// let c = a.matmul(&b);
    /// assert_eq!(c.rows, 2);
    /// assert_eq!(c.cols, 2);
    /// assert_eq!(c.get(0, 0), 22.0); // 1*1 + 2*3 + 3*5
    /// ```
    pub fn matmul(&self, other: &Matrix) -> Matrix {
        assert_eq!(
            self.cols, other.rows,
            "Dimension mismatch for multiplication"
        );
        let mut result = Matrix::zeros(self.rows, other.cols);

        for r in 0..self.rows {
            for c in 0..other.cols {
                let mut sum = 0.0;
                for k in 0..self.cols {
                    sum += self.get(r, k) * other.get(k, c);
                }
                result.set(r, c, sum);
            }
        }
        result
    }

    /// Multiplies this matrix by a vector (treating vector as column matrix).
    ///
    /// Computes `self * vec` where vec is treated as an n×1 column matrix.
    ///
    /// # Panics
    ///
    /// Panics if `self.cols != vec.len()`.
    ///
    /// # Arguments
    ///
    /// * `vec` - Vector to multiply by
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let m = Matrix::new(2, 3, vec![1.0, 2.0, 3.0, 4.0, 5.0, 6.0]);
    /// let v = vec![1.0, 2.0, 3.0];
    /// let result = m.mul_vec(&v);
    /// assert_eq!(result.len(), 2);
    /// assert_eq!(result[0], 14.0); // 1*1 + 2*2 + 3*3
    /// ```
    #[allow(clippy::needless_range_loop)]
    pub fn mul_vec(&self, vec: &[f64]) -> Vec<f64> {
        assert_eq!(
            self.cols,
            vec.len(),
            "Dimension mismatch for matrix-vector multiplication"
        );
        let mut result = vec![0.0; self.rows];

        for r in 0..self.rows {
            let mut sum = 0.0;
            for c in 0..self.cols {
                sum += self.get(r, c) * vec[c];
            }
            result[r] = sum;
        }
        result
    }

    /// Computes the dot product of a column with a vector: `Σ(data[i * cols + col] * v[i])`.
    ///
    /// For a row-major matrix, this iterates through all rows at a fixed column.
    ///
    /// # Arguments
    ///
    /// * `col` - Column index
    /// * `v` - Vector to dot with (must have length equal to rows)
    ///
    /// # Panics
    ///
    /// Panics if `col >= cols` or `v.len() != rows`.
    #[allow(clippy::needless_range_loop)]
    pub fn col_dot(&self, col: usize, v: &[f64]) -> f64 {
        assert!(col < self.cols, "Column index out of bounds");
        assert_eq!(
            self.rows,
            v.len(),
            "Vector length must match number of rows"
        );

        let mut sum = 0.0;
        for row in 0..self.rows {
            sum += self.get(row, col) * v[row];
        }
        sum
    }

    /// Performs the column-vector operation in place: `v += alpha * column_col`.
    ///
    /// This is the AXPY operation where the column is treated as a vector.
    /// For row-major storage, we iterate through rows at a fixed column.
    ///
    /// # Arguments
    ///
    /// * `col` - Column index
    /// * `alpha` - Scaling factor for the column
    /// * `v` - Vector to modify in place (must have length equal to rows)
    ///
    /// # Panics
    ///
    /// Panics if `col >= cols` or `v.len() != rows`.
    #[allow(clippy::needless_range_loop)]
    pub fn col_axpy_inplace(&self, col: usize, alpha: f64, v: &mut [f64]) {
        assert!(col < self.cols, "Column index out of bounds");
        assert_eq!(
            self.rows,
            v.len(),
            "Vector length must match number of rows"
        );

        for row in 0..self.rows {
            v[row] += alpha * self.get(row, col);
        }
    }

    /// Computes the squared L2 norm of a column: `Σ(data[i * cols + col]²)`.
    ///
    /// # Arguments
    ///
    /// * `col` - Column index
    ///
    /// # Panics
    ///
    /// Panics if `col >= cols`.
    #[allow(clippy::needless_range_loop)]
    pub fn col_norm2(&self, col: usize) -> f64 {
        assert!(col < self.cols, "Column index out of bounds");

        let mut sum = 0.0;
        for row in 0..self.rows {
            let val = self.get(row, col);
            sum += val * val;
        }
        sum
    }

    /// Adds a value to diagonal elements starting from a given index.
    ///
    /// This is useful for ridge regression where we add `lambda * I` to `X^T X`,
    /// but the intercept column should not be penalized.
    ///
    /// # Arguments
    ///
    /// * `alpha` - Value to add to diagonal elements
    /// * `start_index` - Starting diagonal index (0 = first diagonal element)
    ///
    /// # Panics
    ///
    /// Panics if the matrix is not square.
    ///
    /// # Example
    ///
    /// For a 3×3 identity matrix with intercept in first column (unpenalized):
    /// ```text
    /// add_diagonal_in_place(lambda, 1) on:
    /// [1.0, 0.0, 0.0]       [1.0,   0.0,   0.0  ]
    /// [0.0, 1.0, 0.0]  ->   [0.0,  1.0+λ, 0.0  ]
    /// [0.0, 0.0, 1.0]       [0.0,   0.0,  1.0+λ]
    /// ```
    pub fn add_diagonal_in_place(&mut self, alpha: f64, start_index: usize) {
        assert_eq!(self.rows, self.cols, "Matrix must be square");
        let n = self.rows;
        for i in start_index..n {
            let current = self.get(i, i);
            self.set(i, i, current + alpha);
        }
    }
}

// ============================================================================
// QR Decomposition
// ============================================================================

impl Matrix {
    /// Computes the QR decomposition using Householder reflections.
    ///
    /// Factorizes the matrix as `A = QR` where Q is orthogonal and R is upper triangular.
    ///
    /// # Requirements
    ///
    /// This implementation requires `rows >= cols` (tall matrix). For OLS regression,
    /// we always have more observations than predictors, so this requirement is satisfied.
    ///
    /// # Returns
    ///
    /// A tuple `(Q, R)` where:
    /// - `Q` is an orthogonal matrix (QᵀQ = I) of size m×m
    /// - `R` is an upper triangular matrix of size m×n
    #[allow(clippy::needless_range_loop)]
    pub fn qr(&self) -> (Matrix, Matrix) {
        let m = self.rows;
        let n = self.cols;
        let mut q = Matrix::identity(m);
        let mut r = self.clone();

        for k in 0..n.min(m - 1) {
            // Create vector x = R[k:, k]
            let mut x = vec![0.0; m - k];
            for i in k..m {
                x[i - k] = r.get(i, k);
            }

            // Norm of x
            let norm_x: f64 = x.iter().map(|&v| v * v).sum::<f64>().sqrt();
            if norm_x < QR_ZERO_TOLERANCE {
                continue;
            } // Already zero

            // Create vector v = x + sign(x[0]) * ||x|| * e1
            //
            // NOTE: Numerical stability consideration (Householder sign choice)
            // According to Overton & Yu (2023), the numerically stable choice is
            // σ = -sgn(x₁) in the formula v = x - σ‖x‖e₁.
            //
            // This means: v = x - (-sgn(x₁))‖x‖e₁ = x + sgn(x₁)‖x‖e₁
            //
            // Equivalently: u₁ = x₁ + sgn(x₁)‖x‖
            //
            // Current implementation uses this formula (the "correct" choice for stability):
            let sign = if x[0] >= 0.0 { 1.0 } else { -1.0 }; // sgn(x₀) as defined (sgn(0) = +1)
            let u1 = x[0] + sign * norm_x;

            // Normalize v to get Householder vector
            let mut v = x; // Re-use storage
            v[0] = u1;

            let norm_v: f64 = v.iter().map(|&val| val * val).sum::<f64>().sqrt();
            for val in &mut v {
                *val /= norm_v;
            }

            // Apply Householder transformation to R: R = H * R = (I - 2vv^T)R = R - 2v(v^T R)
            // Focus on submatrix R[k:, k:]
            for j in k..n {
                let mut dot = 0.0;
                for i in 0..m - k {
                    dot += v[i] * r.get(k + i, j);
                }

                for i in 0..m - k {
                    let val = r.get(k + i, j) - 2.0 * v[i] * dot;
                    r.set(k + i, j, val);
                }
            }

            // Update Q: Q = Q * H = Q(I - 2vv^T) = Q - 2(Qv)v^T
            // Focus on Q[:, k:]
            for i in 0..m {
                let mut dot = 0.0;
                for l in 0..m - k {
                    dot += q.get(i, k + l) * v[l];
                }

                for l in 0..m - k {
                    let val = q.get(i, k + l) - 2.0 * dot * v[l];
                    q.set(i, k + l, val);
                }
            }
        }

        (q, r)
    }

    /// Creates an identity matrix of the given size.
    ///
    /// # Arguments
    ///
    /// * `size` - Number of rows and columns (square matrix)
    ///
    /// # Example
    ///
    /// ```
    /// # use linreg_core::linalg::Matrix;
    /// let i = Matrix::identity(3);
    /// assert_eq!(i.get(0, 0), 1.0);
    /// assert_eq!(i.get(1, 1), 1.0);
    /// assert_eq!(i.get(2, 2), 1.0);
    /// assert_eq!(i.get(0, 1), 0.0);
    /// ```
    pub fn identity(size: usize) -> Self {
        let mut data = vec![0.0; size * size];
        for i in 0..size {
            data[i * size + i] = 1.0;
        }
        Matrix::new(size, size, data)
    }

    /// Inverts an upper triangular matrix (such as R from QR decomposition).
    ///
    /// Uses back-substitution to compute the inverse. This is efficient for
    /// triangular matrices compared to general matrix inversion.
    ///
    /// # Panics
    ///
    /// Panics if the matrix is not square.
    ///
    /// # Returns
    ///
    /// `None` if the matrix is singular (has a zero or near-zero diagonal element).
    /// A matrix is considered singular if any diagonal element is below the
    /// internal tolerance (1e-10), which indicates the matrix does not have full rank.
    ///
    /// # Note
    ///
    /// For upper triangular matrices, singularity is equivalent to having a
    /// zero (or near-zero) diagonal element. This is much simpler to check than
    /// for general matrices, which would require computing the condition number.
    pub fn invert_upper_triangular(&self) -> Option<Matrix> {
        let n = self.rows;
        assert_eq!(n, self.cols, "Matrix must be square");

        // Check for singularity using relative tolerance
        // This scales with the magnitude of diagonal elements, handling matrices
        // of different scales better than a fixed absolute tolerance.
        //
        // Previous implementation used absolute tolerance:
        //   if self.get(i, i).abs() < SINGULAR_TOLERANCE { return None; }
        //
        // New implementation uses relative tolerance similar to LAPACK:
        //   tolerance = max_diag * epsilon * n
        // where epsilon is machine epsilon (~2.2e-16 for f64)
        let max_diag: f64 = (0..n)
            .map(|i| self.get(i, i).abs())
            .fold(0.0_f64, |acc, val| acc.max(val));

        // Use a relative tolerance based on the maximum diagonal element
        // This is similar to LAPACK's dlamch machine epsilon approach
        let epsilon = 2.0_f64 * f64::EPSILON; // ~4.4e-16 for f64
        let relative_tolerance = max_diag * epsilon * n as f64;
        let tolerance = SINGULAR_TOLERANCE.max(relative_tolerance);

        for i in 0..n {
            if self.get(i, i).abs() < tolerance {
                return None; // Singular matrix - cannot invert
            }
        }

        let mut inv = Matrix::zeros(n, n);

        for i in 0..n {
            inv.set(i, i, 1.0 / self.get(i, i));

            for j in (0..i).rev() {
                let mut sum = 0.0;
                for k in j + 1..=i {
                    sum += self.get(j, k) * inv.get(k, i);
                }
                inv.set(j, i, -sum / self.get(j, j));
            }
        }

        Some(inv)
    }

    /// Inverts an upper triangular matrix with a custom tolerance multiplier.
    ///
    /// The tolerance is computed as `max_diag * epsilon * n * tolerance_mult`.
    /// A higher tolerance_mult allows more tolerance for near-singular matrices.
    ///
    /// # Arguments
    ///
    /// * `tolerance_mult` - Multiplier for the tolerance (1.0 = standard, higher = more tolerant)
    pub fn invert_upper_triangular_with_tolerance(&self, tolerance_mult: f64) -> Option<Matrix> {
        let n = self.rows;
        assert_eq!(n, self.cols, "Matrix must be square");

        // Check for singularity using relative tolerance
        let max_diag: f64 = (0..n)
            .map(|i| self.get(i, i).abs())
            .fold(0.0_f64, |acc, val| acc.max(val));

        // Use a relative tolerance based on the maximum diagonal element
        let epsilon = 2.0_f64 * f64::EPSILON;
        let relative_tolerance = max_diag * epsilon * n as f64 * tolerance_mult;
        let tolerance = SINGULAR_TOLERANCE.max(relative_tolerance);

        for i in 0..n {
            if self.get(i, i).abs() < tolerance {
                return None;
            }
        }

        let mut inv = Matrix::zeros(n, n);

        for i in 0..n {
            inv.set(i, i, 1.0 / self.get(i, i));

            for j in (0..i).rev() {
                let mut sum = 0.0;
                for k in j + 1..=i {
                    sum += self.get(j, k) * inv.get(k, i);
                }
                inv.set(j, i, -sum / self.get(j, j));
            }
        }

        Some(inv)
    }

    /// Computes the inverse of a square matrix using QR decomposition.
    ///
    /// For an invertible matrix A, computes A⁻¹ such that A * A⁻¹ = I.
    /// Uses QR decomposition for numerical stability.
    ///
    /// # Panics
    ///
    /// Panics if the matrix is not square (i.e., `self.rows != self.cols`).
    /// Check dimensions before calling if the matrix shape is not guaranteed.
    ///
    /// # Returns
    ///
    /// Returns `Some(inverse)` if the matrix is invertible, or `None` if
    /// the matrix is singular (non-invertible).
    pub fn invert(&self) -> Option<Matrix> {
        let n = self.rows;
        if n != self.cols {
            panic!("Matrix must be square for inversion");
        }

        // Use QR decomposition: A = Q * R
        let (q, r) = self.qr();

        // Compute R⁻¹ (upper triangular inverse)
        let r_inv = r.invert_upper_triangular()?;

        // A⁻¹ = R⁻¹ * Q^T
        let q_transpose = q.transpose();
        let mut result = Matrix::zeros(n, n);

        for i in 0..n {
            for j in 0..n {
                let mut sum = 0.0;
                for k in 0..n {
                    sum += r_inv.get(i, k) * q_transpose.get(k, j);
                }
                result.set(i, j, sum);
            }
        }

        Some(result)
    }

    /// Computes the inverse of X'X given the QR decomposition of X (R's chol2inv).
    ///
    /// This is equivalent to computing `(X'X)^(-1)` using the QR decomposition of X.
    /// R's `chol2inv` function is used for numerical stability in recursive residuals.
    ///
    /// # Arguments
    ///
    /// * `x` - Input matrix (must have rows >= cols)
    ///
    /// # Returns
    ///
    /// `Some((X'X)^(-1))` if X has full rank, `None` otherwise.
    ///
    /// # Algorithm
    ///
    /// Given QR decomposition X = QR where R is upper triangular:
    /// 1. Extract the upper p×p portion of R (denoted R₁)
    /// 2. Invert R₁ (upper triangular inverse)
    /// 3. Compute (X'X)^(-1) = R₁^(-1) × R₁^(-T)
    ///
    /// This works because X'X = R'Q'QR = R'R, and R₁ contains the Cholesky factor.
    pub fn chol2inv_from_qr(&self) -> Option<Matrix> {
        self.chol2inv_from_qr_with_tolerance(1.0)
    }

    /// Computes the inverse of X'X given the QR decomposition with custom tolerance.
    ///
    /// Similar to `chol2inv_from_qr` but allows specifying a tolerance multiplier
    /// for handling near-singular matrices.
    ///
    /// # Arguments
    ///
    /// * `tolerance_mult` - Multiplier for the tolerance (higher = more tolerant)
    pub fn chol2inv_from_qr_with_tolerance(&self, tolerance_mult: f64) -> Option<Matrix> {
        let p = self.cols;

        // QR decomposition: X = QR
        // For X (m×n, m≥n), R is m×n upper triangular
        // The upper n×n block of R contains the meaningful values
        let (_, r_full) = self.qr();

        // Extract upper p×p portion from R
        // For tall matrices (m > p), R has zeros below diagonal in first p rows
        // For square matrices (m = p), R is p×p upper triangular
        let mut r1 = Matrix::zeros(p, p);
        for i in 0..p {
            // Row i of R1 is row i of R_full, columns 0..p
            // But we only copy the upper triangular part (columns i..p)
            for j in i..p {
                r1.set(i, j, r_full.get(i, j));
            }
            // Also copy diagonal if not yet copied
            if i < p {
                r1.set(i, i, r_full.get(i, i));
            }
        }

        // Invert R₁ (upper triangular) with custom tolerance
        let r1_inv = r1.invert_upper_triangular_with_tolerance(tolerance_mult)?;

        // Compute (X'X)^(-1) = R₁^(-1) × R₁^(-T)
        let mut result = Matrix::zeros(p, p);
        for i in 0..p {
            for j in 0..p {
                let mut sum = 0.0;
                // result[i,j] = sum(R1_inv[i,k] * R1_inv[j,k] for k=0..p)
                // R1_inv is upper triangular, but we iterate full range
                for k in 0..p {
                    sum += r1_inv.get(i, k) * r1_inv.get(j, k);
                }
                result.set(i, j, sum);
            }
        }

        Some(result)
    }
}

// ============================================================================
// Vector Helper Functions
// ============================================================================

/// Computes the arithmetic mean of a slice of f64 values.
///
/// Returns 0.0 for empty slices.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_mean;
///
/// assert_eq!(vec_mean(&[1.0, 2.0, 3.0, 4.0, 5.0]), 3.0);
/// assert_eq!(vec_mean(&[]), 0.0);
/// ```
///
/// # Arguments
///
/// * `v` - Slice of values
pub fn vec_mean(v: &[f64]) -> f64 {
    if v.is_empty() {
        return 0.0;
    }
    v.iter().sum::<f64>() / v.len() as f64
}

/// Computes element-wise subtraction of two slices: `a - b`.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_sub;
///
/// let a = vec![5.0, 4.0, 3.0];
/// let b = vec![1.0, 1.0, 1.0];
/// let result = vec_sub(&a, &b);
/// assert_eq!(result, vec![4.0, 3.0, 2.0]);
/// ```
///
/// # Arguments
///
/// * `a` - Minuend slice
/// * `b` - Subtrahend slice
///
/// # Panics
///
/// Panics if slices have different lengths.
pub fn vec_sub(a: &[f64], b: &[f64]) -> Vec<f64> {
    assert_eq!(a.len(), b.len(), "vec_sub: slice lengths must match");
    a.iter().zip(b.iter()).map(|(x, y)| x - y).collect()
}

/// Computes the dot product of two slices: `Σ(a[i] * b[i])`.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_dot;
///
/// let a = vec![1.0, 2.0, 3.0];
/// let b = vec![4.0, 5.0, 6.0];
/// assert_eq!(vec_dot(&a, &b), 32.0);  // 1*4 + 2*5 + 3*6
/// ```
///
/// # Arguments
///
/// * `a` - First slice
/// * `b` - Second slice
///
/// # Panics
///
/// Panics if slices have different lengths.
pub fn vec_dot(a: &[f64], b: &[f64]) -> f64 {
    assert_eq!(a.len(), b.len(), "vec_dot: slice lengths must match");
    a.iter().zip(b.iter()).map(|(x, y)| x * y).sum()
}

/// Computes element-wise addition of two slices: `a + b`.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_add;
///
/// let a = vec![1.0, 2.0, 3.0];
/// let b = vec![4.0, 5.0, 6.0];
/// assert_eq!(vec_add(&a, &b), vec![5.0, 7.0, 9.0]);
/// ```
///
/// # Arguments
///
/// * `a` - First slice
/// * `b` - Second slice
///
/// # Panics
///
/// Panics if slices have different lengths.
pub fn vec_add(a: &[f64], b: &[f64]) -> Vec<f64> {
    assert_eq!(a.len(), b.len(), "vec_add: slice lengths must match");
    a.iter().zip(b.iter()).map(|(x, y)| x + y).collect()
}

/// Computes a scaled vector addition in place: `dst += alpha * src`.
///
/// This is the classic BLAS AXPY operation.
///
/// # Arguments
///
/// * `dst` - Destination slice (modified in place)
/// * `alpha` - Scaling factor for src
/// * `src` - Source slice
///
/// # Panics
///
/// Panics if slices have different lengths.
///
/// # Example
///
/// ```
/// use linreg_core::linalg::vec_axpy_inplace;
///
/// let mut dst = vec![1.0, 2.0, 3.0];
/// let src = vec![0.5, 0.5, 0.5];
/// vec_axpy_inplace(&mut dst, 2.0, &src);
/// assert_eq!(dst, vec![2.0, 3.0, 4.0]);  // [1+2*0.5, 2+2*0.5, 3+2*0.5]
/// ```
pub fn vec_axpy_inplace(dst: &mut [f64], alpha: f64, src: &[f64]) {
    assert_eq!(
        dst.len(),
        src.len(),
        "vec_axpy_inplace: slice lengths must match"
    );
    for (d, &s) in dst.iter_mut().zip(src.iter()) {
        *d += alpha * s;
    }
}

/// Scales a vector in place: `v *= alpha`.
///
/// # Arguments
///
/// * `v` - Vector to scale (modified in place)
/// * `alpha` - Scaling factor
///
/// # Example
///
/// ```
/// use linreg_core::linalg::vec_scale_inplace;
///
/// let mut v = vec![1.0, 2.0, 3.0];
/// vec_scale_inplace(&mut v, 2.5);
/// assert_eq!(v, vec![2.5, 5.0, 7.5]);
/// ```
pub fn vec_scale_inplace(v: &mut [f64], alpha: f64) {
    for val in v.iter_mut() {
        *val *= alpha;
    }
}

/// Returns a scaled copy of a vector: `v * alpha`.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_scale;
///
/// let v = vec![1.0, 2.0, 3.0];
/// let scaled = vec_scale(&v, 2.5);
/// assert_eq!(scaled, vec![2.5, 5.0, 7.5]);
/// // Original is unchanged
/// assert_eq!(v, vec![1.0, 2.0, 3.0]);
/// ```
///
/// # Arguments
///
/// * `v` - Vector to scale
/// * `alpha` - Scaling factor
pub fn vec_scale(v: &[f64], alpha: f64) -> Vec<f64> {
    v.iter().map(|&x| x * alpha).collect()
}

/// Computes the L2 norm (Euclidean norm) of a vector: `sqrt(Σ(v[i]²))`.
///
/// # Examples
///
/// ```
/// use linreg_core::linalg::vec_l2_norm;
///
/// // Pythagorean triple: 3-4-5
/// assert_eq!(vec_l2_norm(&[3.0, 4.0]), 5.0);
/// // Unit vector
/// assert_eq!(vec_l2_norm(&[1.0, 0.0, 0.0]), 1.0);
/// ```
///
/// # Arguments
///
/// * `v` - Vector slice
pub fn vec_l2_norm(v: &[f64]) -> f64 {
    v.iter().map(|&x| x * x).sum::<f64>().sqrt()
}

/// Computes the maximum absolute value in a vector.
///
/// # Arguments
///
/// * `v` - Vector slice
///
/// # Example
///
/// ```
/// use linreg_core::linalg::vec_max_abs;
///
/// assert_eq!(vec_max_abs(&[1.0, -5.0, 3.0]), 5.0);
/// assert_eq!(vec_max_abs(&[-2.5, -1.5]), 2.5);
/// ```
pub fn vec_max_abs(v: &[f64]) -> f64 {
    v.iter().map(|&x| x.abs()).fold(0.0_f64, f64::max)
}

// ============================================================================
// R-Compatible QR Decomposition (LINPACK dqrdc2 with Column Pivoting)
// ============================================================================

/// QR decomposition result using R's LINPACK dqrdc2 algorithm.
///
/// This implements the QR decomposition with column pivoting as used by R's
/// `qr()` function with `LAPACK=FALSE`. The algorithm is a modification of
/// LINPACK's DQRDC that:
/// - Uses Householder transformations
/// - Implements limited column pivoting based on 2-norms of reduced columns
/// - Moves columns with near-zero norm to the right-hand edge
/// - Computes the rank (number of linearly independent columns)
///
/// # Fields
///
/// * `qr` - The QR factorization (upper triangle contains R, below diagonal
///   contains Householder vector information)
/// * `qraux` - Auxiliary information for recovering the orthogonal part Q
/// * `pivot` - Column permutation: `pivot\[j\]` contains the original column index
///   now in column j
/// * `rank` - Number of linearly independent columns (the computed rank)
#[derive(Clone, Debug)]
pub struct QRLinpack {
    /// QR factorization matrix (same dimensions as input)
    pub qr: Matrix,
    /// Auxiliary information for Q recovery
    pub qraux: Vec<f64>,
    /// Column pivot vector (1-based indices like R)
    pub pivot: Vec<usize>,
    /// Computed rank (number of linearly independent columns)
    pub rank: usize,
}

impl Matrix {
    /// Computes QR decomposition using R's LINPACK dqrdc2 algorithm with column pivoting.
    ///
    /// This is a port of R's dqrdc2.f, which is a modification of LINPACK's DQRDC.
    /// The algorithm:
    /// 1. Uses Householder transformations for QR factorization
    /// 2. Implements limited column pivoting based on column 2-norms
    /// 3. Moves columns with near-zero norm to the right-hand edge
    /// 4. Computes the rank (number of linearly independent columns)
    ///
    /// # Arguments
    ///
    /// * `tol` - Tolerance for determining linear independence. Default is 1e-7 (R's default).
    ///   Columns with norm < tol * original_norm are considered negligible.
    ///
    /// # Returns
    ///
    /// A [`QRLinpack`] struct containing the QR factorization, auxiliary information,
    /// pivot vector, and computed rank.
    ///
    /// # Algorithm Details
    ///
    /// The decomposition is A * P = Q * R where:
    /// - P is the permutation matrix coded by `pivot`
    /// - Q is orthogonal (m × m)
    /// - R is upper triangular in the first `rank` rows
    ///
    /// The `qr` matrix contains:
    /// - Upper triangle: R matrix (if pivoting was performed, this is R of the permuted matrix)
    /// - Below diagonal: Householder vector information
    ///
    /// # Reference
    ///
    /// - R source: src/appl/dqrdc2.f
    /// - LINPACK documentation: <https://www.netlib.org/linpack/dqrdc.f>
    pub fn qr_linpack(&self, tol: Option<f64>) -> QRLinpack {
        let n = self.rows;
        let p = self.cols;
        let lup = n.min(p);

        // Default tolerance matches R's qr.default: tol = 1e-07
        let tol = tol.unwrap_or(1e-07);

        // Initialize working matrices
        let mut x = self.clone(); // Working copy that will be modified
        let mut qraux = vec![0.0; p];
        let mut pivot: Vec<usize> = (1..=p).collect(); // 1-based indices like R
        let mut work = vec![(0.0, 0.0); p]; // (work[j,1], work[j,2])

        // Compute the norms of the columns of x (initialization)
        if n > 0 {
            for j in 0..p {
                let mut norm = 0.0;
                for i in 0..n {
                    norm += x.get(i, j) * x.get(i, j);
                }
                norm = norm.sqrt();
                qraux[j] = norm;
                let original_norm = if norm == 0.0 { 1.0 } else { norm };
                work[j] = (norm, original_norm);
            }
        }

        let mut k = p + 1; // Will be decremented to get the final rank

        // Perform the Householder reduction of x
        for l in 0..lup {
            // Cycle columns from l to p until one with non-negligible norm is found
            // A column is negligible if its norm has fallen below tol * original_norm
            while l < k - 1 && qraux[l] < work[l].1 * tol {
                // Move column l to the end (it's negligible)
                let lp1 = l + 1;

                // Shift columns in x: x(i, l..p-1) = x(i, l+1..p)
                for i in 0..n {
                    let t = x.get(i, l);
                    for j in lp1..p {
                        x.set(i, j - 1, x.get(i, j));
                    }
                    x.set(i, p - 1, t);
                }

                // Shift pivot, qraux, and work arrays
                let saved_pivot = pivot[l];
                let saved_qraux = qraux[l];
                let saved_work = work[l];

                for j in lp1..p {
                    pivot[j - 1] = pivot[j];
                    qraux[j - 1] = qraux[j];
                    work[j - 1] = work[j];
                }

                pivot[p - 1] = saved_pivot;
                qraux[p - 1] = saved_qraux;
                work[p - 1] = saved_work;

                k -= 1;
            }

            if l == n - 1 {
                // Last row - skip transformation
                break;
            }

            // Compute the Householder transformation for column l
            // nrmxl = norm of x[l:, l]
            let mut nrmxl = 0.0;
            for i in l..n {
                let val = x.get(i, l);
                nrmxl += val * val;
            }
            nrmxl = nrmxl.sqrt();

            if nrmxl == 0.0 {
                // Zero column - continue to next
                continue;
            }

            // Apply sign for numerical stability
            let x_ll = x.get(l, l);
            if x_ll != 0.0 {
                nrmxl = nrmxl.copysign(x_ll);
            }

            // Scale the column
            let scale = 1.0 / nrmxl;
            for i in l..n {
                x.set(i, l, x.get(i, l) * scale);
            }
            x.set(l, l, 1.0 + x.get(l, l));

            // Apply the transformation to remaining columns, updating the norms
            let lp1 = l + 1;
            if p > lp1 {
                for j in lp1..p {
                    // Compute t = -dot(x[l:, l], x[l:, j]) / x(l, l)
                    let mut dot = 0.0;
                    for i in l..n {
                        dot += x.get(i, l) * x.get(i, j);
                    }
                    let t = -dot / x.get(l, l);

                    // x[l:, j] = x[l:, j] + t * x[l:, l]
                    for i in l..n {
                        let val = x.get(i, j) + t * x.get(i, l);
                        x.set(i, j, val);
                    }

                    // Update the norm
                    if qraux[j] != 0.0 {
                        // tt = 1.0 - (x(l, j) / qraux[j])^2
                        let x_lj = x.get(l, j).abs();
                        let mut tt = 1.0 - (x_lj / qraux[j]).powi(2);
                        tt = tt.max(0.0);

                        // Recompute norm if there is large reduction (BDR mod 9/99)
                        // The tolerance here is on the squared norm
                        if tt.abs() < 1e-6 {
                            // Re-compute norm directly
                            let mut new_norm = 0.0;
                            for i in (l + 1)..n {
                                let val = x.get(i, j);
                                new_norm += val * val;
                            }
                            new_norm = new_norm.sqrt();
                            qraux[j] = new_norm;
                            work[j].0 = new_norm;
                        } else {
                            qraux[j] *= tt.sqrt();
                        }
                    }
                }
            }

            // Save the transformation
            qraux[l] = x.get(l, l);
            x.set(l, l, -nrmxl);
        }

        // Compute final rank
        let rank = k - 1;
        let rank = rank.min(n);

        QRLinpack {
            qr: x,
            qraux,
            pivot,
            rank,
        }
    }

    /// Solves a linear system using the QR decomposition with column pivoting.
    ///
    /// This implements a least squares solver using the pivoted QR decomposition.
    /// For rank-deficient cases, coefficients corresponding to linearly dependent
    /// columns are set to `f64::NAN`.
    ///
    /// # Arguments
    ///
    /// * `qr_result` - QR decomposition from [`Matrix::qr_linpack`]
    /// * `y` - Right-hand side vector
    ///
    /// # Returns
    ///
    /// A vector of coefficients, or `None` if the system is exactly singular.
    ///
    /// # Algorithm
    ///
    /// This solver uses the standard QR decomposition approach:
    /// 1. Compute the QR decomposition of the permuted matrix
    /// 2. Extract R matrix (upper triangular with positive diagonal)
    /// 3. Compute qty = Q^T * y
    /// 4. Solve R * coef = qty using back substitution
    /// 5. Apply the pivot permutation to restore original column order
    ///
    /// # Note
    ///
    /// The LINPACK QR algorithm stores R with mixed signs on the diagonal.
    /// This solver corrects for that by taking the absolute value of R's diagonal.
    #[allow(clippy::needless_range_loop)]
    pub fn qr_solve_linpack(&self, qr_result: &QRLinpack, y: &[f64]) -> Option<Vec<f64>> {
        let n = self.rows;
        let p = self.cols;
        let k = qr_result.rank;

        if y.len() != n {
            return None;
        }

        if k == 0 {
            return None;
        }

        // Step 1: Compute Q^T * y using the Householder vectors directly
        // This is more efficient than reconstructing the full Q matrix
        let mut qty = y.to_vec();

        for j in 0..k {
            // Check if this Householder transformation is valid
            let r_jj = qr_result.qr.get(j, j);
            if r_jj == 0.0 {
                continue;
            }

            // Compute dot = v_j^T * qty[j:]
            // where v_j is the Householder vector stored in qr[j:, j]
            // The storage convention:
            // - qr[j,j] = -nrmxl (after final overwrite)
            // - qr[i,j] for i > j is the scaled Householder vector element
            // - qraux[j] = 1 + original_x[j,j]/nrmxl (the unscaled first element)

            // Reconstruct the Householder vector v_j
            // After scaling by 1/nrmxl, we have:
            // v_scaled[j] = 1 + x[j,j]/nrmxl
            // v_scaled[i] = x[i,j]/nrmxl for i > j
            // The actual unit vector is v = v_scaled / ||v_scaled||

            let mut v = vec![0.0; n - j];
            // Copy the scaled Householder vector from qr
            for i in j..n {
                v[i - j] = qr_result.qr.get(i, j);
            }

            // The j-th element was modified during the QR decomposition
            // We need to reconstruct it from qraux
            let alpha = qr_result.qraux[j];
            if alpha != 0.0 {
                v[0] = alpha;
            }

            // Compute the norm of v
            let v_norm: f64 = v.iter().map(|&x| x * x).sum::<f64>().sqrt();
            if v_norm < 1e-14 {
                continue;
            }

            // Compute dot = v^T * qty[j:]
            let mut dot = 0.0;
            for i in j..n {
                dot += v[i - j] * qty[i];
            }

            // Apply Householder transformation: qty[j:] = qty[j:] - 2 * v * (v^T * qty[j:]) / (v^T * v)
            // Since v is already scaled, we use: t = 2 * dot / (v_norm^2)
            let t = 2.0 * dot / (v_norm * v_norm);

            for i in j..n {
                qty[i] -= t * v[i - j];
            }
        }

        // Step 2: Back substitution on R (solve R * coef = qty)
        // The R matrix is stored in the upper triangle of qr
        // Note: The diagonal elements of R are negative (from -nrmxl)
        // We use them as-is since the signs cancel out in the computation
        let mut coef_permuted = vec![f64::NAN; p];

        for row in (0..k).rev() {
            let r_diag = qr_result.qr.get(row, row);
            // Use relative tolerance for singularity check
            let max_abs = (0..k)
                .map(|i| qr_result.qr.get(i, i).abs())
                .fold(0.0_f64, f64::max);
            let tolerance = 1e-14 * max_abs.max(1.0);

            if r_diag.abs() < tolerance {
                return None; // Singular
            }

            let mut sum = qty[row];
            for col in (row + 1)..k {
                sum -= qr_result.qr.get(row, col) * coef_permuted[col];
            }
            coef_permuted[row] = sum / r_diag;
        }

        // Step 3: Apply pivot permutation to get coefficients in original order
        // pivot[j] is 1-based, indicating which original column is now in position j
        let mut result = vec![0.0; p];
        for j in 0..p {
            let original_col = qr_result.pivot[j] - 1; // Convert to 0-based
            result[original_col] = coef_permuted[j];
        }

        Some(result)
    }
}

/// Performs OLS regression using R's LINPACK QR algorithm.
///
/// This function is a drop-in replacement for `fit_ols` that uses the
/// R-compatible QR decomposition with column pivoting. It handles
/// rank-deficient matrices more gracefully than the standard QR decomposition.
///
/// # Arguments
///
/// * `y` - Response variable (n observations)
/// * `x` - Design matrix (n rows, p columns including intercept)
///
/// # Returns
///
/// * `Some(Vec<f64>)` - OLS coefficient vector (p elements)
/// * `None` - If the matrix is exactly singular or dimensions don't match
///
/// # Note
///
/// For rank-deficient systems, this function uses the pivoted QR which
/// automatically handles multicollinearity by selecting a linearly
/// independent subset of columns.
///
/// # Arguments
///
/// * `y` - Response variable (n observations)
/// * `x` - Design matrix (n rows × p columns, including intercept column)
///
/// # Returns
///
/// * `Some(Vec<f64>)` - OLS coefficient estimates (length p), or `None` if the matrix is singular
///
/// # Notes
///
/// - Coefficients for dropped (collinear) columns are set to `NaN`
/// - This is a convenience wrapper around `Matrix::qr_linpack` and `Matrix::qr_solve_linpack`
///
/// # Example
///
/// ```
/// # use linreg_core::linalg::{fit_ols_linpack, Matrix};
/// let y = vec![2.0, 4.0, 6.0];
/// let x = Matrix::new(3, 2, vec![1.0, 1.0, 1.0, 2.0, 1.0, 3.0]);
///
/// let beta = fit_ols_linpack(&y, &x).unwrap();
/// assert_eq!(beta.len(), 2);  // Intercept and slope
/// ```
pub fn fit_ols_linpack(y: &[f64], x: &Matrix) -> Option<Vec<f64>> {
    let qr_result = x.qr_linpack(None);
    x.qr_solve_linpack(&qr_result, y)
}

// ============================================================================
// SVD Decomposition (Golub-Kahan)
// ============================================================================

/// SVD decomposition result.
///
/// Contains the singular value decomposition A = U * Sigma * V^T where:
/// - U is an m×min(m,n) orthogonal matrix (left singular vectors)
/// - Sigma is a vector of min(m,n) singular values (sorted in descending order)
/// - V is an n×n orthogonal matrix (right singular vectors, stored transposed as V^T)
#[derive(Clone, Debug)]
pub struct SVDResult {
    /// Left singular vectors (m × k matrix where k = min(m,n))
    pub u: Matrix,
    /// Singular values (k elements, sorted in descending order)
    pub sigma: Vec<f64>,
    /// Right singular vectors transposed (n × n matrix, rows are V^T)
    pub v_t: Matrix,
}

impl Matrix {
    /// Computes the Singular Value Decomposition (SVD) using the Golub-Kahan algorithm.
    ///
    /// Factorizes the matrix as `A = U * Sigma * V^T` where:
    /// - U is an m×k orthogonal matrix (k = min(m,n))
    /// - Sigma is a diagonal matrix of singular values (sorted in descending order)
    /// - V is an n×n orthogonal matrix
    ///
    /// This implementation uses a simplified Golub-Kahan bidiagonalization approach
    /// suitable for the small matrices encountered in LOESS local fitting.
    ///
    /// # Algorithm
    ///
    /// The algorithm follows these steps:
    /// 1. Compute A^T * A (smaller symmetric matrix when m >= n)
    /// 2. Eigen-decompose A^T * A using QR iteration
    /// 3. Singular values are sqrt of eigenvalues
    /// 4. V contains eigenvectors of A^T * A
    /// 5. U = A * V * Sigma^(-1)
    ///
    /// # Returns
    ///
    /// A [`SVDResult`] struct containing U, Sigma, and V^T.
    ///
    /// # Note
    ///
    /// This implementation is designed for numerical stability with rank-deficient
    /// matrices, which is essential for LOESS fitting where some neighborhoods may
    /// have collinear points.
    ///
    /// # Alternative
    ///
    /// For potentially higher accuracy on small matrices, see [`Matrix::svd_jacobi`].
    #[allow(clippy::needless_range_loop)]
    pub fn svd(&self) -> SVDResult {
        let m = self.rows;
        let n = self.cols;
        let k = m.min(n);

        // For typical LOESS cases, m >= n (tall matrix)
        // We use the covariance method: A^T * A = V * Sigma^2 * V^T
        // This is more efficient for tall matrices

        // Compute A^T * A (n × n symmetric matrix)
        let mut ata = Matrix::zeros(n, n);
        for i in 0..n {
            for j in i..n {
                // ata[i,j] = sum(A[p,i] * A[p,j] for p in 0..m)
                let mut sum = 0.0;
                for p in 0..m {
                    sum += self.get(p, i) * self.get(p, j);
                }
                ata.set(i, j, sum);
                ata.set(j, i, sum); // Symmetric
            }
        }

        // Eigen-decomposition of A^T * A using QR algorithm
        // Start with identity as V (will converge to eigenvectors)
        let mut v = Matrix::identity(n);
        let mut lambda = Vec::with_capacity(n);

        // QR iteration for symmetric matrices
        // This finds eigenvalues (lambda) and eigenvectors (columns of V)
        let max_iterations = 100;
        let tolerance = 1e-14;

        // Working copy for QR iteration
        let mut a_work = ata.clone();

        for _iter in 0..max_iterations {
            // Check convergence - sum of off-diagonal elements
            let mut off_diag_sum = 0.0;
            for i in 0..n {
                for j in (i + 1)..n {
                    off_diag_sum += a_work.get(i, j).abs();
                }
            }

            if off_diag_sum < tolerance {
                break;
            }

            // QR decomposition with Wilkinson shift for faster convergence
            // For simplicity, we use basic QR without shift
            let (q, r) = a_work.qr();
            a_work = r.matmul(&q);
            v = v.matmul(&q);
        }

        // Extract eigenvalues from diagonal
        for i in 0..n {
            lambda.push(a_work.get(i, i));
        }

        // Sort eigenvalues and corresponding eigenvectors in descending order
        // We need to keep V synchronized
        for i in 0..n {
            for j in (i + 1)..n {
                if lambda[j] > lambda[i] {
                    // Swap eigenvalues
                    lambda.swap(i, j);

                    // Swap corresponding columns of V
                    #[allow(clippy::manual_swap)]
                    for row in 0..n {
                        let temp = v.get(row, i);
                        v.set(row, i, v.get(row, j));
                        v.set(row, j, temp);
                    }
                }
            }
        }

        // Singular values are sqrt of eigenvalues (clamp non-negative)
        let mut sigma = Vec::with_capacity(k);
        for i in 0..k {
            let s = lambda[i].max(0.0).sqrt();
            sigma.push(s);
        }

        // Compute U = A * V * Sigma^(-1)
        // Only compute for non-zero singular values
        let mut u = Matrix::zeros(m, k);
        for j in 0..k {
            if sigma[j] > 1e-14 {
                // U[:,j] = (A * V[:,j]) / sigma[j]
                for i in 0..m {
                    let mut sum = 0.0;
                    for p in 0..n {
                        sum += self.get(i, p) * v.get(p, j);
                    }
                    u.set(i, j, sum / sigma[j]);
                }
            }
        }

        // V^T is the transpose of V
        let v_t = v.transpose();

        SVDResult { u, sigma, v_t }
    }

    /// Solves a least squares problem using SVD with pseudoinverse for rank-deficient matrices.
    ///
    /// This implements the pseudoinverse solution: x = V * Sigma^+ * U^T * b
    /// where Sigma^+ replaces 1/sigma\[i\] with 0 for sigma\[i\] below tolerance.
    ///
    /// # Arguments
    ///
    /// * `svd_result` - SVD decomposition from [`Matrix::svd`]
    /// * `b` - Right-hand side vector (m elements)
    ///
    /// # Returns
    ///
    /// A vector of coefficients (n elements) that minimizes ||Ax - b||.
    ///
    #[allow(clippy::needless_range_loop)]
    pub fn svd_solve(&self, svd_result: &SVDResult, b: &[f64]) -> Vec<f64> {
        let m = self.rows;
        let n = self.cols;
        let k = m.min(n);

        // Tolerance: sigma[0] * 100 * epsilon
        let max_sigma = svd_result.sigma.first().copied().unwrap_or(0.0);
        let tol = if max_sigma > 0.0 {
            max_sigma * 100.0 * f64::EPSILON
        } else {
            1e-14
        };

        // Compute U^T * b
        let mut ut_b = vec![0.0; k];
        for j in 0..k {
            let mut sum = 0.0;
            for i in 0..m {
                sum += svd_result.u.get(i, j) * b[i];
            }
            ut_b[j] = sum;
        }

        // Compute coefficients in V space: c[j] = ut_b[j] / sigma[j] if sigma[j] > tol, else 0
        let mut coeffs_v = vec![0.0; k];
        for j in 0..k {
            if svd_result.sigma[j] > tol {
                coeffs_v[j] = ut_b[j] / svd_result.sigma[j];
            } else {
                coeffs_v[j] = 0.0; // Singular value below threshold - use pseudoinverse (set to 0)
            }
        }

        // Transform back to original space: x = V^T * coeffs_v
        // Since v_t contains rows of V^T, we compute x = v_t^T * coeffs_v
        let mut x = vec![0.0; n];
        for i in 0..n {
            let mut sum = 0.0;
            for j in 0..k {
                // v_t.get(j, i) is element (j, i) of V^T, which is V[i, j]
                sum += svd_result.v_t.get(j, i) * coeffs_v[j];
            }
            x[i] = sum;
        }

        x
    }
}

/// Fits OLS and predicts using R's LINPACK QR with rank-deficient handling.
///
/// This function matches R's `lm.fit` behavior for rank-deficient cases:
/// coefficients for linearly dependent columns are set to NA, and predictions
/// are computed using only the valid (non-NA) coefficients and their corresponding
/// columns. This matches how R handles rank-deficient models in prediction.
///
/// # Arguments
///
/// * `y` - Response variable (n observations)
/// * `x` - Design matrix (n rows, p columns including intercept)
///
/// # Returns
///
/// * `Some(Vec<f64>)` - Predictions (n elements)
/// * `None` - If the matrix is exactly singular or dimensions don't match
///
/// # Algorithm
///
/// For rank-deficient systems (rank < p):
/// 1. Compute QR decomposition with column pivoting
/// 2. Get coefficients (rank-deficient columns will have NaN)
/// 3. Build a reduced design matrix with only pivoted, non-singular columns
/// 4. Compute predictions using only the valid columns
///
/// This matches R's behavior where `predict(lm.fit(...))` handles NA coefficients
/// by excluding the corresponding columns from the prediction.
///
/// # Example
///
/// ```
/// # use linreg_core::linalg::{fit_and_predict_linpack, Matrix};
/// let y = vec![2.0, 4.0, 6.0];
/// let x = Matrix::new(3, 2, vec![1.0, 1.0, 1.0, 2.0, 1.0, 3.0]);
///
/// let preds = fit_and_predict_linpack(&y, &x).unwrap();
/// assert_eq!(preds.len(), 3);  // One prediction per observation
/// ```
#[allow(clippy::needless_range_loop)]
pub fn fit_and_predict_linpack(y: &[f64], x: &Matrix) -> Option<Vec<f64>> {
    let n = x.rows;
    let p = x.cols;

    // Compute QR decomposition
    let qr_result = x.qr_linpack(None);
    let k = qr_result.rank;

    // Solve for coefficients
    let beta_permuted = x.qr_solve_linpack(&qr_result, y)?;

    // Check for rank deficiency
    if k == p {
        // Full rank - use standard prediction
        return Some(x.mul_vec(&beta_permuted));
    }

    // Rank-deficient case: some columns are collinear and have NaN coefficients
    // We compute predictions using only columns with valid (non-NaN) coefficients
    // This matches R's behavior where NA coefficients exclude columns from prediction

    let mut pred = vec![0.0; n];

    for row in 0..n {
        let mut sum = 0.0;
        for j in 0..p {
            let b_val = beta_permuted[j];
            if b_val.is_nan() {
                continue; // Skip collinear columns (matches R's NA coefficient behavior)
            }
            sum += x.get(row, j) * b_val;
        }
        pred[row] = sum;
    }

    Some(pred)
}

// ============================================================================
// SVD and Pseudoinverse for Robust Weighted Least Squares
// ============================================================================

impl Matrix {
    /// Compute SVD decomposition using the eigendecomposition method (Jacobi)
    ///
    /// For a matrix A (m×n), this computes A = U * Σ * V^T where:
    /// - U is m×k orthogonal (left singular vectors, k = min(m,n))
    /// - Σ is k singular values (sorted in descending order)
    /// - V is n×n orthogonal (right singular vectors)
    ///
    /// This implementation uses the method of computing the eigendecomposition
    /// of A^T*A to get V and singular values, then computing U = A * V * Σ^(-1).
    /// The Jacobi method is used for the eigendecomposition, which provides
    /// excellent numerical accuracy for small to medium-sized matrices.
    ///
    /// Returns None if the decomposition fails.
    ///
    /// # Alternative
    ///
    /// For a simpler/faster approach suitable for LOESS, see [`Matrix::svd`].
    pub fn svd_jacobi(&self) -> Option<SVDResult> {
        let m = self.rows;
        let n = self.cols;

        if m < n {
            // For wide matrices, compute SVD of transpose instead
            let at = self.transpose();
            let svd_t = at.svd_jacobi()?;
            // A = U * Σ * V^T = (V' * Σ' * U'^T)^T = U' * Σ' * V'^T
            // So swap U and V, and V becomes V^T
            return Some(SVDResult {
                u: svd_t.v_t.transpose(),  // V' becomes U (need to transpose to get correct format)
                sigma: svd_t.sigma,
                v_t: svd_t.u.transpose(), // U' becomes V (need to transpose to get V^T)
            });
        }

        // Compute A^T * A (n×n symmetric matrix)
        let ata = self.transpose().matmul(self);

        // Compute eigendecomposition of A^T * A using Jacobi method
        let (v, s_sq) = ata.symmetric_eigen()?;

        // Singular values are sqrt of eigenvalues
        let s: Vec<f64> = s_sq.iter().map(|&x| x.sqrt().max(0.0)).collect();

        // Sort by singular values (descending)
        let mut indexed: Vec<(usize, f64)> = s.iter().enumerate()
            .map(|(i, &val)| (i, val))
            .collect();
        indexed.sort_by(|a, b| b.1.partial_cmp(&a.1).unwrap_or(std::cmp::Ordering::Equal));

        // Reorder V according to sorted singular values
        let mut v_sorted = Matrix::zeros(n, n);
        let mut s_sorted = vec![0.0; n];
        for (new_idx, (old_idx, val)) in indexed.iter().enumerate() {
            s_sorted[new_idx] = *val;
            for row in 0..n {
                v_sorted.set(row, new_idx, v.get(row, *old_idx));
            }
        }

        // Compute U = A * V * Σ^(-1)
        let mut u = Matrix::zeros(m, n);
        for i in 0..m {
            for j in 0..n {
                if s_sorted[j] > f64::EPSILON {
                    let mut sum = 0.0;
                    for k in 0..n {
                        sum += self.get(i, k) * v_sorted.get(k, j);
                    }
                    u.set(i, j, sum / s_sorted[j]);
                }
            }
        }

        // V^T is the transpose of V (columns of V are right singular vectors)
        let v_t = v_sorted.transpose();

        Some(SVDResult {
            u,
            sigma: s_sorted,
            v_t,
        })
    }

    /// Compute eigendecomposition of a symmetric matrix using Jacobi method
    ///
    /// For a symmetric matrix A (n×n), computes eigenvalues λ and eigenvectors V
    /// such that A = V * Λ * V^T where Λ is diagonal and V is orthogonal.
    ///
    /// Returns (V, eigenvalues) where eigenvalues\[i\] is the eigenvalue for column i of V.
    fn symmetric_eigen(&self) -> Option<(Matrix, Vec<f64>)> {
        let n = self.rows;
        assert_eq!(n, self.cols, "Matrix must be square");

        let max_iterations = 100;
        let tolerance = 1e-10;

        // Start with identity matrix for eigenvectors
        let mut v = Matrix::identity(n);

        // Working copy of the matrix
        let mut a = self.clone();

        for _iter in 0..max_iterations {
            let mut max_off_diag = 0.0;

            // Find largest off-diagonal element
            let mut p = 0;
            let mut q = 1;
            for i in 0..n {
                for j in (i + 1)..n {
                    let val = a.get(i, j).abs();
                    if val > max_off_diag {
                        max_off_diag = val;
                        p = i;
                        q = j;
                    }
                }
            }

            if max_off_diag < tolerance {
                break;
            }

            // Jacobi rotation to zero out a(p,q)
            let app = a.get(p, p);
            let aqq = a.get(q, q);
            let apq = a.get(p, q);

            if apq.abs() < tolerance {
                continue;
            }

            // Compute rotation angle
            let tau = (aqq - app) / (2.0 * apq);
            let t = if tau >= 0.0 {
                1.0 / (tau + (1.0 + tau * tau).sqrt())
            } else {
                -1.0 / (-tau + (1.0 + tau * tau).sqrt())
            };

            let c = 1.0 / (1.0 + t * t).sqrt();
            let s = t * c;

            // Update matrix A
            for i in 0..n {
                if i != p && i != q {
                    let aip = a.get(i, p);
                    let aiq = a.get(i, q);
                    a.set(i, p, aip - s * (aiq + s * aip));
                    a.set(i, q, aiq + s * (aip - s * aiq));
                }
            }

            a.set(p, p, app - t * apq);
            a.set(q, q, aqq + t * apq);
            a.set(p, q, 0.0);
            a.set(q, p, 0.0);

            // Update eigenvectors V
            for i in 0..n {
                let vip = v.get(i, p);
                let viq = v.get(i, q);
                v.set(i, p, vip - s * (viq + s * vip));
                v.set(i, q, viq + s * (vip - s * viq));
            }
        }

        // Extract eigenvalues from diagonal
        let eigenvalues: Vec<f64> = (0..n).map(|i| a.get(i, i)).collect();

        Some((v, eigenvalues))
    }

    /// Compute Moore-Penrose pseudoinverse using SVD
    ///
    /// For matrix A, computes A^+ = V * Σ^+ * U^T where:
    /// - Σ^+ has 1/σ for σ > tolerance, and 0 otherwise
    ///
    /// This provides a least-squares solution for rank-deficient or singular matrices.
    pub fn pseudo_inverse(&self, tolerance: Option<f64>) -> Option<Matrix> {
        let svd = self.svd_jacobi()?;

        let m = self.rows;
        let n = self.cols;

        // Use provided tolerance or compute based on largest singular value
        let tol = tolerance.unwrap_or_else(|| {
            let max_s = svd.sigma.iter().fold(0.0_f64, |a: f64, &b| a.max(b));
            if max_s > 0.0 {
                max_s * f64::EPSILON * (m.max(n) as f64)
            } else {
                f64::EPSILON
            }
        });

        // Compute Σ^+ (pseudoinverse of singular values)
        let mut s_pinv = vec![0.0; svd.sigma.len()];
        for (i, &s_val) in svd.sigma.iter().enumerate() {
            if s_val > tol {
                s_pinv[i] = 1.0 / s_val;
            }
        }

        // Compute A^+ = V * Σ^+ * U^T
        // First compute Σ^+ * U^T (n×m matrix)
        let mut s_ut = Matrix::zeros(n, m);
        for i in 0..n {
            for j in 0..m {
                s_ut.set(i, j, s_pinv[i] * svd.u.get(j, i));
            }
        }

        // Then compute V * (Σ^+ * U^T)
        // Since v_t is V^T, we transpose it to get V
        let v = svd.v_t.transpose();
        let pseudoinv = v.matmul(&s_ut);

        Some(pseudoinv)
    }
}

// ============================================================================
// SVD Tests
// ============================================================================

#[cfg(test)]
mod svd_tests {
    use super::*;

    #[test]
    fn test_svd_simple_matrix() {
        // Test SVD on a simple 2x2 matrix
        let data = vec![1.0, 2.0, 3.0, 4.0];
        let m = Matrix::new(2, 2, data);
        let svd = m.svd();

        // Verify singular values are sorted descending
        for i in 1..svd.sigma.len() {
            assert!(svd.sigma[i-1] >= svd.sigma[i]);
        }

        // Verify U is orthogonal: U^T * U = I
        let ut = svd.u.transpose();
        let ut_u = ut.matmul(&svd.u);
        assert!((ut_u.get(0, 0) - 1.0).abs() < 1e-10);
        assert!(ut_u.get(0, 1).abs() < 1e-10);
        assert!(ut_u.get(1, 0).abs() < 1e-10);
        assert!((ut_u.get(1, 1) - 1.0).abs() < 1e-10);
    }

    #[test]
    fn test_svd_solve_basic() {
        // Test SVD solver on a simple system
        let data = vec![
            1.0, 1.0,
            1.0, 2.0,
            1.0, 3.0,
        ];
        let m = Matrix::new(3, 2, data);
        let svd = m.svd();

        // Solve: x + y = 2, x + 2y = 4, x + 3y = 6
        // Solution: x = 0, y = 2
        let b = vec![2.0, 4.0, 6.0];
        let x = m.svd_solve(&svd, &b);

        // Check solution
        assert!((x[0] - 0.0).abs() < 1e-10);
        assert!((x[1] - 2.0).abs() < 1e-10);
    }

    #[test]
    fn test_svd_tolerance_formula() {
        // Verify tolerance formula: tol = sigma[0] * 100 * epsilon
        let data = vec![
            1.0, 1.0,
            1.0, 2.0,
            1.0, 3.0,
        ];
        let m = Matrix::new(3, 2, data);
        let svd = m.svd();

        let max_sigma = svd.sigma[0];
        let expected_tol = max_sigma * 100.0 * f64::EPSILON;

        // Verify tolerance is computed correctly
        assert!(expected_tol > 0.0);
        assert!(expected_tol < 1e-10);
    }

    #[test]
    fn test_svd_solve_rank_deficient() {
        // Test SVD solver on rank-deficient matrix
        // Second column is 2x first column
        let data = vec![
            1.0, 2.0,
            2.0, 4.0,
            3.0, 6.0,
        ];
        let m = Matrix::new(3, 2, data);
        let svd = m.svd();

        // One singular value should be near zero
        assert!(svd.sigma[0] > 1e-10);
        assert!(svd.sigma[1] < 1e-10);

        // Solve: should still work with pseudoinverse
        let b = vec![3.0, 6.0, 9.0];
        let x = m.svd_solve(&svd, &b);

        // Check that solution is valid
        assert!(x[0].is_finite());
        assert!(x[1].is_finite());

        // Verify prediction at first row: 1*x0 + 2*x1 ≈ 3
        let pred = m.get(0, 0) * x[0] + m.get(0, 1) * x[1];
        assert!((pred - b[0]).abs() < 1e-6);
    }

    #[test]
    fn test_svd_jacobi_kahan_produce_results() {
        // Verify both Jacobi and Kahan SVD produce valid results
        let data = vec![
            1.0, 2.0, 3.0,
            4.0, 5.0, 6.0,
            7.0, 8.0, 9.0,
        ];
        let m = Matrix::new(3, 3, data);

        // Kahan (default)
        let svd_kahan = m.svd();

        // Jacobi
        let svd_jacobi = m.svd_jacobi().unwrap();

        // Both should produce same number of singular values
        assert_eq!(svd_kahan.sigma.len(), svd_jacobi.sigma.len());

        // Both should have sorted (descending) singular values
        for i in 1..svd_kahan.sigma.len() {
            assert!(svd_kahan.sigma[i-1] >= svd_kahan.sigma[i]);
            assert!(svd_jacobi.sigma[i-1] >= svd_jacobi.sigma[i]);
        }

        // Both should detect the rank deficiency (one near-zero singular value)
        assert!(svd_kahan.sigma[2] < 1e-10);
        assert!(svd_jacobi.sigma[2] < 1e-10);
    }

    #[test]
    fn test_svd_jacobi_rank_deficient() {
        // Test Jacobi SVD on rank-deficient matrix
        let data = vec![
            1.0, 2.0,
            2.0, 4.0,
            3.0, 6.0,
        ];
        let m = Matrix::new(3, 2, data);
        let svd = m.svd_jacobi().unwrap();

        // Should successfully decompose
        assert_eq!(svd.sigma.len(), 2);
        // One singular value should be near zero (rank = 1)
        assert!(svd.sigma[1] < 1e-10);
    }

    #[test]
    fn test_pseudo_inverse_basic() {
        // Test pseudoinverse on simple invertible matrix
        let data = vec![
            1.0, 0.0,
            0.0, 1.0,
        ];
        let m = Matrix::new(2, 2, data);
        let pinv = m.pseudo_inverse(None).unwrap();

        // For identity matrix, pseudoinverse should be identity
        assert!((pinv.get(0, 0) - 1.0).abs() < 1e-10);
        assert!(pinv.get(0, 1).abs() < 1e-10);
        assert!(pinv.get(1, 0).abs() < 1e-10);
        assert!((pinv.get(1, 1) - 1.0).abs() < 1e-10);
    }

    #[test]
    fn test_pseudo_inverse_rank_deficient() {
        // Test pseudoinverse on rank-deficient matrix
        let data = vec![
            1.0, 0.0,
            1.0, 0.0,
            1.0, 0.0,
        ];
        let m = Matrix::new(3, 2, data);
        let pinv = m.pseudo_inverse(None).unwrap();

        // Verify pseudoinverse exists and is finite
        assert_eq!(pinv.rows, 2);
        assert_eq!(pinv.cols, 3);
        for i in 0..pinv.rows {
            for j in 0..pinv.cols {
                assert!(pinv.get(i, j).is_finite());
            }
        }
    }

    #[test]
    fn test_svd_wide_matrix() {
        // Test SVD on wide matrix (more columns than rows)
        let data = vec![
            1.0, 2.0, 3.0, 4.0,
            5.0, 6.0, 7.0, 8.0,
        ];
        let m = Matrix::new(2, 4, data);
        let svd = m.svd();

        // Should produce 2 singular values (min(2,4))
        assert_eq!(svd.sigma.len(), 2);

        // U should be 2x2
        assert_eq!(svd.u.rows, 2);
        assert_eq!(svd.u.cols, 2);

        // V^T should be 4x4
        assert_eq!(svd.v_t.rows, 4);
        assert_eq!(svd.v_t.cols, 4);
    }

    #[test]
    fn test_svd_tall_matrix() {
        // Test SVD on tall matrix (more rows than columns)
        let data = vec![
            1.0, 2.0,
            3.0, 4.0,
            5.0, 6.0,
            7.0, 8.0,
        ];
        let m = Matrix::new(4, 2, data);
        let svd = m.svd();

        // Should produce 2 singular values (min(4,2))
        assert_eq!(svd.sigma.len(), 2);

        // U should be 4x2
        assert_eq!(svd.u.rows, 4);
        assert_eq!(svd.u.cols, 2);

        // V^T should be 2x2
        assert_eq!(svd.v_t.rows, 2);
        assert_eq!(svd.v_t.cols, 2);
    }

    #[test]
    fn test_svd_solve_with_custom_tolerance() {
        // Test that tolerance affects which singular values are kept
        let data = vec![
            1.0, 2.0,
            2.0, 4.0,
            3.0, 6.0,
        ];
        let m = Matrix::new(3, 2, data);
        let svd = m.svd();

        let b = vec![3.0, 6.0, 9.0];

        // Default tolerance
        let x_default = m.svd_solve(&svd, &b);

        // With very high tolerance (more aggressive rejection)
        // This should still produce valid results
        assert!(x_default[0].is_finite());
        assert!(x_default[1].is_finite());
    }
}