rustytorch/archive/legacy_files_backup/mod_original_backup.rs

//! Advanced linear algebra operations for RTX Tensor
//!
//! This module provides numerically stable implementations of linear algebra decompositions:
//! 
//! # Decompositions
//! - **SVD** (Singular Value Decomposition): A = U*S*V^T
//! - **QR** decomposition using Householder reflections: A = Q*R
//! - **Cholesky** decomposition for positive definite matrices: A = L*L^T
//! - **Eigendecomposition** for symmetric matrices: A = V*Λ*V^T
//! - **Pseudo-inverse** using SVD with numerical tolerance
//! - **Truncated SVD** for dimensionality reduction
//!
//! # Features
//! - Numerical stability with proper conditioning
//! - Support for f32 and f64 precision
//! - Error handling for ill-conditioned matrices  
//! - Integration with autograd system
//! - Memory-efficient implementations
//!
//! # Example
//! ```rust
//! use rtx_tensor::{Tensor, Device};
//!
//! let device = Device::cpu();
//! let a = Tensor::from_data(vec![3.0, 1.0, 1.0, 3.0], vec![2, 2], &device).unwrap();
//! 
//! // SVD decomposition
//! let svd = a.svd(true, true).unwrap();
//! println!("Singular values: {:?}", svd.s.to_cpu().unwrap());
//! 
//! // QR decomposition
//! let qr = a.qr().unwrap();
//! 
//! // Eigendecomposition (symmetric matrices only)
//! let eigen = a.symeig(true).unwrap();
//! ```

use crate::{Tensor, TensorError, Result};
use std::fmt;

#[cfg(test)]
mod tests;

#[cfg(test)]
mod simple_test;

#[cfg(test)]
mod matrix_power_tests;

// Re-export test module types for external testing (when testing)
#[cfg(test)]
pub use tests::*;

// Public types are defined below and available for import

/// Result of Singular Value Decomposition
#[derive(Debug, Clone)]
pub struct SVDResult {
    /// Left singular vectors (U matrix)
    pub u: Tensor,
    /// Singular values (diagonal of S matrix)
    pub s: Tensor,
    /// Right singular vectors transposed (V^T matrix)
    pub vt: Tensor,
}

impl fmt::Display for SVDResult {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "SVDResult {{ U: {}, S: {}, V^T: {} }}",
            self.u.shape(),
            self.s.shape(),
            self.vt.shape()
        )
    }
}

/// Result of QR Decomposition
#[derive(Debug, Clone)]
pub struct QRResult {
    /// Orthogonal matrix Q
    pub q: Tensor,
    /// Upper triangular matrix R  
    pub r: Tensor,
}

impl fmt::Display for QRResult {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "QRResult {{ Q: {}, R: {} }}",
            self.q.shape(),
            self.r.shape()
        )
    }
}

/// Result of Eigenvalue Decomposition
#[derive(Debug, Clone)]
pub struct EigenResult {
    /// Eigenvalues (Λ)
    pub eigenvalues: Tensor,
    /// Eigenvectors (V)
    pub eigenvectors: Tensor,
}

/// Result of LU Decomposition
#[derive(Debug, Clone)]
pub struct LUResult {
    /// Lower triangular matrix L (with 1s on diagonal in Doolittle algorithm)
    pub l: Tensor,
    /// Upper triangular matrix U
    pub u: Tensor,
    /// Permutation vector P (indices of row exchanges)
    pub p: Tensor,
    /// Permutation matrix P (optional, returned when requested)
    pub p_matrix: Option<Tensor>,
    /// Determinant of the original matrix (computed from LU)
    pub determinant: f32,
    /// Whether the matrix is invertible (non-singular)
    pub is_invertible: bool,
    /// Condition number estimate
    pub condition_number: f32,
    /// Number of row exchanges performed during pivoting
    pub num_pivots: usize,
}

/// Options for LU decomposition
#[derive(Debug, Clone)]
pub struct LUOptions {
    /// Whether to use partial pivoting for numerical stability
    pub pivoting: bool,
    /// Whether to perform decomposition in-place (memory efficient)
    pub in_place: bool,
    /// Whether to return permutation matrix P in addition to permutation vector
    pub return_permutation_matrix: bool,
}

impl fmt::Display for EigenResult {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "EigenResult {{ eigenvalues: {}, eigenvectors: {} }}",
            self.eigenvalues.shape(),
            self.eigenvectors.shape()
        )
    }
}

impl fmt::Display for LUResult {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "LUResult {{ L: {}, U: {}, det: {:.6}, invertible: {}, pivots: {} }}",
            self.l.shape(),
            self.u.shape(),
            self.determinant,
            self.is_invertible,
            self.num_pivots
        )
    }
}

impl Default for LUOptions {
    fn default() -> Self {
        Self {
            pivoting: true,
            in_place: false,
            return_permutation_matrix: false,
        }
    }
}

/// Matrix structure type for triangular solving
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum TriangularType {
    /// Lower triangular matrix (non-zero entries below or on diagonal)
    Lower,
    /// Upper triangular matrix (non-zero entries above or on diagonal)
    Upper,
}

/// Options for triangular system solving
#[derive(Debug, Clone)]
pub struct TriangularSolveOptions {
    /// Whether the matrix is lower or upper triangular
    pub triangular_type: TriangularType,
    /// Whether to solve A^T x = b instead of A x = b
    pub transpose: bool,
    /// Whether the diagonal elements are assumed to be 1 (unit diagonal)
    pub unit_diagonal: bool,
    /// Whether to validate that the matrix is actually triangular
    pub validate_triangular: bool,
    /// Whether to use strict triangular validation (exact zeros) or relaxed (within tolerance)
    pub strict_validation: bool,
}

/// Method to use for least squares solving
#[derive(Debug, Clone, Copy, PartialEq, Eq)]
pub enum LeastSquaresMethod {
    /// Automatic method selection based on matrix properties
    Auto,
    /// QR decomposition method (best for overdetermined systems)
    QR,
    /// SVD method (most robust, handles rank-deficient matrices)
    SVD,
    /// Normal equations (A^T A)x = A^T b (fastest but least stable)
    NormalEquations,
}

impl Default for LeastSquaresMethod {
    fn default() -> Self {
        Self::Auto
    }
}

/// Options for least squares solving
#[derive(Debug, Clone)]
pub struct LeastSquaresOptions {
    /// Method to use for solving
    pub method: LeastSquaresMethod,
    /// Tikhonov regularization parameter (λ in (A^T A + λI)x = A^T b)
    pub regularization_lambda: Option<f32>,
    /// Diagonal weight matrix for weighted least squares
    pub weights: Option<Tensor>,
    /// Relative condition number threshold for rank determination
    pub rcond: f32,
    /// Maximum number of iterations for iterative methods
    pub max_iterations: Option<usize>,
    /// Convergence tolerance for iterative methods
    pub tolerance: Option<f32>,
}

impl Default for LeastSquaresOptions {
    fn default() -> Self {
        Self {
            method: LeastSquaresMethod::Auto,
            regularization_lambda: None,
            weights: None,
            rcond: 1e-12,
            max_iterations: None,
            tolerance: None,
        }
    }
}

/// Result of least squares solving
#[derive(Debug, Clone)]
pub struct LeastSquaresResult {
    /// Solution vector(s) x
    pub solution: Tensor,
    /// Residual norm ||Ax - b||
    pub residual_norm: f32,
    /// Condition number of the coefficient matrix
    pub condition_number: f32,
    /// Effective rank of the coefficient matrix
    pub rank: usize,
    /// Method actually used for solving
    pub method_used: LeastSquaresMethod,
    /// R-squared goodness-of-fit statistic (if applicable)
    pub r_squared: Option<f32>,
    /// Sum of squared residuals
    pub sum_squared_residuals: f32,
}

impl fmt::Display for LeastSquaresResult {
    fn fmt(&self, f: &mut fmt::Formatter<'_>) -> fmt::Result {
        write!(
            f,
            "LeastSquaresResult {{ solution: {}, residual_norm: {:.6}, condition_number: {:.6}, rank: {}, method: {:?} }}",
            self.solution.shape(),
            self.residual_norm,
            self.condition_number,
            self.rank,
            self.method_used
        )
    }
}

/// Linear algebra operations implementation
impl Tensor {
    // Note: Helper functions unravel_index, ravel_index, and broadcast_index are defined in tensor::utilities
    /// Singular Value Decomposition (SVD)
    ///
    /// Decomposes a matrix A into A = U * S * V^T where:
    /// - U contains the left singular vectors
    /// - S contains the singular values (non-negative, decreasing)
    /// - V^T contains the right singular vectors transposed
    ///
    /// # Arguments
    /// - `compute_u`: Whether to compute the U matrix (left singular vectors)
    /// - `compute_vt`: Whether to compute the V^T matrix (right singular vectors)
    ///
    /// # Returns
    /// `SVDResult` containing U, S, and V^T tensors
    ///
    /// # Numerical Stability
    /// Uses iterative power method with Gram-Schmidt orthogonalization.
    /// Handles rank-deficient and ill-conditioned matrices gracefully.
    pub fn svd(&self, compute_u: bool, compute_vt: bool) -> Result<SVDResult> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("SVD requires 2D tensor"));
        }

        let dims = self.shape().dims();
        let m = dims[0]; // rows
        let n = dims[1]; // cols
        let min_dim = m.min(n);

        if m == 0 || n == 0 {
            return Err(TensorError::shape("SVD requires non-empty matrix"));
        }

        let data = self.to_cpu()?;
        
        // Use power iteration method for SVD
        // This is a simplified implementation - production would use more sophisticated algorithms
        let mut a = data.clone();
        let mut u_data = vec![0.0f32; if compute_u { m * min_dim } else { 0 }];
        let mut s_data = vec![0.0f32; min_dim];
        let mut vt_data = vec![0.0f32; if compute_vt { min_dim * n } else { 0 }];

        // For each singular value/vector pair
        for k in 0..min_dim {
            // Power iteration to find dominant singular value and vectors
            let mut v = vec![1.0f32; n];
            normalize_vector(&mut v);

            const MAX_ITER: usize = 100;
            const TOLERANCE: f32 = 1e-6;

            for _iter in 0..MAX_ITER {
                // v = A^T * (A * v)
                let mut av = vec![0.0f32; m];
                matrix_vector_multiply(&a, &v, &mut av, m, n);
                
                let mut atav = vec![0.0f32; n];
                matrix_transpose_vector_multiply(&a, &av, &mut atav, m, n);

                let old_v = v.clone();
                v = atav;
                normalize_vector(&mut v);

                // Check convergence
                let mut diff = 0.0f32;
                for i in 0..n {
                    diff += (v[i] - old_v[i]).abs();
                }
                if diff < TOLERANCE {
                    break;
                }
            }

            // Compute Av to get left singular vector
            let mut av = vec![0.0f32; m];
            matrix_vector_multiply(&a, &v, &mut av, m, n);
            let sigma = vector_norm(&av);
            s_data[k] = sigma;

            if sigma > 1e-12 {
                // Normalize to get left singular vector
                for i in 0..m {
                    av[i] /= sigma;
                }

                // Store singular vectors
                if compute_u {
                    for i in 0..m {
                        u_data[i * min_dim + k] = av[i];
                    }
                }
                
                if compute_vt {
                    for j in 0..n {
                        vt_data[k * n + j] = v[j];
                    }
                }

                // Deflate matrix: A = A - sigma * u * v^T
                for i in 0..m {
                    for j in 0..n {
                        a[i * n + j] -= sigma * av[i] * v[j];
                    }
                }
            } else {
                // Zero singular value - fill with zeros
                if compute_u {
                    for i in 0..m {
                        u_data[i * min_dim + k] = 0.0;
                    }
                }
                
                if compute_vt {
                    for j in 0..n {
                        vt_data[k * n + j] = 0.0;
                    }
                }
            }
        }

        // Create result tensors
        let u = if compute_u {
            if min_dim < m {
                // Thin SVD - U is m x min_dim  
                Tensor::from_data(u_data, vec![m, min_dim], &self.device)?
            } else {
                // Complete orthogonal basis for U (pad with random orthogonal vectors)
                let mut full_u_data = vec![0.0f32; m * m];
                
                // Copy existing U data
                for i in 0..m {
                    for j in 0..min_dim {
                        full_u_data[i * m + j] = u_data[i * min_dim + j];
                    }
                }
                
                // Fill remaining columns with orthogonal vectors using Gram-Schmidt
                for col in min_dim..m {
                    let mut new_col = vec![0.0f32; m];
                    if col < m {
                        new_col[col] = 1.0; // Start with standard basis vector
                    }
                    
                    // Gram-Schmidt orthogonalization against existing columns
                    for prev_col in 0..col {
                        let mut dot = 0.0f32;
                        for i in 0..m {
                            dot += new_col[i] * full_u_data[i * m + prev_col];
                        }
                        for i in 0..m {
                            new_col[i] -= dot * full_u_data[i * m + prev_col];
                        }
                    }
                    
                    // Normalize
                    let norm = (new_col.iter().map(|x| x * x).sum::<f32>()).sqrt();
                    if norm > 1e-12 {
                        for i in 0..m {
                            new_col[i] /= norm;
                            full_u_data[i * m + col] = new_col[i];
                        }
                    }
                }
                
                Tensor::from_data(full_u_data, vec![m, m], &self.device)?
            }
        } else {
            Tensor::zeros([m, min_dim], &self.device)?
        };

        let s = Tensor::from_data(s_data, vec![min_dim], &self.device)?;

        let vt = if compute_vt {
            if min_dim < n {
                // Thin SVD - V^T is min_dim x n
                Tensor::from_data(vt_data, vec![min_dim, n], &self.device)?
            } else {
                // Complete orthogonal basis for V^T
                let mut full_vt_data = vec![0.0f32; n * n];
                
                // Copy existing V^T data  
                for i in 0..min_dim {
                    for j in 0..n {
                        full_vt_data[i * n + j] = vt_data[i * n + j];
                    }
                }
                
                // Fill remaining rows with orthogonal vectors
                for row in min_dim..n {
                    let mut new_row = vec![0.0f32; n];
                    if row < n {
                        new_row[row] = 1.0;
                    }
                    
                    // Gram-Schmidt orthogonalization
                    for prev_row in 0..row {
                        let mut dot = 0.0f32;
                        for j in 0..n {
                            dot += new_row[j] * full_vt_data[prev_row * n + j];
                        }
                        for j in 0..n {
                            new_row[j] -= dot * full_vt_data[prev_row * n + j];
                        }
                    }
                    
                    // Normalize
                    let norm = (new_row.iter().map(|x| x * x).sum::<f32>()).sqrt();
                    if norm > 1e-12 {
                        for j in 0..n {
                            new_row[j] /= norm;
                            full_vt_data[row * n + j] = new_row[j];
                        }
                    }
                }
                
                Tensor::from_data(full_vt_data, vec![n, n], &self.device)?
            }
        } else {
            Tensor::zeros([min_dim, n], &self.device)?
        };

        Ok(SVDResult { u, s, vt })
    }

    /// Truncated SVD for dimensionality reduction
    ///
    /// Computes only the k largest singular values and corresponding vectors.
    /// More efficient than full SVD when only the dominant components are needed.
    ///
    /// # Arguments
    /// - `k`: Number of singular values/vectors to compute
    ///
    /// # Returns
    /// `SVDResult` with dimensions reduced to rank k
    pub fn svd_truncated(&self, k: usize) -> Result<SVDResult> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Truncated SVD requires 2D tensor"));
        }

        let dims = self.shape().dims();
        let m = dims[0];
        let n = dims[1];
        let min_dim = m.min(n);

        if k > min_dim {
            return Err(TensorError::shape(format!(
                "k ({}) cannot exceed min(m, n) ({})", k, min_dim
            )));
        }

        let data = self.to_cpu()?;
        
        // Use power iteration to find top k singular values/vectors
        let mut a = data.clone();
        let mut u_data = vec![0.0f32; m * k];
        let mut s_data = vec![0.0f32; k];
        let mut vt_data = vec![0.0f32; k * n];

        for i in 0..k {
            // Power iteration for i-th singular value
            let mut v = vec![1.0f32; n];
            normalize_vector(&mut v);

            for _iter in 0..50 {
                let mut av = vec![0.0f32; m];
                matrix_vector_multiply(&a, &v, &mut av, m, n);
                
                let mut atav = vec![0.0f32; n];
                matrix_transpose_vector_multiply(&a, &av, &mut atav, m, n);

                v = atav;
                normalize_vector(&mut v);
            }

            // Compute singular value and vectors
            let mut av = vec![0.0f32; m];
            matrix_vector_multiply(&a, &v, &mut av, m, n);
            let sigma = vector_norm(&av);
            s_data[i] = sigma;

            if sigma > 1e-12 {
                // Normalize left singular vector
                for j in 0..m {
                    av[j] /= sigma;
                    u_data[j * k + i] = av[j];
                }

                // Store right singular vector
                for j in 0..n {
                    vt_data[i * n + j] = v[j];
                }

                // Deflate matrix
                for row in 0..m {
                    for col in 0..n {
                        a[row * n + col] -= sigma * av[row] * v[col];
                    }
                }
            }
        }

        let u = Tensor::from_data(u_data, vec![m, k], &self.device)?;
        let s = Tensor::from_data(s_data, vec![k], &self.device)?;
        let vt = Tensor::from_data(vt_data, vec![k, n], &self.device)?;

        Ok(SVDResult { u, s, vt })
    }

    /// QR Decomposition using Householder reflections
    ///
    /// Decomposes matrix A into A = Q * R where:
    /// - Q is an orthogonal matrix
    /// - R is an upper triangular matrix
    ///
    /// # Returns
    /// `QRResult` containing Q and R tensors
    ///
    /// # Numerical Stability
    /// Uses Householder reflections for better numerical stability than Gram-Schmidt.
    pub fn qr(&self) -> Result<QRResult> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("QR requires 2D tensor"));
        }

        let dims = self.shape().dims();
        let m = dims[0]; // rows
        let n = dims[1]; // cols

        if m == 0 || n == 0 {
            return Err(TensorError::shape("QR requires non-empty matrix"));
        }

        let data = self.to_cpu()?;
        let mut r = data.clone(); // Will become R matrix
        let mut q = vec![0.0f32; m * m]; // Full Q matrix
        
        // Initialize Q as identity matrix
        for i in 0..m {
            q[i * m + i] = 1.0;
        }

        let k = n.min(m); // Number of Householder reflections

        // Apply Householder reflections
        for col in 0..k {
            // Extract column vector starting from diagonal
            let mut x = vec![0.0f32; m - col];
            for i in col..m {
                x[i - col] = r[i * n + col];
            }

            let norm_x = vector_norm(&x);
            if norm_x < 1e-12 {
                continue; // Skip near-zero columns
            }

            // Create Householder vector
            let mut v = x.clone();
            let sign = if v[0] >= 0.0 { 1.0 } else { -1.0 };
            v[0] += sign * norm_x;
            
            let norm_v = vector_norm(&v);
            if norm_v < 1e-12 {
                continue;
            }
            
            // Normalize Householder vector
            for i in 0..v.len() {
                v[i] /= norm_v;
            }

            let tau = 2.0;

            // Apply Householder reflection to R: R = (I - tau * v * v^T) * R
            // Only update the submatrix starting from (col, col)
            for j in col..n {
                let mut dot = 0.0f32;
                for i in 0..(m - col) {
                    dot += v[i] * r[(col + i) * n + j];
                }
                
                for i in 0..(m - col) {
                    r[(col + i) * n + j] -= tau * dot * v[i];
                }
            }

            // Apply Householder reflection to Q: Q = Q * (I - tau * v * v^T)
            // This builds up the Q matrix from the right
            for i in 0..m {
                let mut dot = 0.0f32;
                for j in 0..(m - col) {
                    dot += q[i * m + (col + j)] * v[j];
                }
                
                for j in 0..(m - col) {
                    q[i * m + (col + j)] -= tau * dot * v[j];
                }
            }
        }

        // For rectangular matrices, we want reduced QR
        let q_result = if m > n {
            // Extract first n columns of Q
            let mut q_reduced = vec![0.0f32; m * n];
            for i in 0..m {
                for j in 0..n {
                    q_reduced[i * n + j] = q[i * m + j];
                }
            }
            Tensor::from_data(q_reduced, vec![m, n], &self.device)?
        } else {
            Tensor::from_data(q, vec![m, m], &self.device)?
        };

        let r_result = if m > n {
            // Extract first n rows of R
            let mut r_reduced = vec![0.0f32; n * n];
            for i in 0..n {
                for j in 0..n {
                    r_reduced[i * n + j] = r[i * n + j];
                }
            }
            Tensor::from_data(r_reduced, vec![n, n], &self.device)?
        } else {
            Tensor::from_data(r, vec![m, n], &self.device)?
        };

        Ok(QRResult { q: q_result, r: r_result })
    }

    /// Cholesky Decomposition
    ///
    /// Decomposes a positive definite matrix A into A = L * L^T where L is lower triangular.
    ///
    /// # Arguments  
    /// - `upper`: If true, return upper triangular U such that A = U^T * U
    ///
    /// # Returns
    /// Lower triangular matrix L (or upper triangular U if upper=true)
    ///
    /// # Errors
    /// Returns error if matrix is not positive definite
    pub fn cholesky(&self, upper: bool) -> Result<Tensor> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Cholesky requires 2D tensor"));
        }

        let dims = self.shape().dims();
        if dims[0] != dims[1] {
            return Err(TensorError::shape("Cholesky requires square matrix"));
        }

        let n = dims[0];
        let data = self.to_cpu()?;
        let mut l = vec![0.0f32; n * n];

        // Cholesky-Banachiewicz algorithm
        for i in 0..n {
            for j in 0..=i {
                if i == j {
                    // Diagonal elements: L[i,i] = sqrt(A[i,i] - sum(L[i,k]^2 for k < i))
                    let mut sum = 0.0f32;
                    for k in 0..i {
                        sum += l[i * n + k] * l[i * n + k];
                    }
                    
                    let diagonal_val = data[i * n + i] - sum;
                    if diagonal_val <= 0.0 {
                        return Err(TensorError::numerical(
                            "Matrix is not positive definite - negative diagonal element in Cholesky"
                        ));
                    }
                    
                    l[i * n + i] = diagonal_val.sqrt();
                } else {
                    // Off-diagonal elements: L[i,j] = (A[i,j] - sum(L[i,k]*L[j,k] for k < j)) / L[j,j]
                    let mut sum = 0.0f32;
                    for k in 0..j {
                        sum += l[i * n + k] * l[j * n + k];
                    }
                    
                    if l[j * n + j].abs() < 1e-12 {
                        return Err(TensorError::numerical(
                            "Matrix is not positive definite - zero diagonal element in Cholesky"
                        ));
                    }
                    
                    l[i * n + j] = (data[i * n + j] - sum) / l[j * n + j];
                }
            }
        }

        if upper {
            // Return upper triangular matrix U = L^T
            let mut u = vec![0.0f32; n * n];
            for i in 0..n {
                for j in i..n {
                    u[i * n + j] = l[j * n + i];
                }
            }
            Tensor::from_data(u, vec![n, n], &self.device)
        } else {
            Tensor::from_data(l, vec![n, n], &self.device)
        }
    }

    /// Symmetric Eigenvalue Decomposition
    ///
    /// Computes eigenvalues and eigenvectors for symmetric matrices.
    /// For symmetric matrix A: A = V * Λ * V^T
    ///
    /// # Arguments
    /// - `eigenvectors`: Whether to compute eigenvectors (true) or just eigenvalues (false)
    ///
    /// # Returns
    /// `EigenResult` containing eigenvalues and optionally eigenvectors
    ///
    /// # Note
    /// Only works for symmetric matrices. Eigenvalues are returned in ascending order.
    pub fn symeig(&self, eigenvectors: bool) -> Result<EigenResult> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Eigendecomposition requires 2D tensor"));
        }

        let dims = self.shape().dims();
        if dims[0] != dims[1] {
            return Err(TensorError::shape("Eigendecomposition requires square matrix"));
        }

        let n = dims[0];
        let data = self.to_cpu()?;

        // Check symmetry
        for i in 0..n {
            for j in 0..n {
                if (data[i * n + j] - data[j * n + i]).abs() > 1e-8 {
                    return Err(TensorError::shape("symeig requires symmetric matrix"));
                }
            }
        }

        // Use power iteration method for eigenvalues/eigenvectors
        // This is simplified - production code would use QR algorithm or Jacobi method
        let mut eigenvalues = vec![0.0f32; n];
        let mut eigenvectors_data = if eigenvectors {
            vec![0.0f32; n * n]
        } else {
            vec![]
        };

        let mut a = data.clone();

        // Find eigenvalues using deflation
        for k in 0..n {
            // Power iteration for dominant eigenvalue
            let mut v = vec![1.0f32; n - k];
            for i in 1..(n - k) {
                v[i] = i as f32; // Different starting vector for each eigenvalue
            }
            normalize_vector(&mut v);

            let mut lambda = 0.0f32;
            
            for _iter in 0..100 {
                let mut av = vec![0.0f32; n - k];
                
                // Matrix-vector multiply with current submatrix
                for i in 0..(n - k) {
                    for j in 0..(n - k) {
                        av[i] += a[(k + i) * n + (k + j)] * v[j];
                    }
                }

                // Rayleigh quotient
                let mut vav = 0.0f32;
                let mut vv = 0.0f32;
                for i in 0..(n - k) {
                    vav += v[i] * av[i];
                    vv += v[i] * v[i];
                }

                let new_lambda = if vv > 1e-12 { vav / vv } else { 0.0 };
                
                // Normalize av to get new v
                normalize_vector(&mut av);
                
                // Check convergence
                if (new_lambda - lambda).abs() < 1e-8 {
                    lambda = new_lambda;
                    v = av;
                    break;
                }
                
                lambda = new_lambda;
                v = av;
            }

            eigenvalues[k] = lambda;

            if eigenvectors {
                // Store eigenvector (pad with zeros for deflated dimensions)
                for i in 0..n {
                    for j in 0..n {
                        if i >= k && j == k && i - k < v.len() {
                            eigenvectors_data[i * n + j] = v[i - k];
                        } else if i < k && j == k {
                            eigenvectors_data[i * n + j] = 0.0;
                        }
                    }
                }
            }

            // Deflate matrix: A = A - lambda * v * v^T
            if k < n - 1 {
                for i in 0..(n - k) {
                    for j in 0..(n - k) {
                        a[(k + i) * n + (k + j)] -= lambda * v[i] * v[j];
                    }
                }
            }
        }

        // Sort eigenvalues in ascending order (and reorder eigenvectors accordingly)
        let mut indices: Vec<usize> = (0..n).collect();
        indices.sort_by(|&a, &b| eigenvalues[a].partial_cmp(&eigenvalues[b]).unwrap());

        let mut sorted_eigenvalues = vec![0.0f32; n];
        for (i, &idx) in indices.iter().enumerate() {
            sorted_eigenvalues[i] = eigenvalues[idx];
        }

        let sorted_eigenvectors = if eigenvectors {
            let mut sorted = vec![0.0f32; n * n];
            for (new_col, &old_col) in indices.iter().enumerate() {
                for row in 0..n {
                    sorted[row * n + new_col] = eigenvectors_data[row * n + old_col];
                }
            }
            Tensor::from_data(sorted, vec![n, n], &self.device)?
        } else {
            Tensor::zeros([n, n], &self.device)?
        };

        Ok(EigenResult {
            eigenvalues: Tensor::from_data(sorted_eigenvalues, vec![n], &self.device)?,
            eigenvectors: sorted_eigenvectors,
        })
    }

    /// Moore-Penrose Pseudo-inverse using SVD
    ///
    /// Computes the pseudo-inverse A^+ such that:
    /// - If A has full row rank: A^+ = A^T(AA^T)^(-1) 
    /// - If A has full column rank: A^+ = (A^TA)^(-1)A^T
    /// - General case: A^+ = V * S^+ * U^T (via SVD)
    ///
    /// # Arguments
    /// - `rcond`: Relative condition number. Singular values below rcond*max(S) are set to zero
    ///
    /// # Returns
    /// Pseudo-inverse tensor with dimensions transposed relative to input
    pub fn pinv(&self, rcond: f32) -> Result<Tensor> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Pseudo-inverse requires 2D tensor"));
        }

        // Use SVD to compute pseudo-inverse
        let svd_result = self.svd(true, true)?;
        let s_data = svd_result.s.to_cpu()?;
        
        // Find cutoff threshold
        let s_max = s_data.iter().fold(0.0f32, |a, &b| a.max(b));
        let cutoff = rcond * s_max;
        
        // Compute reciprocal of singular values (with threshold)
        let mut s_inv_data = vec![0.0f32; s_data.len()];
        for (i, &s_val) in s_data.iter().enumerate() {
            s_inv_data[i] = if s_val > cutoff { 1.0 / s_val } else { 0.0 };
        }
        
        let s_inv = Tensor::from_data(s_inv_data, svd_result.s.shape().dims().to_vec(), &self.device)?;
        
        // A^+ = V * S^+ * U^T
        // Note: SVD gives us V^T, so we need to transpose it
        let v = svd_result.vt.transpose(0, 1)?; // V^T -> V
        let u_t = svd_result.u.transpose(0, 1)?; // U -> U^T
        let s_inv_diag = Tensor::diag(&s_inv)?;
        
        let result = v.matmul(&s_inv_diag)?.matmul(&u_t)?;
        Ok(result)
    }

    /// Compute matrix condition number (ratio of largest to smallest singular value)
    pub fn condition_number(&self) -> Result<f32> {
        let svd_result = self.svd(false, false)?;
        let s_data = svd_result.s.to_cpu()?;
        
        if s_data.is_empty() {
            return Ok(1.0);
        }
        
        let s_max = s_data[0]; // Singular values are sorted in descending order
        let s_min = s_data[s_data.len() - 1];
        
        if s_min < 1e-12 {
            Ok(f32::INFINITY)
        } else {
            Ok(s_max / s_min)
        }
    }

    /// Compute matrix rank (number of non-zero singular values)
    pub fn matrix_rank(&self, tol: f32) -> Result<usize> {
        let svd_result = self.svd(false, false)?;
        let s_data = svd_result.s.to_cpu()?;
        
        let s_max = s_data.iter().fold(0.0f32, |a, &b| a.max(b));
        let cutoff = tol * s_max;
        
        let rank = s_data.iter().filter(|&&s| s > cutoff).count();
        Ok(rank)
    }

    /// Compute matrix norm
    pub fn matrix_norm(&self, ord: &str) -> Result<f32> {
        match ord {
            "fro" => {
                // Frobenius norm: sqrt(sum of squares of all elements)
                let data = self.to_cpu()?;
                let sum_squares: f32 = data.iter().map(|&x| x * x).sum();
                Ok(sum_squares.sqrt())
            }
            "2" => {
                // Spectral norm: largest singular value
                let svd_result = self.svd(false, false)?;
                let s_data = svd_result.s.to_cpu()?;
                Ok(s_data.get(0).copied().unwrap_or(0.0))
            }
            _ => Err(TensorError::value(format!("Unsupported matrix norm: {}", ord)))
        }
    }

    /// Compute determinant using LU decomposition
    pub fn det(&self) -> Result<f32> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Determinant requires 2D tensor"));
        }

        let dims = self.shape().dims();
        if dims[0] != dims[1] {
            return Err(TensorError::shape("Determinant requires square matrix"));
        }

        let n = dims[0];
        if n == 0 {
            return Ok(1.0);
        }
        if n == 1 {
            let data = self.to_cpu()?;
            return Ok(data[0]);
        }
        if n == 2 {
            let data = self.to_cpu()?;
            return Ok(data[0] * data[3] - data[1] * data[2]);
        }

        // For larger matrices, use LU decomposition
        let data = self.to_cpu()?;
        let mut a = data.clone();
        let mut det = 1.0f32;
        
        // Gaussian elimination with partial pivoting
        for i in 0..n {
            // Find pivot
            let mut max_val = 0.0f32;
            let mut pivot_row = i;
            for k in i..n {
                let val = a[k * n + i].abs();
                if val > max_val {
                    max_val = val;
                    pivot_row = k;
                }
            }
            
            if max_val < 1e-12 {
                return Ok(0.0); // Singular matrix
            }
            
            // Swap rows if needed
            if pivot_row != i {
                for j in 0..n {
                    let temp = a[i * n + j];
                    a[i * n + j] = a[pivot_row * n + j];
                    a[pivot_row * n + j] = temp;
                }
                det = -det; // Row swap changes sign
            }
            
            det *= a[i * n + i];
            
            // Eliminate below diagonal
            for k in (i + 1)..n {
                let factor = a[k * n + i] / a[i * n + i];
                for j in i..n {
                    a[k * n + j] -= factor * a[i * n + j];
                }
            }
        }
        
        Ok(det)
    }

    /// Compute trace (sum of diagonal elements)
    pub fn trace(&self) -> Result<f32> {
        if self.ndim() != 2 {
            return Err(TensorError::shape("Trace requires 2D tensor"));
        }

        let dims = self.shape().dims();
        let n = dims[0].min(dims[1]);
        let data = self.to_cpu()?;
        let cols = dims[1];
        
        let mut trace = 0.0f32;
        for i in 0..n {
            trace += data[i * cols + i];
        }
        
        Ok(trace)
    }

    /// Create diagonal matrix from vector
    pub fn diag(diagonal: &Tensor) -> Result<Tensor> {
        if diagonal.ndim() != 1 {
            return Err(TensorError::shape("diag requires 1D tensor"));
        }

        let n = diagonal.shape().dims()[0];
        let diag_data = diagonal.to_cpu()?;
        let mut matrix_data = vec![0.0f32; n * n];
        
        for i in 0..n {
            matrix_data[i * n + i] = diag_data[i];
        }
        
        Tensor::from_data(matrix_data, vec![n, n], &diagonal.device)
    }

    // Note: eye function is already defined in tensor::core

    /// Safe reciprocal with threshold for pseudo-inverse computation
    pub fn reciprocal_safe(&self, threshold: f32) -> Result<Tensor> {
        let data = self.to_cpu()?;
        let result_data: Vec<f32> = data.iter()
            .map(|&x| if x.abs() > threshold { 1.0 / x } else { 0.0 })
            .collect();
        
        Tensor::from_data(result_data, self.shape().dims().to_vec(), &self.device)
    }

    /// Solve triangular systems of linear equations: A x = b
    /// 
    /// Efficiently solves triangular systems using forward or back substitution.
    /// This is much faster than general matrix solve for triangular matrices.
    ///
    /// # Arguments
    /// * `b` - Right-hand side tensor(s). Can be 1D (single RHS) or 2D (multiple RHS)
    /// * `options` - Configuration for the solve operation
    ///
    /// # Returns  
    /// Solution tensor x such that A x = b
    ///
    /// # Algorithm Complexity
    /// - Forward substitution (lower): O(n²) for single RHS, O(n²m) for m RHS
    /// - Back substitution (upper): O(n²) for single RHS, O(n²m) for m RHS
    ///
    /// # Numerical Stability
    /// - Checks for singular matrices (zero diagonal elements)
    /// - Optional triangular validation with configurable tolerance
    /// - Handles unit diagonal matrices efficiently
    pub fn solve_triangular(&self, b: &Tensor, options: &TriangularSolveOptions) -> Result<Tensor> {
        // Validate inputs
        if self.ndim() != 2 {
            return Err(TensorError::shape("solve_triangular requires 2D matrix"));
        }
        
        let matrix_dims = self.shape().dims();
        if matrix_dims[0] != matrix_dims[1] {
            return Err(TensorError::shape("solve_triangular requires square matrix"));
        }
        
        let n = matrix_dims[0];
        if n == 0 {
            return Err(TensorError::shape("solve_triangular requires non-empty matrix"));
        }
        
        // Validate RHS dimensions
        let b_dims = b.shape().dims();
        let is_multiple_rhs = b.ndim() == 2;
        let num_rhs = if is_multiple_rhs { b_dims[1] } else { 1 };
        
        if (is_multiple_rhs && b_dims[0] != n) || (!is_multiple_rhs && b.ndim() != 1) || (!is_multiple_rhs && b_dims[0] != n) {
            return Err(TensorError::shape(format!(
                "RHS dimensions {} do not match matrix dimensions [{}x{}]", 
                format!("{:?}", b_dims), n, n
            )));
        }

        // Check device compatibility
        if self.device != b.device {
            return Err(TensorError::device(format!(
                "Matrix and RHS must be on same device: {:?} vs {:?}",
                self.device, b.device
            )));
        }

        // Get data for computation
        let matrix_data = self.to_cpu()?;
        let b_data = b.to_cpu()?;
        
        // Triangular validation if requested
        if options.validate_triangular {
            validate_triangular_matrix(&matrix_data, n, options.triangular_type, options.strict_validation)?;
        }

        // Prepare result tensor shape and data
        let result_shape = b.shape().dims().to_vec();
        let mut result_data = vec![0.0f32; b.numel()];

        // Choose algorithm based on options
        if options.transpose {
            // Solve A^T x = b
            match options.triangular_type {
                TriangularType::Lower => {
                    // L^T is upper triangular, use back substitution
                    solve_upper_triangular_batch(&matrix_data, &b_data, &mut result_data, n, num_rhs, true, options.unit_diagonal)?;
                }
                TriangularType::Upper => {
                    // U^T is lower triangular, use forward substitution  
                    solve_lower_triangular_batch(&matrix_data, &b_data, &mut result_data, n, num_rhs, true, options.unit_diagonal)?;
                }
            }
        } else {
            // Solve A x = b
            match options.triangular_type {
                TriangularType::Lower => {
                    solve_lower_triangular_batch(&matrix_data, &b_data, &mut result_data, n, num_rhs, false, options.unit_diagonal)?;
                }
                TriangularType::Upper => {
                    solve_upper_triangular_batch(&matrix_data, &b_data, &mut result_data, n, num_rhs, false, options.unit_diagonal)?;
                }
            }
        }

        // Create result tensor and copy data back
        Tensor::from_data(result_data, result_shape, &self.device)
    }

    /// Solve least squares problem min ||Ax - b||² 
    ///
    /// Solves linear least squares problems for overdetermined and underdetermined systems:
    /// - Overdetermined (m > n): Finds x that minimizes ||Ax - b||²
    /// - Underdetermined (m < n): Finds minimum norm solution x such that Ax = b
    /// - Square well-conditioned: Exact solution if it exists
    /// - Rank-deficient: Minimum norm least squares solution
    ///
    /// # Arguments
    /// * `b` - Right-hand side vector or matrix (multiple RHS)
    /// * `options` - Optional solving configuration
    ///
    /// # Returns
    /// `LeastSquaresResult` containing solution and diagnostic information
    ///
    /// # Methods
    /// - **QR**: Best for overdetermined systems (m > n), numerically stable
    /// - **SVD**: Most robust, handles rank-deficient matrices, slower
    /// - **Normal Equations**: Fastest for well-conditioned problems, less stable
    /// - **Auto**: Automatically selects best method based on matrix properties
    ///
    /// # Features
    /// - Tikhonov regularization: (A^T A + λI)x = A^T b
    /// - Weighted least squares with diagonal weight matrix
    /// - Automatic condition number estimation and rank detection
    /// - Residual computation and goodness-of-fit metrics
    ///
    /// # Example
    /// ```rust
    /// use rtx_tensor::{Tensor, Device};
    /// use rtx_tensor::linalg::LeastSquaresOptions;
    ///
    /// let device = Device::cpu();
    /// let a = Tensor::from_data(vec![1.0, 1.0, 2.0, 1.0, 3.0, 1.0], vec![3, 2], &device).unwrap();
    /// let b = Tensor::from_data(vec![6.0, 8.0, 10.0], vec![3], &device).unwrap();
    /// 
    /// let result = a.least_squares(&b, None).unwrap();
    /// println!("Solution: {:?}", result.solution.to_cpu().unwrap());
    /// println!("Residual norm: {}", result.residual_norm);
    /// ```
    pub fn least_squares(&self, b: &Tensor, options: Option<&LeastSquaresOptions>) -> Result<LeastSquaresResult> {
        // Validate inputs
        if self.ndim() != 2 {
            return Err(TensorError::shape("least_squares requires 2D coefficient matrix"));
        }
        
        let matrix_dims = self.shape().dims();
        let m = matrix_dims[0]; // number of equations
        let n = matrix_dims[1]; // number of unknowns
        
        if m == 0 || n == 0 {
            return Err(TensorError::shape("least_squares requires non-empty matrix"));
        }
        
        // Validate RHS dimensions
        let b_dims = b.shape().dims();
        let is_multiple_rhs = b.ndim() == 2;
        let num_rhs = if is_multiple_rhs { b_dims[1] } else { 1 };
        
        if (is_multiple_rhs && b_dims[0] != m) || (!is_multiple_rhs && b.ndim() != 1) || (!is_multiple_rhs && b_dims[0] != m) {
            return Err(TensorError::shape(format!(
                "RHS dimensions {:?} do not match matrix rows {}", 
                b_dims, m
            )));
        }

        // Check device compatibility
        if self.device != b.device {
            return Err(TensorError::device(format!(
                "Matrix and RHS must be on same device: {:?} vs {:?}",
                self.device, b.device
            )));
        }

        // Use default options if none provided
        let default_opts = LeastSquaresOptions::default();
        let opts = options.unwrap_or(&default_opts);
        
        // Determine method to use
        let method_to_use = match opts.method {
            LeastSquaresMethod::Auto => {
                // Auto-select based on matrix properties
                if m >= n {
                    // Overdetermined or square - use QR for stability
                    LeastSquaresMethod::QR
                } else {
                    // Underdetermined - use SVD for minimum norm solution
                    LeastSquaresMethod::SVD
                }
            }
            specified_method => specified_method,
        };

        // Solve based on selected method
        match method_to_use {
            LeastSquaresMethod::QR => {
                self.solve_least_squares_qr(b, opts, num_rhs, is_multiple_rhs)
            }
            LeastSquaresMethod::SVD => {
                self.solve_least_squares_svd(b, opts, num_rhs, is_multiple_rhs)
            }
            LeastSquaresMethod::NormalEquations => {
                self.solve_least_squares_normal_equations(b, opts, num_rhs, is_multiple_rhs)
            }
            LeastSquaresMethod::Auto => unreachable!(), // Already handled above
        }
    }

    /// Solve least squares using QR decomposition
    /// Best for overdetermined systems (m >= n)
    fn solve_least_squares_qr(&self, b: &Tensor, opts: &LeastSquaresOptions, num_rhs: usize, is_multiple_rhs: bool) -> Result<LeastSquaresResult> {
        // Apply weights if provided
        let (weighted_a, weighted_b) = if let Some(ref weights) = opts.weights {
            self.apply_weights(b, weights)?
        } else {
            (self.clone(), b.clone())
        };

        // Compute QR decomposition
        let qr_result = weighted_a.qr()?;
        
        // Solve Rx = Q^T b using back substitution
        let qt = qr_result.q.transpose(0, 1)?;
        let qtb = qt.matmul(&weighted_b)?;
        
        // Solve upper triangular system R x = Q^T b
        let options = TriangularSolveOptions {
            triangular_type: TriangularType::Upper,
            transpose: false,
            unit_diagonal: false,
            validate_triangular: false, // We trust QR decomposition
            strict_validation: false,
        };
        
        let solution = qr_result.r.solve_triangular(&qtb, &options)?;
        
        // Compute residual and diagnostic information
        let residual_result = self.compute_residuals_and_metrics(&solution, b, &weighted_a)?;
        
        Ok(LeastSquaresResult {
            solution,
            residual_norm: residual_result.residual_norm,
            condition_number: residual_result.condition_number,
            rank: residual_result.rank,
            method_used: LeastSquaresMethod::QR,
            r_squared: residual_result.r_squared,
            sum_squared_residuals: residual_result.sum_squared_residuals,
        })
    }

    /// Solve least squares using SVD decomposition
    /// Most robust method, handles rank-deficient matrices
    fn solve_least_squares_svd(&self, b: &Tensor, opts: &LeastSquaresOptions, num_rhs: usize, is_multiple_rhs: bool) -> Result<LeastSquaresResult> {
        // Apply weights if provided
        let (weighted_a, weighted_b) = if let Some(ref weights) = opts.weights {
            self.apply_weights(b, weights)?
        } else {
            (self.clone(), b.clone())
        };

        // Compute SVD: A = U S V^T
        let svd_result = weighted_a.svd(true, true)?;
        
        // Compute pseudo-inverse using SVD with rcond threshold
        let s_data = svd_result.s.to_cpu()?;
        let s_max = s_data.iter().fold(0.0f32, |a, &b| a.max(b));
        let cutoff = opts.rcond * s_max;
        
        // Create S^+ (pseudo-inverse of singular values)
        let mut s_pinv_data = vec![0.0f32; s_data.len()];
        let mut rank = 0;
        for (i, &s_val) in s_data.iter().enumerate() {
            if s_val > cutoff {
                s_pinv_data[i] = 1.0 / s_val;
                rank += 1;
            } else {
                s_pinv_data[i] = 0.0;
            }
        }
        
        let s_pinv = Tensor::from_data(s_pinv_data, svd_result.s.shape().dims().to_vec(), &self.device)?;
        let s_pinv_diag = Tensor::diag(&s_pinv)?;
        
        // Compute x = V S^+ U^T b
        let ut = svd_result.u.transpose(0, 1)?;
        let v = svd_result.vt.transpose(0, 1)?;
        
        let utb = ut.matmul(&weighted_b)?;
        let s_pinv_utb = s_pinv_diag.matmul(&utb)?;
        let solution = v.matmul(&s_pinv_utb)?;
        
        // Compute residual and diagnostic information
        let residual_result = self.compute_residuals_and_metrics(&solution, b, &weighted_a)?;
        
        Ok(LeastSquaresResult {
            solution,
            residual_norm: residual_result.residual_norm,
            condition_number: if s_data[s_data.len() - 1] > 1e-15 { s_max / s_data[s_data.len() - 1] } else { f32::INFINITY },
            rank,
            method_used: LeastSquaresMethod::SVD,
            r_squared: residual_result.r_squared,
            sum_squared_residuals: residual_result.sum_squared_residuals,
        })
    }

    /// Solve least squares using normal equations (A^T A)x = A^T b
    /// Fastest but least numerically stable method
    fn solve_least_squares_normal_equations(&self, b: &Tensor, opts: &LeastSquaresOptions, num_rhs: usize, is_multiple_rhs: bool) -> Result<LeastSquaresResult> {
        // Apply weights if provided
        let (weighted_a, weighted_b) = if let Some(ref weights) = opts.weights {
            self.apply_weights(b, weights)?
        } else {
            (self.clone(), b.clone())
        };

        // Compute A^T A and A^T b
        let at = weighted_a.transpose(0, 1)?;
        let mut ata = at.matmul(&weighted_a)?;
        let atb = at.matmul(&weighted_b)?;
        
        // Apply Tikhonov regularization if specified
        if let Some(lambda) = opts.regularization_lambda {
            let n = ata.shape().dims()[0];
            let identity = Tensor::eye(n, &self.device)?;
            let regularization = identity.mul_scalar(lambda)?;
            ata = ata.add(&regularization)?;
        }
        
        // Solve (A^T A + λI)x = A^T b using Cholesky decomposition
        let l = ata.cholesky(false)?;
        
        // Forward substitution: solve L y = A^T b
        let forward_opts = TriangularSolveOptions {
            triangular_type: TriangularType::Lower,
            transpose: false,
            unit_diagonal: false,
            validate_triangular: false,
            strict_validation: false,
        };
        let y = l.solve_triangular(&atb, &forward_opts)?;
        
        // Back substitution: solve L^T x = y
        let back_opts = TriangularSolveOptions {
            triangular_type: TriangularType::Lower,
            transpose: true,
            unit_diagonal: false,
            validate_triangular: false,
            strict_validation: false,
        };
        let solution = l.solve_triangular(&y, &back_opts)?;
        
        // Compute residual and diagnostic information
        let residual_result = self.compute_residuals_and_metrics(&solution, b, &weighted_a)?;
        
        Ok(LeastSquaresResult {
            solution,
            residual_norm: residual_result.residual_norm,
            condition_number: residual_result.condition_number,
            rank: residual_result.rank,
            method_used: LeastSquaresMethod::NormalEquations,
            r_squared: residual_result.r_squared,
            sum_squared_residuals: residual_result.sum_squared_residuals,
        })
    }

    /// Matrix power (exponentiation) A^n
    ///
    /// Computes matrix exponentiation using efficient algorithms:
    /// - For n = 0: Returns identity matrix
    /// - For n > 0: Uses binary exponentiation (repeated squaring)  
    /// - For n < 0: Computes A^(-|n|) = (A^(-1))^|n| using matrix inverse
    ///
    /// # Arguments
    /// - `n`: Integer power (can be positive, negative, or zero)
    ///
    /// # Returns
    /// Matrix A raised to power n
    ///
    /// # Algorithm
    /// Uses binary exponentiation for efficiency:
    /// - If n = 2k: A^n = (A^k)^2
    /// - If n = 2k+1: A^n = A * (A^k)^2
    /// - Complexity: O(log|n|) matrix multiplications
    ///
    /// # Errors
    /// - Returns error if matrix is not square
    /// - Returns error if n < 0 and matrix is singular (not invertible)
    pub fn matrix_power(&self, n: i32) -> Result<Tensor> {
        // Input validation
        if self.ndim() != 2 {
            return Err(TensorError::shape("matrix_power requires 2D tensor"));
        }

        let dims = self.shape().dims();
        if dims[0] != dims[1] {
            return Err(TensorError::shape("matrix_power requires square matrix"));
        }

        let size = dims[0];
        if size == 0 {
            return Err(TensorError::shape("matrix_power requires non-empty matrix"));
        }

        // Special case: A^0 = I (identity matrix)
        if n == 0 {
            return Tensor::eye(size, &self.device);
        }

        // Special case: A^1 = A
        if n == 1 {
            return Ok(self.clone());
        }

        // Handle negative powers: A^n = (A^-1)^|n|
        if n < 0 {
            // Check if matrix is invertible by computing determinant
            let det = self.det()?;
            if det.abs() < 1e-12 {
                return Err(TensorError::numerical(
                    "Cannot compute negative power of singular matrix (determinant ≈ 0)"
                ));
            }

            // Compute inverse and then raise to positive power
            let inverse = self.pinv(1e-12)?;
            return inverse.matrix_power((-n) as i32);
        }

        // For positive powers, use binary exponentiation
        self.matrix_power_binary_exponentiation(n as u32)
    }

    /// Matrix power with floating-point exponent using eigendecomposition
    ///
    /// Computes A^p for real power p using eigendecomposition.
    /// For matrix A with eigendecomposition A = V*D*V^-1,
    /// we have A^p = V*D^p*V^-1 where D^p applies power element-wise.
    ///
    /// # Arguments
    /// - `p`: Real-valued power
    ///
    /// # Returns  
    /// Matrix A raised to power p
    ///
    /// # Requirements
    /// - Matrix must be square and diagonalizable
    /// - For non-integer powers, matrix should have positive eigenvalues for real result
    ///
    /// # Note
    /// This is more expensive than integer powers but supports fractional exponents.
    pub fn matrix_power_f32(&self, p: f32) -> Result<Tensor> {
        // Input validation
        if self.ndim() != 2 {
            return Err(TensorError::shape("matrix_power_f32 requires 2D tensor"));
        }

        let dims = self.shape().dims();
        if dims[0] != dims[1] {
            return Err(TensorError::shape("matrix_power_f32 requires square matrix"));
        }

        // Special case: p = 0 gives identity
        if (p - 0.0).abs() < 1e-12 {
            return Tensor::eye(dims[0], &self.device);
        }

        // Special case: p = 1 gives original matrix
        if (p - 1.0).abs() < 1e-12 {
            return Ok(self.clone());
        }

        // For integer powers, use more efficient integer algorithm
        if (p.fract()).abs() < 1e-12 {
            return self.matrix_power(p as i32);
        }

        // Use eigendecomposition for fractional powers
        // For now, implement simplified version for diagonal matrices
        // Full eigendecomposition would be more complex
        
        // Check if matrix is diagonal (simplified case)
        if self.is_diagonal()? {
            return self.matrix_power_diagonal_fractional(p);
        }

        // For general matrices, use eigendecomposition via symmetric case
        // This is a simplified implementation - production would handle non-symmetric matrices
        let eigen_result = self.symeig(true)?;
        
        // Compute D^p (eigenvalues raised to power p)
        let eigenvals_data = eigen_result.eigenvalues.to_cpu()?;
        let mut powered_eigenvals = vec![0.0f32; eigenvals_data.len()];
        
        for (i, &lambda) in eigenvals_data.iter().enumerate() {
            if lambda < 0.0 && (p.fract()).abs() > 1e-12 {
                return Err(TensorError::numerical(
                    "Fractional power of matrix with negative eigenvalues not supported"
                ));
            }
            powered_eigenvals[i] = lambda.powf(p);
        }
        
        let powered_eigenvals_tensor = Tensor::from_data(powered_eigenvals, vec![eigenvals_data.len()], &self.device)?;
        let d_powered = Tensor::diag(&powered_eigenvals_tensor)?;
        
        // Reconstruct: A^p = V * D^p * V^-1
        let v_inv = eigen_result.eigenvectors.pinv(1e-12)?;
        let result = eigen_result.eigenvectors.matmul(&d_powered)?.matmul(&v_inv)?;
        
        Ok(result)
    }

    /// Binary exponentiation for positive integer powers
    /// Implements efficient O(log n) algorithm using repeated squaring
    fn matrix_power_binary_exponentiation(&self, mut n: u32) -> Result<Tensor> {
        if n == 0 {
            return Tensor::eye(self.shape().dims()[0], &self.device);
        }

        let mut base = self.clone();
        let mut result = Tensor::eye(self.shape().dims()[0], &self.device)?;

        while n > 0 {
            // If n is odd, multiply result by current base
            if n % 2 == 1 {
                result = result.matmul(&base)?;
            }
            
            // Square the base and halve the exponent
            if n > 1 {
                base = base.matmul(&base)?;
            }
            n /= 2;
        }

        Ok(result)
    }

    /// Check if matrix is diagonal
    fn is_diagonal(&self) -> Result<bool> {
        let dims = self.shape().dims();
        let n = dims[0];
        let data = self.to_cpu()?;
        
        for i in 0..n {
            for j in 0..n {
                if i != j && data[i * n + j].abs() > 1e-12 {
                    return Ok(false);
                }
            }
        }
        
        Ok(true)
    }

    /// Compute fractional power for diagonal matrix
    fn matrix_power_diagonal_fractional(&self, p: f32) -> Result<Tensor> {
        let dims = self.shape().dims();
        let n = dims[0];
        let data = self.to_cpu()?;
        let mut result_data = vec![0.0f32; n * n];
        
        // For diagonal matrix, just raise each diagonal element to power p
        for i in 0..n {
            let diagonal_val = data[i * n + i];
            if diagonal_val < 0.0 && (p.fract()).abs() > 1e-12 {
                return Err(TensorError::numerical(
                    "Fractional power of negative diagonal element not supported"
                ));
            }
            result_data[i * n + i] = diagonal_val.powf(p);
        }
        
        Tensor::from_data(result_data, dims.to_vec(), &self.device)
    }

    /// LU Decomposition with partial pivoting
    ///
    /// Decomposes a matrix A into P*A = L*U where:
    /// - P is a permutation matrix representing row exchanges
    /// - L is a lower triangular matrix with 1s on the diagonal (Doolittle algorithm)
    /// - U is an upper triangular matrix
    ///
    /// # Arguments
    /// - `options`: Configuration options for the decomposition
    ///
    /// # Returns
    /// `LUResult` containing L, U, permutation information, and computed properties
    ///
    /// # Algorithms
    /// Uses Gaussian elimination with partial pivoting for numerical stability.
    /// The algorithm selects the largest element in each column as the pivot
    /// to minimize roundoff errors.
    ///
    /// # Features
    /// - Partial pivoting for numerical stability
    /// - Supports rectangular matrices (m×n)
    /// - Efficient determinant computation
    /// - Singularity detection
    /// - In-place option for memory efficiency
    ///
    /// # Example
    /// ```rust
    /// use rtx_tensor::{Tensor, Device};
    /// use rtx_tensor::linalg::LUOptions;
    ///
    /// let device = Device::cpu();
    /// let a = Tensor::from_data(vec![2.0, 1.0, 1.0, 1.0], vec![2, 2], &device).unwrap();
    /// 
    /// let options = LUOptions::default();
    /// let lu_result = a.lu_decomposition(&options).unwrap();
    /// 
    /// println!("L matrix: {:?}", lu_result.l.to_cpu().unwrap());
    /// println!("U matrix: {:?}", lu_result.u.to_cpu().unwrap());
    /// println!("Determinant: {}", lu_result.determinant);
    /// ```
    pub fn lu_decomposition(&self, options: &LUOptions) -> Result<LUResult> {
        // Input validation
        if self.ndim() != 2 {
            return Err(TensorError::shape("LU decomposition requires 2D tensor"));
        }

        let dims = self.shape().dims();
        let m = dims[0]; // rows
        let n = dims[1]; // cols
        
        if m == 0 || n == 0 {
            return Err(TensorError::shape("LU decomposition requires non-empty matrix"));
        }

        // Get matrix data
        let data = self.to_cpu()?;
        
        // Choose algorithm based on options
        if options.pivoting {
            self.lu_decomposition_with_pivoting(&data, m, n, options)
        } else {
            self.lu_decomposition_without_pivoting(&data, m, n, options)
        }
    }

    /// Batch LU decomposition for multiple matrices
    ///
    /// Performs LU decomposition on a batch of matrices efficiently.
    /// Input should be a tensor of shape (batch_size, m, n).
    ///
    /// # Arguments
    /// - `options`: Configuration options applied to all matrices in the batch
    ///
    /// # Returns
    /// `LUResult` with batched L, U, and permutation tensors
    pub fn lu_decomposition_batch(&self, options: &LUOptions) -> Result<LUResult> {
        // This will be implemented later
        Err(TensorError::not_implemented("Batch LU decomposition not yet implemented"))
    }

    /// LU decomposition without pivoting using Doolittle algorithm
    /// 
    /// Performs A = L*U decomposition where L has 1s on diagonal
    /// This method assumes the matrix doesn't need pivoting for numerical stability
    fn lu_decomposition_without_pivoting(&self, data: &[f32], m: usize, n: usize, options: &LUOptions) -> Result<LUResult> {
        let min_dim = m.min(n);
        
        // Create working matrix (copy of input)
        let mut a = data.to_vec();
        
        // Initialize L and U matrices
        let mut l_data = vec![0.0f32; m * min_dim];
        let mut u_data = if m <= n {
            a.clone() // U will be stored in-place in upper part
        } else {
            vec![0.0f32; min_dim * n] // U is smaller for tall matrices
        };
        
        // Perform Doolittle LU decomposition: A = L*U
        // L has 1s on diagonal, U has the computed values
        for k in 0..min_dim {
            // Check for zero pivot (singularity)
            if a[k * n + k].abs() < 1e-15 {
                // Matrix is singular - continue but mark as non-invertible
            }
            
            // Set L diagonal to 1 (Doolittle)
            l_data[k * min_dim + k] = 1.0;
            
            // Compute L column k (below diagonal)
            for i in (k + 1)..m {
                if a[k * n + k].abs() > 1e-15 {
                    l_data[i * min_dim + k] = a[i * n + k] / a[k * n + k];
                } else {
                    l_data[i * min_dim + k] = 0.0;
                }
            }
            
            // Update remaining submatrix: A[i,j] -= L[i,k] * U[k,j]
            for i in (k + 1)..m {
                for j in (k + 1)..n {
                    a[i * n + j] -= l_data[i * min_dim + k] * a[k * n + j];
                }
            }
        }
        
        // Extract U matrix from upper triangular part of A
        if m <= n {
            // Square or wide matrix: U is min_dim x n
            for i in 0..min_dim {
                for j in i..n {
                    u_data[i * n + j] = a[i * n + j];
                }
                // Zero out below diagonal
                for j in 0..i {
                    u_data[i * n + j] = 0.0;
                }
            }
        } else {
            // Tall matrix: U is min_dim x n
            for i in 0..min_dim {
                for j in 0..n {
                    if i <= j {
                        u_data[i * n + j] = a[i * n + j];
                    } else {
                        u_data[i * n + j] = 0.0;
                    }
                }
            }
        }
        
        // Create result tensors
        let l_shape = vec![m, min_dim];
        let u_shape = if m <= n { vec![min_dim, n] } else { vec![min_dim, n] };
        
        let l_tensor = Tensor::from_data(l_data, l_shape, &self.device)?;
        let u_tensor = Tensor::from_data(u_data, u_shape, &self.device)?;
        
        // Create identity permutation (no row swaps)
        let p_data: Vec<f32> = (0..m).map(|i| i as f32).collect();
        let p_tensor = Tensor::from_data(p_data, vec![m], &self.device)?;
        
        // Compute determinant (product of U diagonal elements)
        let mut determinant = 1.0f32;
        for i in 0..min_dim.min(n) {
            determinant *= a[i * n + i];
        }
        
        // Check if matrix is invertible (non-singular)
        let is_invertible = determinant.abs() > 1e-12 && m == n;
        
        // Estimate condition number (simplified)
        let condition_number = self.condition_number().unwrap_or(1.0);
        
        Ok(LUResult {
            l: l_tensor,
            u: u_tensor,
            p: p_tensor,
            p_matrix: if options.return_permutation_matrix {
                Some(Tensor::eye(m, &self.device)?) // Identity matrix for no pivoting
            } else {
                None
            },
            determinant,
            is_invertible,
            condition_number,
            num_pivots: 0, // No pivots in this algorithm
        })
    }
    
    /// LU decomposition with partial pivoting using Doolittle algorithm
    /// 
    /// Performs P*A = L*U decomposition with row pivoting for numerical stability
    fn lu_decomposition_with_pivoting(&self, data: &[f32], m: usize, n: usize, options: &LUOptions) -> Result<LUResult> {
        let min_dim = m.min(n);
        
        // Create working matrix (copy of input)
        let mut a = data.to_vec();
        
        // Track permutation
        let mut permutation: Vec<usize> = (0..m).collect();
        let mut num_pivots = 0;
        
        // Initialize L matrix
        let mut l_data = vec![0.0f32; m * min_dim];
        
        // Perform LU decomposition with partial pivoting
        for k in 0..min_dim {
            // Find the best pivot in column k (largest absolute value)
            let mut pivot_row = k;
            let mut max_val = 0.0f32;
            
            for i in k..m {
                let val = a[i * n + k].abs();
                if val > max_val {
                    max_val = val;
                    pivot_row = i;
                }
            }
            
            // Check for singularity
            if max_val < 1e-15 {
                // Column is effectively zero - matrix is singular
                // Continue with zero pivot but mark as non-invertible
            }
            
            // Swap rows if needed
            if pivot_row != k {
                // Swap rows in matrix A
                for j in 0..n {
                    let temp = a[k * n + j];
                    a[k * n + j] = a[pivot_row * n + j];
                    a[pivot_row * n + j] = temp;
                }
                
                // Swap rows in L matrix (for already computed part)
                for j in 0..k {
                    let temp = l_data[k * min_dim + j];
                    l_data[k * min_dim + j] = l_data[pivot_row * min_dim + j];
                    l_data[pivot_row * min_dim + j] = temp;
                }
                
                // Update permutation
                permutation.swap(k, pivot_row);
                num_pivots += 1;
            }
            
            // Set L diagonal to 1 (Doolittle)
            l_data[k * min_dim + k] = 1.0;
            
            // Compute L column k (below diagonal)
            for i in (k + 1)..m {
                if a[k * n + k].abs() > 1e-15 {
                    l_data[i * min_dim + k] = a[i * n + k] / a[k * n + k];
                } else {
                    l_data[i * min_dim + k] = 0.0;
                }
            }
            
            // Update remaining submatrix: A[i,j] -= L[i,k] * U[k,j]
            for i in (k + 1)..m {
                for j in (k + 1)..n {
                    a[i * n + j] -= l_data[i * min_dim + k] * a[k * n + j];
                }
            }
        }
        
        // Extract U matrix from upper triangular part of A
        let u_shape = if m <= n { vec![min_dim, n] } else { vec![min_dim, n] };
        let mut u_data = vec![0.0f32; u_shape[0] * u_shape[1]];
        
        for i in 0..min_dim {
            for j in i..n {
                if j < n {
                    u_data[i * n + j] = a[i * n + j];
                }
            }
        }
        
        // Create result tensors
        let l_tensor = Tensor::from_data(l_data, vec![m, min_dim], &self.device)?;
        let u_tensor = Tensor::from_data(u_data, u_shape, &self.device)?;
        
        // Create permutation vector
        let p_data: Vec<f32> = permutation.iter().map(|&i| i as f32).collect();
        let p_tensor = Tensor::from_data(p_data, vec![m], &self.device)?;
        
        // Create permutation matrix if requested
        let p_matrix = if options.return_permutation_matrix {
            let mut p_matrix_data = vec![0.0f32; m * m];
            for (i, &perm_i) in permutation.iter().enumerate() {
                p_matrix_data[i * m + perm_i] = 1.0;
            }
            Some(Tensor::from_data(p_matrix_data, vec![m, m], &self.device)?)
        } else {
            None
        };
        
        // Compute determinant (product of U diagonal * sign of permutation)
        let mut determinant = 1.0f32;
        for i in 0..min_dim.min(n) {
            determinant *= a[i * n + i];
        }
        // Adjust for row swaps (each swap changes sign)
        if num_pivots % 2 == 1 {
            determinant = -determinant;
        }
        
        // Check if matrix is invertible
        let is_invertible = determinant.abs() > 1e-12 && m == n;
        
        // Estimate condition number
        let condition_number = self.condition_number().unwrap_or(1.0);
        
        Ok(LUResult {
            l: l_tensor,
            u: u_tensor,
            p: p_tensor,
            p_matrix,
            determinant,
            is_invertible,
            condition_number,
            num_pivots,
        })
    }

    /// Apply diagonal weights to the least squares problem
    /// Transform Ax = b to sqrt(W)A x = sqrt(W)b where W is diagonal weight matrix
    fn apply_weights(&self, b: &Tensor, weights: &Tensor) -> Result<(Tensor, Tensor)> {
        // Weights should be a 1D tensor matching number of rows
        if weights.ndim() != 1 {
            return Err(TensorError::shape("Weights must be 1D tensor"));
        }
        
        let m = self.shape().dims()[0];
        if weights.shape().dims()[0] != m {
            return Err(TensorError::shape("Weights length must match number of equations"));
        }
        
        // Compute sqrt of weights for scaling
        let weights_data = weights.to_cpu()?;
        let sqrt_weights_data: Vec<f32> = weights_data.iter()
            .map(|&w| {
                if w < 0.0 {
                    panic!("Weights must be non-negative");
                }
                w.sqrt()
            })
            .collect();
        
        let sqrt_weights = Tensor::from_data(sqrt_weights_data, vec![m], &self.device)?;
        
        // Scale each row of A and b by sqrt(weight)
        let a_data = self.to_cpu()?;
        let b_data = b.to_cpu()?;
        let n = self.shape().dims()[1];
        
        let mut weighted_a_data = vec![0.0f32; m * n];
        let mut weighted_b_data = vec![0.0f32; b.numel()];
        
        let sqrt_weights_cpu = sqrt_weights.to_cpu()?;
        
        for i in 0..m {
            let weight = sqrt_weights_cpu[i];
            // Scale row i of A
            for j in 0..n {
                weighted_a_data[i * n + j] = a_data[i * n + j] * weight;
            }
            // Scale row i of b
            if b.ndim() == 1 {
                weighted_b_data[i] = b_data[i] * weight;
            } else {
                // Multiple RHS case
                let b_cols = b.shape().dims()[1];
                for j in 0..b_cols {
                    weighted_b_data[i * b_cols + j] = b_data[i * b_cols + j] * weight;
                }
            }
        }
        
        let weighted_a = Tensor::from_data(weighted_a_data, self.shape().dims().to_vec(), &self.device)?;
        let weighted_b = Tensor::from_data(weighted_b_data, b.shape().dims().to_vec(), &self.device)?;
        
        Ok((weighted_a, weighted_b))
    }
}

impl LUResult {
    /// Apply permutation to a matrix or vector
    ///
    /// Computes P*A where P is the permutation matrix from LU decomposition.
    /// This is used to verify the decomposition: P*A = L*U
    pub fn apply_permutation(&self, matrix: &Tensor) -> Result<Tensor> {
        if matrix.ndim() == 0 || matrix.ndim() > 2 {
            return Err(TensorError::shape("Permutation can only be applied to 1D or 2D tensors"));
        }

        let matrix_data = matrix.to_cpu()?;
        let p_data = self.p.to_cpu()?;
        let permutation: Vec<usize> = p_data.iter().map(|&x| x as usize).collect();
        
        if matrix.ndim() == 1 {
            // Apply permutation to vector
            let n = matrix.shape().dims()[0];
            if n != permutation.len() {
                return Err(TensorError::shape("Vector length doesn't match permutation size"));
            }
            
            let mut result_data = vec![0.0f32; n];
            for (i, &perm_i) in permutation.iter().enumerate() {
                if perm_i < n {
                    result_data[i] = matrix_data[perm_i];
                } else {
                    return Err(TensorError::shape("Invalid permutation index"));
                }
            }
            
            Tensor::from_data(result_data, vec![n], &matrix.device)
        } else {
            // Apply permutation to matrix (permute rows)
            let dims = matrix.shape().dims();
            let m = dims[0];
            let n = dims[1];
            
            if m != permutation.len() {
                return Err(TensorError::shape("Matrix rows don't match permutation size"));
            }
            
            let mut result_data = vec![0.0f32; m * n];
            for (new_row, &old_row) in permutation.iter().enumerate() {
                if old_row < m {
                    for j in 0..n {
                        result_data[new_row * n + j] = matrix_data[old_row * n + j];
                    }
                } else {
                    return Err(TensorError::shape("Invalid permutation index"));
                }
            }
            
            Tensor::from_data(result_data, dims.to_vec(), &matrix.device)
        }
    }

    /// Apply permutation for batch operations
    ///
    /// Applies permutation from batch index i to the corresponding matrix
    pub fn apply_permutation_batch(&self, matrix: &Tensor, batch_idx: usize) -> Result<Tensor> {
        // This will be implemented
        Err(TensorError::not_implemented("Batch permutation application not yet implemented"))
    }

    /// Solve linear system using LU decomposition
    ///
    /// Solves Ax = b using the precomputed LU decomposition.
    /// This is much faster than recomputing the decomposition for each solve.
    ///
    /// # Algorithm
    /// 1. Solve Ly = Pb using forward substitution
    /// 2. Solve Ux = y using back substitution
    ///
    /// # Arguments
    /// - `b`: Right-hand side vector or matrix (multiple RHS supported)
    ///
    /// # Returns
    /// Solution tensor x such that Ax = b
    pub fn solve(&self, b: &Tensor) -> Result<Tensor> {
        if b.ndim() == 0 || b.ndim() > 2 {
            return Err(TensorError::shape("RHS must be 1D or 2D tensor"));
        }

        let b_dims = b.shape().dims();
        let n = b_dims[0];
        
        // Check dimensions compatibility
        let l_dims = self.l.shape().dims();
        let u_dims = self.u.shape().dims();
        
        if l_dims[0] != n {
            return Err(TensorError::shape(format!(
                "L matrix rows {} don't match RHS rows {}",
                l_dims[0], n
            )));
        }
        
        if u_dims[0] != u_dims[1] {
            return Err(TensorError::shape("Cannot solve with rectangular U matrix - need square system"));
        }

        // Step 1: Apply permutation to b to get Pb
        let pb = self.apply_permutation(b)?;
        
        // Step 2: Solve Ly = Pb using forward substitution
        let forward_options = TriangularSolveOptions {
            triangular_type: TriangularType::Lower,
            transpose: false,
            unit_diagonal: true, // L has 1s on diagonal (Doolittle)
            validate_triangular: false,
            strict_validation: false,
        };
        
        let y = self.l.solve_triangular(&pb, &forward_options)?;
        
        // Step 3: Solve Ux = y using back substitution
        let back_options = TriangularSolveOptions {
            triangular_type: TriangularType::Upper,
            transpose: false,
            unit_diagonal: false, // U has computed values on diagonal
            validate_triangular: false,
            strict_validation: false,
        };
        
        let x = self.u.solve_triangular(&y, &back_options)?;
        
        Ok(x)
    }

    /// Compute matrix inverse using LU decomposition
    ///
    /// Computes A^(-1) using the LU decomposition by solving AX = I
    /// where I is the identity matrix.
    pub fn inverse(&self) -> Result<Tensor> {
        // This will be implemented
        Err(TensorError::not_implemented("LU inverse not yet implemented"))
    }
}

// Helper struct for residual computation results
struct ResidualMetrics {
    residual_norm: f32,
    condition_number: f32,
    rank: usize,
    r_squared: Option<f32>,
    sum_squared_residuals: f32,
}

impl Tensor {
    /// Compute residuals and goodness-of-fit metrics
    fn compute_residuals_and_metrics(&self, solution: &Tensor, b_original: &Tensor, a_used: &Tensor) -> Result<ResidualMetrics> {
        // Compute residual r = b - A*x
        let ax = self.matmul(solution)?;
        let residual = b_original.sub(&ax)?;
        
        // Compute residual norm
        let residual_data = residual.to_cpu()?;
        let residual_norm = (residual_data.iter().map(|&r| r * r).sum::<f32>()).sqrt();
        let sum_squared_residuals = residual_data.iter().map(|&r| r * r).sum::<f32>();
        
        // Estimate condition number using existing method
        let condition_number = self.condition_number()?;
        
        // Estimate rank using existing method
        let rank = self.matrix_rank(1e-12)?;
        
        // Compute R-squared if this is not a weighted problem
        let r_squared = if b_original.ndim() == 1 {
            let b_data = b_original.to_cpu()?;
            let b_mean = b_data.iter().sum::<f32>() / b_data.len() as f32;
            let total_sum_squares: f32 = b_data.iter().map(|&b_val| (b_val - b_mean) * (b_val - b_mean)).sum();
            
            if total_sum_squares > 1e-15 {
                Some(1.0 - sum_squared_residuals / total_sum_squares)
            } else {
                None
            }
        } else {
            None // R-squared not computed for multiple RHS
        };
        
        Ok(ResidualMetrics {
            residual_norm,
            condition_number,
            rank,
            r_squared,
            sum_squared_residuals,
        })
    }
}

// Helper functions for linear algebra computations

fn normalize_vector(v: &mut [f32]) {
    let norm = vector_norm(v);
    if norm > 1e-12 {
        for x in v.iter_mut() {
            *x /= norm;
        }
    }
}

fn vector_norm(v: &[f32]) -> f32 {
    v.iter().map(|&x| x * x).sum::<f32>().sqrt()
}

fn matrix_vector_multiply(matrix: &[f32], vector: &[f32], result: &mut [f32], rows: usize, cols: usize) {
    for i in 0..rows {
        result[i] = 0.0;
        for j in 0..cols {
            result[i] += matrix[i * cols + j] * vector[j];
        }
    }
}

fn matrix_transpose_vector_multiply(matrix: &[f32], vector: &[f32], result: &mut [f32], rows: usize, cols: usize) {
    for j in 0..cols {
        result[j] = 0.0;
        for i in 0..rows {
            result[j] += matrix[i * cols + j] * vector[i];
        }
    }
}

/// Validate that a matrix is triangular according to the specified type
fn validate_triangular_matrix(data: &[f32], n: usize, tri_type: TriangularType, strict: bool) -> Result<()> {
    let tolerance = if strict { 0.0 } else { 1e-9 };
    
    match tri_type {
        TriangularType::Lower => {
            // Check that upper triangle is zero (or within tolerance)
            for i in 0..n {
                for j in (i + 1)..n {
                    let idx = i * n + j;
                    if data[idx].abs() > tolerance {
                        return Err(TensorError::value(format!(
                            "Matrix is not lower triangular: element at [{},{}] = {} exceeds tolerance {}",
                            i, j, data[idx], tolerance
                        )));
                    }
                }
            }
        }
        TriangularType::Upper => {
            // Check that lower triangle is zero (or within tolerance)
            for i in 0..n {
                for j in 0..i {
                    let idx = i * n + j;
                    if data[idx].abs() > tolerance {
                        return Err(TensorError::value(format!(
                            "Matrix is not upper triangular: element at [{},{}] = {} exceeds tolerance {}",
                            i, j, data[idx], tolerance
                        )));
                    }
                }
            }
        }
    }
    
    Ok(())
}

/// Forward substitution for lower triangular systems (with batch support)
/// Solves L x = b or L^T x = b (if transpose=true)
fn solve_lower_triangular_batch(
    matrix: &[f32], 
    b: &[f32], 
    result: &mut [f32], 
    n: usize, 
    num_rhs: usize,
    transpose: bool,
    unit_diagonal: bool
) -> Result<()> {
    
    for rhs_idx in 0..num_rhs {
        if transpose {
            // Solve L^T x = b (L^T is upper triangular)
            // Back substitution from bottom to top
            for i in (0..n).rev() {
                let b_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                let result_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                
                let mut sum = 0.0f32;
                
                // Sum contributions from already solved variables
                for j in (i + 1)..n {
                    let matrix_idx = j * n + i; // L^T[i,j] = L[j,i] 
                    let x_idx = if num_rhs == 1 { j } else { j * num_rhs + rhs_idx };
                    sum += matrix[matrix_idx] * result[x_idx];
                }
                
                // Solve for current variable
                let diagonal_val = if unit_diagonal { 
                    1.0 
                } else { 
                    matrix[i * n + i] 
                };
                
                if !unit_diagonal && diagonal_val.abs() < 1e-12 {
                    return Err(TensorError::numerical(format!(
                        "Matrix is singular: diagonal element at [{},{}] = {}",
                        i, i, diagonal_val
                    )));
                }
                
                result[result_idx] = (b[b_idx] - sum) / diagonal_val;
            }
        } else {
            // Standard forward substitution for L x = b
            for i in 0..n {
                let b_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                let result_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                
                let mut sum = 0.0f32;
                
                // Sum contributions from already solved variables
                for j in 0..i {
                    let matrix_idx = i * n + j;
                    let x_idx = if num_rhs == 1 { j } else { j * num_rhs + rhs_idx };
                    sum += matrix[matrix_idx] * result[x_idx];
                }
                
                // Solve for current variable
                let diagonal_val = if unit_diagonal { 
                    1.0 
                } else { 
                    matrix[i * n + i] 
                };
                
                if !unit_diagonal && diagonal_val.abs() < 1e-12 {
                    return Err(TensorError::numerical(format!(
                        "Matrix is singular: diagonal element at [{},{}] = {}",
                        i, i, diagonal_val
                    )));
                }
                
                result[result_idx] = (b[b_idx] - sum) / diagonal_val;
            }
        }
    }
    
    Ok(())
}

/// Back substitution for upper triangular systems (with batch support)
/// Solves U x = b or U^T x = b (if transpose=true)
fn solve_upper_triangular_batch(
    matrix: &[f32], 
    b: &[f32], 
    result: &mut [f32], 
    n: usize, 
    num_rhs: usize,
    transpose: bool,
    unit_diagonal: bool
) -> Result<()> {
    
    for rhs_idx in 0..num_rhs {
        if transpose {
            // Solve U^T x = b (U^T is lower triangular)
            // Forward substitution from top to bottom
            for i in 0..n {
                let b_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                let result_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                
                let mut sum = 0.0f32;
                
                // Sum contributions from already solved variables
                for j in 0..i {
                    let matrix_idx = j * n + i; // U^T[i,j] = U[j,i]
                    let x_idx = if num_rhs == 1 { j } else { j * num_rhs + rhs_idx };
                    sum += matrix[matrix_idx] * result[x_idx];
                }
                
                // Solve for current variable
                let diagonal_val = if unit_diagonal { 
                    1.0 
                } else { 
                    matrix[i * n + i] 
                };
                
                if !unit_diagonal && diagonal_val.abs() < 1e-12 {
                    return Err(TensorError::numerical(format!(
                        "Matrix is singular: diagonal element at [{},{}] = {}",
                        i, i, diagonal_val
                    )));
                }
                
                result[result_idx] = (b[b_idx] - sum) / diagonal_val;
            }
        } else {
            // Standard back substitution for U x = b
            for i in (0..n).rev() {
                let b_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                let result_idx = if num_rhs == 1 { i } else { i * num_rhs + rhs_idx };
                
                let mut sum = 0.0f32;
                
                // Sum contributions from already solved variables
                for j in (i + 1)..n {
                    let matrix_idx = i * n + j;
                    let x_idx = if num_rhs == 1 { j } else { j * num_rhs + rhs_idx };
                    sum += matrix[matrix_idx] * result[x_idx];
                }
                
                // Solve for current variable
                let diagonal_val = if unit_diagonal { 
                    1.0 
                } else { 
                    matrix[i * n + i] 
                };
                
                if !unit_diagonal && diagonal_val.abs() < 1e-12 {
                    return Err(TensorError::numerical(format!(
                        "Matrix is singular: diagonal element at [{},{}] = {}",
                        i, i, diagonal_val
                    )));
                }
                
                result[result_idx] = (b[b_idx] - sum) / diagonal_val;
            }
        }
    }
    
    Ok(())
}