numr 0.5.1

//! Core linear algebra algorithm trait
//!
//! Defines the mathematical contract that all backends must implement
//! for consistent linear algebra operations.

use super::super::MatrixNormOrder;
use super::super::decompositions::*;
use crate::error::{Error, Result};
use crate::runtime::Runtime;
use crate::tensor::Tensor;

/// Algorithmic contract for linear algebra operations
///
/// All backends implementing linear algebra MUST implement this trait
/// using the EXACT SAME ALGORITHMS to ensure numerical parity.
///
/// # Implementation Requirements
///
/// Backends may differ in:
/// - Parallelization strategy (threads, SIMD, GPU blocks)
/// - Memory access patterns (blocking, tiling)
///
/// Backends MUST match in:
/// - Mathematical formula (same operations in same order)
/// - Pivot selection criteria (partial pivoting for LU)
/// - Special case handling (singular matrices, non-positive-definite)
pub trait LinearAlgebraAlgorithms<R: Runtime> {
    /// LU Decomposition with partial pivoting: PA = LU
    fn lu_decompose(&self, a: &Tensor<R>) -> Result<LuDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::lu_decompose",
        })
    }

    /// Cholesky Decomposition: A = LL^T
    fn cholesky_decompose(&self, a: &Tensor<R>) -> Result<CholeskyDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::cholesky_decompose",
        })
    }

    /// QR Decomposition using Householder reflections: A = QR
    fn qr_decompose(&self, a: &Tensor<R>) -> Result<QrDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::qr_decompose",
        })
    }

    /// Thin QR Decomposition: A = QR where Q is `[m, k]` and R is `[k, n]`
    fn qr_decompose_thin(&self, a: &Tensor<R>) -> Result<QrDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::qr_decompose_thin",
        })
    }

    /// Solve linear system Ax = b using LU decomposition
    fn solve(&self, a: &Tensor<R>, b: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = (a, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::solve",
        })
    }

    /// Solve triangular system Lx = b (forward substitution)
    fn solve_triangular_lower(
        &self,
        l: &Tensor<R>,
        b: &Tensor<R>,
        unit_diagonal: bool,
    ) -> Result<Tensor<R>> {
        let _ = (l, b, unit_diagonal);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::solve_triangular_lower",
        })
    }

    /// Solve triangular system Ux = b (backward substitution)
    fn solve_triangular_upper(&self, u: &Tensor<R>, b: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = (u, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::solve_triangular_upper",
        })
    }

    /// Least squares solution: minimize ||Ax - b||²
    fn lstsq(&self, a: &Tensor<R>, b: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = (a, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::lstsq",
        })
    }

    /// Matrix inverse using LU decomposition
    fn inverse(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::inverse",
        })
    }

    /// Matrix determinant using LU decomposition
    fn det(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::det",
        })
    }

    /// Matrix trace: sum of diagonal elements
    fn trace(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::trace",
        })
    }

    /// Extract diagonal elements
    fn diag(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::diag",
        })
    }

    /// Create diagonal matrix from 1D tensor
    fn diagflat(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::diagflat",
        })
    }

    /// Kronecker product: A ⊗ B
    ///
    /// Computes the Kronecker product of two matrices. For A of shape [m, n]
    /// and B of shape [p, q], the result has shape [m*p, n*q].
    ///
    /// # Definition
    ///
    /// ```text
    /// (A ⊗ B)[i*p + k, j*q + l] = A[i, j] * B[k, l]
    /// ```
    ///
    /// Equivalently, the Kronecker product replaces each element a_ij of A
    /// with the block a_ij * B.
    ///
    /// # Properties
    ///
    /// - `(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)` (associative)
    /// - `A ⊗ (B + C) = A ⊗ B + A ⊗ C` (distributive)
    /// - `(A ⊗ B)^T = A^T ⊗ B^T`
    /// - `(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)` (mixed-product property)
    /// - `det(A ⊗ B) = det(A)^q * det(B)^m` for square matrices
    ///
    /// # Use Cases
    ///
    /// - Quantum computing (tensor products of quantum states)
    /// - Control theory (Sylvester/Lyapunov equation solvers)
    /// - Signal processing (2D filtering)
    /// - Graph theory (graph products)
    fn kron(&self, a: &Tensor<R>, b: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = (a, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::kron",
        })
    }

    /// Upper triangular part of a matrix
    ///
    /// Returns a copy of the matrix with elements below the k-th diagonal zeroed.
    /// Supports all numeric dtypes (not just F32/F64).
    fn triu(&self, a: &Tensor<R>, diagonal: i64) -> Result<Tensor<R>> {
        let _ = (a, diagonal);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::triu",
        })
    }

    /// Lower triangular part of a matrix
    ///
    /// Returns a copy of the matrix with elements above the k-th diagonal zeroed.
    /// Supports all numeric dtypes (not just F32/F64).
    fn tril(&self, a: &Tensor<R>, diagonal: i64) -> Result<Tensor<R>> {
        let _ = (a, diagonal);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::tril",
        })
    }

    /// Sign and log-absolute-determinant
    ///
    /// Computes sign(det(A)) and log(|det(A)|) separately for numerical stability.
    fn slogdet(&self, a: &Tensor<R>) -> Result<SlogdetResult<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::slogdet",
        })
    }

    /// Solve banded linear system Ab*x = b
    ///
    /// Uses LAPACK-style band storage where `ab` has shape `[kl + ku + 1, n]`
    /// and `ab[ku + i - j, j] = A[i, j]` for `max(0, j-ku) <= i <= min(n-1, j+kl)`.
    ///
    /// For tridiagonal systems (kl=1, ku=1), uses Thomas algorithm O(n).
    /// For general banded systems, uses banded LU with partial pivoting.
    ///
    /// # Arguments
    ///
    /// * `ab` - Band matrix in LAPACK band storage `[kl + ku + 1, n]`
    /// * `b` - Right-hand side vector `[n]` or matrix `[n, nrhs]`
    /// * `kl` - Number of subdiagonals
    /// * `ku` - Number of superdiagonals
    fn solve_banded(
        &self,
        ab: &Tensor<R>,
        b: &Tensor<R>,
        kl: usize,
        ku: usize,
    ) -> Result<Tensor<R>> {
        let _ = (ab, b, kl, ku);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::solve_banded",
        })
    }

    /// Khatri-Rao product (column-wise Kronecker product): A ⊙ B
    ///
    /// Computes the column-wise Kronecker product of two matrices.
    /// For A of shape [m, k] and B of shape [n, k], the result has shape [m*n, k].
    ///
    /// # Definition
    ///
    /// ```text
    /// (A ⊙ B)[:, j] = A[:, j] ⊗ B[:, j]
    /// ```
    ///
    /// Each column of the result is the Kronecker product of the corresponding
    /// columns of A and B.
    ///
    /// # Properties
    ///
    /// - `(A ⊙ B)^T (A ⊙ B) = (A^T A) * (B^T B)` (Hadamard/element-wise product)
    /// - Essential for CP/PARAFAC tensor decomposition
    /// - Related to mode-n unfolding operations
    ///
    /// # Use Cases
    ///
    /// - CP/PARAFAC tensor decomposition (ALS updates)
    /// - Tucker decomposition
    /// - Multi-linear algebra
    /// - Compressed sensing
    fn khatri_rao(&self, a: &Tensor<R>, b: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = (a, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::khatri_rao",
        })
    }

    /// Matrix rank via SVD
    fn matrix_rank(&self, a: &Tensor<R>, tol: Option<f64>) -> Result<Tensor<R>> {
        let _ = (a, tol);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::matrix_rank",
        })
    }

    /// Matrix norm (Frobenius, Spectral, or Nuclear)
    fn matrix_norm(&self, a: &Tensor<R>, ord: MatrixNormOrder) -> Result<Tensor<R>> {
        let _ = (a, ord);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::matrix_norm",
        })
    }

    /// Singular Value Decomposition: A = U @ diag(S) @ V^T
    fn svd_decompose(&self, a: &Tensor<R>) -> Result<SvdDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::svd_decompose",
        })
    }

    /// Eigendecomposition for symmetric matrices: A = V @ diag(λ) @ V^T
    fn eig_decompose_symmetric(&self, a: &Tensor<R>) -> Result<EigenDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::eig_decompose_symmetric",
        })
    }

    /// General Eigendecomposition for non-symmetric matrices
    fn eig_decompose(&self, a: &Tensor<R>) -> Result<GeneralEigenDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::eig_decompose",
        })
    }

    /// Schur Decomposition: A = Z @ T @ Z^T
    fn schur_decompose(&self, a: &Tensor<R>) -> Result<SchurDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::schur_decompose",
        })
    }

    /// Moore-Penrose pseudo-inverse via SVD
    fn pinverse(&self, a: &Tensor<R>, rcond: Option<f64>) -> Result<Tensor<R>> {
        let _ = (a, rcond);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::pinverse",
        })
    }

    /// Matrix condition number via SVD
    fn cond(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::cond",
        })
    }

    /// Covariance matrix
    fn cov(&self, a: &Tensor<R>, ddof: Option<usize>) -> Result<Tensor<R>> {
        let _ = (a, ddof);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::cov",
        })
    }

    /// Correlation coefficient matrix (Pearson correlation)
    fn corrcoef(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::corrcoef",
        })
    }

    /// Convert Real Schur form to Complex Schur form: rsf2csf
    ///
    /// Transforms the real Schur decomposition (with 2×2 blocks for complex
    /// eigenvalue pairs) into the complex Schur form where T is truly upper
    /// triangular with complex eigenvalues on the diagonal.
    ///
    /// # Algorithm
    ///
    /// For each 2×2 block in T corresponding to complex conjugate eigenvalues a ± bi:
    /// 1. Compute eigenvalues of the 2×2 block
    /// 2. Construct a 2×2 unitary rotation that diagonalizes the block
    /// 3. Apply the rotation to T and accumulate into Z
    ///
    /// # Input
    ///
    /// Takes the output of `schur_decompose` (real Schur form).
    ///
    /// # Output
    ///
    /// Returns complex Schur form where:
    /// - T is upper triangular (not quasi-triangular)
    /// - Eigenvalues appear on diagonal of T
    /// - Z is unitary
    fn rsf2csf(&self, schur: &SchurDecomposition<R>) -> Result<ComplexSchurDecomposition<R>> {
        let _ = schur;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::rsf2csf",
        })
    }

    /// Generalized Schur (QZ) decomposition for matrix pencil (A, B)
    ///
    /// Computes the QZ decomposition: A = Q @ S @ Z^H, B = Q @ T @ Z^H
    ///
    /// # Algorithm
    ///
    /// 1. Reduce (A, B) to Hessenberg-triangular form (H, T)
    /// 2. Apply QZ iteration to reduce H to quasi-triangular S
    /// 3. Extract generalized eigenvalues from diagonal blocks
    ///
    /// # Generalized Eigenvalues
    ///
    /// The generalized eigenvalues `λ` satisfy: `det(A - λB) = 0`
    /// Computed as: `λ_i = alpha_i / beta_i` where `alpha` and `beta` are
    /// extracted from the diagonal of `S` and `T` respectively.
    ///
    /// # Requirements
    ///
    /// Both A and B must be square matrices of the same size.
    fn qz_decompose(
        &self,
        a: &Tensor<R>,
        b: &Tensor<R>,
    ) -> Result<GeneralizedSchurDecomposition<R>> {
        let _ = (a, b);
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::qz_decompose",
        })
    }

    /// Polar decomposition: A = U @ P
    ///
    /// Decomposes matrix A into a unitary matrix U and a positive semi-definite
    /// Hermitian matrix P.
    ///
    /// # Algorithm
    ///
    /// Uses the Newton iteration for matrix sign function:
    /// 1. Compute SVD: A = U_svd @ S @ V^H
    /// 2. U = U_svd @ V^H (unitary factor)
    /// 3. P = V @ S @ V^H (positive semi-definite factor)
    ///
    /// Alternative: Newton iteration X_{k+1} = (X_k + X_k^{-H}) / 2
    ///
    /// # Properties
    ///
    /// - For invertible A: `U` is the closest unitary matrix to A (in Frobenius norm)
    /// - `P` is unique and equals `sqrt(A^H @ A)`
    /// - For real matrices: `U` is orthogonal, `P` is symmetric positive semi-definite
    fn polar_decompose(&self, a: &Tensor<R>) -> Result<PolarDecomposition<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "LinearAlgebraAlgorithms::polar_decompose",
        })
    }
}