numr 0.5.1

//! Matrix function operations (expm, logm, sqrtm, etc.)
//!
//! Matrix functions extend scalar functions to matrices. For a scalar function `f`
//! and a diagonalizable matrix `A = V @ diag(λ) @ V^{-1}`, the matrix function is:
//!
//! `f(A) = V @ diag(f(λ)) @ V^{-1}`
//!
//! For non-diagonalizable matrices, the Schur decomposition is used:
//! `A = Z @ T @ Z^T`, then `f(A) = Z @ f(T) @ Z^T`
//!
//! where f(T) is computed using special formulas for quasi-triangular matrices.

use crate::error::{Error, Result};
use crate::runtime::Runtime;
use crate::tensor::Tensor;

/// Trait for matrix function operations (expm, logm, sqrtm)
///
/// Matrix functions extend scalar functions to matrices. For a scalar function `f`
/// and a diagonalizable matrix `A = V @ diag(λ) @ V^{-1}`, the matrix function is:
///
/// `f(A) = V @ diag(f(λ)) @ V^{-1}`
///
/// For non-diagonalizable matrices, the Schur decomposition is used:
/// `A = Z @ T @ Z^T`, then `f(A) = Z @ f(T) @ Z^T`
///
/// where f(T) is computed using special formulas for quasi-triangular matrices.
///
/// # Implemented Functions
///
/// - **expm**: Matrix exponential e^A
/// - **logm**: Matrix logarithm (principal branch)
/// - **sqrtm**: Matrix square root (principal branch)
///
/// # Use Cases
///
/// - **expm**: Solving linear ODEs `dx/dt = Ax` → `x(t) = e^{At} x(0)`
/// - **logm**: Computing matrix powers `A^p = e^{p*log(A)}`
/// - **sqrtm**: Polar decomposition, control theory
pub trait MatrixFunctionsAlgorithms<R: Runtime> {
    /// Matrix exponential: e^A
    ///
    /// Computes the matrix exponential using the Schur-Parlett algorithm:
    /// 1. Compute Schur decomposition: `A = Z @ T @ Z^T`
    /// 2. Compute `exp(T)` for quasi-triangular `T`
    /// 3. Reconstruct: `exp(A) = Z @ exp(T) @ Z^T`
    ///
    /// # Algorithm for Quasi-Triangular T
    ///
    /// For 1×1 diagonal blocks: `exp(t_ii) = e^{t_ii}`
    ///
    /// For 2×2 diagonal blocks (complex conjugate eigenvalues `a ± bi`):
    /// ```text
    /// exp([a, b; -b, a]) = e^a * [cos(b), sin(b); -sin(b), cos(b)]
    /// ```
    ///
    /// For off-diagonal elements, use the formula:
    /// ```text
    /// exp(T)[i,j] = (exp(T[i,i]) - exp(T[j,j])) / (T[i,i] - T[j,j]) * T[i,j]
    ///              when T[i,i] ≠ T[j,j]
    /// exp(T)[i,j] = exp(T[i,i]) * T[i,j]  when T[i,i] = T[j,j]
    /// ```
    ///
    /// # Properties
    ///
    /// - `exp(0) = I` (identity)
    /// - `exp(A + B) = exp(A) @ exp(B)` if `AB = BA`
    /// - `det(exp(A)) = e^{tr(A)}`
    /// - `exp(A)^{-1} = exp(-A)`
    fn expm(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::expm",
        })
    }

    /// Matrix logarithm: log(A) (principal branch)
    ///
    /// Computes the principal matrix logarithm using Schur decomposition
    /// with inverse scaling and squaring.
    ///
    /// # Requirements
    ///
    /// The matrix A must have no eigenvalues on the closed negative real axis.
    fn logm(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::logm",
        })
    }

    /// Matrix square root: A^{1/2} (principal branch)
    ///
    /// Computes the principal square root using Denman-Beavers iteration:
    /// ```text
    /// Y_0 = A, Z_0 = I
    /// REPEAT:
    ///   Y_{k+1} = (Y_k + Z_k^{-1}) / 2
    ///   Z_{k+1} = (Z_k + Y_k^{-1}) / 2
    /// UNTIL convergence
    /// sqrt(A) = Y_∞
    /// ```
    ///
    /// # Requirements
    ///
    /// The matrix A must have no eigenvalues on the closed negative real axis.
    fn sqrtm(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::sqrtm",
        })
    }

    /// Matrix sign function: sign(A)
    ///
    /// Computes the matrix sign function using Newton iteration:
    /// ```text
    /// X_0 = A
    /// REPEAT:
    ///   X_{k+1} = (X_k + X_k^{-1}) / 2
    /// UNTIL convergence
    /// sign(A) = X_∞
    /// ```
    ///
    /// # Properties
    ///
    /// - sign(A)^2 = I (involutory)
    /// - Eigenvalues of sign(A) are +1 or -1
    fn signm(&self, a: &Tensor<R>) -> Result<Tensor<R>> {
        let _ = a;
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::signm",
        })
    }

    /// Fractional matrix power: A^p for any real p
    ///
    /// Computes `A^p` using: `A^p = exp(p * log(A))`
    ///
    /// # Special Cases
    ///
    /// - p = 0: Returns identity matrix
    /// - p = 1: Returns A unchanged
    /// - p = -1: Returns matrix inverse
    /// - p = 0.5: Equivalent to sqrtm(A)
    /// - Integer p: Uses repeated squaring
    fn fractional_matrix_power(&self, a: &Tensor<R>, p: f64) -> Result<Tensor<R>> {
        let _ = (a, p);
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::fractional_matrix_power",
        })
    }

    /// General matrix function: f(A) for any scalar function f
    ///
    /// Computes f(A) using the Schur-Parlett algorithm.
    ///
    /// # Closure Requirements
    ///
    /// The closure `f` must be `Send + Sync` to support GPU backends (CUDA, WebGPU).
    /// This allows the function to be safely captured and potentially executed across
    /// GPU threads or transferred to device memory contexts.
    ///
    /// # Example
    ///
    /// ```ignore
    /// // Custom matrix function: f(x) = sin(x)
    /// let result = client.funm(&matrix, |x| x.sin())?;
    /// ```
    fn funm<F>(&self, a: &Tensor<R>, f: F) -> Result<Tensor<R>>
    where
        F: Fn(f64) -> f64 + Send + Sync,
    {
        let _ = (a, f);
        Err(Error::NotImplemented {
            feature: "MatrixFunctionsAlgorithms::funm",
        })
    }
}