numr 0.5.2

High-performance numerical computing with multi-backend GPU acceleration (CPU/CUDA/WebGPU)
Documentation 
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//! Schur decomposition using QR iteration with Wilkinson shift

use super::super::jacobi::LinalgElement;
use super::super::{CpuClient, CpuRuntime};
use crate::algorithm::linalg::{
    SchurDecomposition, linalg_demote, linalg_promote, validate_linalg_dtype,
    validate_square_matrix,
};
use crate::dtype::{DType, Element};
use crate::error::Result;
use crate::runtime::RuntimeClient;
use crate::tensor::Tensor;

/// Schur decomposition using QR iteration
///
/// Computes A = Z @ T @ Z^T where Z is orthogonal and T is upper quasi-triangular.
pub fn schur_decompose_impl(
    client: &CpuClient,
    a: &Tensor<CpuRuntime>,
) -> Result<SchurDecomposition<CpuRuntime>> {
    validate_linalg_dtype(a.dtype())?;
    let (a, original_dtype) = linalg_promote(client, a)?;
    let n = validate_square_matrix(a.shape())?;

    let result = match a.dtype() {
        DType::F32 => schur_decompose_typed::<f32>(client, &a, n),
        DType::F64 => schur_decompose_typed::<f64>(client, &a, n),
        _ => unreachable!(),
    }?;

    Ok(SchurDecomposition {
        z: linalg_demote(client, result.z, original_dtype)?,
        t: linalg_demote(client, result.t, original_dtype)?,
    })
}

fn schur_decompose_typed<T: Element + LinalgElement>(
    client: &CpuClient,
    a: &Tensor<CpuRuntime>,
    n: usize,
) -> Result<SchurDecomposition<CpuRuntime>> {
    let device = client.device();

    // Handle trivial cases
    if n == 0 {
        return Ok(SchurDecomposition {
            z: Tensor::<CpuRuntime>::from_slice(&[] as &[T], &[0, 0], device),
            t: Tensor::<CpuRuntime>::from_slice(&[] as &[T], &[0, 0], device),
        });
    }

    if n == 1 {
        let data: Vec<T> = a.to_vec();
        return Ok(SchurDecomposition {
            z: Tensor::<CpuRuntime>::from_slice(&[T::one()], &[1, 1], device),
            t: Tensor::<CpuRuntime>::from_slice(&data, &[1, 1], device),
        });
    }

    // Copy input to working matrix T (will become the Schur form)
    let mut t_data: Vec<T> = a.to_vec();

    // Initialize Z as identity matrix
    let mut z_data: Vec<T> = vec![T::zero(); n * n];
    for i in 0..n {
        z_data[i * n + i] = T::one();
    }

    // Step 1: Reduce to upper Hessenberg form using Householder reflections
    // H = Q0^T @ A @ Q0, accumulate Q0 into Z
    hessenberg_reduction::<T>(&mut t_data, &mut z_data, n);

    // Step 2: QR iteration to get upper quasi-triangular form
    let max_iter = 30 * n;
    let eps = T::epsilon_val();

    for _iter in 0..max_iter {
        // Check convergence: subdiagonal elements should be small
        let mut converged = true;
        for i in 0..(n - 1) {
            let h_ii = t_data[i * n + i].to_f64().abs();
            let h_ip1 = t_data[(i + 1) * n + (i + 1)].to_f64().abs();
            let threshold = eps * (h_ii + h_ip1).max(1.0);

            let subdiag = t_data[(i + 1) * n + i].to_f64().abs();
            if subdiag > threshold {
                converged = false;
                break;
            }
        }

        if converged {
            break;
        }

        // Perform one QR iteration with implicit shift
        qr_iteration_step::<T>(&mut t_data, &mut z_data, n);
    }

    // Clean up small subdiagonal elements to make T exactly quasi-triangular
    for i in 0..(n - 1) {
        let h_ii = t_data[i * n + i].to_f64().abs();
        let h_ip1 = t_data[(i + 1) * n + (i + 1)].to_f64().abs();
        let threshold = eps * (h_ii + h_ip1).max(1.0);

        let subdiag = t_data[(i + 1) * n + i].to_f64();
        if subdiag.abs() <= threshold {
            t_data[(i + 1) * n + i] = T::zero();
        }
    }

    // Clear lower triangular part (except 2x2 blocks for complex eigenvalues)
    for i in 2..n {
        for j in 0..(i - 1) {
            t_data[i * n + j] = T::zero();
        }
    }

    Ok(SchurDecomposition {
        z: Tensor::<CpuRuntime>::from_slice(&z_data, &[n, n], device),
        t: Tensor::<CpuRuntime>::from_slice(&t_data, &[n, n], device),
    })
}

/// Reduce matrix to upper Hessenberg form using Householder reflections.
/// Modifies H in-place and accumulates transformations into Q.
pub fn hessenberg_reduction<T: Element + LinalgElement>(h: &mut [T], q: &mut [T], n: usize) {
    for k in 0..(n - 2) {
        // Compute Householder vector for column k, rows k+1 to n-1
        let mut v = vec![T::zero(); n - k - 1];
        let mut norm_sq = 0.0;

        for i in (k + 1)..n {
            let val = h[i * n + k].to_f64();
            v[i - k - 1] = T::from_f64(val);
            norm_sq += val * val;
        }

        if norm_sq < T::epsilon_val() {
            continue;
        }

        let norm = norm_sq.sqrt();
        let x0 = v[0].to_f64();
        let alpha = if x0 >= 0.0 { -norm } else { norm };

        v[0] = T::from_f64(x0 - alpha);

        // Normalize v
        let mut v_norm_sq = 0.0;
        for vi in &v {
            v_norm_sq += vi.to_f64() * vi.to_f64();
        }
        if v_norm_sq < T::epsilon_val() {
            continue;
        }
        let v_norm = v_norm_sq.sqrt();
        for vi in &mut v {
            *vi = T::from_f64(vi.to_f64() / v_norm);
        }

        // Apply Householder: H = H - 2 * v * (v^T * H) for rows k+1..n
        // Left multiplication: H[k+1:n, :] = H[k+1:n, :] - 2 * v * (v^T @ H[k+1:n, :])
        for j in 0..n {
            let mut dot = 0.0;
            for i in 0..v.len() {
                dot += v[i].to_f64() * h[(k + 1 + i) * n + j].to_f64();
            }
            for i in 0..v.len() {
                let old_val = h[(k + 1 + i) * n + j].to_f64();
                h[(k + 1 + i) * n + j] = T::from_f64(old_val - 2.0 * v[i].to_f64() * dot);
            }
        }

        // Right multiplication: H[:, k+1:n] = H[:, k+1:n] - 2 * (H[:, k+1:n] @ v) * v^T
        for i in 0..n {
            let mut dot = 0.0;
            for j in 0..v.len() {
                dot += h[i * n + (k + 1 + j)].to_f64() * v[j].to_f64();
            }
            for j in 0..v.len() {
                let old_val = h[i * n + (k + 1 + j)].to_f64();
                h[i * n + (k + 1 + j)] = T::from_f64(old_val - 2.0 * dot * v[j].to_f64());
            }
        }

        // Accumulate into Q: Q = Q @ H_k (where H_k is the Householder reflection)
        // Q[:, k+1:n] = Q[:, k+1:n] - 2 * (Q[:, k+1:n] @ v) * v^T
        for i in 0..n {
            let mut dot = 0.0;
            for j in 0..v.len() {
                dot += q[i * n + (k + 1 + j)].to_f64() * v[j].to_f64();
            }
            for j in 0..v.len() {
                let old_val = q[i * n + (k + 1 + j)].to_f64();
                q[i * n + (k + 1 + j)] = T::from_f64(old_val - 2.0 * dot * v[j].to_f64());
            }
        }
    }
}

/// Perform one QR iteration step with implicit Wilkinson shift.
pub fn qr_iteration_step<T: Element + LinalgElement>(h: &mut [T], q: &mut [T], n: usize) {
    // Compute Wilkinson shift from bottom 2x2 block
    let a = h[(n - 2) * n + (n - 2)].to_f64();
    let b = h[(n - 2) * n + (n - 1)].to_f64();
    let c = h[(n - 1) * n + (n - 2)].to_f64();
    let d = h[(n - 1) * n + (n - 1)].to_f64();

    // Wilkinson shift: eigenvalue of 2x2 block closest to d
    let trace = a + d;
    let det = a * d - b * c;
    let disc = trace * trace - 4.0 * det;

    let shift = if disc >= 0.0 {
        let sqrt_disc = disc.sqrt();
        let lambda1 = (trace + sqrt_disc) / 2.0;
        let lambda2 = (trace - sqrt_disc) / 2.0;
        if (lambda1 - d).abs() < (lambda2 - d).abs() {
            lambda1
        } else {
            lambda2
        }
    } else {
        // Complex eigenvalues, use trace/2 as shift
        trace / 2.0
    };

    // Apply shift: H = H - shift * I
    for i in 0..n {
        h[i * n + i] = T::from_f64(h[i * n + i].to_f64() - shift);
    }

    // QR factorization using Givens rotations
    // Process each subdiagonal element
    for i in 0..(n - 1) {
        let a_val = h[i * n + i].to_f64();
        let b_val = h[(i + 1) * n + i].to_f64();

        if b_val.abs() < T::epsilon_val() {
            continue;
        }

        // Compute Givens rotation to zero out b_val
        let r = (a_val * a_val + b_val * b_val).sqrt();
        let c = a_val / r;
        let s = -b_val / r;

        // Apply rotation from the left: rows i and i+1
        for j in 0..n {
            let t1 = h[i * n + j].to_f64();
            let t2 = h[(i + 1) * n + j].to_f64();
            h[i * n + j] = T::from_f64(c * t1 - s * t2);
            h[(i + 1) * n + j] = T::from_f64(s * t1 + c * t2);
        }

        // Apply rotation from the right: cols i and i+1
        for k in 0..n {
            let t1 = h[k * n + i].to_f64();
            let t2 = h[k * n + (i + 1)].to_f64();
            h[k * n + i] = T::from_f64(c * t1 - s * t2);
            h[k * n + (i + 1)] = T::from_f64(s * t1 + c * t2);
        }

        // Accumulate into Q
        for k in 0..n {
            let t1 = q[k * n + i].to_f64();
            let t2 = q[k * n + (i + 1)].to_f64();
            q[k * n + i] = T::from_f64(c * t1 - s * t2);
            q[k * n + (i + 1)] = T::from_f64(s * t1 + c * t2);
        }
    }

    // Remove shift: H = H + shift * I
    for i in 0..n {
        h[i * n + i] = T::from_f64(h[i * n + i].to_f64() + shift);
    }
}