schur.rs - source
Description: Source of the Rust file `/Users/erlendbasso/.cargo/registry/src/github.com-1ecc6299db9ec823/nalgebra-0.32.1/src/linalg/schur.rs`.
Keywords: rust, rustlang, rust-lang

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    #![allow(clippy::suspicious_operation_groupings)]
    #[cfg(feature = "serde-serialize-no-std")]
    use serde::{Deserialize, Serialize};

    use approx::AbsDiffEq;
    use num_complex::Complex as NumComplex;
    use simba::scalar::{ComplexField, RealField};
    use std::cmp;

    use crate::allocator::Allocator;
    use crate::base::dimension::{Const, Dim, DimDiff, DimSub, Dyn, U1, U2};
    use crate::base::storage::Storage;
    use crate::base::{DefaultAllocator, OMatrix, OVector, SquareMatrix, Unit, Vector2, Vector3};

    use crate::geometry::Reflection;
    use crate::linalg::givens::GivensRotation;
    use crate::linalg::householder;
    use crate::linalg::Hessenberg;
    use crate::{Matrix, UninitVector};
    use std::mem::MaybeUninit;

    /// Schur decomposition of a square matrix.
    ///
    /// If this is a real matrix, this will be a `RealField` Schur decomposition.
    #[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
    #[cfg_attr(
        feature = "serde-serialize-no-std",
        serde(bound(serialize = "DefaultAllocator: Allocator<T, D, D>,
             OMatrix<T, D, D>: Serialize"))
    )]
    #[cfg_attr(
        feature = "serde-serialize-no-std",
        serde(bound(deserialize = "DefaultAllocator: Allocator<T, D, D>,
             OMatrix<T, D, D>: Deserialize<'de>"))
    )]
    #[derive(Clone, Debug)]
    pub struct Schur<T: ComplexField, D: Dim>
    where
        DefaultAllocator: Allocator<T, D, D>,
    {
        q: OMatrix<T, D, D>,
        t: OMatrix<T, D, D>,
    }

    impl<T: ComplexField, D: Dim> Copy for Schur<T, D>
    where
        DefaultAllocator: Allocator<T, D, D>,
        OMatrix<T, D, D>: Copy,
    {
    }

    impl<T: ComplexField, D: Dim> Schur<T, D>
    where
        D: DimSub<U1>, // For Hessenberg.
        DefaultAllocator: Allocator<T, D, DimDiff<D, U1>>
            + Allocator<T, DimDiff<D, U1>>
            + Allocator<T, D, D>
            + Allocator<T, D>,
    {
        /// Computes the Schur decomposition of a square matrix.
        pub fn new(m: OMatrix<T, D, D>) -> Self {
            Self::try_new(m, T::RealField::default_epsilon(), 0).unwrap()
        }

        /// Attempts to compute the Schur decomposition of a square matrix.
        ///
        /// If only eigenvalues are needed, it is more efficient to call the matrix method
        /// `.eigenvalues()` instead.
        ///
        /// # Arguments
        ///
        /// * `eps`       − tolerance used to determine when a value converged to 0.
        /// * `max_niter` − maximum total number of iterations performed by the algorithm. If this
        /// number of iteration is exceeded, `None` is returned. If `niter == 0`, then the algorithm
        /// continues indefinitely until convergence.
        pub fn try_new(m: OMatrix<T, D, D>, eps: T::RealField, max_niter: usize) -> Option<Self> {
            let mut work = Matrix::zeros_generic(m.shape_generic().0, Const::<1>);

            Self::do_decompose(m, &mut work, eps, max_niter, true)
                .map(|(q, t)| Schur { q: q.unwrap(), t })
        }

        fn do_decompose(
            mut m: OMatrix<T, D, D>,
            work: &mut OVector<T, D>,
            eps: T::RealField,
            max_niter: usize,
            compute_q: bool,
        ) -> Option<(Option<OMatrix<T, D, D>>, OMatrix<T, D, D>)> {
            assert!(
                m.is_square(),
                "Unable to compute the eigenvectors and eigenvalues of a non-square matrix."
            );

            let dim = m.shape_generic().0;

            // Specialization would make this easier.
            if dim.value() == 0 {
                let vecs = Some(OMatrix::from_element_generic(dim, dim, T::zero()));
                let vals = OMatrix::from_element_generic(dim, dim, T::zero());
                return Some((vecs, vals));
            } else if dim.value() == 1 {
                if compute_q {
                    let q = OMatrix::from_element_generic(dim, dim, T::one());
                    return Some((Some(q), m));
                } else {
                    return Some((None, m));
                }
            } else if dim.value() == 2 {
                return decompose_2x2(m, compute_q);
            }

            let amax_m = m.camax();
            m.unscale_mut(amax_m.clone());

            let hess = Hessenberg::new_with_workspace(m, work);
            let mut q;
            let mut t;

            if compute_q {
                // TODO: could we work without unpacking? Using only the internal representation of
                // hessenberg decomposition.
                let (vecs, vals) = hess.unpack();
                q = Some(vecs);
                t = vals;
            } else {
                q = None;
                t = hess.unpack_h()
            }

            // Implicit double-shift QR method.
            let mut niter = 0;
            let (mut start, mut end) = Self::delimit_subproblem(&mut t, eps.clone(), dim.value() - 1);

            while end != start {
                let subdim = end - start + 1;

                if subdim > 2 {
                    let m = end - 1;
                    let n = end;

                    let h11 = t[(start, start)].clone();
                    let h12 = t[(start, start + 1)].clone();
                    let h21 = t[(start + 1, start)].clone();
                    let h22 = t[(start + 1, start + 1)].clone();
                    let h32 = t[(start + 2, start + 1)].clone();

                    let hnn = t[(n, n)].clone();
                    let hmm = t[(m, m)].clone();
                    let hnm = t[(n, m)].clone();
                    let hmn = t[(m, n)].clone();

                    let tra = hnn.clone() + hmm.clone();
                    let det = hnn * hmm - hnm * hmn;

                    let mut axis = Vector3::new(
                        h11.clone() * h11.clone() + h12 * h21.clone() - tra.clone() * h11.clone() + det,
                        h21.clone() * (h11 + h22 - tra),
                        h21 * h32,
                    );

                    for k in start..n - 1 {
                        let (norm, not_zero) = householder::reflection_axis_mut(&mut axis);

                        if not_zero {
                            if k > start {
                                t[(k, k - 1)] = norm;
                                t[(k + 1, k - 1)] = T::zero();
                                t[(k + 2, k - 1)] = T::zero();
                            }

                            let refl = Reflection::new(Unit::new_unchecked(axis.clone()), T::zero());

                            {
                                let krows = cmp::min(k + 4, end + 1);
                                let mut work = work.rows_mut(0, krows);
                                refl.reflect(
                                    &mut t.generic_view_mut((k, k), (Const::<3>, Dyn(dim.value() - k))),
                                );
                                refl.reflect_rows(
                                    &mut t.generic_view_mut((0, k), (Dyn(krows), Const::<3>)),
                                    &mut work,
                                );
                            }

                            if let Some(ref mut q) = q {
                                refl.reflect_rows(
                                    &mut q.generic_view_mut((0, k), (dim, Const::<3>)),
                                    work,
                                );
                            }
                        }

                        axis.x = t[(k + 1, k)].clone();
                        axis.y = t[(k + 2, k)].clone();

                        if k < n - 2 {
                            axis.z = t[(k + 3, k)].clone();
                        }
                    }

                    let mut axis = Vector2::new(axis.x.clone(), axis.y.clone());
                    let (norm, not_zero) = householder::reflection_axis_mut(&mut axis);

                    if not_zero {
                        let refl = Reflection::new(Unit::new_unchecked(axis), T::zero());

                        t[(m, m - 1)] = norm;
                        t[(n, m - 1)] = T::zero();

                        {
                            let mut work = work.rows_mut(0, end + 1);
                            refl.reflect(
                                &mut t.generic_view_mut((m, m), (Const::<2>, Dyn(dim.value() - m))),
                            );
                            refl.reflect_rows(
                                &mut t.generic_view_mut((0, m), (Dyn(end + 1), Const::<2>)),
                                &mut work,
                            );
                        }

                        if let Some(ref mut q) = q {
                            refl.reflect_rows(&mut q.generic_view_mut((0, m), (dim, Const::<2>)), work);
                        }
                    }
                } else {
                    // Decouple the 2x2 block if it has real eigenvalues.
                    if let Some(rot) = compute_2x2_basis(&t.fixed_view::<2, 2>(start, start)) {
                        let inv_rot = rot.inverse();
                        inv_rot.rotate(
                            &mut t.generic_view_mut(
                                (start, start),
                                (Const::<2>, Dyn(dim.value() - start)),
                            ),
                        );
                        rot.rotate_rows(
                            &mut t.generic_view_mut((0, start), (Dyn(end + 1), Const::<2>)),
                        );
                        t[(end, start)] = T::zero();

                        if let Some(ref mut q) = q {
                            rot.rotate_rows(&mut q.generic_view_mut((0, start), (dim, Const::<2>)));
                        }
                    }

                    // Check if we reached the beginning of the matrix.
                    if end > 2 {
                        end -= 2;
                    } else {
                        break;
                    }
                }

                let sub = Self::delimit_subproblem(&mut t, eps.clone(), end);

                start = sub.0;
                end = sub.1;

                niter += 1;
                if niter == max_niter {
                    return None;
                }
            }

            t.scale_mut(amax_m);

            Some((q, t))
        }

        /// Computes the eigenvalues of the decomposed matrix.
        fn do_eigenvalues(t: &OMatrix<T, D, D>, out: &mut OVector<T, D>) -> bool {
            let dim = t.nrows();
            let mut m = 0;

            while m < dim - 1 {
                let n = m + 1;

                if t[(n, m)].is_zero() {
                    out[m] = t[(m, m)].clone();
                    m += 1;
                } else {
                    // Complex eigenvalue.
                    return false;
                }
            }

            if m == dim - 1 {
                out[m] = t[(m, m)].clone();
            }

            true
        }

        /// Computes the complex eigenvalues of the decomposed matrix.
        fn do_complex_eigenvalues(t: &OMatrix<T, D, D>, out: &mut UninitVector<NumComplex<T>, D>)
        where
            T: RealField,
            DefaultAllocator: Allocator<NumComplex<T>, D>,
        {
            let dim = t.nrows();
            let mut m = 0;

            while m < dim - 1 {
                let n = m + 1;

                if t[(n, m)].is_zero() {
                    out[m] = MaybeUninit::new(NumComplex::new(t[(m, m)].clone(), T::zero()));
                    m += 1;
                } else {
                    // Solve the 2x2 eigenvalue subproblem.
                    let hmm = t[(m, m)].clone();
                    let hnm = t[(n, m)].clone();
                    let hmn = t[(m, n)].clone();
                    let hnn = t[(n, n)].clone();

                    // NOTE: use the same algorithm as in compute_2x2_eigvals.
                    let val = (hmm.clone() - hnn.clone()) * crate::convert(0.5);
                    let discr = hnm * hmn + val.clone() * val;

                    // All 2x2 blocks have negative discriminant because we already decoupled those
                    // with positive eigenvalues.
                    let sqrt_discr = NumComplex::new(T::zero(), (-discr).sqrt());

                    let half_tra = (hnn + hmm) * crate::convert(0.5);
                    out[m] = MaybeUninit::new(
                        NumComplex::new(half_tra.clone(), T::zero()) + sqrt_discr.clone(),
                    );
                    out[m + 1] =
                        MaybeUninit::new(NumComplex::new(half_tra, T::zero()) - sqrt_discr.clone());

                    m += 2;
                }
            }

            if m == dim - 1 {
                out[m] = MaybeUninit::new(NumComplex::new(t[(m, m)].clone(), T::zero()));
            }
        }

        fn delimit_subproblem(t: &mut OMatrix<T, D, D>, eps: T::RealField, end: usize) -> (usize, usize)
        where
            D: DimSub<U1>,
            DefaultAllocator: Allocator<T, DimDiff<D, U1>>,
        {
            let mut n = end;

            while n > 0 {
                let m = n - 1;

                if t[(n, m)].clone().norm1()
                    <= eps.clone() * (t[(n, n)].clone().norm1() + t[(m, m)].clone().norm1())
                {
                    t[(n, m)] = T::zero();
                } else {
                    break;
                }

                n -= 1;
            }

            if n == 0 {
                return (0, 0);
            }

            let mut new_start = n - 1;
            while new_start > 0 {
                let m = new_start - 1;

                let off_diag = t[(new_start, m)].clone();
                if off_diag.is_zero()
                    || off_diag.norm1()
                        <= eps.clone()
                            * (t[(new_start, new_start)].clone().norm1() + t[(m, m)].clone().norm1())
                {
                    t[(new_start, m)] = T::zero();
                    break;
                }

                new_start -= 1;
            }

            (new_start, n)
        }

        /// Retrieves the unitary matrix `Q` and the upper-quasitriangular matrix `T` such that the
        /// decomposed matrix equals `Q * T * Q.transpose()`.
        pub fn unpack(self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
            (self.q, self.t)
        }

        /// Computes the real eigenvalues of the decomposed matrix.
        ///
        /// Return `None` if some eigenvalues are complex.
        #[must_use]
        pub fn eigenvalues(&self) -> Option<OVector<T, D>> {
            let mut out = Matrix::zeros_generic(self.t.shape_generic().0, Const::<1>);
            if Self::do_eigenvalues(&self.t, &mut out) {
                Some(out)
            } else {
                None
            }
        }

        /// Computes the complex eigenvalues of the decomposed matrix.
        #[must_use]
        pub fn complex_eigenvalues(&self) -> OVector<NumComplex<T>, D>
        where
            T: RealField,
            DefaultAllocator: Allocator<NumComplex<T>, D>,
        {
            let mut out = Matrix::uninit(self.t.shape_generic().0, Const::<1>);
            Self::do_complex_eigenvalues(&self.t, &mut out);
            // Safety: out has been fully initialized by do_complex_eigenvalues.
            unsafe { out.assume_init() }
        }
    }

    fn decompose_2x2<T: ComplexField, D: Dim>(
        mut m: OMatrix<T, D, D>,
        compute_q: bool,
    ) -> Option<(Option<OMatrix<T, D, D>>, OMatrix<T, D, D>)>
    where
        DefaultAllocator: Allocator<T, D, D>,
    {
        let dim = m.shape_generic().0;
        let mut q = None;
        match compute_2x2_basis(&m.fixed_view::<2, 2>(0, 0)) {
            Some(rot) => {
                let mut m = m.fixed_view_mut::<2, 2>(0, 0);
                let inv_rot = rot.inverse();
                inv_rot.rotate(&mut m);
                rot.rotate_rows(&mut m);
                m[(1, 0)] = T::zero();

                if compute_q {
                    // XXX: we have to build the matrix manually because
                    // rot.to_rotation_matrix().unwrap() causes an ICE.
                    let c = T::from_real(rot.c());
                    q = Some(OMatrix::from_column_slice_generic(
                        dim,
                        dim,
                        &[c.clone(), rot.s(), -rot.s().conjugate(), c],
                    ));
                }
            }
            None => {
                if compute_q {
                    q = Some(OMatrix::identity_generic(dim, dim));
                }
            }
        };

        Some((q, m))
    }

    fn compute_2x2_eigvals<T: ComplexField, S: Storage<T, U2, U2>>(
        m: &SquareMatrix<T, U2, S>,
    ) -> Option<(T, T)> {
        // Solve the 2x2 eigenvalue subproblem.
        let h00 = m[(0, 0)].clone();
        let h10 = m[(1, 0)].clone();
        let h01 = m[(0, 1)].clone();
        let h11 = m[(1, 1)].clone();

        // NOTE: this discriminant computation is more stable than the
        // one based on the trace and determinant: 0.25 * tra * tra - det
        // because it ensures positiveness for symmetric matrices.
        let val = (h00.clone() - h11.clone()) * crate::convert(0.5);
        let discr = h10 * h01 + val.clone() * val;

        discr.try_sqrt().map(|sqrt_discr| {
            let half_tra = (h00 + h11) * crate::convert(0.5);
            (half_tra.clone() + sqrt_discr.clone(), half_tra - sqrt_discr)
        })
    }

    // Computes the 2x2 transformation that upper-triangulates a 2x2 matrix with real eigenvalues.
    /// Computes the singular vectors for a 2x2 matrix.
    ///
    /// Returns `None` if the matrix has complex eigenvalues, or is upper-triangular. In both case,
    /// the basis is the identity.
    fn compute_2x2_basis<T: ComplexField, S: Storage<T, U2, U2>>(
        m: &SquareMatrix<T, U2, S>,
    ) -> Option<GivensRotation<T>> {
        let h10 = m[(1, 0)].clone();

        if h10.is_zero() {
            return None;
        }

        if let Some((eigval1, eigval2)) = compute_2x2_eigvals(m) {
            let x1 = eigval1 - m[(1, 1)].clone();
            let x2 = eigval2 - m[(1, 1)].clone();

            // NOTE: Choose the one that yields a larger x component.
            // This is necessary for numerical stability of the normalization of the complex
            // number.
            if x1.clone().norm1() > x2.clone().norm1() {
                Some(GivensRotation::new(x1, h10).0)
            } else {
                Some(GivensRotation::new(x2, h10).0)
            }
        } else {
            None
        }
    }

    impl<T: ComplexField, D: Dim, S: Storage<T, D, D>> SquareMatrix<T, D, S>
    where
        D: DimSub<U1>, // For Hessenberg.
        DefaultAllocator: Allocator<T, D, DimDiff<D, U1>>
            + Allocator<T, DimDiff<D, U1>>
            + Allocator<T, D, D>
            + Allocator<T, D>,
    {
        /// Computes the eigenvalues of this matrix.
        #[must_use]
        pub fn eigenvalues(&self) -> Option<OVector<T, D>> {
            assert!(
                self.is_square(),
                "Unable to compute eigenvalues of a non-square matrix."
            );

            let mut work = Matrix::zeros_generic(self.shape_generic().0, Const::<1>);

            // Special case for 2x2 matrices.
            if self.nrows() == 2 {
                // TODO: can we avoid this slicing
                // (which is needed here just to transform D to U2)?
                let me = self.fixed_view::<2, 2>(0, 0);
                return match compute_2x2_eigvals(&me) {
                    Some((a, b)) => {
                        work[0] = a;
                        work[1] = b;
                        Some(work)
                    }
                    None => None,
                };
            }

            // TODO: add balancing?
            let schur = Schur::do_decompose(
                self.clone_owned(),
                &mut work,
                T::RealField::default_epsilon(),
                0,
                false,
            )
            .unwrap();

            if Schur::do_eigenvalues(&schur.1, &mut work) {
                Some(work)
            } else {
                None
            }
        }

        /// Computes the eigenvalues of this matrix.
        #[must_use]
        pub fn complex_eigenvalues(&self) -> OVector<NumComplex<T>, D>
        // TODO: add balancing?
        where
            T: RealField,
            DefaultAllocator: Allocator<NumComplex<T>, D>,
        {
            let dim = self.shape_generic().0;
            let mut work = Matrix::zeros_generic(dim, Const::<1>);

            let schur = Schur::do_decompose(
                self.clone_owned(),
                &mut work,
                T::default_epsilon(),
                0,
                false,
            )
            .unwrap();
            let mut eig = Matrix::uninit(dim, Const::<1>);
            Schur::do_complex_eigenvalues(&schur.1, &mut eig);
            // Safety: eig has been fully initialized by do_complex_eigenvalues.
            unsafe { eig.assume_init() }
        }
    }