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

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

    use num::{One, Zero};
    use simba::scalar::ComplexField;
    use simba::simd::SimdComplexField;

    use crate::allocator::Allocator;
    use crate::base::{Const, DefaultAllocator, Matrix, OMatrix, Vector};
    use crate::constraint::{SameNumberOfRows, ShapeConstraint};
    use crate::dimension::{Dim, DimAdd, DimDiff, DimSub, DimSum, U1};
    use crate::storage::{Storage, StorageMut};

    /// The Cholesky decomposition of a symmetric-definite-positive matrix.
    #[cfg_attr(feature = "serde-serialize-no-std", derive(Serialize, Deserialize))]
    #[cfg_attr(
        feature = "serde-serialize-no-std",
        serde(bound(serialize = "DefaultAllocator: Allocator<T, D>,
             OMatrix<T, D, D>: Serialize"))
    )]
    #[cfg_attr(
        feature = "serde-serialize-no-std",
        serde(bound(deserialize = "DefaultAllocator: Allocator<T, D>,
             OMatrix<T, D, D>: Deserialize<'de>"))
    )]
    #[derive(Clone, Debug)]
    pub struct Cholesky<T: SimdComplexField, D: Dim>
    where
        DefaultAllocator: Allocator<T, D, D>,
    {
        chol: OMatrix<T, D, D>,
    }

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

    impl<T: SimdComplexField, D: Dim> Cholesky<T, D>
    where
        DefaultAllocator: Allocator<T, D, D>,
    {
        /// Computes the Cholesky decomposition of `matrix` without checking that the matrix is definite-positive.
        ///
        /// If the input matrix is not definite-positive, the decomposition may contain trash values (Inf, NaN, etc.)
        pub fn new_unchecked(mut matrix: OMatrix<T, D, D>) -> Self {
            assert!(matrix.is_square(), "The input matrix must be square.");

            let n = matrix.nrows();

            for j in 0..n {
                for k in 0..j {
                    let factor = unsafe { -matrix.get_unchecked((j, k)).clone() };

                    let (mut col_j, col_k) = matrix.columns_range_pair_mut(j, k);
                    let mut col_j = col_j.rows_range_mut(j..);
                    let col_k = col_k.rows_range(j..);
                    col_j.axpy(factor.simd_conjugate(), &col_k, T::one());
                }

                let diag = unsafe { matrix.get_unchecked((j, j)).clone() };
                let denom = diag.simd_sqrt();

                unsafe {
                    *matrix.get_unchecked_mut((j, j)) = denom.clone();
                }

                let mut col = matrix.view_range_mut(j + 1.., j);
                col /= denom;
            }

            Cholesky { chol: matrix }
        }

        /// Uses the given matrix as-is without any checks or modifications as the
        /// Cholesky decomposition.
        ///
        /// It is up to the user to ensure all invariants hold.
        pub fn pack_dirty(matrix: OMatrix<T, D, D>) -> Self {
            Cholesky { chol: matrix }
        }

        /// Retrieves the lower-triangular factor of the Cholesky decomposition with its strictly
        /// upper-triangular part filled with zeros.
        pub fn unpack(mut self) -> OMatrix<T, D, D> {
            self.chol.fill_upper_triangle(T::zero(), 1);
            self.chol
        }

        /// Retrieves the lower-triangular factor of the Cholesky decomposition, without zeroing-out
        /// its strict upper-triangular part.
        ///
        /// The values of the strict upper-triangular part are garbage and should be ignored by further
        /// computations.
        pub fn unpack_dirty(self) -> OMatrix<T, D, D> {
            self.chol
        }

        /// Retrieves the lower-triangular factor of the Cholesky decomposition with its strictly
        /// uppen-triangular part filled with zeros.
        #[must_use]
        pub fn l(&self) -> OMatrix<T, D, D> {
            self.chol.lower_triangle()
        }

        /// Retrieves the lower-triangular factor of the Cholesky decomposition, without zeroing-out
        /// its strict upper-triangular part.
        ///
        /// This is an allocation-less version of `self.l()`. The values of the strict upper-triangular
        /// part are garbage and should be ignored by further computations.
        #[must_use]
        pub fn l_dirty(&self) -> &OMatrix<T, D, D> {
            &self.chol
        }

        /// Solves the system `self * x = b` where `self` is the decomposed matrix and `x` the unknown.
        ///
        /// The result is stored on `b`.
        pub fn solve_mut<R2: Dim, C2: Dim, S2>(&self, b: &mut Matrix<T, R2, C2, S2>)
        where
            S2: StorageMut<T, R2, C2>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
        {
            self.chol.solve_lower_triangular_unchecked_mut(b);
            self.chol.ad_solve_lower_triangular_unchecked_mut(b);
        }

        /// Returns the solution of the system `self * x = b` where `self` is the decomposed matrix and
        /// `x` the unknown.
        #[must_use = "Did you mean to use solve_mut()?"]
        pub fn solve<R2: Dim, C2: Dim, S2>(&self, b: &Matrix<T, R2, C2, S2>) -> OMatrix<T, R2, C2>
        where
            S2: Storage<T, R2, C2>,
            DefaultAllocator: Allocator<T, R2, C2>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
        {
            let mut res = b.clone_owned();
            self.solve_mut(&mut res);
            res
        }

        /// Computes the inverse of the decomposed matrix.
        #[must_use]
        pub fn inverse(&self) -> OMatrix<T, D, D> {
            let shape = self.chol.shape_generic();
            let mut res = OMatrix::identity_generic(shape.0, shape.1);

            self.solve_mut(&mut res);
            res
        }

        /// Computes the determinant of the decomposed matrix.
        #[must_use]
        pub fn determinant(&self) -> T::SimdRealField {
            let dim = self.chol.nrows();
            let mut prod_diag = T::one();
            for i in 0..dim {
                prod_diag *= unsafe { self.chol.get_unchecked((i, i)).clone() };
            }
            prod_diag.simd_modulus_squared()
        }

        /// Computes the natural logarithm of determinant of the decomposed matrix.
        ///
        /// This method is more robust than `.determinant()` to very small or very
        /// large determinants since it returns the natural logarithm of the
        /// determinant rather than the determinant itself.
        #[must_use]
        pub fn ln_determinant(&self) -> T::SimdRealField {
            let dim = self.chol.nrows();
            let mut sum_diag = T::SimdRealField::zero();
            for i in 0..dim {
                sum_diag += unsafe {
                    self.chol
                        .get_unchecked((i, i))
                        .clone()
                        .simd_modulus_squared()
                        .simd_ln()
                };
            }
            sum_diag
        }
    }

    impl<T: ComplexField, D: Dim> Cholesky<T, D>
    where
        DefaultAllocator: Allocator<T, D, D>,
    {
        /// Attempts to compute the Cholesky decomposition of `matrix`.
        ///
        /// Returns `None` if the input matrix is not definite-positive. The input matrix is assumed
        /// to be symmetric and only the lower-triangular part is read.
        pub fn new(matrix: OMatrix<T, D, D>) -> Option<Self> {
            Self::new_internal(matrix, None)
        }

        /// Attempts to approximate the Cholesky decomposition of `matrix` by
        /// replacing non-positive values on the diagonals during the decomposition
        /// with the given `substitute`.
        ///
        /// [`try_sqrt`](ComplexField::try_sqrt) will be applied to the `substitute`
        /// when it has to be used.
        ///
        /// If your input matrix results only in positive values on the diagonals
        /// during the decomposition, `substitute` is unused and the result is just
        /// the same as if you used [`new`](Cholesky::new).
        ///
        /// This method allows to compensate for matrices with very small or even
        /// negative values due to numerical errors but necessarily results in only
        /// an approximation: it is basically a hack. If you don't specifically need
        /// Cholesky, it may be better to consider alternatives like the
        /// [`LU`](crate::linalg::LU) decomposition/factorization.
        pub fn new_with_substitute(matrix: OMatrix<T, D, D>, substitute: T) -> Option<Self> {
            Self::new_internal(matrix, Some(substitute))
        }

        /// Common implementation for `new` and `new_with_substitute`.
        fn new_internal(mut matrix: OMatrix<T, D, D>, substitute: Option<T>) -> Option<Self> {
            assert!(matrix.is_square(), "The input matrix must be square.");

            let n = matrix.nrows();

            for j in 0..n {
                for k in 0..j {
                    let factor = unsafe { -matrix.get_unchecked((j, k)).clone() };

                    let (mut col_j, col_k) = matrix.columns_range_pair_mut(j, k);
                    let mut col_j = col_j.rows_range_mut(j..);
                    let col_k = col_k.rows_range(j..);

                    col_j.axpy(factor.conjugate(), &col_k, T::one());
                }

                let sqrt_denom = |v: T| {
                    if v.is_zero() {
                        return None;
                    }
                    v.try_sqrt()
                };

                let diag = unsafe { matrix.get_unchecked((j, j)).clone() };

                if let Some(denom) =
                    sqrt_denom(diag).or_else(|| substitute.clone().and_then(sqrt_denom))
                {
                    unsafe {
                        *matrix.get_unchecked_mut((j, j)) = denom.clone();
                    }

                    let mut col = matrix.view_range_mut(j + 1.., j);
                    col /= denom;
                    continue;
                }

                // The diagonal element is either zero or its square root could not
                // be taken (e.g. for negative real numbers).
                return None;
            }

            Some(Cholesky { chol: matrix })
        }

        /// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `v`,
        /// performs a rank one update such that we end up with the decomposition of `M + sigma * (v * v.adjoint())`.
        #[inline]
        pub fn rank_one_update<R2: Dim, S2>(&mut self, x: &Vector<T, R2, S2>, sigma: T::RealField)
        where
            S2: Storage<T, R2, U1>,
            DefaultAllocator: Allocator<T, R2, U1>,
            ShapeConstraint: SameNumberOfRows<R2, D>,
        {
            Self::xx_rank_one_update(&mut self.chol, &mut x.clone_owned(), sigma)
        }

        /// Updates the decomposition such that we get the decomposition of a matrix with the given column `col` in the `j`th position.
        /// Since the matrix is square, an identical row will be added in the `j`th row.
        pub fn insert_column<R2, S2>(
            &self,
            j: usize,
            col: Vector<T, R2, S2>,
        ) -> Cholesky<T, DimSum<D, U1>>
        where
            D: DimAdd<U1>,
            R2: Dim,
            S2: Storage<T, R2, U1>,
            DefaultAllocator: Allocator<T, DimSum<D, U1>, DimSum<D, U1>> + Allocator<T, R2>,
            ShapeConstraint: SameNumberOfRows<R2, DimSum<D, U1>>,
        {
            let mut col = col.into_owned();
            // for an explanation of the formulas, see https://en.wikipedia.org/wiki/Cholesky_decomposition#Updating_the_decomposition
            let n = col.nrows();
            assert_eq!(
                n,
                self.chol.nrows() + 1,
                "The new column must have the size of the factored matrix plus one."
            );
            assert!(j < n, "j needs to be within the bound of the new matrix.");

            // loads the data into a new matrix with an additional jth row/column
            // TODO: would it be worth it to avoid the zero-initialization?
            let mut chol = Matrix::zeros_generic(
                self.chol.shape_generic().0.add(Const::<1>),
                self.chol.shape_generic().1.add(Const::<1>),
            );
            chol.view_range_mut(..j, ..j)
                .copy_from(&self.chol.view_range(..j, ..j));
            chol.view_range_mut(..j, j + 1..)
                .copy_from(&self.chol.view_range(..j, j..));
            chol.view_range_mut(j + 1.., ..j)
                .copy_from(&self.chol.view_range(j.., ..j));
            chol.view_range_mut(j + 1.., j + 1..)
                .copy_from(&self.chol.view_range(j.., j..));

            // update the jth row
            let top_left_corner = self.chol.view_range(..j, ..j);

            let col_j = col[j].clone();
            let (mut new_rowj_adjoint, mut new_colj) = col.rows_range_pair_mut(..j, j + 1..);
            assert!(
                top_left_corner.solve_lower_triangular_mut(&mut new_rowj_adjoint),
                "Cholesky::insert_column : Unable to solve lower triangular system!"
            );

            new_rowj_adjoint.adjoint_to(&mut chol.view_range_mut(j, ..j));

            // update the center element
            let center_element = T::sqrt(col_j - T::from_real(new_rowj_adjoint.norm_squared()));
            chol[(j, j)] = center_element.clone();

            // update the jth column
            let bottom_left_corner = self.chol.view_range(j.., ..j);
            // new_colj = (col_jplus - bottom_left_corner * new_rowj.adjoint()) / center_element;
            new_colj.gemm(
                -T::one() / center_element.clone(),
                &bottom_left_corner,
                &new_rowj_adjoint,
                T::one() / center_element,
            );
            chol.view_range_mut(j + 1.., j).copy_from(&new_colj);

            // update the bottom right corner
            let mut bottom_right_corner = chol.view_range_mut(j + 1.., j + 1..);
            Self::xx_rank_one_update(
                &mut bottom_right_corner,
                &mut new_colj,
                -T::RealField::one(),
            );

            Cholesky { chol }
        }

        /// Updates the decomposition such that we get the decomposition of the factored matrix with its `j`th column removed.
        /// Since the matrix is square, the `j`th row will also be removed.
        #[must_use]
        pub fn remove_column(&self, j: usize) -> Cholesky<T, DimDiff<D, U1>>
        where
            D: DimSub<U1>,
            DefaultAllocator: Allocator<T, DimDiff<D, U1>, DimDiff<D, U1>> + Allocator<T, D>,
        {
            let n = self.chol.nrows();
            assert!(n > 0, "The matrix needs at least one column.");
            assert!(j < n, "j needs to be within the bound of the matrix.");

            // loads the data into a new matrix except for the jth row/column
            // TODO: would it be worth it to avoid this zero initialization?
            let mut chol = Matrix::zeros_generic(
                self.chol.shape_generic().0.sub(Const::<1>),
                self.chol.shape_generic().1.sub(Const::<1>),
            );
            chol.view_range_mut(..j, ..j)
                .copy_from(&self.chol.view_range(..j, ..j));
            chol.view_range_mut(..j, j..)
                .copy_from(&self.chol.view_range(..j, j + 1..));
            chol.view_range_mut(j.., ..j)
                .copy_from(&self.chol.view_range(j + 1.., ..j));
            chol.view_range_mut(j.., j..)
                .copy_from(&self.chol.view_range(j + 1.., j + 1..));

            // updates the bottom right corner
            let mut bottom_right_corner = chol.view_range_mut(j.., j..);
            let mut workspace = self.chol.column(j).clone_owned();
            let mut old_colj = workspace.rows_range_mut(j + 1..);
            Self::xx_rank_one_update(&mut bottom_right_corner, &mut old_colj, T::RealField::one());

            Cholesky { chol }
        }

        /// Given the Cholesky decomposition of a matrix `M`, a scalar `sigma` and a vector `x`,
        /// performs a rank one update such that we end up with the decomposition of `M + sigma * (x * x.adjoint())`.
        ///
        /// This helper method is called by `rank_one_update` but also `insert_column` and `remove_column`
        /// where it is used on a square view of the decomposition
        fn xx_rank_one_update<Dm, Sm, Rx, Sx>(
            chol: &mut Matrix<T, Dm, Dm, Sm>,
            x: &mut Vector<T, Rx, Sx>,
            sigma: T::RealField,
        ) where
            //T: ComplexField,
            Dm: Dim,
            Rx: Dim,
            Sm: StorageMut<T, Dm, Dm>,
            Sx: StorageMut<T, Rx, U1>,
        {
            // heavily inspired by Eigen's `llt_rank_update_lower` implementation https://eigen.tuxfamily.org/dox/LLT_8h_source.html
            let n = x.nrows();
            assert_eq!(
                n,
                chol.nrows(),
                "The input vector must be of the same size as the factorized matrix."
            );

            let mut beta = crate::one::<T::RealField>();

            for j in 0..n {
                // updates the diagonal
                let diag = T::real(unsafe { chol.get_unchecked((j, j)).clone() });
                let diag2 = diag.clone() * diag.clone();
                let xj = unsafe { x.get_unchecked(j).clone() };
                let sigma_xj2 = sigma.clone() * T::modulus_squared(xj.clone());
                let gamma = diag2.clone() * beta.clone() + sigma_xj2.clone();
                let new_diag = (diag2.clone() + sigma_xj2.clone() / beta.clone()).sqrt();
                unsafe { *chol.get_unchecked_mut((j, j)) = T::from_real(new_diag.clone()) };
                beta += sigma_xj2 / diag2;
                // updates the terms of L
                let mut xjplus = x.rows_range_mut(j + 1..);
                let mut col_j = chol.view_range_mut(j + 1.., j);
                // temp_jplus -= (wj / T::from_real(diag)) * col_j;
                xjplus.axpy(-xj.clone() / T::from_real(diag.clone()), &col_j, T::one());
                if gamma != crate::zero::<T::RealField>() {
                    // col_j = T::from_real(nljj / diag) * col_j  + (T::from_real(nljj * sigma / gamma) * T::conjugate(wj)) * temp_jplus;
                    col_j.axpy(
                        T::from_real(new_diag.clone() * sigma.clone() / gamma) * T::conjugate(xj),
                        &xjplus,
                        T::from_real(new_diag / diag),
                    );
                }
            }
        }
    }