nalgebra_sparse/factorization/

cholesky.rs

use crate::csc::CscMatrix;
use crate::ops::Op;
use crate::ops::serial::spsolve_csc_lower_triangular;
use crate::pattern::SparsityPattern;
use core::{iter, mem};
use nalgebra::{DMatrix, DMatrixView, DMatrixViewMut, RealField};
use std::fmt::{Display, Formatter};

/// A symbolic sparse Cholesky factorization of a CSC matrix.
///
/// The symbolic factorization computes the sparsity pattern of `L`, the Cholesky factor.
#[derive(Debug, Clone, PartialEq, Eq)]
pub struct CscSymbolicCholesky {
    // Pattern of the original matrix that was decomposed
    m_pattern: SparsityPattern,
    l_pattern: SparsityPattern,
    // u in this context is L^T, so that M = L L^T
    u_pattern: SparsityPattern,
}

impl CscSymbolicCholesky {
    /// Compute the symbolic factorization for a sparsity pattern belonging to a CSC matrix.
    ///
    /// The sparsity pattern must be symmetric. However, this is not enforced, and it is the
    /// responsibility of the user to ensure that this property holds.
    ///
    /// # Panics
    ///
    /// Panics if the sparsity pattern is not square.
    pub fn factor(pattern: SparsityPattern) -> Self {
        assert_eq!(
            pattern.major_dim(),
            pattern.minor_dim(),
            "Major and minor dimensions must be the same (square matrix)."
        );
        let (l_pattern, u_pattern) = nonzero_pattern(&pattern);
        Self {
            m_pattern: pattern,
            l_pattern,
            u_pattern,
        }
    }

    /// The pattern of the Cholesky factor `L`.
    #[must_use]
    pub fn l_pattern(&self) -> &SparsityPattern {
        &self.l_pattern
    }
}

/// A sparse Cholesky factorization `A = L L^T` of a [`CscMatrix`].
///
/// The factor `L` is a sparse, lower-triangular matrix. See the article on [Wikipedia] for
/// more information.
///
/// The implementation is a port of the `CsCholesky` implementation in `nalgebra`. It is similar
/// to Tim Davis' [`CSparse`]. The current implementation performs no fill-in reduction, and can
/// therefore be expected to produce much too dense Cholesky factors for many matrices.
/// It is therefore not currently recommended to use this implementation for serious projects.
///
/// [`CSparse`]: https://epubs.siam.org/doi/book/10.1137/1.9780898718881
/// [Wikipedia]: https://en.wikipedia.org/wiki/Cholesky_decomposition
// TODO: We should probably implement PartialEq/Eq, but in that case we'd probably need a
// custom implementation, due to the need to exclude the workspace arrays
#[derive(Debug, Clone)]
pub struct CscCholesky<T> {
    // Pattern of the original matrix
    m_pattern: SparsityPattern,
    l_factor: CscMatrix<T>,
    u_pattern: SparsityPattern,
    work_x: Vec<T>,
    work_c: Vec<usize>,
}

#[derive(Debug, PartialEq, Eq, Copy, Clone)]
#[non_exhaustive]
/// Possible errors produced by the Cholesky factorization.
pub enum CholeskyError {
    /// The matrix is not positive definite.
    NotPositiveDefinite,
}

impl Display for CholeskyError {
    fn fmt(&self, f: &mut Formatter<'_>) -> std::fmt::Result {
        write!(f, "Matrix is not positive definite")
    }
}

impl std::error::Error for CholeskyError {}

impl<T: RealField> CscCholesky<T> {
    /// Computes the numerical Cholesky factorization associated with the given
    /// symbolic factorization and the provided values.
    ///
    /// The values correspond to the non-zero values of the CSC matrix for which the
    /// symbolic factorization was computed.
    ///
    /// # Errors
    ///
    /// Returns an error if the numerical factorization fails. This can occur if the matrix is not
    /// symmetric positive definite.
    ///
    /// # Panics
    ///
    /// Panics if the number of values differ from the number of non-zeros of the sparsity pattern
    /// of the matrix that was symbolically factored.
    pub fn factor_numerical(
        symbolic: CscSymbolicCholesky,
        values: &[T],
    ) -> Result<Self, CholeskyError> {
        assert_eq!(
            symbolic.l_pattern.nnz(),
            symbolic.u_pattern.nnz(),
            "u is just the transpose of l, so should have the same nnz"
        );

        let l_nnz = symbolic.l_pattern.nnz();
        let l_values = vec![T::zero(); l_nnz];
        let l_factor =
            CscMatrix::try_from_pattern_and_values(symbolic.l_pattern, l_values).unwrap();

        let (nrows, ncols) = (l_factor.nrows(), l_factor.ncols());

        let mut factorization = CscCholesky {
            m_pattern: symbolic.m_pattern,
            l_factor,
            u_pattern: symbolic.u_pattern,
            work_x: vec![T::zero(); nrows],
            // Fill with MAX so that things hopefully totally fail if values are not
            // overwritten. Might be easier to debug this way
            work_c: vec![usize::MAX, ncols],
        };

        factorization.refactor(values)?;
        Ok(factorization)
    }

    /// Computes the Cholesky factorization of the provided matrix.
    ///
    /// The matrix must be symmetric positive definite. Symmetry is not checked, and it is up
    /// to the user to enforce this property.
    ///
    /// # Errors
    ///
    /// Returns an error if the numerical factorization fails. This can occur if the matrix is not
    /// symmetric positive definite.
    ///
    /// # Panics
    ///
    /// Panics if the matrix is not square.
    pub fn factor(matrix: &CscMatrix<T>) -> Result<Self, CholeskyError> {
        let symbolic = CscSymbolicCholesky::factor(matrix.pattern().clone());
        Self::factor_numerical(symbolic, matrix.values())
    }

    /// Re-computes the factorization for a new set of non-zero values.
    ///
    /// This is useful when the values of a matrix changes, but the sparsity pattern remains
    /// constant.
    ///
    /// # Errors
    ///
    /// Returns an error if the numerical factorization fails. This can occur if the matrix is not
    /// symmetric positive definite.
    ///
    /// # Panics
    ///
    /// Panics if the number of values does not match the number of non-zeros in the sparsity
    /// pattern.
    pub fn refactor(&mut self, values: &[T]) -> Result<(), CholeskyError> {
        self.decompose_left_looking(values)
    }

    /// Returns a reference to the Cholesky factor `L`.
    #[must_use]
    pub fn l(&self) -> &CscMatrix<T> {
        &self.l_factor
    }

    /// Returns the Cholesky factor `L`.
    pub fn take_l(self) -> CscMatrix<T> {
        self.l_factor
    }

    /// Perform a numerical left-looking cholesky decomposition of a matrix with the same structure as the
    /// one used to initialize `self`, but with different non-zero values provided by `values`.
    fn decompose_left_looking(&mut self, values: &[T]) -> Result<(), CholeskyError> {
        assert!(
            values.len() >= self.m_pattern.nnz(),
            // TODO: Improve error message
            "The set of values is too small."
        );

        let n = self.l_factor.nrows();

        // Reset `work_c` to the column pointers of `l`.
        self.work_c.clear();
        self.work_c.extend_from_slice(self.l_factor.col_offsets());

        unsafe {
            for k in 0..n {
                // Scatter the k-th column of the original matrix with the values provided.
                let range_begin = *self.m_pattern.major_offsets().get_unchecked(k);
                let range_end = *self.m_pattern.major_offsets().get_unchecked(k + 1);
                let range_k = range_begin..range_end;

                *self.work_x.get_unchecked_mut(k) = T::zero();
                for p in range_k.clone() {
                    let irow = *self.m_pattern.minor_indices().get_unchecked(p);

                    if irow >= k {
                        *self.work_x.get_unchecked_mut(irow) = values.get_unchecked(p).clone();
                    }
                }

                for &j in self.u_pattern.lane(k) {
                    let factor = -self
                        .l_factor
                        .values()
                        .get_unchecked(*self.work_c.get_unchecked(j))
                        .clone();
                    *self.work_c.get_unchecked_mut(j) += 1;

                    if j < k {
                        let col_j = self.l_factor.col(j);
                        let col_j_entries = col_j.row_indices().iter().zip(col_j.values());
                        for (&z, val) in col_j_entries {
                            if z >= k {
                                *self.work_x.get_unchecked_mut(z) += val.clone() * factor.clone();
                            }
                        }
                    }
                }

                let diag = self.work_x.get_unchecked(k).clone();

                if diag > T::zero() {
                    let denom = diag.sqrt();

                    {
                        let (offsets, _, values) = self.l_factor.csc_data_mut();
                        *values.get_unchecked_mut(*offsets.get_unchecked(k)) = denom.clone();
                    }

                    let mut col_k = self.l_factor.col_mut(k);
                    let (col_k_rows, col_k_values) = col_k.rows_and_values_mut();
                    let col_k_entries = col_k_rows.iter().zip(col_k_values);
                    for (&p, val) in col_k_entries {
                        *val = self.work_x.get_unchecked(p).clone() / denom.clone();
                        *self.work_x.get_unchecked_mut(p) = T::zero();
                    }
                } else {
                    return Err(CholeskyError::NotPositiveDefinite);
                }
            }
        }

        Ok(())
    }

    /// Solves the system `A X = B`, where `X` and `B` are dense matrices.
    ///
    /// # Panics
    ///
    /// Panics if `B` is not square.
    #[must_use = "Did you mean to use solve_mut()?"]
    pub fn solve<'a>(&'a self, b: impl Into<DMatrixView<'a, T>>) -> DMatrix<T> {
        let b = b.into();
        let mut output = b.clone_owned();
        self.solve_mut(&mut output);
        output
    }

    /// Solves the system `AX = B`, where `X` and `B` are dense matrices.
    ///
    /// The result is stored in-place in `b`.
    ///
    /// # Panics
    ///
    /// Panics if `b` is not square.
    pub fn solve_mut<'a>(&'a self, b: impl Into<DMatrixViewMut<'a, T>>) {
        let expect_msg = "If the Cholesky factorization succeeded,\
            then the triangular solve should never fail";
        // Solve LY = B
        let mut y = b.into();
        spsolve_csc_lower_triangular(Op::NoOp(self.l()), &mut y).expect(expect_msg);

        // Solve L^T X = Y
        let mut x = y;
        spsolve_csc_lower_triangular(Op::Transpose(self.l()), &mut x).expect(expect_msg);
    }
}

fn reach(
    pattern: &SparsityPattern,
    j: usize,
    max_j: usize,
    tree: &[usize],
    marks: &mut Vec<bool>,
    out: &mut Vec<usize>,
) {
    marks.clear();
    marks.resize(tree.len(), false);

    // TODO: avoid all those allocations.
    let mut tmp = Vec::new();
    let mut res = Vec::new();

    for &irow in pattern.lane(j) {
        let mut curr = irow;
        while curr != usize::MAX && curr <= max_j && !marks[curr] {
            marks[curr] = true;
            tmp.push(curr);
            curr = tree[curr];
        }

        tmp.append(&mut res);
        mem::swap(&mut tmp, &mut res);
    }

    res.sort_unstable();

    out.append(&mut res);
}

fn nonzero_pattern(m: &SparsityPattern) -> (SparsityPattern, SparsityPattern) {
    let etree = elimination_tree(m);
    // Note: We assume CSC, therefore rows == minor and cols == major
    let (nrows, ncols) = (m.minor_dim(), m.major_dim());
    let mut rows = Vec::with_capacity(m.nnz());
    let mut col_offsets = Vec::with_capacity(ncols + 1);
    let mut marks = Vec::new();

    // NOTE: the following will actually compute the non-zero pattern of
    // the transpose of l.
    col_offsets.push(0);
    for i in 0..nrows {
        reach(m, i, i, &etree, &mut marks, &mut rows);
        col_offsets.push(rows.len());
    }

    let u_pattern =
        SparsityPattern::try_from_offsets_and_indices(nrows, ncols, col_offsets, rows).unwrap();

    // TODO: Avoid this transpose?
    let l_pattern = u_pattern.transpose();

    (l_pattern, u_pattern)
}

fn elimination_tree(pattern: &SparsityPattern) -> Vec<usize> {
    // Note: The pattern is assumed to of a CSC matrix, so the number of rows is
    // given by the minor dimension
    let nrows = pattern.minor_dim();
    let mut forest: Vec<_> = iter::repeat(usize::MAX).take(nrows).collect();
    let mut ancestor: Vec<_> = iter::repeat(usize::MAX).take(nrows).collect();

    for k in 0..nrows {
        for &irow in pattern.lane(k) {
            let mut i = irow;

            while i < k {
                let i_ancestor = ancestor[i];
                ancestor[i] = k;

                if i_ancestor == usize::MAX {
                    forest[i] = k;
                    break;
                }

                i = i_ancestor;
            }
        }
    }

    forest
}