ndarray_linalg/

solveh.rs

//! Solve Hermitian (or real symmetric) linear problems and invert Hermitian
//! (or real symmetric) matrices
//!
//! **Note that only the upper triangular portion of the matrix is used.**
//!
//! # Examples
//!
//! Solve `A * x = b`, where `A` is a Hermitian (or real symmetric) matrix:
//!
//! ```
//! use ndarray::prelude::*;
//! use ndarray_linalg::SolveH;
//!
//! let a: Array2<f64> = array![
//!     [3., 2., -1.],
//!     [2., -2., 4.],
//!     [-1., 4., 5.]
//! ];
//! let b: Array1<f64> = array![11., -12., 1.];
//! let x = a.solveh_into(b).unwrap();
//! assert!(x.abs_diff_eq(&array![1., 3., -2.], 1e-9));
//! ```
//!
//! If you are solving multiple systems of linear equations with the same
//! Hermitian or real symmetric coefficient matrix `A`, it's faster to compute
//! the factorization once at the beginning than solving directly using `A`:
//!
//! ```
//! use ndarray::prelude::*;
//! use ndarray_linalg::*;
//!
//! /// Use fixed algorithm and seed of PRNG for reproducible test
//! let mut rng = rand_pcg::Mcg128Xsl64::new(0xcafef00dd15ea5e5);
//!
//! let a: Array2<f64> = random_using((3, 3), &mut rng);
//! let f = a.factorizeh_into().unwrap(); // Factorize A (A is consumed)
//! for _ in 0..10 {
//!     let b: Array1<f64> = random_using(3, &mut rng);
//!     let x = f.solveh_into(b).unwrap(); // Solve A * x = b using the factorization
//! }
//! ```

use ndarray::*;
use num_traits::{Float, One, Zero};

use crate::convert::*;
use crate::error::*;
use crate::layout::*;
use crate::types::*;

pub use lax::{Pivot, UPLO};

/// An interface for solving systems of Hermitian (or real symmetric) linear equations.
///
/// If you plan to solve many equations with the same Hermitian (or real
/// symmetric) coefficient matrix `A` but different `b` vectors, it's faster to
/// factor the `A` matrix once using the `FactorizeH` trait, and then solve
/// using the `BKFactorized` struct.
pub trait SolveH<A: Scalar> {
    /// Solves a system of linear equations `A * x = b` with Hermitian (or real
    /// symmetric) matrix `A`, where `A` is `self`, `b` is the argument, and
    /// `x` is the successful result.
    ///
    /// # Panics
    ///
    /// Panics if the length of `b` is not the equal to the number of columns
    /// of `A`.
    fn solveh(&self, b: &ArrayRef<A, Ix1>) -> Result<Array1<A>> {
        let mut b = replicate(b);
        self.solveh_inplace(&mut b)?;
        Ok(b)
    }

    /// Solves a system of linear equations `A * x = b` with Hermitian (or real
    /// symmetric) matrix `A`, where `A` is `self`, `b` is the argument, and
    /// `x` is the successful result.
    ///
    /// # Panics
    ///
    /// Panics if the length of `b` is not the equal to the number of columns
    /// of `A`.
    fn solveh_into<S: DataMut<Elem = A>>(
        &self,
        mut b: ArrayBase<S, Ix1>,
    ) -> Result<ArrayBase<S, Ix1>> {
        self.solveh_inplace(&mut b)?;
        Ok(b)
    }

    /// Solves a system of linear equations `A * x = b` with Hermitian (or real
    /// symmetric) matrix `A`, where `A` is `self`, `b` is the argument, and
    /// `x` is the successful result. The value of `x` is also assigned to the
    /// argument.
    ///
    /// # Panics
    ///
    /// Panics if the length of `b` is not the equal to the number of columns
    /// of `A`.
    fn solveh_inplace<'a>(&self, b: &'a mut ArrayRef<A, Ix1>) -> Result<&'a mut ArrayRef<A, Ix1>>;
}

/// Represents the Bunch–Kaufman factorization of a Hermitian (or real
/// symmetric) matrix as `A = P * U * D * U^H * P^T`.
pub struct BKFactorized<S: Data> {
    pub a: ArrayBase<S, Ix2>,
    pub ipiv: Pivot,
}

impl<A, S> SolveH<A> for BKFactorized<S>
where
    A: Scalar + Lapack,
    S: Data<Elem = A>,
{
    fn solveh_inplace<'a>(
        &self,
        rhs: &'a mut ArrayRef<A, Ix1>,
    ) -> Result<&'a mut ArrayRef<A, Ix1>> {
        assert_eq!(
            rhs.len(),
            self.a.len_of(Axis(1)),
            "The length of `rhs` must be compatible with the shape of the factored matrix.",
        );
        A::solveh(
            self.a.square_layout()?,
            UPLO::Upper,
            self.a.as_allocated()?,
            &self.ipiv,
            rhs.as_slice_mut().unwrap(),
        )?;
        Ok(rhs)
    }
}

impl<A> SolveH<A> for ArrayRef<A, Ix2>
where
    A: Scalar + Lapack,
{
    fn solveh_inplace<'a>(
        &self,
        rhs: &'a mut ArrayRef<A, Ix1>,
    ) -> Result<&'a mut ArrayRef<A, Ix1>> {
        let f = self.factorizeh()?;
        f.solveh_inplace(rhs)
    }
}

/// An interface for computing the Bunch–Kaufman factorization of Hermitian (or
/// real symmetric) matrix refs.
pub trait FactorizeH<S: Data> {
    /// Computes the Bunch–Kaufman factorization of a Hermitian (or real
    /// symmetric) matrix.
    fn factorizeh(&self) -> Result<BKFactorized<S>>;
}

/// An interface for computing the Bunch–Kaufman factorization of Hermitian (or
/// real symmetric) matrices.
pub trait FactorizeHInto<S: Data> {
    /// Computes the Bunch–Kaufman factorization of a Hermitian (or real
    /// symmetric) matrix.
    fn factorizeh_into(self) -> Result<BKFactorized<S>>;
}

impl<A, S> FactorizeHInto<S> for ArrayBase<S, Ix2>
where
    A: Scalar + Lapack,
    S: DataMut<Elem = A>,
{
    fn factorizeh_into(mut self) -> Result<BKFactorized<S>> {
        let ipiv = A::bk(self.square_layout()?, UPLO::Upper, self.as_allocated_mut()?)?;
        Ok(BKFactorized { a: self, ipiv })
    }
}

impl<A> FactorizeH<OwnedRepr<A>> for ArrayRef<A, Ix2>
where
    A: Scalar + Lapack,
{
    fn factorizeh(&self) -> Result<BKFactorized<OwnedRepr<A>>> {
        let mut a: Array2<A> = replicate(self);
        let ipiv = A::bk(a.square_layout()?, UPLO::Upper, a.as_allocated_mut()?)?;
        Ok(BKFactorized { a, ipiv })
    }
}

/// An interface for inverting Hermitian (or real symmetric) matrix refs.
pub trait InverseH {
    type Output;
    /// Computes the inverse of the Hermitian (or real symmetric) matrix.
    fn invh(&self) -> Result<Self::Output>;
}

/// An interface for inverting Hermitian (or real symmetric) matrices.
pub trait InverseHInto {
    type Output;
    /// Computes the inverse of the Hermitian (or real symmetric) matrix.
    fn invh_into(self) -> Result<Self::Output>;
}

impl<A, S> InverseHInto for BKFactorized<S>
where
    A: Scalar + Lapack,
    S: DataMut<Elem = A>,
{
    type Output = ArrayBase<S, Ix2>;

    fn invh_into(mut self) -> Result<ArrayBase<S, Ix2>> {
        A::invh(
            self.a.square_layout()?,
            UPLO::Upper,
            self.a.as_allocated_mut()?,
            &self.ipiv,
        )?;
        triangular_fill_hermitian(&mut self.a, UPLO::Upper);
        Ok(self.a)
    }
}

impl<A, S> InverseH for BKFactorized<S>
where
    A: Scalar + Lapack,
    S: Data<Elem = A>,
{
    type Output = Array2<A>;

    fn invh(&self) -> Result<Self::Output> {
        let f = BKFactorized {
            a: replicate(&self.a),
            ipiv: self.ipiv.clone(),
        };
        f.invh_into()
    }
}

impl<A, S> InverseHInto for ArrayBase<S, Ix2>
where
    A: Scalar + Lapack,
    S: DataMut<Elem = A>,
{
    type Output = Self;

    fn invh_into(self) -> Result<Self::Output> {
        let f = self.factorizeh_into()?;
        f.invh_into()
    }
}

impl<A> InverseH for ArrayRef<A, Ix2>
where
    A: Scalar + Lapack,
{
    type Output = Array2<A>;

    fn invh(&self) -> Result<Self::Output> {
        let f = self.factorizeh()?;
        f.invh_into()
    }
}

/// An interface for calculating determinants of Hermitian (or real symmetric) matrix refs.
pub trait DeterminantH {
    /// The element type of the matrix.
    type Elem: Scalar;

    /// Computes the determinant of the Hermitian (or real symmetric) matrix.
    fn deth(&self) -> Result<<Self::Elem as Scalar>::Real>;

    /// Computes the `(sign, natural_log)` of the determinant of the Hermitian
    /// (or real symmetric) matrix.
    ///
    /// The `natural_log` is the natural logarithm of the absolute value of the
    /// determinant. If the determinant is zero, `sign` is 0 and `natural_log`
    /// is negative infinity.
    ///
    /// To obtain the determinant, you can compute `sign * natural_log.exp()`
    /// or just call `.deth()` instead.
    ///
    /// This method is more robust than `.deth()` to very small or very large
    /// determinants since it returns the natural logarithm of the determinant
    /// rather than the determinant itself.
    fn sln_deth(&self) -> Result<(<Self::Elem as Scalar>::Real, <Self::Elem as Scalar>::Real)>;
}

/// An interface for calculating determinants of Hermitian (or real symmetric) matrices.
pub trait DeterminantHInto {
    /// The element type of the matrix.
    type Elem: Scalar;

    /// Computes the determinant of the Hermitian (or real symmetric) matrix.
    fn deth_into(self) -> Result<<Self::Elem as Scalar>::Real>;

    /// Computes the `(sign, natural_log)` of the determinant of the Hermitian
    /// (or real symmetric) matrix.
    ///
    /// The `natural_log` is the natural logarithm of the absolute value of the
    /// determinant. If the determinant is zero, `sign` is 0 and `natural_log`
    /// is negative infinity.
    ///
    /// To obtain the determinant, you can compute `sign * natural_log.exp()`
    /// or just call `.deth_into()` instead.
    ///
    /// This method is more robust than `.deth_into()` to very small or very
    /// large determinants since it returns the natural logarithm of the
    /// determinant rather than the determinant itself.
    fn sln_deth_into(self) -> Result<(<Self::Elem as Scalar>::Real, <Self::Elem as Scalar>::Real)>;
}

/// Returns the sign and natural log of the determinant.
fn bk_sln_det<P, A>(uplo: UPLO, ipiv_iter: P, a: &ArrayRef<A, Ix2>) -> (A::Real, A::Real)
where
    P: Iterator<Item = i32>,
    A: Scalar + Lapack,
{
    let layout = a.layout().unwrap();
    let mut sign = A::Real::one();
    let mut ln_det = A::Real::zero();
    let mut ipiv_enum = ipiv_iter.enumerate();
    while let Some((k, ipiv_k)) = ipiv_enum.next() {
        debug_assert!(k < a.nrows() && k < a.ncols());
        if ipiv_k > 0 {
            // 1x1 block at k, must be real.
            let elem = unsafe { a.uget((k, k)) }.re();
            debug_assert_eq!(elem.im(), Zero::zero());
            sign *= elem.signum();
            ln_det += Float::ln(Float::abs(elem));
        } else {
            // 2x2 block at k..k+2.

            // Upper left diagonal elem, must be real.
            let upper_diag = unsafe { a.uget((k, k)) }.re();
            debug_assert_eq!(upper_diag.im(), Zero::zero());

            // Lower right diagonal elem, must be real.
            let lower_diag = unsafe { a.uget((k + 1, k + 1)) }.re();
            debug_assert_eq!(lower_diag.im(), Zero::zero());

            // Off-diagonal elements, can be complex.
            let off_diag = match layout {
                MatrixLayout::C { .. } => match uplo {
                    UPLO::Upper => unsafe { a.uget((k + 1, k)) },
                    UPLO::Lower => unsafe { a.uget((k, k + 1)) },
                },
                MatrixLayout::F { .. } => match uplo {
                    UPLO::Upper => unsafe { a.uget((k, k + 1)) },
                    UPLO::Lower => unsafe { a.uget((k + 1, k)) },
                },
            };

            // Determinant of 2x2 block.
            let block_det = upper_diag * lower_diag - off_diag.square();
            sign *= block_det.signum();
            ln_det += Float::ln(Float::abs(block_det));

            // Skip the k+1 ipiv value.
            ipiv_enum.next();
        }
    }
    (sign, ln_det)
}

impl<A, S> BKFactorized<S>
where
    A: Scalar + Lapack,
    S: Data<Elem = A>,
{
    /// Computes the determinant of the factorized Hermitian (or real
    /// symmetric) matrix.
    pub fn deth(&self) -> A::Real {
        let (sign, ln_det) = self.sln_deth();
        sign * Float::exp(ln_det)
    }

    /// Computes the `(sign, natural_log)` of the determinant of the factorized
    /// Hermitian (or real symmetric) matrix.
    ///
    /// The `natural_log` is the natural logarithm of the absolute value of the
    /// determinant. If the determinant is zero, `sign` is 0 and `natural_log`
    /// is negative infinity.
    ///
    /// To obtain the determinant, you can compute `sign * natural_log.exp()`
    /// or just call `.deth()` instead.
    ///
    /// This method is more robust than `.deth()` to very small or very large
    /// determinants since it returns the natural logarithm of the determinant
    /// rather than the determinant itself.
    pub fn sln_deth(&self) -> (A::Real, A::Real) {
        bk_sln_det(UPLO::Upper, self.ipiv.iter().cloned(), &self.a)
    }

    /// Computes the determinant of the factorized Hermitian (or real
    /// symmetric) matrix.
    pub fn deth_into(self) -> A::Real {
        let (sign, ln_det) = self.sln_deth_into();
        sign * Float::exp(ln_det)
    }

    /// Computes the `(sign, natural_log)` of the determinant of the factorized
    /// Hermitian (or real symmetric) matrix.
    ///
    /// The `natural_log` is the natural logarithm of the absolute value of the
    /// determinant. If the determinant is zero, `sign` is 0 and `natural_log`
    /// is negative infinity.
    ///
    /// To obtain the determinant, you can compute `sign * natural_log.exp()`
    /// or just call `.deth_into()` instead.
    ///
    /// This method is more robust than `.deth_into()` to very small or very
    /// large determinants since it returns the natural logarithm of the
    /// determinant rather than the determinant itself.
    pub fn sln_deth_into(self) -> (A::Real, A::Real) {
        bk_sln_det(UPLO::Upper, self.ipiv.into_iter(), &self.a)
    }
}

impl<A> DeterminantH for ArrayRef<A, Ix2>
where
    A: Scalar + Lapack,
{
    type Elem = A;

    fn deth(&self) -> Result<A::Real> {
        let (sign, ln_det) = self.sln_deth()?;
        Ok(sign * Float::exp(ln_det))
    }

    fn sln_deth(&self) -> Result<(A::Real, A::Real)> {
        match self.factorizeh() {
            Ok(fac) => Ok(fac.sln_deth()),
            Err(LinalgError::Lapack(e))
                if matches!(e, lax::error::Error::LapackComputationalFailure { .. }) =>
            {
                // Determinant is zero.
                Ok((A::Real::zero(), A::Real::neg_infinity()))
            }
            Err(err) => Err(err),
        }
    }
}

impl<A, S> DeterminantHInto for ArrayBase<S, Ix2>
where
    A: Scalar + Lapack,
    S: DataMut<Elem = A>,
{
    type Elem = A;

    fn deth_into(self) -> Result<A::Real> {
        let (sign, ln_det) = self.sln_deth_into()?;
        Ok(sign * Float::exp(ln_det))
    }

    fn sln_deth_into(self) -> Result<(A::Real, A::Real)> {
        match self.factorizeh_into() {
            Ok(fac) => Ok(fac.sln_deth_into()),
            Err(LinalgError::Lapack(e))
                if matches!(e, lax::error::Error::LapackComputationalFailure { .. }) =>
            {
                // Determinant is zero.
                Ok((A::Real::zero(), A::Real::neg_infinity()))
            }
            Err(err) => Err(err),
        }
    }
}