nalgebra/linalg/decomposition.rs
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use crate::storage::Storage;
use crate::{
Allocator, Bidiagonal, Cholesky, ColPivQR, ComplexField, DefaultAllocator, Dim, DimDiff,
DimMin, DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, OMatrix, RealField, Schur,
SymmetricEigen, SymmetricTridiagonal, LU, QR, SVD, U1, UDU,
};
/// # Rectangular matrix decomposition
///
/// This section contains the methods for computing some common decompositions of rectangular
/// matrices with real or complex components. The following are currently supported:
///
/// | Decomposition | Factors | Details |
/// | -------------------------|---------------------|--------------|
/// | QR | `Q * R` | `Q` is an unitary matrix, and `R` is upper-triangular. |
/// | QR with column pivoting | `Q * R * P⁻¹` | `Q` is an unitary matrix, and `R` is upper-triangular. `P` is a permutation matrix. |
/// | LU with partial pivoting | `P⁻¹ * L * U` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` is a permutation matrix. |
/// | LU with full pivoting | `P⁻¹ * L * U * Q⁻¹` | `L` is lower-triangular with a diagonal filled with `1` and `U` is upper-triangular. `P` and `Q` are permutation matrices. |
/// | SVD | `U * Σ * Vᵀ` | `U` and `V` are two orthogonal matrices and `Σ` is a diagonal matrix containing the singular values. |
/// | Polar (Left Polar) | `P' * U` | `U` is semi-unitary/unitary and `P'` is a positive semi-definite Hermitian Matrix
impl<T: ComplexField, R: Dim, C: Dim, S: Storage<T, R, C>> Matrix<T, R, C, S> {
/// Computes the bidiagonalization using householder reflections.
pub fn bidiagonalize(self) -> Bidiagonal<T, R, C>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>,
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
Bidiagonal::new(self.into_owned())
}
/// Computes the LU decomposition with full pivoting of `matrix`.
///
/// This effectively computes `P, L, U, Q` such that `P * matrix * Q = LU`.
pub fn full_piv_lu(self) -> FullPivLU<T, R, C>
where
R: DimMin<C>,
DefaultAllocator: Allocator<R, C> + Allocator<DimMinimum<R, C>>,
{
FullPivLU::new(self.into_owned())
}
/// Computes the LU decomposition with partial (row) pivoting of `matrix`.
pub fn lu(self) -> LU<T, R, C>
where
R: DimMin<C>,
DefaultAllocator: Allocator<R, C> + Allocator<DimMinimum<R, C>>,
{
LU::new(self.into_owned())
}
/// Computes the QR decomposition of this matrix.
pub fn qr(self) -> QR<T, R, C>
where
R: DimMin<C>,
DefaultAllocator: Allocator<R, C> + Allocator<R> + Allocator<DimMinimum<R, C>>,
{
QR::new(self.into_owned())
}
/// Computes the QR decomposition (with column pivoting) of this matrix.
pub fn col_piv_qr(self) -> ColPivQR<T, R, C>
where
R: DimMin<C>,
DefaultAllocator: Allocator<R, C>
+ Allocator<R>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>,
{
ColPivQR::new(self.into_owned())
}
/// Computes the Singular Value Decomposition using implicit shift.
/// The singular values are guaranteed to be sorted in descending order.
/// If this order is not required consider using `svd_unordered`.
pub fn svd(self, compute_u: bool, compute_v: bool) -> SVD<T, R, C>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>,
{
SVD::new(self.into_owned(), compute_u, compute_v)
}
/// Computes the Singular Value Decomposition using implicit shift.
/// The singular values are not guaranteed to be sorted in any particular order.
/// If a descending order is required, consider using `svd` instead.
pub fn svd_unordered(self, compute_u: bool, compute_v: bool) -> SVD<T, R, C>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>,
{
SVD::new_unordered(self.into_owned(), compute_u, compute_v)
}
/// Attempts to compute the Singular Value Decomposition of `matrix` using implicit shift.
/// The singular values are guaranteed to be sorted in descending order.
/// If this order is not required consider using `try_svd_unordered`.
///
/// # Arguments
///
/// * `compute_u` − set this to `true` to enable the computation of left-singular vectors.
/// * `compute_v` − set this to `true` to enable the computation of right-singular vectors.
/// * `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_svd(
self,
compute_u: bool,
compute_v: bool,
eps: T::RealField,
max_niter: usize,
) -> Option<SVD<T, R, C>>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>,
{
SVD::try_new(self.into_owned(), compute_u, compute_v, eps, max_niter)
}
/// Attempts to compute the Singular Value Decomposition of `matrix` using implicit shift.
/// The singular values are not guaranteed to be sorted in any particular order.
/// If a descending order is required, consider using `try_svd` instead.
///
/// # Arguments
///
/// * `compute_u` − set this to `true` to enable the computation of left-singular vectors.
/// * `compute_v` − set this to `true` to enable the computation of right-singular vectors.
/// * `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_svd_unordered(
self,
compute_u: bool,
compute_v: bool,
eps: T::RealField,
max_niter: usize,
) -> Option<SVD<T, R, C>>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
SVD::try_new_unordered(self.into_owned(), compute_u, compute_v, eps, max_niter)
}
/// Computes the Polar Decomposition of a `matrix` (indirectly uses SVD).
pub fn polar(self) -> (OMatrix<T, R, R>, OMatrix<T, R, C>)
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<DimMinimum<R, C>, R>
+ Allocator<DimMinimum<R, C>>
+ Allocator<R, R>
+ Allocator<DimMinimum<R, C>, DimMinimum<R, C>>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
SVD::new_unordered(self.into_owned(), true, true)
.to_polar()
.unwrap()
}
/// Attempts to compute the Polar Decomposition of a `matrix` (indirectly uses SVD).
///
/// # Arguments
///
/// * `eps` − tolerance used to determine when a value converged to 0 when computing the SVD.
/// * `max_niter` − maximum total number of iterations performed by the SVD computation algorithm.
pub fn try_polar(
self,
eps: T::RealField,
max_niter: usize,
) -> Option<(OMatrix<T, R, R>, OMatrix<T, R, C>)>
where
R: DimMin<C>,
DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
DefaultAllocator: Allocator<R, C>
+ Allocator<DimMinimum<R, C>, R>
+ Allocator<DimMinimum<R, C>>
+ Allocator<R, R>
+ Allocator<DimMinimum<R, C>, DimMinimum<R, C>>
+ Allocator<C>
+ Allocator<R>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>
+ Allocator<DimMinimum<R, C>, C>
+ Allocator<R, DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimMinimum<R, C>>
+ Allocator<DimDiff<DimMinimum<R, C>, U1>>,
{
SVD::try_new_unordered(self.into_owned(), true, true, eps, max_niter)
.and_then(|svd| svd.to_polar())
}
}
/// # Square matrix decomposition
///
/// This section contains the methods for computing some common decompositions of square
/// matrices with real or complex components. The following are currently supported:
///
/// | Decomposition | Factors | Details |
/// | -------------------------|---------------------------|--------------|
/// | Hessenberg | `Q * H * Qᵀ` | `Q` is a unitary matrix and `H` an upper-Hessenberg matrix. |
/// | Cholesky | `L * Lᵀ` | `L` is a lower-triangular matrix. |
/// | UDU | `U * D * Uᵀ` | `U` is a upper-triangular matrix, and `D` a diagonal matrix. |
/// | Schur decomposition | `Q * T * Qᵀ` | `Q` is an unitary matrix and `T` a quasi-upper-triangular matrix. |
/// | Symmetric eigendecomposition | `Q ~ Λ ~ Qᵀ` | `Q` is an unitary matrix, and `Λ` is a real diagonal matrix. |
/// | Symmetric tridiagonalization | `Q ~ T ~ Qᵀ` | `Q` is an unitary matrix, and `T` is a tridiagonal matrix. |
impl<T: ComplexField, D: Dim, S: Storage<T, D, D>> Matrix<T, D, D, S> {
/// Attempts to compute the Cholesky decomposition of this 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 cholesky(self) -> Option<Cholesky<T, D>>
where
DefaultAllocator: Allocator<D, D>,
{
Cholesky::new(self.into_owned())
}
/// Attempts to compute the UDU decomposition of this matrix.
///
/// The input matrix `self` is assumed to be symmetric and this decomposition will only read
/// the upper-triangular part of `self`.
pub fn udu(self) -> Option<UDU<T, D>>
where
T: RealField,
DefaultAllocator: Allocator<D> + Allocator<D, D>,
{
UDU::new(self.into_owned())
}
/// Computes the Hessenberg decomposition of this matrix using householder reflections.
pub fn hessenberg(self) -> Hessenberg<T, D>
where
D: DimSub<U1>,
DefaultAllocator: Allocator<D, D> + Allocator<D> + Allocator<DimDiff<D, U1>>,
{
Hessenberg::new(self.into_owned())
}
/// Computes the Schur decomposition of a square matrix.
pub fn schur(self) -> Schur<T, D>
where
D: DimSub<U1>, // For Hessenberg.
DefaultAllocator: Allocator<D, DimDiff<D, U1>>
+ Allocator<DimDiff<D, U1>>
+ Allocator<D, D>
+ Allocator<D>,
{
Schur::new(self.into_owned())
}
/// 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_schur(self, eps: T::RealField, max_niter: usize) -> Option<Schur<T, D>>
where
D: DimSub<U1>, // For Hessenberg.
DefaultAllocator: Allocator<D, DimDiff<D, U1>>
+ Allocator<DimDiff<D, U1>>
+ Allocator<D, D>
+ Allocator<D>,
{
Schur::try_new(self.into_owned(), eps, max_niter)
}
/// Computes the eigendecomposition of this symmetric matrix.
///
/// Only the lower-triangular part (including the diagonal) of `m` is read.
pub fn symmetric_eigen(self) -> SymmetricEigen<T, D>
where
D: DimSub<U1>,
DefaultAllocator:
Allocator<D, D> + Allocator<DimDiff<D, U1>> + Allocator<D> + Allocator<DimDiff<D, U1>>,
{
SymmetricEigen::new(self.into_owned())
}
/// Computes the eigendecomposition of the given symmetric matrix with user-specified
/// convergence parameters.
///
/// Only the lower-triangular part (including the diagonal) of `m` is read.
///
/// # 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_symmetric_eigen(
self,
eps: T::RealField,
max_niter: usize,
) -> Option<SymmetricEigen<T, D>>
where
D: DimSub<U1>,
DefaultAllocator:
Allocator<D, D> + Allocator<DimDiff<D, U1>> + Allocator<D> + Allocator<DimDiff<D, U1>>,
{
SymmetricEigen::try_new(self.into_owned(), eps, max_niter)
}
/// Computes the tridiagonalization of this symmetric matrix.
///
/// Only the lower-triangular part (including the diagonal) of `m` is read.
pub fn symmetric_tridiagonalize(self) -> SymmetricTridiagonal<T, D>
where
D: DimSub<U1>,
DefaultAllocator: Allocator<D, D> + Allocator<DimDiff<D, U1>>,
{
SymmetricTridiagonal::new(self.into_owned())
}
}