Merge pull request #1050 from metric-space/polar-decomposition-take-2
Take-2 of polar-decomposition
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99ac8c4032
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@ -1,8 +1,8 @@
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use crate::storage::Storage;
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use crate::{
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Allocator, Bidiagonal, Cholesky, ColPivQR, ComplexField, DefaultAllocator, Dim, DimDiff,
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DimMin, DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, RealField, Schur, SymmetricEigen,
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SymmetricTridiagonal, LU, QR, SVD, U1, UDU,
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DimMin, DimMinimum, DimSub, FullPivLU, Hessenberg, Matrix, OMatrix, RealField, Schur,
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SymmetricEigen, SymmetricTridiagonal, LU, QR, SVD, U1, UDU,
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};
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/// # Rectangular matrix decomposition
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@ -17,6 +17,7 @@ use crate::{
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/// | 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. |
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/// | 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. |
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/// | SVD | `U * Σ * Vᵀ` | `U` and `V` are two orthogonal matrices and `Σ` is a diagonal matrix containing the singular values. |
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/// | Polar (Left Polar) | `P' * U` | `U` is semi-unitary/unitary and `P'` is a positive semi-definite Hermitian Matrix
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impl<T: ComplexField, R: Dim, C: Dim, S: Storage<T, R, C>> Matrix<T, R, C, S> {
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/// Computes the bidiagonalization using householder reflections.
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pub fn bidiagonalize(self) -> Bidiagonal<T, R, C>
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@ -186,6 +187,62 @@ impl<T: ComplexField, R: Dim, C: Dim, S: Storage<T, R, C>> Matrix<T, R, C, S> {
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{
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SVD::try_new_unordered(self.into_owned(), compute_u, compute_v, eps, max_niter)
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}
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/// Computes the Polar Decomposition of a `matrix` (indirectly uses SVD).
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pub fn polar(self) -> (OMatrix<T, R, R>, OMatrix<T, R, C>)
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where
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R: DimMin<C>,
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DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
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DefaultAllocator: Allocator<T, R, C>
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+ Allocator<T, DimMinimum<R, C>, R>
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+ Allocator<T, DimMinimum<R, C>>
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+ Allocator<T, R, R>
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+ Allocator<T, DimMinimum<R, C>, DimMinimum<R, C>>
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+ Allocator<T, C>
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+ Allocator<T, R>
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+ Allocator<T, DimDiff<DimMinimum<R, C>, U1>>
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+ Allocator<T, DimMinimum<R, C>, C>
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+ Allocator<T, R, DimMinimum<R, C>>
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+ Allocator<T, DimMinimum<R, C>>
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+ Allocator<T::RealField, DimMinimum<R, C>>
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+ Allocator<T::RealField, DimDiff<DimMinimum<R, C>, U1>>,
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{
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SVD::new_unordered(self.into_owned(), true, true)
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.to_polar()
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.unwrap()
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}
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/// Attempts to compute the Polar Decomposition of a `matrix` (indirectly uses SVD).
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///
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/// # Arguments
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///
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/// * `eps` − tolerance used to determine when a value converged to 0 when computing the SVD.
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/// * `max_niter` − maximum total number of iterations performed by the SVD computation algorithm.
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pub fn try_polar(
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self,
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eps: T::RealField,
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max_niter: usize,
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) -> Option<(OMatrix<T, R, R>, OMatrix<T, R, C>)>
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where
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R: DimMin<C>,
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DimMinimum<R, C>: DimSub<U1>, // for Bidiagonal.
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DefaultAllocator: Allocator<T, R, C>
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+ Allocator<T, DimMinimum<R, C>, R>
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+ Allocator<T, DimMinimum<R, C>>
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+ Allocator<T, R, R>
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+ Allocator<T, DimMinimum<R, C>, DimMinimum<R, C>>
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+ Allocator<T, C>
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+ Allocator<T, R>
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+ Allocator<T, DimDiff<DimMinimum<R, C>, U1>>
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+ Allocator<T, DimMinimum<R, C>, C>
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+ Allocator<T, R, DimMinimum<R, C>>
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+ Allocator<T, DimMinimum<R, C>>
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+ Allocator<T::RealField, DimMinimum<R, C>>
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+ Allocator<T::RealField, DimDiff<DimMinimum<R, C>, U1>>,
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{
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SVD::try_new_unordered(self.into_owned(), true, true, eps, max_niter)
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.and_then(|svd| svd.to_polar())
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}
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}
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/// # Square matrix decomposition
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@ -641,6 +641,28 @@ where
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}
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}
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}
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/// converts SVD results to Polar decomposition form of the original Matrix: `A = P' * U`.
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///
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/// The polar decomposition used here is Left Polar Decomposition (or Reverse Polar Decomposition)
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/// Returns None if the singular vectors of the SVD haven't been calculated
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pub fn to_polar(&self) -> Option<(OMatrix<T, R, R>, OMatrix<T, R, C>)>
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where
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DefaultAllocator: Allocator<T, R, C> //result
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+ Allocator<T, DimMinimum<R, C>, R> // adjoint
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+ Allocator<T, DimMinimum<R, C>> // mapped vals
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+ Allocator<T, R, R> // result
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+ Allocator<T, DimMinimum<R, C>, DimMinimum<R, C>>, // square matrix
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{
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match (&self.u, &self.v_t) {
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(Some(u), Some(v_t)) => Some((
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u * OMatrix::from_diagonal(&self.singular_values.map(|e| T::from_real(e)))
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* u.adjoint(),
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u * v_t,
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)),
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_ => None,
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}
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}
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}
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impl<T: ComplexField, R: DimMin<C>, C: Dim> SVD<T, R, C>
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@ -153,6 +153,25 @@ mod proptest_tests {
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prop_assert!(relative_eq!(&m * &sol2, b2, epsilon = 1.0e-6));
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}
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}
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#[test]
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fn svd_polar_decomposition(m in dmatrix_($scalar)) {
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let svd = m.clone().svd_unordered(true, true);
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let (p, u) = svd.to_polar().unwrap();
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assert_relative_eq!(m, &p* &u, epsilon = 1.0e-5);
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// semi-unitary check
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assert!(u.is_orthogonal(1.0e-5) || u.transpose().is_orthogonal(1.0e-5));
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// hermitian check
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assert_relative_eq!(p, p.adjoint(), epsilon = 1.0e-5);
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/*
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* Same thing, but using the method instead of calling the SVD explicitly.
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*/
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let (p2, u2) = m.clone().polar();
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assert_eq!(p, p2);
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assert_eq!(u, u2);
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}
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}
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}
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}
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