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#[cfg(feature = "serde-serialize")]
use serde::{Deserialize, Serialize};
use num::Zero;
use num_complex::Complex;
use simba::scalar::RealField;
use crate::ComplexHelper;
use na::allocator::Allocator;
use na::dimension::{Const, Dim};
use na::{DefaultAllocator, Matrix, OMatrix, OVector, Scalar};
use lapack;
/// Eigendecomposition of a real square matrix with real eigenvalues.
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr(
feature = "serde-serialize",
serde(bound(serialize = "DefaultAllocator: Allocator<D, D> + Allocator<D>,
OVector<T, D>: Serialize,
OMatrix<T, D, D>: Serialize"))
)]
#[cfg_attr(
feature = "serde-serialize",
serde(bound(deserialize = "DefaultAllocator: Allocator<D, D> + Allocator<D>,
OVector<T, D>: Deserialize<'de>,
OMatrix<T, D, D>: Deserialize<'de>"))
)]
#[derive(Clone, Debug)]
pub struct Schur<T: Scalar, D: Dim>
where
DefaultAllocator: Allocator<D> + Allocator<D, D>,
{
re: OVector<T, D>,
im: OVector<T, D>,
t: OMatrix<T, D, D>,
q: OMatrix<T, D, D>,
}
impl<T: Scalar + Copy, D: Dim> Copy for Schur<T, D>
where
DefaultAllocator: Allocator<D, D> + Allocator<D>,
OMatrix<T, D, D>: Copy,
OVector<T, D>: Copy,
{
}
impl<T: SchurScalar + RealField, D: Dim> Schur<T, D>
where
DefaultAllocator: Allocator<D, D> + Allocator<D>,
{
/// Computes the eigenvalues and real Schur form of the matrix `m`.
///
/// Panics if the method did not converge.
pub fn new(m: OMatrix<T, D, D>) -> Self {
Self::try_new(m).expect("Schur decomposition: convergence failed.")
}
/// Computes the eigenvalues and real Schur form of the matrix `m`.
///
/// Returns `None` if the method did not converge.
pub fn try_new(mut m: OMatrix<T, D, D>) -> Option<Self> {
assert!(
m.is_square(),
"Unable to compute the eigenvalue decomposition of a non-square matrix."
);
let (nrows, ncols) = m.shape_generic();
let n = nrows.value();
let lda = n as i32;
let mut info = 0;
let mut wr = Matrix::zeros_generic(nrows, Const::<1>);
let mut wi = Matrix::zeros_generic(nrows, Const::<1>);
let mut q = Matrix::zeros_generic(nrows, ncols);
// Placeholders:
let mut bwork = [0i32];
let mut unused = 0;
let lwork = T::xgees_work_size(
b'V',
//@note(geo-ant): this interface is bad because we could pass
// 'S' here, but we cannot give a function to SELECT, which
// isn't exposed here.
// https://www.netlib.org/lapack/explore-html/d5/d38/group__gees.html
b'N',
n as i32,
m.as_mut_slice(),
lda,
&mut unused,
wr.as_mut_slice(),
wi.as_mut_slice(),
q.as_mut_slice(),
n as i32,
&mut bwork,
&mut info,
);
lapack_check!(info);
let mut work = vec![T::zero(); lwork as usize];
T::xgees(
b'V',
b'N',
n as i32,
m.as_mut_slice(),
lda,
&mut unused,
wr.as_mut_slice(),
wi.as_mut_slice(),
q.as_mut_slice(),
n as i32,
&mut work,
lwork,
&mut bwork,
&mut info,
);
lapack_check!(info);
Some(Schur {
re: wr,
im: wi,
t: m,
q,
})
}
/// Retrieves the unitary matrix `Q` and the upper-quasitriangular matrix `T` such that the
/// decomposed matrix equals `Q * T * Q.transpose()`.
pub fn unpack(self) -> (OMatrix<T, D, D>, OMatrix<T, D, D>) {
(self.q, self.t)
}
/// Computes the real eigenvalues of the decomposed matrix.
///
/// Return `None` if some eigenvalues are complex.
#[must_use]
pub fn eigenvalues(&self) -> Option<OVector<T, D>> {
if self.im.iter().all(|e| e.is_zero()) {
Some(self.re.clone())
} else {
None
}
}
/// Computes the complex eigenvalues of the decomposed matrix.
#[must_use]
pub fn complex_eigenvalues(&self) -> OVector<Complex<T>, D>
where
DefaultAllocator: Allocator<D>,
{
let mut out = Matrix::zeros_generic(self.t.shape_generic().0, Const::<1>);
for i in 0..out.len() {
out[i] = Complex::new(self.re[i].clone(), self.im[i].clone())
}
out
}
}
/*
*
* Lapack functions dispatch.
*
*/
/// Trait implemented by scalars for which Lapack implements the RealField Schur decomposition.
pub trait SchurScalar: Scalar {
#[allow(missing_docs)]
fn xgees(
jobvs: u8,
sort: u8,
//@note(geo-ant) see other comments in this file on this method
// select: ???
n: i32,
a: &mut [Self],
lda: i32,
sdim: &mut i32,
wr: &mut [Self],
wi: &mut [Self],
vs: &mut [Self],
ldvs: i32,
work: &mut [Self],
lwork: i32,
bwork: &mut [i32],
info: &mut i32,
);
#[allow(missing_docs)]
fn xgees_work_size(
jobvs: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [Self],
lda: i32,
sdim: &mut i32,
wr: &mut [Self],
wi: &mut [Self],
vs: &mut [Self],
ldvs: i32,
bwork: &mut [i32],
info: &mut i32,
) -> i32;
}
macro_rules! real_eigensystem_scalar_impl (
($N: ty, $xgees: path) => (
impl SchurScalar for $N {
#[inline]
fn xgees(jobvs: u8,
sort: u8,
//@note(geo-ant) see the documentation for xGEES here:
// https://www.netlib.org/lapack/explore-html/d5/d38/group__gees.html
// If we pass a sort argument of b'S', then we must offer a
// SELECT implementation.
// select: ???
n: i32,
a: &mut [$N],
lda: i32,
sdim: &mut i32,
wr: &mut [$N],
wi: &mut [$N],
vs: &mut [$N],
ldvs: i32,
work: &mut [$N],
lwork: i32,
bwork: &mut [i32],
info: &mut i32) {
unsafe { $xgees(jobvs, sort, None, n, a, lda, sdim, wr, wi, vs, ldvs, work, lwork, bwork, info); }
}
#[inline]
fn xgees_work_size(jobvs: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [$N],
lda: i32,
sdim: &mut i32,
wr: &mut [$N],
wi: &mut [$N],
vs: &mut [$N],
ldvs: i32,
bwork: &mut [i32],
info: &mut i32)
-> i32 {
let mut work = [ Zero::zero() ];
let lwork = -1 as i32;
unsafe { $xgees(jobvs, sort, None, n, a, lda, sdim, wr, wi, vs, ldvs, &mut work, lwork, bwork, info); }
ComplexHelper::real_part(work[0]) as i32
}
}
)
);
real_eigensystem_scalar_impl!(f32, lapack::sgees);
real_eigensystem_scalar_impl!(f64, lapack::dgees);