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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;
/// QZ decomposition of a pair of N*N square matrices.
///
/// Retrieves the left and right matrices of Schur Vectors (VSL and VSR)
/// the upper-quasitriangular matrix `S` and upper triangular matrix `T` such that the
/// decomposed input matrix `a` equals `VSL * S * VSL.transpose()` and
/// decomposed input matrix `b` equals `VSL * T * VSL.transpose()`.
#[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 QZ<T: Scalar, D: Dim>
where
DefaultAllocator: Allocator<D> + Allocator<D, D>,
{
alphar: OVector<T, D>,
alphai: OVector<T, D>,
beta: OVector<T, D>,
vsl: OMatrix<T, D, D>,
s: OMatrix<T, D, D>,
vsr: OMatrix<T, D, D>,
t: OMatrix<T, D, D>,
}
impl<T: Scalar + Copy, D: Dim> Copy for QZ<T, D>
where
DefaultAllocator: Allocator<D, D> + Allocator<D>,
OMatrix<T, D, D>: Copy,
OVector<T, D>: Copy,
{
}
impl<T: QZScalar + RealField, D: Dim> QZ<T, D>
where
DefaultAllocator: Allocator<D, D> + Allocator<D>,
{
/// Attempts to compute the QZ decomposition of input real square matrices `a` and `b`.
///
/// i.e retrieves the left and right matrices of Schur Vectors (VSL and VSR)
/// the upper-quasitriangular matrix `S` and upper triangular matrix `T` such that the
/// decomposed matrix `a` equals `VSL * S * VSL.transpose()` and
/// decomposed matrix `b` equals `VSL * T * VSL.transpose()`.
///
/// Panics if the method did not converge.
pub fn new(a: OMatrix<T, D, D>, b: OMatrix<T, D, D>) -> Self {
Self::try_new(a, b).expect("QZ decomposition: convergence failed.")
}
/// Computes the decomposition of input matrices `a` and `b` into a pair of matrices of Schur vectors
/// , a quasi-upper triangular matrix and an upper-triangular matrix .
///
/// Returns `None` if the method did not converge.
pub fn try_new(mut a: OMatrix<T, D, D>, mut b: OMatrix<T, D, D>) -> Option<Self> {
assert!(
a.is_square() && b.is_square(),
"Unable to compute the qz decomposition of non-square matrices."
);
assert!(
a.shape_generic() == b.shape_generic(),
"Unable to compute the qz decomposition of two square matrices of different dimensions."
);
let (nrows, ncols) = a.shape_generic();
let n = nrows.value();
let mut info = 0;
let mut alphar = Matrix::zeros_generic(nrows, Const::<1>);
let mut alphai = Matrix::zeros_generic(nrows, Const::<1>);
let mut beta = Matrix::zeros_generic(nrows, Const::<1>);
let mut vsl = Matrix::zeros_generic(nrows, ncols);
let mut vsr = Matrix::zeros_generic(nrows, ncols);
// Placeholders:
let mut bwork = [0i32];
let mut unused = 0;
let lwork = T::xgges_work_size(
b'V',
b'V',
b'N',
n as i32,
a.as_mut_slice(),
n as i32,
b.as_mut_slice(),
n as i32,
&mut unused,
alphar.as_mut_slice(),
alphai.as_mut_slice(),
beta.as_mut_slice(),
vsl.as_mut_slice(),
n as i32,
vsr.as_mut_slice(),
n as i32,
&mut bwork,
&mut info,
);
lapack_check!(info);
let mut work = vec![T::zero(); lwork as usize];
T::xgges(
b'V',
b'V',
b'N',
n as i32,
a.as_mut_slice(),
n as i32,
b.as_mut_slice(),
n as i32,
&mut unused,
alphar.as_mut_slice(),
alphai.as_mut_slice(),
beta.as_mut_slice(),
vsl.as_mut_slice(),
n as i32,
vsr.as_mut_slice(),
n as i32,
&mut work,
lwork,
&mut bwork,
&mut info,
);
lapack_check!(info);
Some(QZ {
alphar,
alphai,
beta,
vsl,
s: a,
vsr,
t: b,
})
}
/// Retrieves the left and right matrices of Schur Vectors (VSL and VSR)
/// the upper-quasitriangular matrix `S` and upper triangular matrix `T` such that the
/// decomposed input matrix `a` equals `VSL * S * VSL.transpose()` and
/// decomposed input matrix `b` equals `VSL * T * VSL.transpose()`.
pub fn unpack(
self,
) -> (
OMatrix<T, D, D>,
OMatrix<T, D, D>,
OMatrix<T, D, D>,
OMatrix<T, D, D>,
) {
(self.vsl, self.s, self.t, self.vsr)
}
/// outputs the unprocessed (almost) version of generalized eigenvalues ((alphar, alpai), beta)
/// straight from LAPACK
#[must_use]
pub fn raw_eigenvalues(&self) -> OVector<(Complex<T>, T), D>
where
DefaultAllocator: Allocator<D>,
{
let mut out = Matrix::from_element_generic(
self.vsl.shape_generic().0,
Const::<1>,
(Complex::zero(), T::RealField::zero()),
);
for i in 0..out.len() {
out[i] = (
Complex::new(self.alphar[i].clone(), self.alphai[i].clone()),
self.beta[i].clone(),
)
}
out
}
}
/*
*
* Lapack functions dispatch.
*
*/
/// Trait implemented by scalars for which Lapack implements the RealField QZ decomposition.
pub trait QZScalar: Scalar {
#[allow(missing_docs)]
fn xgges(
jobvsl: u8,
jobvsr: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [Self],
lda: i32,
b: &mut [Self],
ldb: i32,
sdim: &mut i32,
alphar: &mut [Self],
alphai: &mut [Self],
beta: &mut [Self],
vsl: &mut [Self],
ldvsl: i32,
vsr: &mut [Self],
ldvsr: i32,
work: &mut [Self],
lwork: i32,
bwork: &mut [i32],
info: &mut i32,
);
#[allow(missing_docs)]
fn xgges_work_size(
jobvsl: u8,
jobvsr: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [Self],
lda: i32,
b: &mut [Self],
ldb: i32,
sdim: &mut i32,
alphar: &mut [Self],
alphai: &mut [Self],
beta: &mut [Self],
vsl: &mut [Self],
ldvsl: i32,
vsr: &mut [Self],
ldvsr: i32,
bwork: &mut [i32],
info: &mut i32,
) -> i32;
}
macro_rules! qz_scalar_impl (
($N: ty, $xgges: path) => (
impl QZScalar for $N {
#[inline]
fn xgges(jobvsl: u8,
jobvsr: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [$N],
lda: i32,
b: &mut [$N],
ldb: i32,
sdim: &mut i32,
alphar: &mut [$N],
alphai: &mut [$N],
beta : &mut [$N],
vsl: &mut [$N],
ldvsl: i32,
vsr: &mut [$N],
ldvsr: i32,
work: &mut [$N],
lwork: i32,
bwork: &mut [i32],
info: &mut i32) {
unsafe { $xgges(jobvsl, jobvsr, sort, None, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, bwork, info); }
}
#[inline]
fn xgges_work_size(jobvsl: u8,
jobvsr: u8,
sort: u8,
// select: ???
n: i32,
a: &mut [$N],
lda: i32,
b: &mut [$N],
ldb: i32,
sdim: &mut i32,
alphar: &mut [$N],
alphai: &mut [$N],
beta : &mut [$N],
vsl: &mut [$N],
ldvsl: i32,
vsr: &mut [$N],
ldvsr: i32,
bwork: &mut [i32],
info: &mut i32)
-> i32 {
let mut work = [ Zero::zero() ];
let lwork = -1 as i32;
unsafe { $xgges(jobvsl, jobvsr, sort, None, n, a, lda, b, ldb, sdim, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, &mut work, lwork, bwork, info); }
ComplexHelper::real_part(work[0]) as i32
}
}
)
);
qz_scalar_impl!(f32, lapack::sgges);
qz_scalar_impl!(f64, lapack::dgges);