351 lines
11 KiB
Rust
351 lines
11 KiB
Rust
#[cfg(feature = "serde-serialize")]
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use serde::{Deserialize, Serialize};
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use num::Zero;
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use num_complex::Complex;
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use simba::scalar::RealField;
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use crate::ComplexHelper;
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use na::allocator::Allocator;
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use na::dimension::{Const, Dim};
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use na::{DefaultAllocator, Matrix, OMatrix, OVector, Scalar};
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use lapack;
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/// Generalized eigenvalues and generalized eigenvectors (left and right) of a pair of N*N real square matrices.
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///
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/// Each generalized eigenvalue (lambda) satisfies determinant(A - lambda*B) = 0
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///
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/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
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/// of (A,B) satisfies
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///
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/// A * v(j) = lambda(j) * B * v(j).
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///
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/// The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
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/// of (A,B) satisfies
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///
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/// u(j)**H * A = lambda(j) * u(j)**H * B .
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/// where u(j)**H is the conjugate-transpose of u(j).
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#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
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#[cfg_attr(
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feature = "serde-serialize",
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serde(
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bound(serialize = "DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
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OVector<T, D>: Serialize,
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OMatrix<T, D, D>: Serialize")
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)
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)]
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#[cfg_attr(
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feature = "serde-serialize",
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serde(
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bound(deserialize = "DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
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OVector<T, D>: Deserialize<'de>,
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OMatrix<T, D, D>: Deserialize<'de>")
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)
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)]
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#[derive(Clone, Debug)]
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pub struct GeneralizedEigen<T: Scalar, D: Dim>
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where
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DefaultAllocator: Allocator<T, D> + Allocator<T, D, D>,
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{
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alphar: OVector<T, D>,
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alphai: OVector<T, D>,
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beta: OVector<T, D>,
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vsl: OMatrix<T, D, D>,
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vsr: OMatrix<T, D, D>,
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}
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impl<T: Scalar + Copy, D: Dim> Copy for GeneralizedEigen<T, D>
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where
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DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
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OMatrix<T, D, D>: Copy,
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OVector<T, D>: Copy,
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{
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}
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impl<T: GeneralizedEigenScalar + RealField + Copy, D: Dim> GeneralizedEigen<T, D>
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where
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DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
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{
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/// Attempts to compute the generalized eigenvalues, and left and right associated eigenvectors
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/// via the raw returns from LAPACK's dggev and sggev routines
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///
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/// Panics if the method did not converge.
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pub fn new(a: OMatrix<T, D, D>, b: OMatrix<T, D, D>) -> Self {
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Self::try_new(a, b).expect("Calculation of generalized eigenvalues failed.")
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}
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/// Attempts to compute the generalized eigenvalues (and eigenvectors) via the raw returns from LAPACK's
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/// dggev and sggev routines
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///
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/// Returns `None` if the method did not converge.
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pub fn try_new(mut a: OMatrix<T, D, D>, mut b: OMatrix<T, D, D>) -> Option<Self> {
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assert!(
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a.is_square() && b.is_square(),
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"Unable to compute the generalized eigenvalues of non-square matrices."
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);
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assert!(
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a.shape_generic() == b.shape_generic(),
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"Unable to compute the generalized eigenvalues of two square matrices of different dimensions."
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);
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let (nrows, ncols) = a.shape_generic();
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let n = nrows.value();
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let mut info = 0;
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let mut alphar = Matrix::zeros_generic(nrows, Const::<1>);
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let mut alphai = Matrix::zeros_generic(nrows, Const::<1>);
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let mut beta = Matrix::zeros_generic(nrows, Const::<1>);
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let mut vsl = Matrix::zeros_generic(nrows, ncols);
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let mut vsr = Matrix::zeros_generic(nrows, ncols);
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let lwork = T::xggev_work_size(
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b'V',
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b'V',
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n as i32,
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a.as_mut_slice(),
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n as i32,
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b.as_mut_slice(),
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n as i32,
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alphar.as_mut_slice(),
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alphai.as_mut_slice(),
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beta.as_mut_slice(),
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vsl.as_mut_slice(),
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n as i32,
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vsr.as_mut_slice(),
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n as i32,
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&mut info,
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);
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lapack_check!(info);
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let mut work = vec![T::zero(); lwork as usize];
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T::xggev(
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b'V',
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b'V',
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n as i32,
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a.as_mut_slice(),
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n as i32,
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b.as_mut_slice(),
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n as i32,
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alphar.as_mut_slice(),
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alphai.as_mut_slice(),
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beta.as_mut_slice(),
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vsl.as_mut_slice(),
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n as i32,
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vsr.as_mut_slice(),
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n as i32,
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&mut work,
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lwork,
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&mut info,
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);
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lapack_check!(info);
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Some(GeneralizedEigen {
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alphar,
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alphai,
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beta,
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vsl,
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vsr,
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})
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}
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/// Calculates the generalized eigenvectors (left and right) associated with the generalized eigenvalues
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///
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/// Outputs two matrices.
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/// The first output matrix contains the left eigenvectors of the generalized eigenvalues
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/// as columns.
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/// The second matrix contains the right eigenvectors of the generalized eigenvalues
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/// as columns.
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pub fn eigenvectors(&self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>)
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where
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DefaultAllocator:
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Allocator<Complex<T>, D, D> + Allocator<Complex<T>, D> + Allocator<(Complex<T>, T), D>,
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{
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/*
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How the eigenvectors are built up:
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Since the input entries are all real, the generalized eigenvalues if complex come in pairs
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as a consequence of the [complex conjugate root thorem](https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem)
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The Lapack routine output reflects this by expecting the user to unpack the real and complex eigenvalues associated
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eigenvectors from the real matrix output via the following procedure
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(Note: VL stands for the lapack real matrix output containing the left eigenvectors as columns,
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VR stands for the lapack real matrix output containing the right eigenvectors as columns)
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If the j-th and (j+1)-th eigenvalues form a complex conjugate pair,
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then
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u(j) = VL(:,j)+i*VL(:,j+1)
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u(j+1) = VL(:,j)-i*VL(:,j+1)
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and
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u(j) = VR(:,j)+i*VR(:,j+1)
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v(j+1) = VR(:,j)-i*VR(:,j+1).
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*/
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let n = self.vsl.shape().0;
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let mut l = self.vsl.map(|x| Complex::new(x, T::RealField::zero()));
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let mut r = self.vsr.map(|x| Complex::new(x, T::RealField::zero()));
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let eigenvalues = self.raw_eigenvalues();
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let mut c = 0;
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while c < n {
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if eigenvalues[c].0.im.abs() != T::RealField::zero() && c + 1 < n {
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// taking care of the left eigenvector matrix
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l.column_mut(c).zip_apply(&self.vsl.column(c + 1), |r, i| {
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*r = Complex::new(r.re.clone(), i.clone());
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});
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l.column_mut(c + 1).zip_apply(&self.vsl.column(c), |i, r| {
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*i = Complex::new(r.clone(), -i.re.clone());
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});
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// taking care of the right eigenvector matrix
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r.column_mut(c).zip_apply(&self.vsr.column(c + 1), |r, i| {
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*r = Complex::new(r.re.clone(), i.clone());
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});
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r.column_mut(c + 1).zip_apply(&self.vsr.column(c), |i, r| {
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*i = Complex::new(r.clone(), -i.re.clone());
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});
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c += 2;
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} else {
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c += 1;
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}
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}
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(l, r)
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}
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/// Outputs the unprocessed (almost) version of generalized eigenvalues ((alphar, alphai), beta)
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/// straight from LAPACK
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#[must_use]
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pub fn raw_eigenvalues(&self) -> OVector<(Complex<T>, T), D>
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where
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DefaultAllocator: Allocator<(Complex<T>, T), D>,
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{
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let mut out = Matrix::from_element_generic(
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self.vsl.shape_generic().0,
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Const::<1>,
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(Complex::zero(), T::RealField::zero()),
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);
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for i in 0..out.len() {
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out[i] = (Complex::new(self.alphar[i], self.alphai[i]), self.beta[i])
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}
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out
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}
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}
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/*
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*
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* Lapack functions dispatch.
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*
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*/
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/// Trait implemented by scalars for which Lapack implements the RealField GeneralizedEigen decomposition.
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pub trait GeneralizedEigenScalar: Scalar {
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#[allow(missing_docs)]
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fn xggev(
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jobvsl: u8,
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jobvsr: u8,
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n: i32,
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a: &mut [Self],
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lda: i32,
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b: &mut [Self],
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ldb: i32,
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alphar: &mut [Self],
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alphai: &mut [Self],
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beta: &mut [Self],
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vsl: &mut [Self],
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ldvsl: i32,
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vsr: &mut [Self],
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ldvsr: i32,
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work: &mut [Self],
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lwork: i32,
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info: &mut i32,
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);
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#[allow(missing_docs)]
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fn xggev_work_size(
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jobvsl: u8,
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jobvsr: u8,
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n: i32,
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a: &mut [Self],
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lda: i32,
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b: &mut [Self],
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ldb: i32,
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alphar: &mut [Self],
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alphai: &mut [Self],
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beta: &mut [Self],
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vsl: &mut [Self],
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ldvsl: i32,
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vsr: &mut [Self],
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ldvsr: i32,
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info: &mut i32,
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) -> i32;
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}
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macro_rules! generalized_eigen_scalar_impl (
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($N: ty, $xggev: path) => (
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impl GeneralizedEigenScalar for $N {
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#[inline]
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fn xggev(jobvsl: u8,
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jobvsr: u8,
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n: i32,
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a: &mut [$N],
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lda: i32,
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b: &mut [$N],
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ldb: i32,
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alphar: &mut [$N],
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alphai: &mut [$N],
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beta : &mut [$N],
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vsl: &mut [$N],
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ldvsl: i32,
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vsr: &mut [$N],
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ldvsr: i32,
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work: &mut [$N],
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lwork: i32,
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info: &mut i32) {
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unsafe { $xggev(jobvsl, jobvsr, n, a, lda, b, ldb, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, info); }
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}
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#[inline]
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fn xggev_work_size(jobvsl: u8,
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jobvsr: u8,
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n: i32,
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a: &mut [$N],
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lda: i32,
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b: &mut [$N],
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ldb: i32,
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alphar: &mut [$N],
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alphai: &mut [$N],
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beta : &mut [$N],
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vsl: &mut [$N],
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ldvsl: i32,
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vsr: &mut [$N],
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ldvsr: i32,
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info: &mut i32)
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-> i32 {
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let mut work = [ Zero::zero() ];
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let lwork = -1 as i32;
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unsafe { $xggev(jobvsl, jobvsr, n, a, lda, b, ldb, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, &mut work, lwork, info); }
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ComplexHelper::real_part(work[0]) as i32
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}
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}
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)
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);
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generalized_eigen_scalar_impl!(f32, lapack::sggev);
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generalized_eigen_scalar_impl!(f64, lapack::dggev);
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