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Minimal post-processing and fix to documentation

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metric-space 2022-02-27 17:17:31 -05:00 committed by Saurabh
parent 73543b2121
commit 6a22e74c00
1 changed files with 36 additions and 46 deletions
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82
nalgebra-lapack/src/generalized_eigenvalues.rs
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@ -180,25 +180,44 @@ where
}
/// Calculates the generalized eigenvectors (left and right) associated with the generalized eigenvalues
/// Outputs two matrices, the first one containing the left eigenvectors of the generalized eigenvalues
/// as columns and the second matrix contains the right eigenvectors of the generalized eigenvalues
/// as columns
/// Outputs two matrices.
/// The first output matix contains the left eigenvectors of the generalized eigenvalues
/// as columns.
/// The second matrix contains the right eigenvectors of the generalized eigenvalues
/// as columns.
///
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// A * v(j) = lambda(j) * B * v(j).
/// A * v(j) = lambda(j) * B * v(j)
///
/// The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
/// The left eigenvector u(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// u(j)**H * A = lambda(j) * u(j)**H * B .
/// where u(j)**H is the conjugate-transpose of u(j).
/// u(j)**H * A = lambda(j) * u(j)**H * B
/// where u(j)**H is the conjugate-transpose of u(j).
///
/// How the eigenvectors are build up:
///
/// Since the input entries are all real, the generalized eigenvalues if complex come in pairs
/// as a consequence of <https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem>
/// The Lapack routine output reflects this by expecting the user to unpack the complex eigenvalues associated
/// eigenvectors from the real matrix output via the following procedure
///
/// (Note: VL stands for the lapack real matrix output containing the left eigenvectors as columns,
/// VR stands for the lapack real matrix output containing the right eigenvectors as columns)
///
/// If the j-th and (j+1)-th eigenvalues form a complex conjugate pair,
/// then
///
/// u(j) = VL(:,j)+i*VL(:,j+1)
/// u(j+1) = VL(:,j)-i*VL(:,j+1)
///
/// and
///
/// u(j) = VR(:,j)+i*VR(:,j+1)
/// v(j+1) = VR(:,j)-i*VR(:,j+1).
///
/// What is going on below?
/// If the j-th and (j+1)-th eigenvalues form a complex conjugate pair,
/// then u(j) = VSL(:,j)+i*VSL(:,j+1) and u(j+1) = VSL(:,j)-i*VSL(:,j+1).
/// and then v(j) = VSR(:,j)+i*VSR(:,j+1) and v(j+1) = VSR(:,j)-i*VSR(:,j+1).
pub fn eigenvectors(self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>)
where
DefaultAllocator:
@ -216,18 +235,14 @@ where
.clone()
.map(|x| Complex::new(x, T::RealField::zero()));
let eigenvalues = &self.eigenvalues();
let eigenvalues = &self.raw_eigenvalues();
let mut c = 0;
let epsilon = T::RealField::default_epsilon();
while c < n {
if eigenvalues[c].im.abs() > epsilon && c + 1 < n && {
let e_conj = eigenvalues[c].conj();
let e = eigenvalues[c + 1];
(&e_conj.re).ulps_eq(&e.re, epsilon, 6) && (&e_conj.im).ulps_eq(&e.im, epsilon, 6)
} {
if eigenvalues[c].0.im.abs() > epsilon && c + 1 < n {
// taking care of the left eigenvector matrix
l.column_mut(c).zip_apply(&self.vsl.column(c + 1), |r, i| {
*r = Complex::new(r.re.clone(), i.clone());
@ -253,32 +268,7 @@ where
(l, r)
}
// only used for internal calculation for assembling eigenvectors based on realness of
// eigenvalues and complex-conjugate checks of subsequent non-real eigenvalues
fn eigenvalues(&self) -> OVector<Complex<T>, D>
where
DefaultAllocator: Allocator<Complex<T>, D>,
{
let mut out = Matrix::zeros_generic(self.vsl.shape_generic().0, Const::<1>);
let epsilon = T::RealField::default_epsilon();
for i in 0..out.len() {
out[i] = if self.beta[i].clone().abs() < epsilon
|| (self.alphai[i].clone().abs() < epsilon
&& self.alphar[i].clone().abs() < epsilon)
{
Complex::zero()
} else {
Complex::new(self.alphar[i].clone(), self.alphai[i].clone())
* (Complex::new(self.beta[i].clone(), T::RealField::zero()).inv())
}
}
out
}
/// outputs the unprocessed (almost) version of generalized eigenvalues ((alphar, alpai), beta)
/// outputs the unprocessed (almost) version of generalized eigenvalues ((alphar, alphai), beta)
/// straight from LAPACK
#[must_use]
pub fn raw_eigenvalues(&self) -> OVector<(Complex<T>, T), D>