New wrapper for generalized eigenvalues and associated eigenvectors via LAPACK routines sggev/dggev
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nalgebra-lapack/src/generalized_eigenvalues.rs
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339
nalgebra-lapack/src/generalized_eigenvalues.rs
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@ -0,0 +1,339 @@
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#[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 square matrices.
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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>: Serialize,
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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 GE<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 GE<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: GEScalar + RealField + Copy, D: Dim> GE<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 eigenvectors) via the raw returns from LAPACK's
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/// dggev and sggev routines
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///
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/// For each e in generalized eigenvalues and the associated eigenvectors e_l and e_r (left andf right)
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/// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r
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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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/// For each e in generalized eigenvalues and the associated eigenvectors e_l and e_r (left andf right)
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/// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r
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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(GE {
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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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pub fn eigenvectors(self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>)
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where
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DefaultAllocator: Allocator<Complex<T>, D, D> + Allocator<Complex<T>, D>,
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{
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let n = self.vsl.shape().0;
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let mut l = self
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.vsl
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.clone()
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.map(|x| Complex::new(x, T::RealField::zero()));
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let mut r = self
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.vsr
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.clone()
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.map(|x| Complex::new(x, T::RealField::zero()));
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let eigenvalues = &self.eigenvalues();
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let mut ll;
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let mut c = 0;
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while c < n {
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if eigenvalues[c].im.abs() > T::RealField::default_epsilon() && c + 1 < n && {
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let e_conj = eigenvalues[c].conj();
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let e = eigenvalues[c + 1];
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((e_conj.re - e.re).abs() < T::RealField::default_epsilon())
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&& ((e_conj.im - e.im).abs() < T::RealField::default_epsilon())
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} {
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ll = l.column(c + 1).into_owned();
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l.column_mut(c).zip_apply(&ll, |r, i| {
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*r = Complex::new(r.re.clone(), i.re);
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});
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ll.copy_from(&l.column(c));
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l.column_mut(c + 1).zip_apply(&ll, |r, i| {
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*r = i.conj();
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});
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ll.copy_from(&r.column(c + 1));
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r.column_mut(c).zip_apply(&ll, |r, i| {
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*r = Complex::new(r.re, i.re);
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});
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ll.copy_from(&r.column(c));
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r.column_mut(c + 1).zip_apply(&ll, |r, i| {
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*r = i.conj();
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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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/// computes the generalized eigenvalues
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#[must_use]
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pub fn eigenvalues(&self) -> OVector<Complex<T>, D>
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where
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DefaultAllocator: Allocator<Complex<T>, D>,
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{
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let mut out = Matrix::zeros_generic(self.vsl.shape_generic().0, Const::<1>);
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for i in 0..out.len() {
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out[i] = if self.beta[i].clone().abs() < T::RealField::default_epsilon() {
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Complex::zero()
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} else {
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let mut cr = self.alphar[i].clone();
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let mut ci = self.alphai[i].clone();
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let b = self.beta[i].clone();
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if cr.clone().abs() < T::RealField::default_epsilon() {
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cr = T::RealField::zero()
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} else {
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cr = cr / b.clone()
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};
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if ci.clone().abs() < T::RealField::default_epsilon() {
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ci = T::RealField::zero()
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} else {
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ci = ci / b
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};
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Complex::new(cr, ci)
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}
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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 GE decomposition.
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pub trait GEScalar: 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! real_eigensystem_scalar_impl (
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($N: ty, $xggev: path) => (
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impl GEScalar 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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real_eigensystem_scalar_impl!(f32, lapack::sggev);
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real_eigensystem_scalar_impl!(f64, lapack::dggev);
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@ -87,6 +87,7 @@ mod hessenberg;
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mod lu;
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mod qr;
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mod qz;
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mod generalized_eigenvalues;
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mod schur;
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mod svd;
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mod symmetric_eigen;
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@ -99,6 +100,7 @@ pub use self::hessenberg::Hessenberg;
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pub use self::lu::{LUScalar, LU};
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pub use self::qr::QR;
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pub use self::qz::QZ;
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pub use self::generalized_eigenvalues::GE;
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pub use self::schur::Schur;
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pub use self::svd::SVD;
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pub use self::symmetric_eigen::SymmetricEigen;
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59
nalgebra-lapack/tests/linalg/generalized_eigenvalues.rs
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59
nalgebra-lapack/tests/linalg/generalized_eigenvalues.rs
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@ -0,0 +1,59 @@
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use na::dimension::{Const, Dynamic};
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use na::{DMatrix, EuclideanNorm, Norm, OMatrix};
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use nl::GE;
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use num_complex::Complex;
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use simba::scalar::ComplexField;
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use std::cmp;
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use crate::proptest::*;
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use proptest::{prop_assert, proptest};
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proptest! {
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#[test]
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fn ge(n in PROPTEST_MATRIX_DIM) {
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let n = cmp::max(1, cmp::min(n, 10));
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let a = DMatrix::<f64>::new_random(n, n);
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let b = DMatrix::<f64>::new_random(n, n);
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let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a)));
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let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b)));
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if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0 {
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let a_c =a.clone().map(|x| Complex::new(x, 0.0));
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let b_c = b.clone().map(|x| Complex::new(x, 0.0));
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let ge = GE::new(a.clone(), b.clone());
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let (vsl,vsr) = ge.clone().eigenvectors();
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let eigenvalues = ge.clone().eigenvalues();
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for i in 0..n {
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let left_eigenvector = &vsl.column(i);
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prop_assert!(relative_eq!((left_eigenvector.transpose()*&a_c - left_eigenvector.transpose()*&b_c*eigenvalues[i]).map(|x| x.modulus()), OMatrix::zeros_generic(Const::<1>,Dynamic::new(n)) ,epsilon = 1.0e-7));
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let right_eigenvector = &vsr.column(i);
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prop_assert!(relative_eq!((&a_c*right_eigenvector - &b_c*right_eigenvector*eigenvalues[i]).map(|x| x.modulus()), OMatrix::zeros_generic(Dynamic::new(n), Const::<1>) ,epsilon = 1.0e-7));
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};
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};
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}
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#[test]
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fn ge_static(a in matrix4(), b in matrix4()) {
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let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a)));
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let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b)));
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if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0{
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let ge = GE::new(a.clone(), b.clone());
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let a_c =a.clone().map(|x| Complex::new(x, 0.0));
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let b_c = b.clone().map(|x| Complex::new(x, 0.0));
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let (vsl,vsr) = ge.eigenvectors();
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let eigenvalues = ge.eigenvalues();
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for i in 0..4 {
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let left_eigenvector = &vsl.column(i);
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prop_assert!(relative_eq!((left_eigenvector.transpose()*&a_c - left_eigenvector.transpose()*&b_c*eigenvalues[i]).map(|x| x.modulus()), OMatrix::zeros_generic(Const::<1>,Const::<4>) ,epsilon = 1.0e-7));
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let right_eigenvector = &vsr.column(i);
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prop_assert!(relative_eq!((&a_c*right_eigenvector - &b_c*right_eigenvector*eigenvalues[i]).map(|x| x.modulus()), OMatrix::zeros_generic(Const::<4>, Const::<1>) ,epsilon = 1.0e-7));
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};
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};
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}
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}
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@ -2,6 +2,7 @@ mod cholesky;
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mod lu;
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mod qr;
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mod qz;
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mod generalized_eigenvalues;
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mod real_eigensystem;
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mod schur;
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mod svd;
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