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New code and modified tests for generalized_eigenvalues

This commit is contained in:
metric-space 2022-02-03 06:36:10 -05:00 committed by Saurabh
parent a439121641
commit 714f2ac987
2 changed files with 118 additions and 34 deletions
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89
nalgebra-lapack/src/generalized_eigenvalues.rs
View File

@ -4,7 +4,7 @@ use serde::{Deserialize, Serialize};
use num::Zero; use num::Zero;
use num_complex::Complex; use num_complex::Complex;
use simba::scalar:: RealField; use simba::scalar::RealField;
use crate::ComplexHelper; use crate::ComplexHelper;
use na::allocator::Allocator; use na::allocator::Allocator;
@ -14,6 +14,19 @@ use na::{DefaultAllocator, Matrix, OMatrix, OVector, Scalar};
use lapack; use lapack;
/// Generalized eigenvalues and generalized eigenvectors(left and right) of a pair of N*N square matrices. /// Generalized eigenvalues and generalized eigenvectors(left and right) of a pair of N*N square matrices.
///
/// Each generalized eigenvalue (lambda) satisfies determinant(A - lambda*B) = 0
///
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// A * v(j) = lambda(j) * B * v(j).
///
/// 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).
#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))] #[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
#[cfg_attr( #[cfg_attr(
feature = "serde-serialize", feature = "serde-serialize",
@ -55,11 +68,21 @@ impl<T: GEScalar + RealField + Copy, D: Dim> GE<T, D>
where where
DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>, DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
{ {
/// Attempts to compute the generalized eigenvalues (and eigenvectors) via the raw returns from LAPACK's /// Attempts to compute the generalized eigenvalues, and left and right associated eigenvectors
/// dggev and sggev routines /// via the raw returns from LAPACK's dggev and sggev routines
/// ///
/// For each e in generalized eigenvalues and the associated eigenvectors e_l and e_r (left andf right) /// Each generalized eigenvalue (lambda) satisfies determinant(A - lambda*B) = 0
/// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r ///
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// A * v(j) = lambda(j) * B * v(j).
///
/// 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).
/// ///
/// Panics if the method did not converge. /// Panics if the method did not converge.
pub fn new(a: OMatrix<T, D, D>, b: OMatrix<T, D, D>) -> Self { pub fn new(a: OMatrix<T, D, D>, b: OMatrix<T, D, D>) -> Self {
@ -69,8 +92,18 @@ where
/// Attempts to compute the generalized eigenvalues (and eigenvectors) via the raw returns from LAPACK's /// Attempts to compute the generalized eigenvalues (and eigenvectors) via the raw returns from LAPACK's
/// dggev and sggev routines /// dggev and sggev routines
/// ///
/// For each e in generalized eigenvalues and the associated eigenvectors e_l and e_r (left andf right) /// Each generalized eigenvalue (lambda) satisfies determinant(A - lambda*B) = 0
/// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r ///
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// A * v(j) = lambda(j) * B * v(j).
///
/// 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).
/// ///
/// Returns `None` if the method did not converge. /// 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> { pub fn try_new(mut a: OMatrix<T, D, D>, mut b: OMatrix<T, D, D>) -> Option<Self> {
@ -147,9 +180,24 @@ where
} }
/// Calculates the generalized eigenvectors (left and right) associated with the generalized eigenvalues /// 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
///
/// The right eigenvector v(j) corresponding to the eigenvalue lambda(j)
/// of (A,B) satisfies
///
/// A * v(j) = lambda(j) * B * v(j).
///
/// 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).
pub fn eigenvectors(self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>) pub fn eigenvectors(self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>)
where where
DefaultAllocator: Allocator<Complex<T>, D, D> + Allocator<Complex<T>, D>, DefaultAllocator:
Allocator<Complex<T>, D, D> + Allocator<Complex<T>, D> + Allocator<(Complex<T>, T), D>,
{ {
let n = self.vsl.shape().0; let n = self.vsl.shape().0;
let mut l = self let mut l = self
@ -199,9 +247,10 @@ where
(l, r) (l, r)
} }
/// computes the generalized eigenvalues /// computes the generalized eigenvalues i.e values of lambda that satisfy the following equation
/// determinant(A - lambda* B) = 0
#[must_use] #[must_use]
pub fn eigenvalues(&self) -> OVector<Complex<T>, D> fn eigenvalues(&self) -> OVector<Complex<T>, D>
where where
DefaultAllocator: Allocator<Complex<T>, D>, DefaultAllocator: Allocator<Complex<T>, D>,
{ {
@ -233,6 +282,26 @@ where
out out
} }
/// 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<(Complex<T>, T), 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], self.alphai[i]), self.beta[i])
}
out
}
} }
/* /*

63
nalgebra-lapack/tests/linalg/generalized_eigenvalues.rs
View File

@ -17,21 +17,29 @@ proptest! {
let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a))); let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a)));
let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b))); let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b)));
if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0 { if a_condition_no.unwrap_or(200000.0) < 5.0 && b_condition_no.unwrap_or(200000.0) < 5.0 {
let a_c =a.clone().map(|x| Complex::new(x, 0.0)); let a_c = a.clone().map(|x| Complex::new(x, 0.0));
let b_c = b.clone().map(|x| Complex::new(x, 0.0)); let b_c = b.clone().map(|x| Complex::new(x, 0.0));
let ge = GE::new(a.clone(), b.clone()); let ge = GE::new(a.clone(), b.clone());
let (vsl,vsr) = ge.clone().eigenvectors(); let (vsl,vsr) = ge.clone().eigenvectors();
let eigenvalues = ge.clone().eigenvalues();
for i in 0..n { for (i,(alpha,beta)) in ge.raw_eigenvalues().iter().enumerate() {
let left_eigenvector = &vsl.column(i); let l_a = a_c.clone() * Complex::new(*beta, 0.0);
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)); let l_b = b_c.clone() * *alpha;
let right_eigenvector = &vsr.column(i); prop_assert!(
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)); relative_eq!(
}; ((&l_a - &l_b)*vsr.column(i)).map(|x| x.modulus()),
OMatrix::zeros_generic(Dynamic::new(n), Const::<1>),
epsilon = 1.0e-7));
prop_assert!(
relative_eq!(
(vsl.column(i).adjoint()*(&l_a - &l_b)).map(|x| x.modulus()),
OMatrix::zeros_generic(Const::<1>, Dynamic::new(n)),
epsilon = 1.0e-7))
};
}; };
} }
@ -40,20 +48,27 @@ proptest! {
let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a))); let a_condition_no = a.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&a)));
let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b))); let b_condition_no = b.clone().try_inverse().and_then(|x| Some(EuclideanNorm.norm(&x)* EuclideanNorm.norm(&b)));
if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0{ if a_condition_no.unwrap_or(200000.0) < 5.0 && b_condition_no.unwrap_or(200000.0) < 5.0 {
let ge = GE::new(a.clone(), b.clone()); let ge = GE::new(a.clone(), b.clone());
let a_c =a.clone().map(|x| Complex::new(x, 0.0)); let a_c =a.clone().map(|x| Complex::new(x, 0.0));
let b_c = b.clone().map(|x| Complex::new(x, 0.0)); let b_c = b.clone().map(|x| Complex::new(x, 0.0));
let (vsl,vsr) = ge.eigenvectors(); let (vsl,vsr) = ge.eigenvectors();
let eigenvalues = ge.eigenvalues(); let eigenvalues = ge.raw_eigenvalues();
for i in 0..4 { for (i,(alpha,beta)) in eigenvalues.iter().enumerate() {
let left_eigenvector = &vsl.column(i); let l_a = a_c.clone() * Complex::new(*beta, 0.0);
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)); let l_b = b_c.clone() * *alpha;
let right_eigenvector = &vsr.column(i); prop_assert!(
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)); relative_eq!(
}; ((&l_a - &l_b)*vsr.column(i)).map(|x| x.modulus()),
OMatrix::zeros_generic(Const::<4>, Const::<1>),
epsilon = 1.0e-7));
prop_assert!(
relative_eq!((vsl.column(i).adjoint()*(&l_a - &l_b)).map(|x| x.modulus()),
OMatrix::zeros_generic(Const::<1>, Const::<4>),
epsilon = 1.0e-7))
}
}; };
} }
} }