Merge pull request #1106 from geoeo/dev
enabled complex eigenvectors for lapack
This commit is contained in:
Sébastien Crozet
GitHub
commit
0d9adec0ab
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@ -44,3 +44,4 @@ proptest = { version = "1", default-features = false, features = ["std"] }
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quickcheck = "1"
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approx = "0.5"
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rand = "0.8"
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@ -7,13 +7,12 @@ 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 na::{allocator::Allocator, DefaultAllocator, Matrix, OMatrix, OVector, Scalar};
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use lapack;
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/// Eigendecomposition of a real square matrix with real eigenvalues.
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/// Eigendecomposition of a real square matrix with real or complex eigenvalues.
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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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@ -36,8 +35,10 @@ pub struct Eigen<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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/// The eigenvalues of the decomposed matrix.
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pub eigenvalues: OVector<T, D>,
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/// The real parts of eigenvalues of the decomposed matrix.
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pub eigenvalues_re: OVector<T, D>,
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/// The imaginary parts of the eigenvalues of the decomposed matrix.
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pub eigenvalues_im: OVector<T, D>,
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/// The (right) eigenvectors of the decomposed matrix.
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pub eigenvectors: Option<OMatrix<T, D, D>>,
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/// The left eigenvectors of the decomposed matrix.
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@ -69,8 +70,8 @@ where
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"Unable to compute the eigenvalue decomposition of a non-square matrix."
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);
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let ljob = if left_eigenvectors { b'V' } else { b'T' };
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let rjob = if eigenvectors { b'V' } else { b'T' };
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let ljob = if left_eigenvectors { b'V' } else { b'N' };
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let rjob = if eigenvectors { b'V' } else { b'N' };
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let (nrows, ncols) = m.shape_generic();
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let n = nrows.value();
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@ -104,213 +105,232 @@ where
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lapack_check!(info);
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let mut work = vec![T::zero(); lwork as usize];
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let mut vl = if left_eigenvectors {
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Some(Matrix::zeros_generic(nrows, ncols))
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} else {
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None
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};
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let mut vr = if eigenvectors {
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Some(Matrix::zeros_generic(nrows, ncols))
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} else {
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None
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};
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match (left_eigenvectors, eigenvectors) {
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(true, true) => {
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// TODO: avoid the initializations?
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let mut vl = Matrix::zeros_generic(nrows, ncols);
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let mut vr = Matrix::zeros_generic(nrows, ncols);
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T::xgeev(
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ljob,
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rjob,
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut vl.as_mut_slice(),
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n as i32,
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&mut vr.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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if wi.iter().all(|e| e.is_zero()) {
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return Some(Self {
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eigenvalues: wr,
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left_eigenvectors: Some(vl),
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eigenvectors: Some(vr),
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});
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}
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}
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(true, false) => {
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// TODO: avoid the initialization?
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let mut vl = Matrix::zeros_generic(nrows, ncols);
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T::xgeev(
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ljob,
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rjob,
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut vl.as_mut_slice(),
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n as i32,
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&mut placeholder2,
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1 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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if wi.iter().all(|e| e.is_zero()) {
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return Some(Self {
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eigenvalues: wr,
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left_eigenvectors: Some(vl),
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eigenvectors: None,
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});
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}
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}
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(false, true) => {
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// TODO: avoid the initialization?
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let mut vr = Matrix::zeros_generic(nrows, ncols);
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T::xgeev(
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ljob,
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rjob,
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut placeholder1,
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1 as i32,
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&mut vr.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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if wi.iter().all(|e| e.is_zero()) {
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return Some(Self {
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eigenvalues: wr,
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left_eigenvectors: None,
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eigenvectors: Some(vr),
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});
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}
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}
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(false, false) => {
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T::xgeev(
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ljob,
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rjob,
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut placeholder1,
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1 as i32,
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&mut placeholder2,
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1 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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if wi.iter().all(|e| e.is_zero()) {
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return Some(Self {
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eigenvalues: wr,
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left_eigenvectors: None,
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eigenvectors: None,
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});
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}
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}
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}
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None
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}
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/// The complex eigenvalues of the given matrix.
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///
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/// Panics if the eigenvalue computation does not converge.
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pub fn complex_eigenvalues(mut m: OMatrix<T, D, D>) -> 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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assert!(
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m.is_square(),
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"Unable to compute the eigenvalue decomposition of a non-square matrix."
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);
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let nrows = m.shape_generic().0;
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let n = nrows.value();
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let lda = n as i32;
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// TODO: avoid the initialization?
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let mut wr = Matrix::zeros_generic(nrows, Const::<1>);
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let mut wi = Matrix::zeros_generic(nrows, Const::<1>);
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let mut info = 0;
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let mut placeholder1 = [T::zero()];
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let mut placeholder2 = [T::zero()];
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let lwork = T::xgeev_work_size(
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b'T',
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b'T',
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut placeholder1,
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n as i32,
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&mut placeholder2,
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n as i32,
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&mut info,
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);
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lapack_panic!(info);
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let mut work = vec![T::zero(); lwork as usize];
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let vl_ref = vl
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.as_mut()
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.map(|m| m.as_mut_slice())
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.unwrap_or(&mut placeholder1);
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let vr_ref = vr
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.as_mut()
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.map(|m| m.as_mut_slice())
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.unwrap_or(&mut placeholder2);
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T::xgeev(
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b'T',
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b'T',
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ljob,
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rjob,
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n as i32,
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m.as_mut_slice(),
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lda,
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wr.as_mut_slice(),
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wi.as_mut_slice(),
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&mut placeholder1,
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1 as i32,
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&mut placeholder2,
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1 as i32,
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vl_ref,
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if left_eigenvectors { n as i32 } else { 1 },
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vr_ref,
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if eigenvectors { n as i32 } else { 1 },
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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_panic!(info);
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lapack_check!(info);
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let mut res = Matrix::zeros_generic(nrows, Const::<1>);
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Some(Self {
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eigenvalues_re: wr,
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eigenvalues_im: wi,
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left_eigenvectors: vl,
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eigenvectors: vr,
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})
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}
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for i in 0..res.len() {
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res[i] = Complex::new(wr[i].clone(), wi[i].clone());
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}
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res
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/// Returns `true` if all the eigenvalues are real.
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pub fn eigenvalues_are_real(&self) -> bool {
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self.eigenvalues_im.iter().all(|e| e.is_zero())
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}
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/// The determinant of the decomposed matrix.
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#[inline]
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#[must_use]
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pub fn determinant(&self) -> T {
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let mut det = T::one();
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for e in self.eigenvalues.iter() {
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det *= e.clone();
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pub fn determinant(&self) -> Complex<T> {
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let mut det: Complex<T> = na::one();
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for (re, im) in self.eigenvalues_re.iter().zip(self.eigenvalues_im.iter()) {
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det *= Complex::new(re.clone(), im.clone());
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}
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det
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}
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/// Returns a tuple of vectors. The elements of the tuple are the real parts of the eigenvalues, left eigenvectors and right eigenvectors respectively.
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pub fn get_real_elements(
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&self,
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) -> (
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Vec<T>,
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Option<Vec<OVector<T, D>>>,
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Option<Vec<OVector<T, D>>>,
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)
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where
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DefaultAllocator: Allocator<T, D>,
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{
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let (number_of_elements, _) = self.eigenvalues_re.shape_generic();
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let number_of_elements_value = number_of_elements.value();
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let mut eigenvalues = Vec::<T>::with_capacity(number_of_elements_value);
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let mut eigenvectors = match self.eigenvectors.is_some() {
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true => Some(Vec::<OVector<T, D>>::with_capacity(
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number_of_elements_value,
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)),
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false => None,
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};
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let mut left_eigenvectors = match self.left_eigenvectors.is_some() {
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true => Some(Vec::<OVector<T, D>>::with_capacity(
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number_of_elements_value,
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)),
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false => None,
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};
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let mut c = 0;
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while c < number_of_elements_value {
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eigenvalues.push(self.eigenvalues_re[c].clone());
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if eigenvectors.is_some() {
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eigenvectors.as_mut().unwrap().push(
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(&self.eigenvectors.as_ref())
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.unwrap()
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.column(c)
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.into_owned(),
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);
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}
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if left_eigenvectors.is_some() {
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left_eigenvectors.as_mut().unwrap().push(
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(&self.left_eigenvectors.as_ref())
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.unwrap()
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.column(c)
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.into_owned(),
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);
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}
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if self.eigenvalues_im[c] != T::zero() {
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//skip next entry
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c += 1;
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}
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c += 1;
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}
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(eigenvalues, left_eigenvectors, eigenvectors)
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}
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/// Returns a tuple of vectors. The elements of the tuple are the complex eigenvalues, complex left eigenvectors and complex right eigenvectors respectively.
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/// The elements appear as conjugate pairs within each vector, with the positive of the pair always being first.
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pub fn get_complex_elements(
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&self,
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) -> (
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Option<Vec<Complex<T>>>,
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Option<Vec<OVector<Complex<T>, D>>>,
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Option<Vec<OVector<Complex<T>, D>>>,
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)
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where
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DefaultAllocator: Allocator<Complex<T>, D>,
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{
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match self.eigenvalues_are_real() {
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true => (None, None, None),
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false => {
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let (number_of_elements, _) = self.eigenvalues_re.shape_generic();
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let number_of_elements_value = number_of_elements.value();
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let number_of_complex_entries =
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self.eigenvalues_im
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.iter()
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.fold(0, |acc, e| if !e.is_zero() { acc + 1 } else { acc });
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let mut eigenvalues = Vec::<Complex<T>>::with_capacity(number_of_complex_entries);
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let mut eigenvectors = match self.eigenvectors.is_some() {
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true => Some(Vec::<OVector<Complex<T>, D>>::with_capacity(
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number_of_complex_entries,
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)),
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false => None,
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};
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let mut left_eigenvectors = match self.left_eigenvectors.is_some() {
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true => Some(Vec::<OVector<Complex<T>, D>>::with_capacity(
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number_of_complex_entries,
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)),
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false => None,
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};
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let mut c = 0;
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while c < number_of_elements_value {
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if self.eigenvalues_im[c] != T::zero() {
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//Complex conjugate pairs of eigenvalues appear consecutively with the eigenvalue having the positive imaginary part first.
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eigenvalues.push(Complex::<T>::new(
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self.eigenvalues_re[c].clone(),
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self.eigenvalues_im[c].clone(),
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));
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eigenvalues.push(Complex::<T>::new(
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self.eigenvalues_re[c + 1].clone(),
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self.eigenvalues_im[c + 1].clone(),
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));
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if eigenvectors.is_some() {
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let mut vec = OVector::<Complex<T>, D>::zeros_generic(
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number_of_elements,
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Const::<1>,
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);
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let mut vec_conj = OVector::<Complex<T>, D>::zeros_generic(
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number_of_elements,
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Const::<1>,
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);
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for r in 0..number_of_elements_value {
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vec[r] = Complex::<T>::new(
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(&self.eigenvectors.as_ref()).unwrap()[(r, c)].clone(),
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(&self.eigenvectors.as_ref()).unwrap()[(r, c + 1)].clone(),
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);
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vec_conj[r] = Complex::<T>::new(
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(&self.eigenvectors.as_ref()).unwrap()[(r, c)].clone(),
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(&self.eigenvectors.as_ref()).unwrap()[(r, c + 1)].clone(),
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);
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}
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eigenvectors.as_mut().unwrap().push(vec);
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eigenvectors.as_mut().unwrap().push(vec_conj);
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}
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if left_eigenvectors.is_some() {
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let mut vec = OVector::<Complex<T>, D>::zeros_generic(
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number_of_elements,
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Const::<1>,
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);
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let mut vec_conj = OVector::<Complex<T>, D>::zeros_generic(
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number_of_elements,
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Const::<1>,
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);
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for r in 0..number_of_elements_value {
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vec[r] = Complex::<T>::new(
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(&self.left_eigenvectors.as_ref()).unwrap()[(r, c)].clone(),
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(&self.left_eigenvectors.as_ref()).unwrap()[(r, c + 1)].clone(),
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);
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vec_conj[r] = Complex::<T>::new(
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(&self.left_eigenvectors.as_ref()).unwrap()[(r, c)].clone(),
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(&self.left_eigenvectors.as_ref()).unwrap()[(r, c + 1)].clone(),
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);
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}
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left_eigenvectors.as_mut().unwrap().push(vec);
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left_eigenvectors.as_mut().unwrap().push(vec_conj);
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}
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//skip next entry
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c += 1;
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}
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c += 1;
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}
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(Some(eigenvalues), left_eigenvectors, eigenvectors)
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}
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}
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}
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}
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/*
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|
|
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|
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@ -58,8 +58,8 @@ proptest! {
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let sol1 = chol.solve(&b1).unwrap();
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let sol2 = chol.solve(&b2).unwrap();
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prop_assert!(relative_eq!(m * sol1, b1, epsilon = 1.0e-7));
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prop_assert!(relative_eq!(m * sol2, b2, epsilon = 1.0e-7));
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prop_assert!(relative_eq!(m * sol1, b1, epsilon = 1.0e-4));
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prop_assert!(relative_eq!(m * sol2, b2, epsilon = 1.0e-4));
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}
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}
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|
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@ -84,7 +84,7 @@ proptest! {
|
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let id1 = &m * &m1;
|
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let id2 = &m1 * &m;
|
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|
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prop_assert!(id1.is_identity(1.0e-5) && id2.is_identity(1.0e-5))
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prop_assert!(id1.is_identity(1.0e-4) && id2.is_identity(1.0e-4))
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}
|
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}
|
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}
|
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|
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|
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@ -0,0 +1,47 @@
|
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use na::Matrix3;
|
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use nalgebra_lapack::Eigen;
|
||||
use num_complex::Complex;
|
||||
|
||||
#[test]
|
||||
fn complex_eigen() {
|
||||
let m = Matrix3::<f64>::new(
|
||||
4.0 / 5.0,
|
||||
-3.0 / 5.0,
|
||||
0.0,
|
||||
3.0 / 5.0,
|
||||
4.0 / 5.0,
|
||||
0.0,
|
||||
1.0,
|
||||
2.0,
|
||||
2.0,
|
||||
);
|
||||
let eigen = Eigen::new(m, true, true).expect("Eigen Creation Failed!");
|
||||
let (some_eigenvalues, some_left_vec, some_right_vec) = eigen.get_complex_elements();
|
||||
let eigenvalues = some_eigenvalues.expect("Eigenvalues Failed");
|
||||
let _left_eigenvectors = some_left_vec.expect("Left Eigenvectors Failed");
|
||||
let eigenvectors = some_right_vec.expect("Right Eigenvectors Failed");
|
||||
|
||||
assert_relative_eq!(
|
||||
eigenvalues[0].re,
|
||||
Complex::<f64>::new(4.0 / 5.0, 3.0 / 5.0).re
|
||||
);
|
||||
assert_relative_eq!(
|
||||
eigenvalues[0].im,
|
||||
Complex::<f64>::new(4.0 / 5.0, 3.0 / 5.0).im
|
||||
);
|
||||
assert_relative_eq!(
|
||||
eigenvalues[1].re,
|
||||
Complex::<f64>::new(4.0 / 5.0, -3.0 / 5.0).re
|
||||
);
|
||||
assert_relative_eq!(
|
||||
eigenvalues[1].im,
|
||||
Complex::<f64>::new(4.0 / 5.0, -3.0 / 5.0).im
|
||||
);
|
||||
|
||||
assert_relative_eq!(eigenvectors[0][0].re, -12.0 / 32.7871926215100059134410999);
|
||||
assert_relative_eq!(eigenvectors[0][0].im, -9.0 / 32.7871926215100059134410999);
|
||||
assert_relative_eq!(eigenvectors[0][1].re, -9.0 / 32.7871926215100059134410999);
|
||||
assert_relative_eq!(eigenvectors[0][1].im, 12.0 / 32.7871926215100059134410999);
|
||||
assert_relative_eq!(eigenvectors[0][2].re, 25.0 / 32.7871926215100059134410999);
|
||||
assert_relative_eq!(eigenvectors[0][2].im, 0.0);
|
||||
}
|
||||
|
|
@ -51,10 +51,10 @@ proptest! {
|
|||
let tr_sol1 = lup.solve_transpose(&b1).unwrap();
|
||||
let tr_sol2 = lup.solve_transpose(&b2).unwrap();
|
||||
|
||||
prop_assert!(relative_eq!(&m * sol1, b1, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(&m * sol2, b2, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol1, b1, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol2, b2, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(&m * sol1, b1, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(&m * sol2, b2, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol1, b1, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol2, b2, epsilon = 1.0e-5));
|
||||
}
|
||||
|
||||
#[test]
|
||||
|
|
@ -68,10 +68,10 @@ proptest! {
|
|||
let tr_sol1 = lup.solve_transpose(&b1).unwrap();
|
||||
let tr_sol2 = lup.solve_transpose(&b2).unwrap();
|
||||
|
||||
prop_assert!(relative_eq!(m * sol1, b1, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m * sol2, b2, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol1, b1, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol2, b2, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(m * sol1, b1, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(m * sol2, b2, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol1, b1, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(m.transpose() * tr_sol2, b2, epsilon = 1.0e-5));
|
||||
}
|
||||
|
||||
#[test]
|
||||
|
|
|
|||
|
|
@ -1,4 +1,5 @@
|
|||
mod cholesky;
|
||||
mod complex_eigen;
|
||||
mod generalized_eigenvalues;
|
||||
mod lu;
|
||||
mod qr;
|
||||
|
|
|
|||
|
|
@ -13,30 +13,36 @@ proptest! {
|
|||
let m = DMatrix::<f64>::new_random(n, n);
|
||||
|
||||
if let Some(eig) = Eigen::new(m.clone(), true, true) {
|
||||
let eigvals = DMatrix::from_diagonal(&eig.eigenvalues);
|
||||
let transformed_eigvectors = &m * eig.eigenvectors.as_ref().unwrap();
|
||||
let scaled_eigvectors = eig.eigenvectors.as_ref().unwrap() * &eigvals;
|
||||
// TODO: test the complex case too.
|
||||
if eig.eigenvalues_are_real() {
|
||||
let eigvals = DMatrix::from_diagonal(&eig.eigenvalues_re);
|
||||
let transformed_eigvectors = &m * eig.eigenvectors.as_ref().unwrap();
|
||||
let scaled_eigvectors = eig.eigenvectors.as_ref().unwrap() * &eigvals;
|
||||
|
||||
let transformed_left_eigvectors = m.transpose() * eig.left_eigenvectors.as_ref().unwrap();
|
||||
let scaled_left_eigvectors = eig.left_eigenvectors.as_ref().unwrap() * &eigvals;
|
||||
let transformed_left_eigvectors = m.transpose() * eig.left_eigenvectors.as_ref().unwrap();
|
||||
let scaled_left_eigvectors = eig.left_eigenvectors.as_ref().unwrap() * &eigvals;
|
||||
|
||||
prop_assert!(relative_eq!(transformed_eigvectors, scaled_eigvectors, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(transformed_left_eigvectors, scaled_left_eigvectors, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(transformed_eigvectors, scaled_eigvectors, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(transformed_left_eigvectors, scaled_left_eigvectors, epsilon = 1.0e-5));
|
||||
}
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn eigensystem_static(m in matrix4()) {
|
||||
if let Some(eig) = Eigen::new(m, true, true) {
|
||||
let eigvals = Matrix4::from_diagonal(&eig.eigenvalues);
|
||||
let transformed_eigvectors = m * eig.eigenvectors.unwrap();
|
||||
let scaled_eigvectors = eig.eigenvectors.unwrap() * eigvals;
|
||||
// TODO: test the complex case too.
|
||||
if eig.eigenvalues_are_real() {
|
||||
let eigvals = Matrix4::from_diagonal(&eig.eigenvalues_re);
|
||||
let transformed_eigvectors = m * eig.eigenvectors.unwrap();
|
||||
let scaled_eigvectors = eig.eigenvectors.unwrap() * eigvals;
|
||||
|
||||
let transformed_left_eigvectors = m.transpose() * eig.left_eigenvectors.unwrap();
|
||||
let scaled_left_eigvectors = eig.left_eigenvectors.unwrap() * eigvals;
|
||||
let transformed_left_eigvectors = m.transpose() * eig.left_eigenvectors.unwrap();
|
||||
let scaled_left_eigvectors = eig.left_eigenvectors.unwrap() * eigvals;
|
||||
|
||||
prop_assert!(relative_eq!(transformed_eigvectors, scaled_eigvectors, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(transformed_left_eigvectors, scaled_left_eigvectors, epsilon = 1.0e-7));
|
||||
prop_assert!(relative_eq!(transformed_eigvectors, scaled_eigvectors, epsilon = 1.0e-5));
|
||||
prop_assert!(relative_eq!(transformed_left_eigvectors, scaled_left_eigvectors, epsilon = 1.0e-5));
|
||||
}
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
|
|||
|
|
@ -11,14 +11,17 @@ proptest! {
|
|||
let n = cmp::max(1, cmp::min(n, 10));
|
||||
let m = DMatrix::<f64>::new_random(n, n);
|
||||
|
||||
let (vecs, vals) = Schur::new(m.clone()).unpack();
|
||||
|
||||
prop_assert!(relative_eq!(&vecs * vals * vecs.transpose(), m, epsilon = 1.0e-7))
|
||||
if let Some(schur) = Schur::try_new(m.clone()) {
|
||||
let (vecs, vals) = schur.unpack();
|
||||
prop_assert!(relative_eq!(&vecs * vals * vecs.transpose(), m, epsilon = 1.0e-5))
|
||||
}
|
||||
}
|
||||
|
||||
#[test]
|
||||
fn schur_static(m in matrix4()) {
|
||||
let (vecs, vals) = Schur::new(m.clone()).unpack();
|
||||
prop_assert!(relative_eq!(vecs * vals * vecs.transpose(), m, epsilon = 1.0e-7))
|
||||
if let Some(schur) = Schur::try_new(m.clone()) {
|
||||
let (vecs, vals) = schur.unpack();
|
||||
prop_assert!(relative_eq!(vecs * vals * vecs.transpose(), m, epsilon = 1.0e-5))
|
||||
}
|
||||
}
|
||||
}
|
||||
|
|
|
|||
Loading…
Reference in New Issue