New wrapper for generalized eigenvalues and associated eigenvectors via LAPACK routines sggev/dggev

nalgebra-lapack/src/generalized_eigenvalues.rs

@@ -0,0 +1,339 @@
+#[cfg(feature = "serde-serialize")]
+use serde::{Deserialize, Serialize};
+
+use num::Zero;
+use num_complex::Complex;
+
+use simba::scalar:: RealField;
+
+use crate::ComplexHelper;
+use na::allocator::Allocator;
+use na::dimension::{Const, Dim};
+use na::{DefaultAllocator, Matrix, OMatrix, OVector, Scalar};
+
+use lapack;
+
+/// Generalized eigenvalues and generalized eigenvectors(left and right) of a pair of N*N square matrices.
+#[cfg_attr(feature = "serde-serialize", derive(Serialize, Deserialize))]
+#[cfg_attr(
+    feature = "serde-serialize",
+    serde(
+        bound(serialize = "DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+         OVector<T, D>: Serialize,
+         OMatrix<T, D, D>: Serialize")
+    )
+)]
+#[cfg_attr(
+    feature = "serde-serialize",
+    serde(
+        bound(deserialize = "DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+         OVector<T, D>: Serialize,
+         OMatrix<T, D, D>: Deserialize<'de>")
+    )
+)]
+#[derive(Clone, Debug)]
+pub struct GE<T: Scalar, D: Dim>
+where
+    DefaultAllocator: Allocator<T, D> + Allocator<T, D, D>,
+{
+    alphar: OVector<T, D>,
+    alphai: OVector<T, D>,
+    beta: OVector<T, D>,
+    vsl: OMatrix<T, D, D>,
+    vsr: OMatrix<T, D, D>,
+}
+
+impl<T: Scalar + Copy, D: Dim> Copy for GE<T, D>
+where
+    DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+    OMatrix<T, D, D>: Copy,
+    OVector<T, D>: Copy,
+{
+}
+
+impl<T: GEScalar + RealField + Copy, D: Dim> GE<T, D>
+where
+    DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+{
+    /// Attempts to compute the generalized eigenvalues (and eigenvectors) 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)
+    /// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r
+    ///
+    /// Panics if the method did not converge.
+    pub fn new(a: OMatrix<T, D, D>, b: OMatrix<T, D, D>) -> Self {
+        Self::try_new(a, b).expect("Calculation of generalized eigenvalues failed.")
+    }
+
+    /// Attempts to compute the generalized eigenvalues (and eigenvectors) 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)
+    /// it satisfies e_l*a = e*e_l*b and a*e_r = e*b*e_r
+    ///
+    /// 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> {
+        assert!(
+            a.is_square() && b.is_square(),
+            "Unable to compute the generalized eigenvalues of non-square matrices."
+        );
+
+        assert!(
+            a.shape_generic() ==  b.shape_generic(),
+            "Unable to compute the generalized eigenvalues of two square matrices of different dimensions."
+        );
+
+        let (nrows, ncols) = a.shape_generic();
+        let n = nrows.value();
+
+        let mut info = 0;
+
+        let mut alphar = Matrix::zeros_generic(nrows, Const::<1>);
+        let mut alphai = Matrix::zeros_generic(nrows, Const::<1>);
+        let mut beta = Matrix::zeros_generic(nrows, Const::<1>);
+        let mut vsl = Matrix::zeros_generic(nrows, ncols);
+        let mut vsr = Matrix::zeros_generic(nrows, ncols);
+
+        let lwork = T::xggev_work_size(
+            b'V',
+            b'V',
+            n as i32,
+            a.as_mut_slice(),
+            n as i32,
+            b.as_mut_slice(),
+            n as i32,
+            alphar.as_mut_slice(),
+            alphai.as_mut_slice(),
+            beta.as_mut_slice(),
+            vsl.as_mut_slice(),
+            n as i32,
+            vsr.as_mut_slice(),
+            n as i32,
+            &mut info,
+        );
+        lapack_check!(info);
+
+        let mut work = vec![T::zero(); lwork as usize];
+
+        T::xggev(
+            b'V',
+            b'V',
+            n as i32,
+            a.as_mut_slice(),
+            n as i32,
+            b.as_mut_slice(),
+            n as i32,
+            alphar.as_mut_slice(),
+            alphai.as_mut_slice(),
+            beta.as_mut_slice(),
+            vsl.as_mut_slice(),
+            n as i32,
+            vsr.as_mut_slice(),
+            n as i32,
+            &mut work,
+            lwork,
+            &mut info,
+        );
+        lapack_check!(info);
+
+        Some(GE {
+            alphar,
+            alphai,
+            beta,
+            vsl,
+            vsr,
+        })
+    }
+
+    /// Calculates the generalized eigenvectors (left and right) associated with the generalized eigenvalues
+    pub fn eigenvectors(self) -> (OMatrix<Complex<T>, D, D>, OMatrix<Complex<T>, D, D>)
+    where
+        DefaultAllocator: Allocator<Complex<T>, D, D> + Allocator<Complex<T>, D>,
+    {
+        let n = self.vsl.shape().0;
+        let mut l = self
+            .vsl
+            .clone()
+            .map(|x| Complex::new(x, T::RealField::zero()));
+        let mut r = self
+            .vsr
+            .clone()
+            .map(|x| Complex::new(x, T::RealField::zero()));
+
+        let eigenvalues = &self.eigenvalues();
+
+        let mut ll;
+        let mut c = 0;
+        while c < n {
+            if eigenvalues[c].im.abs() > T::RealField::default_epsilon() && c + 1 < n && {
+                let e_conj = eigenvalues[c].conj();
+                let e = eigenvalues[c + 1];
+                ((e_conj.re - e.re).abs() < T::RealField::default_epsilon())
+                    && ((e_conj.im - e.im).abs() < T::RealField::default_epsilon())
+            } {
+                ll = l.column(c + 1).into_owned();
+                l.column_mut(c).zip_apply(&ll, |r, i| {
+                    *r = Complex::new(r.re.clone(), i.re);
+                });
+                ll.copy_from(&l.column(c));
+                l.column_mut(c + 1).zip_apply(&ll, |r, i| {
+                    *r = i.conj();
+                });
+
+                ll.copy_from(&r.column(c + 1));
+                r.column_mut(c).zip_apply(&ll, |r, i| {
+                    *r = Complex::new(r.re, i.re);
+                });
+                ll.copy_from(&r.column(c));
+                r.column_mut(c + 1).zip_apply(&ll, |r, i| {
+                    *r = i.conj();
+                });
+
+                c += 2;
+            } else {
+                c += 1;
+            }
+        }
+
+        (l, r)
+    }
+
+    /// computes the generalized eigenvalues
+    #[must_use]
+    pub 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>);
+
+        for i in 0..out.len() {
+            out[i] = if self.beta[i].clone().abs() < T::RealField::default_epsilon() {
+                Complex::zero()
+            } else {
+                let mut cr = self.alphar[i].clone();
+                let mut ci = self.alphai[i].clone();
+                let b = self.beta[i].clone();
+
+                if cr.clone().abs() < T::RealField::default_epsilon() {
+                    cr = T::RealField::zero()
+                } else {
+                    cr = cr / b.clone()
+                };
+
+                if ci.clone().abs() < T::RealField::default_epsilon() {
+                    ci = T::RealField::zero()
+                } else {
+                    ci = ci / b
+                };
+
+                Complex::new(cr, ci)
+            }
+        }
+
+        out
+    }
+}
+
+/*
+ *
+ * Lapack functions dispatch.
+ *
+ */
+/// Trait implemented by scalars for which Lapack implements the RealField GE decomposition.
+pub trait GEScalar: Scalar {
+    #[allow(missing_docs)]
+    fn xggev(
+        jobvsl: u8,
+        jobvsr: u8,
+        n: i32,
+        a: &mut [Self],
+        lda: i32,
+        b: &mut [Self],
+        ldb: i32,
+        alphar: &mut [Self],
+        alphai: &mut [Self],
+        beta: &mut [Self],
+        vsl: &mut [Self],
+        ldvsl: i32,
+        vsr: &mut [Self],
+        ldvsr: i32,
+        work: &mut [Self],
+        lwork: i32,
+        info: &mut i32,
+    );
+
+    #[allow(missing_docs)]
+    fn xggev_work_size(
+        jobvsl: u8,
+        jobvsr: u8,
+        n: i32,
+        a: &mut [Self],
+        lda: i32,
+        b: &mut [Self],
+        ldb: i32,
+        alphar: &mut [Self],
+        alphai: &mut [Self],
+        beta: &mut [Self],
+        vsl: &mut [Self],
+        ldvsl: i32,
+        vsr: &mut [Self],
+        ldvsr: i32,
+        info: &mut i32,
+    ) -> i32;
+}
+
+macro_rules! real_eigensystem_scalar_impl (
+    ($N: ty, $xggev: path) => (
+        impl GEScalar for $N {
+            #[inline]
+            fn xggev(jobvsl:  u8,
+                     jobvsr:  u8,
+                     n:      i32,
+                     a:      &mut [$N],
+                     lda:    i32,
+                     b:      &mut [$N],
+                     ldb:    i32,
+                     alphar: &mut [$N],
+                     alphai: &mut [$N],
+                     beta  : &mut [$N],
+                     vsl:    &mut [$N],
+                     ldvsl:  i32,
+                     vsr:    &mut [$N],
+                     ldvsr:  i32,
+                     work:   &mut [$N],
+                     lwork:  i32,
+                     info:   &mut i32) {
+                unsafe { $xggev(jobvsl, jobvsr, n, a, lda, b, ldb, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, work, lwork, info); }
+            }
+
+
+            #[inline]
+            fn xggev_work_size(jobvsl:  u8,
+                               jobvsr:  u8,
+                               n:      i32,
+                               a:      &mut [$N],
+                               lda:    i32,
+                               b:      &mut [$N],
+                               ldb:    i32,
+                               alphar: &mut [$N],
+                               alphai: &mut [$N],
+                               beta  : &mut [$N],
+                               vsl:    &mut [$N],
+                               ldvsl:  i32,
+                               vsr:    &mut [$N],
+                               ldvsr:  i32,
+                               info:   &mut i32)
+                               -> i32 {
+                let mut work = [ Zero::zero() ];
+                let lwork    = -1 as i32;
+
+                unsafe { $xggev(jobvsl, jobvsr, n, a, lda, b, ldb, alphar, alphai, beta, vsl, ldvsl, vsr, ldvsr, &mut work, lwork, info); }
+                ComplexHelper::real_part(work[0]) as i32
+            }
+        }
+    )
+);
+
+real_eigensystem_scalar_impl!(f32, lapack::sggev);
+real_eigensystem_scalar_impl!(f64, lapack::dggev);

nalgebra-lapack/src/lib.rs

@@ -87,6 +87,7 @@ mod hessenberg;
 mod lu;
 mod qr;
 mod qz;
+mod generalized_eigenvalues;
 mod schur;
 mod svd;
 mod symmetric_eigen;
@@ -99,6 +100,7 @@ pub use self::hessenberg::Hessenberg;
 pub use self::lu::{LUScalar, LU};
 pub use self::qr::QR;
 pub use self::qz::QZ;
+pub use self::generalized_eigenvalues::GE;
 pub use self::schur::Schur;
 pub use self::svd::SVD;
 pub use self::symmetric_eigen::SymmetricEigen;

nalgebra-lapack/tests/linalg/generalized_eigenvalues.rs

@@ -0,0 +1,59 @@
+use na::dimension::{Const, Dynamic};
+use na::{DMatrix, EuclideanNorm, Norm, OMatrix};
+use nl::GE;
+use num_complex::Complex;
+use simba::scalar::ComplexField;
+use std::cmp;
+
+use crate::proptest::*;
+use proptest::{prop_assert, proptest};
+
+proptest! {
+    #[test]
+    fn ge(n in PROPTEST_MATRIX_DIM) {
+        let n = cmp::max(1, cmp::min(n, 10));
+        let a = DMatrix::<f64>::new_random(n, n);
+        let b = DMatrix::<f64>::new_random(n, n);
+        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)));
+
+        if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0 {
+        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 ge = GE::new(a.clone(), b.clone());
+        let (vsl,vsr) = ge.clone().eigenvectors();
+        let eigenvalues = ge.clone().eigenvalues();
+
+        for i in 0..n {
+            let left_eigenvector = &vsl.column(i);
+            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 right_eigenvector = &vsr.column(i);
+            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));
+           };
+        };
+    }
+
+    #[test]
+    fn ge_static(a in matrix4(), b in matrix4()) {
+        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)));
+
+        if a_condition_no.unwrap_or(200000.0) < 10.0 && b_condition_no.unwrap_or(200000.0) < 10.0{
+        let ge = GE::new(a.clone(), b.clone());
+        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 (vsl,vsr) = ge.eigenvectors();
+        let eigenvalues = ge.eigenvalues();
+
+        for i in 0..4 {
+                let left_eigenvector = &vsl.column(i);
+                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 right_eigenvector = &vsr.column(i);
+                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));
+            };
+        };
+    }
+}

nalgebra-lapack/tests/linalg/mod.rs

@@ -2,6 +2,7 @@ mod cholesky;
 mod lu;
 mod qr;
 mod qz;
+mod generalized_eigenvalues;
 mod real_eigensystem;
 mod schur;
 mod svd;