QZ-decomposition

nalgebra-lapack/src/generalized_eigenvalues.rs

@@ -0,0 +1,350 @@
+#[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 real 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",
+    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>: Deserialize<'de>,
+         OMatrix<T, D, D>: Deserialize<'de>")
+    )
+)]
+#[derive(Clone, Debug)]
+pub struct GeneralizedEigen<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 GeneralizedEigen<T, D>
+where
+    DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+    OMatrix<T, D, D>: Copy,
+    OVector<T, D>: Copy,
+{
+}
+
+impl<T: GeneralizedEigenScalar + RealField + Copy, D: Dim> GeneralizedEigen<T, D>
+where
+    DefaultAllocator: Allocator<T, D, D> + Allocator<T, D>,
+{
+    /// Attempts to compute the generalized eigenvalues, and left and right associated eigenvectors
+    /// via the raw returns from LAPACK's dggev and sggev routines
+    ///
+    /// 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
+    ///
+    /// 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(GeneralizedEigen {
+            alphar,
+            alphai,
+            beta,
+            vsl,
+            vsr,
+        })
+    }
+
+    /// Calculates the generalized eigenvectors (left and right) associated with the generalized eigenvalues
+    ///
+    /// Outputs two matrices.
+    /// The first output matrix contains the left eigenvectors of the generalized eigenvalues
+    /// as columns.
+    /// The second matrix contains the right eigenvectors of the generalized eigenvalues
+    /// as columns.
+    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> + Allocator<(Complex<T>, T), D>,
+    {
+        /*
+         How the eigenvectors are built up:
+
+         Since the input entries are all real, the generalized eigenvalues if complex come in pairs
+         as a consequence of the [complex conjugate root thorem](https://en.wikipedia.org/wiki/Complex_conjugate_root_theorem)
+         The Lapack routine output reflects this by expecting the user to unpack the real and complex eigenvalues associated
+         eigenvectors from the real matrix output via the following procedure
+
+         (Note: VL stands for the lapack real matrix output containing the left eigenvectors as columns,
+                VR stands for the lapack real matrix output containing the right eigenvectors as columns)
+
+         If the j-th and (j+1)-th eigenvalues form a complex conjugate pair,
+         then
+
+          u(j) = VL(:,j)+i*VL(:,j+1)
+          u(j+1) = VL(:,j)-i*VL(:,j+1)
+
+         and
+
+          u(j) = VR(:,j)+i*VR(:,j+1)
+          v(j+1) = VR(:,j)-i*VR(:,j+1).
+        */
+
+        let n = self.vsl.shape().0;
+
+        let mut l = self.vsl.map(|x| Complex::new(x, T::RealField::zero()));
+
+        let mut r = self.vsr.map(|x| Complex::new(x, T::RealField::zero()));
+
+        let eigenvalues = self.raw_eigenvalues();
+
+        let mut c = 0;
+
+        while c < n {
+            if eigenvalues[c].0.im.abs() != T::RealField::zero() && c + 1 < n {
+                // taking care of the left eigenvector matrix
+                l.column_mut(c).zip_apply(&self.vsl.column(c + 1), |r, i| {
+                    *r = Complex::new(r.re.clone(), i.clone());
+                });
+                l.column_mut(c + 1).zip_apply(&self.vsl.column(c), |i, r| {
+                    *i = Complex::new(r.clone(), -i.re.clone());
+                });
+
+                // taking care of the right eigenvector matrix
+                r.column_mut(c).zip_apply(&self.vsr.column(c + 1), |r, i| {
+                    *r = Complex::new(r.re.clone(), i.clone());
+                });
+                r.column_mut(c + 1).zip_apply(&self.vsr.column(c), |i, r| {
+                    *i = Complex::new(r.clone(), -i.re.clone());
+                });
+
+                c += 2;
+            } else {
+                c += 1;
+            }
+        }
+
+        (l, r)
+    }
+
+    /// Outputs the unprocessed (almost) version of  generalized eigenvalues ((alphar, alphai), beta)
+    /// straight from LAPACK
+    #[must_use]
+    pub fn raw_eigenvalues(&self) -> OVector<(Complex<T>, T), D>
+    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
+    }
+}
+
+/*
+ *
+ * Lapack functions dispatch.
+ *
+ */
+/// Trait implemented by scalars for which Lapack implements the RealField GeneralizedEigen decomposition.
+pub trait GeneralizedEigenScalar: 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! generalized_eigen_scalar_impl (
+    ($N: ty, $xggev: path) => (
+        impl GeneralizedEigenScalar 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
+            }
+        }
+    )
+);
+
+generalized_eigen_scalar_impl!(f32, lapack::sggev);
+generalized_eigen_scalar_impl!(f64, lapack::dggev);

nalgebra-lapack/src/lib.rs

@@ -83,9 +83,11 @@ mod lapack_check;
 
 mod cholesky;
 mod eigen;
+mod generalized_eigenvalues;
 mod hessenberg;
 mod lu;
 mod qr;
+mod qz;
 mod schur;
 mod svd;
 mod symmetric_eigen;
@@ -94,9 +96,11 @@ use num_complex::Complex;
 
 pub use self::cholesky::{Cholesky, CholeskyScalar};
 pub use self::eigen::Eigen;
+pub use self::generalized_eigenvalues::GeneralizedEigen;
 pub use self::hessenberg::Hessenberg;
 pub use self::lu::{LUScalar, LU};
 pub use self::qr::QR;
+pub use self::qz::QZ;
 pub use self::schur::Schur;
 pub use self::svd::SVD;
 pub use self::symmetric_eigen::SymmetricEigen;