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//! Eigenvalue decomposition for general matrices
use crate::{error::*, layout::MatrixLayout, *};
use cauchy::*;
use num_traits::{ToPrimitive, Zero};
/// Wraps `*geev` for general matrices
pub trait Eig_: Scalar {
/// Calculate Right eigenvalue
fn eig(
calc_v: bool,
l: MatrixLayout,
a: &mut [Self],
) -> Result<(Vec<Self::Complex>, Vec<Self::Complex>)>;
}
macro_rules! impl_eig_complex {
($scalar:ty, $ev:path) => {
impl Eig_ for $scalar {
fn eig(
calc_v: bool,
l: MatrixLayout,
mut a: &mut [Self],
) -> Result<(Vec<Self::Complex>, Vec<Self::Complex>)> {
let (n, _) = l.size();
// LAPACK assumes a column-major input. A row-major input can
// be interpreted as the transpose of a column-major input. So,
// for row-major inputs, we we want to solve the following,
// given the column-major input `A`:
//
// A^T V = V Λ ⟺ V^T A = Λ V^T ⟺ conj(V)^H A = Λ conj(V)^H
//
// So, in this case, the right eigenvectors are the conjugates
// of the left eigenvectors computed with `A`, and the
// eigenvalues are the eigenvalues computed with `A`.
let (jobvl, jobvr) = if calc_v {
match l {
MatrixLayout::C { .. } => (b'V', b'N'),
MatrixLayout::F { .. } => (b'N', b'V'),
}
} else {
(b'N', b'N')
};
let mut eigs = unsafe { vec_uninit(n as usize) };
let mut rwork = unsafe { vec_uninit(2 * n as usize) };
let mut vl = if jobvl == b'V' {
Some(unsafe { vec_uninit((n * n) as usize) })
} else {
None
};
let mut vr = if jobvr == b'V' {
Some(unsafe { vec_uninit((n * n) as usize) })
} else {
None
};
// calc work size
let mut info = 0;
let mut work_size = [Self::zero()];
unsafe {
$ev(
jobvl,
jobvr,
n,
&mut a,
n,
&mut eigs,
&mut vl.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut vr.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut work_size,
-1,
&mut rwork,
&mut info,
)
};
info.as_lapack_result()?;
// actal ev
let lwork = work_size[0].to_usize().unwrap();
let mut work = unsafe { vec_uninit(lwork) };
unsafe {
$ev(
jobvl,
jobvr,
n,
&mut a,
n,
&mut eigs,
&mut vl.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut vr.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut work,
lwork as i32,
&mut rwork,
&mut info,
)
};
info.as_lapack_result()?;
// Hermite conjugate
if jobvl == b'V' {
for c in vl.as_mut().unwrap().iter_mut() {
c.im = -c.im
}
}
Ok((eigs, vr.or(vl).unwrap_or(Vec::new())))
}
}
};
}
impl_eig_complex!(c64, lapack::zgeev);
impl_eig_complex!(c32, lapack::cgeev);
macro_rules! impl_eig_real {
($scalar:ty, $ev:path) => {
impl Eig_ for $scalar {
fn eig(
calc_v: bool,
l: MatrixLayout,
mut a: &mut [Self],
) -> Result<(Vec<Self::Complex>, Vec<Self::Complex>)> {
let (n, _) = l.size();
// LAPACK assumes a column-major input. A row-major input can
// be interpreted as the transpose of a column-major input. So,
// for row-major inputs, we we want to solve the following,
// given the column-major input `A`:
//
// A^T V = V Λ ⟺ V^T A = Λ V^T ⟺ conj(V)^H A = Λ conj(V)^H
//
// So, in this case, the right eigenvectors are the conjugates
// of the left eigenvectors computed with `A`, and the
// eigenvalues are the eigenvalues computed with `A`.
//
// We could conjugate the eigenvalues instead of the
// eigenvectors, but we have to reconstruct the eigenvectors
// into new matrices anyway, and by not modifying the
// eigenvalues, we preserve the nice ordering specified by
// `sgeev`/`dgeev`.
let (jobvl, jobvr) = if calc_v {
match l {
MatrixLayout::C { .. } => (b'V', b'N'),
MatrixLayout::F { .. } => (b'N', b'V'),
}
} else {
(b'N', b'N')
};
let mut eig_re = unsafe { vec_uninit(n as usize) };
let mut eig_im = unsafe { vec_uninit(n as usize) };
let mut vl = if jobvl == b'V' {
Some(unsafe { vec_uninit((n * n) as usize) })
} else {
None
};
let mut vr = if jobvr == b'V' {
Some(unsafe { vec_uninit((n * n) as usize) })
} else {
None
};
// calc work size
let mut info = 0;
let mut work_size = [0.0];
unsafe {
$ev(
jobvl,
jobvr,
n,
&mut a,
n,
&mut eig_re,
&mut eig_im,
vl.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
vr.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut work_size,
-1,
&mut info,
)
};
info.as_lapack_result()?;
// actual ev
let lwork = work_size[0].to_usize().unwrap();
let mut work = unsafe { vec_uninit(lwork) };
unsafe {
$ev(
jobvl,
jobvr,
n,
&mut a,
n,
&mut eig_re,
&mut eig_im,
vl.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
vr.as_mut().map(|v| v.as_mut_slice()).unwrap_or(&mut []),
n,
&mut work,
lwork as i32,
&mut info,
)
};
info.as_lapack_result()?;
// reconstruct eigenvalues
let eigs: Vec<Self::Complex> = eig_re
.iter()
.zip(eig_im.iter())
.map(|(&re, &im)| Self::complex(re, im))
.collect();
if !calc_v {
return Ok((eigs, Vec::new()));
}
// Reconstruct eigenvectors into complex-array
// --------------------------------------------
//
// From LAPACK API https://software.intel.com/en-us/node/469230
//
// - If the j-th eigenvalue is real,
// - v(j) = VR(:,j), the j-th column of VR.
//
// - If the j-th and (j+1)-st eigenvalues form a complex conjugate pair,
// - v(j) = VR(:,j) + i*VR(:,j+1)
// - v(j+1) = VR(:,j) - i*VR(:,j+1).
//
// In the C-layout case, we need the conjugates of the left
// eigenvectors, so the signs should be reversed.
let n = n as usize;
let v = vr.or(vl).unwrap();
let mut eigvecs = unsafe { vec_uninit(n * n) };
let mut col = 0;
while col < n {
if eig_im[col] == 0. {
// The corresponding eigenvalue is real.
for row in 0..n {
let re = v[row + col * n];
eigvecs[row + col * n] = Self::complex(re, 0.);
}
col += 1;
} else {
// This is a complex conjugate pair.
assert!(col + 1 < n);
for row in 0..n {
let re = v[row + col * n];
let mut im = v[row + (col + 1) * n];
if jobvl == b'V' {
im = -im;
}
eigvecs[row + col * n] = Self::complex(re, im);
eigvecs[row + (col + 1) * n] = Self::complex(re, -im);
}
col += 2;
}
}
Ok((eigs, eigvecs))
}
}
};
}
impl_eig_real!(f64, lapack::dgeev);
impl_eig_real!(f32, lapack::sgeev);