lax/eig.rs
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//! Eigenvalue problem for general matricies
//!
//! LAPACK correspondance
//! ----------------------
//!
//! | f32 | f64 | c32 | c64 |
//! |:------|:------|:------|:------|
//! | sgeev | dgeev | cgeev | zgeev |
//!
use crate::{error::*, layout::MatrixLayout, *};
use cauchy::*;
use num_traits::{ToPrimitive, Zero};
#[cfg_attr(doc, katexit::katexit)]
/// Eigenvalue problem for general matrix
///
/// To manage memory more strictly, use [EigWork].
///
/// Right and Left eigenvalue problem
/// ----------------------------------
/// LAPACK can solve both right eigenvalue problem
/// $$
/// AV_R = V_R \Lambda
/// $$
/// where $V_R = \left( v_R^1, \cdots, v_R^n \right)$ are right eigenvectors
/// and left eigenvalue problem
/// $$
/// V_L^\dagger A = V_L^\dagger \Lambda
/// $$
/// where $V_L = \left( v_L^1, \cdots, v_L^n \right)$ are left eigenvectors
/// and eigenvalues
/// $$
/// \Lambda = \begin{pmatrix}
/// \lambda_1 & & 0 \\\\
/// & \ddots & \\\\
/// 0 & & \lambda_n
/// \end{pmatrix}
/// $$
/// which satisfies $A v_R^i = \lambda_i v_R^i$ and
/// $\left(v_L^i\right)^\dagger A = \lambda_i \left(v_L^i\right)^\dagger$
/// for column-major matrices, although row-major matrices are not supported.
/// Since a row-major matrix can be interpreted
/// as a transpose of a column-major matrix,
/// this transforms right eigenvalue problem to left one:
///
/// $$
/// A^\dagger V = V Λ ⟺ V^\dagger A = Λ V^\dagger
/// $$
///
#[non_exhaustive]
pub struct EigWork<T: Scalar> {
/// Problem size
pub n: i32,
/// Compute right eigenvectors or not
pub jobvr: JobEv,
/// Compute left eigenvectors or not
pub jobvl: JobEv,
/// Eigenvalues
pub eigs: Vec<MaybeUninit<T::Complex>>,
/// Real part of eigenvalues used in real routines
pub eigs_re: Option<Vec<MaybeUninit<T::Real>>>,
/// Imaginary part of eigenvalues used in real routines
pub eigs_im: Option<Vec<MaybeUninit<T::Real>>>,
/// Left eigenvectors
pub vc_l: Option<Vec<MaybeUninit<T::Complex>>>,
/// Left eigenvectors used in real routines
pub vr_l: Option<Vec<MaybeUninit<T::Real>>>,
/// Right eigenvectors
pub vc_r: Option<Vec<MaybeUninit<T::Complex>>>,
/// Right eigenvectors used in real routines
pub vr_r: Option<Vec<MaybeUninit<T::Real>>>,
/// Working memory
pub work: Vec<MaybeUninit<T>>,
/// Working memory with `T::Real`
pub rwork: Option<Vec<MaybeUninit<T::Real>>>,
}
impl<T> EigWork<T>
where
T: Scalar,
EigWork<T>: EigWorkImpl<Elem = T>,
{
/// Create new working memory for eigenvalues compution.
pub fn new(calc_v: bool, l: MatrixLayout) -> Result<Self> {
EigWorkImpl::new(calc_v, l)
}
/// Compute eigenvalues and vectors on this working memory.
pub fn calc(&mut self, a: &mut [T]) -> Result<EigRef<T>> {
EigWorkImpl::calc(self, a)
}
/// Compute eigenvalues and vectors by consuming this working memory.
pub fn eval(self, a: &mut [T]) -> Result<EigOwned<T>> {
EigWorkImpl::eval(self, a)
}
}
/// Owned result of eigenvalue problem by [EigWork::eval]
#[derive(Debug, Clone, PartialEq)]
pub struct EigOwned<T: Scalar> {
/// Eigenvalues
pub eigs: Vec<T::Complex>,
/// Right eigenvectors
pub vr: Option<Vec<T::Complex>>,
/// Left eigenvectors
pub vl: Option<Vec<T::Complex>>,
}
/// Reference result of eigenvalue problem by [EigWork::calc]
#[derive(Debug, Clone, PartialEq)]
pub struct EigRef<'work, T: Scalar> {
/// Eigenvalues
pub eigs: &'work [T::Complex],
/// Right eigenvectors
pub vr: Option<&'work [T::Complex]>,
/// Left eigenvectors
pub vl: Option<&'work [T::Complex]>,
}
/// Helper trait for implementing [EigWork] methods
pub trait EigWorkImpl: Sized {
type Elem: Scalar;
fn new(calc_v: bool, l: MatrixLayout) -> Result<Self>;
fn calc<'work>(&'work mut self, a: &mut [Self::Elem]) -> Result<EigRef<'work, Self::Elem>>;
fn eval(self, a: &mut [Self::Elem]) -> Result<EigOwned<Self::Elem>>;
}
macro_rules! impl_eig_work_c {
($c:ty, $ev:path) => {
impl EigWorkImpl for EigWork<$c> {
type Elem = $c;
fn new(calc_v: bool, l: MatrixLayout) -> Result<Self> {
let (n, _) = l.size();
let (jobvl, jobvr) = if calc_v {
match l {
MatrixLayout::C { .. } => (JobEv::All, JobEv::None),
MatrixLayout::F { .. } => (JobEv::None, JobEv::All),
}
} else {
(JobEv::None, JobEv::None)
};
let mut eigs = vec_uninit(n as usize);
let mut rwork = vec_uninit(2 * n as usize);
let mut vc_l = jobvl.then(|| vec_uninit((n * n) as usize));
let mut vc_r = jobvr.then(|| vec_uninit((n * n) as usize));
// calc work size
let mut info = 0;
let mut work_size = [<$c>::zero()];
unsafe {
$ev(
jobvl.as_ptr(),
jobvr.as_ptr(),
&n,
std::ptr::null_mut(),
&n,
AsPtr::as_mut_ptr(&mut eigs),
AsPtr::as_mut_ptr(vc_l.as_deref_mut().unwrap_or(&mut [])),
&n,
AsPtr::as_mut_ptr(vc_r.as_deref_mut().unwrap_or(&mut [])),
&n,
AsPtr::as_mut_ptr(&mut work_size),
&(-1),
AsPtr::as_mut_ptr(&mut rwork),
&mut info,
)
};
info.as_lapack_result()?;
let lwork = work_size[0].to_usize().unwrap();
let work: Vec<MaybeUninit<$c>> = vec_uninit(lwork);
Ok(Self {
n,
jobvl,
jobvr,
eigs,
eigs_re: None,
eigs_im: None,
rwork: Some(rwork),
vc_l,
vc_r,
vr_l: None,
vr_r: None,
work,
})
}
fn calc<'work>(
&'work mut self,
a: &mut [Self::Elem],
) -> Result<EigRef<'work, Self::Elem>> {
let lwork = self.work.len().to_i32().unwrap();
let mut info = 0;
unsafe {
$ev(
self.jobvl.as_ptr(),
self.jobvr.as_ptr(),
&self.n,
AsPtr::as_mut_ptr(a),
&self.n,
AsPtr::as_mut_ptr(&mut self.eigs),
AsPtr::as_mut_ptr(self.vc_l.as_deref_mut().unwrap_or(&mut [])),
&self.n,
AsPtr::as_mut_ptr(self.vc_r.as_deref_mut().unwrap_or(&mut [])),
&self.n,
AsPtr::as_mut_ptr(&mut self.work),
&lwork,
AsPtr::as_mut_ptr(self.rwork.as_mut().unwrap()),
&mut info,
)
};
info.as_lapack_result()?;
// Hermite conjugate
if let Some(vl) = self.vc_l.as_mut() {
for value in vl {
let value = unsafe { value.assume_init_mut() };
value.im = -value.im;
}
}
Ok(EigRef {
eigs: unsafe { self.eigs.slice_assume_init_ref() },
vl: self
.vc_l
.as_ref()
.map(|v| unsafe { v.slice_assume_init_ref() }),
vr: self
.vc_r
.as_ref()
.map(|v| unsafe { v.slice_assume_init_ref() }),
})
}
fn eval(mut self, a: &mut [Self::Elem]) -> Result<EigOwned<Self::Elem>> {
let _eig_ref = self.calc(a)?;
Ok(EigOwned {
eigs: unsafe { self.eigs.assume_init() },
vl: self.vc_l.map(|v| unsafe { v.assume_init() }),
vr: self.vc_r.map(|v| unsafe { v.assume_init() }),
})
}
}
};
}
impl_eig_work_c!(c32, lapack_sys::cgeev_);
impl_eig_work_c!(c64, lapack_sys::zgeev_);
macro_rules! impl_eig_work_r {
($f:ty, $ev:path) => {
impl EigWorkImpl for EigWork<$f> {
type Elem = $f;
fn new(calc_v: bool, l: MatrixLayout) -> Result<Self> {
let (n, _) = l.size();
let (jobvl, jobvr) = if calc_v {
match l {
MatrixLayout::C { .. } => (JobEv::All, JobEv::None),
MatrixLayout::F { .. } => (JobEv::None, JobEv::All),
}
} else {
(JobEv::None, JobEv::None)
};
let mut eigs_re = vec_uninit(n as usize);
let mut eigs_im = vec_uninit(n as usize);
let mut vr_l = jobvl.then(|| vec_uninit((n * n) as usize));
let mut vr_r = jobvr.then(|| vec_uninit((n * n) as usize));
let vc_l = jobvl.then(|| vec_uninit((n * n) as usize));
let vc_r = jobvr.then(|| vec_uninit((n * n) as usize));
// calc work size
let mut info = 0;
let mut work_size: [$f; 1] = [0.0];
unsafe {
$ev(
jobvl.as_ptr(),
jobvr.as_ptr(),
&n,
std::ptr::null_mut(),
&n,
AsPtr::as_mut_ptr(&mut eigs_re),
AsPtr::as_mut_ptr(&mut eigs_im),
AsPtr::as_mut_ptr(vr_l.as_deref_mut().unwrap_or(&mut [])),
&n,
AsPtr::as_mut_ptr(vr_r.as_deref_mut().unwrap_or(&mut [])),
&n,
AsPtr::as_mut_ptr(&mut work_size),
&(-1),
&mut info,
)
};
info.as_lapack_result()?;
// actual ev
let lwork = work_size[0].to_usize().unwrap();
let work = vec_uninit(lwork);
Ok(Self {
n,
jobvr,
jobvl,
eigs: vec_uninit(n as usize),
eigs_re: Some(eigs_re),
eigs_im: Some(eigs_im),
rwork: None,
vr_l,
vr_r,
vc_l,
vc_r,
work,
})
}
fn calc<'work>(
&'work mut self,
a: &mut [Self::Elem],
) -> Result<EigRef<'work, Self::Elem>> {
let lwork = self.work.len().to_i32().unwrap();
let mut info = 0;
unsafe {
$ev(
self.jobvl.as_ptr(),
self.jobvr.as_ptr(),
&self.n,
AsPtr::as_mut_ptr(a),
&self.n,
AsPtr::as_mut_ptr(self.eigs_re.as_mut().unwrap()),
AsPtr::as_mut_ptr(self.eigs_im.as_mut().unwrap()),
AsPtr::as_mut_ptr(self.vr_l.as_deref_mut().unwrap_or(&mut [])),
&self.n,
AsPtr::as_mut_ptr(self.vr_r.as_deref_mut().unwrap_or(&mut [])),
&self.n,
AsPtr::as_mut_ptr(&mut self.work),
&lwork,
&mut info,
)
};
info.as_lapack_result()?;
let eigs_re = self
.eigs_re
.as_ref()
.map(|e| unsafe { e.slice_assume_init_ref() })
.unwrap();
let eigs_im = self
.eigs_im
.as_ref()
.map(|e| unsafe { e.slice_assume_init_ref() })
.unwrap();
reconstruct_eigs(eigs_re, eigs_im, &mut self.eigs);
if let Some(v) = self.vr_l.as_ref() {
let v = unsafe { v.slice_assume_init_ref() };
reconstruct_eigenvectors(true, eigs_im, v, self.vc_l.as_mut().unwrap());
}
if let Some(v) = self.vr_r.as_ref() {
let v = unsafe { v.slice_assume_init_ref() };
reconstruct_eigenvectors(false, eigs_im, v, self.vc_r.as_mut().unwrap());
}
Ok(EigRef {
eigs: unsafe { self.eigs.slice_assume_init_ref() },
vl: self
.vc_l
.as_ref()
.map(|v| unsafe { v.slice_assume_init_ref() }),
vr: self
.vc_r
.as_ref()
.map(|v| unsafe { v.slice_assume_init_ref() }),
})
}
fn eval(mut self, a: &mut [Self::Elem]) -> Result<EigOwned<Self::Elem>> {
let _eig_ref = self.calc(a)?;
Ok(EigOwned {
eigs: unsafe { self.eigs.assume_init() },
vl: self.vc_l.map(|v| unsafe { v.assume_init() }),
vr: self.vc_r.map(|v| unsafe { v.assume_init() }),
})
}
}
};
}
impl_eig_work_r!(f32, lapack_sys::sgeev_);
impl_eig_work_r!(f64, lapack_sys::dgeev_);
/// 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.
fn reconstruct_eigenvectors<T: Scalar>(
take_hermite_conjugate: bool,
eig_im: &[T],
vr: &[T],
vc: &mut [MaybeUninit<T::Complex>],
) {
let n = eig_im.len();
assert_eq!(vr.len(), n * n);
assert_eq!(vc.len(), n * n);
let mut col = 0;
while col < n {
if eig_im[col].is_zero() {
// The corresponding eigenvalue is real.
for row in 0..n {
let re = vr[row + col * n];
vc[row + col * n].write(T::complex(re, T::zero()));
}
col += 1;
} else {
// This is a complex conjugate pair.
assert!(col + 1 < n);
for row in 0..n {
let re = vr[row + col * n];
let mut im = vr[row + (col + 1) * n];
if take_hermite_conjugate {
im = -im;
}
vc[row + col * n].write(T::complex(re, im));
vc[row + (col + 1) * n].write(T::complex(re, -im));
}
col += 2;
}
}
}
/// Create complex eigenvalues from real and imaginary parts.
fn reconstruct_eigs<T: Scalar>(re: &[T], im: &[T], eigs: &mut [MaybeUninit<T::Complex>]) {
let n = eigs.len();
assert_eq!(re.len(), n);
assert_eq!(im.len(), n);
for i in 0..n {
eigs[i].write(T::complex(re[i], im[i]));
}
}