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//! Implementation of ?trsyl - solve a triangular Sylvester equation.
use lapack::{ctrsyl, dtrsyl, strsyl, ztrsyl};
use crate::base_types::{LapackError, LapackResult, c32, c64};
use crate::dense::linalg::lapack::interface::{lapack_int, matrix_len};
use crate::traits::rlst_num::RlstScalar;
/// Operation applied to a Schur-form coefficient matrix.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum TrsylTranspose {
/// Use the matrix unchanged.
NoTranspose,
/// Use its transpose for real values and conjugate transpose for complex values.
ConjugateTranspose,
}
/// Sign of the second product in the Sylvester equation.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum TrsylSign {
/// Add `X op(B)`.
Plus,
/// Subtract `X op(B)`.
Minus,
}
/// Whether LAPACK solved the requested equation or a nearby perturbed equation.
#[derive(Clone, Copy, Debug, PartialEq, Eq)]
pub enum TrsylStatus {
/// No perturbation was reported.
Success,
/// Close or common eigenvalues were perturbed.
Perturbed,
}
/// Result metadata from TRSYL.
#[derive(Clone, Copy, Debug, PartialEq)]
pub struct TrsylOutcome<Real> {
/// Scale in `op(A) X + sign X op(B) = scale C`.
pub scale: Real,
/// LAPACK perturbation status.
pub status: TrsylStatus,
}
trait TrsylChar {
fn trans_char(mode: TrsylTranspose) -> u8;
}
macro_rules! impl_char {
($t:ty,$c:expr) => {
impl TrsylChar for $t {
fn trans_char(mode: TrsylTranspose) -> u8 {
match mode {
TrsylTranspose::NoTranspose => b'N',
TrsylTranspose::ConjugateTranspose => $c,
}
}
}
};
}
impl_char!(f32, b'T');
impl_char!(f64, b'T');
impl_char!(c32, b'C');
impl_char!(c64, b'C');
/// Solve `op(A) X +/- X op(B) = scale C`, overwriting C with X.
///
/// `a` and `b` must be in real Schur form for real values and triangular for complex values.
pub trait Trsyl: RlstScalar {
/// Solve the triangular Sylvester equation and report scaling and perturbation.
///
/// A is `m x m`, B is `n x n`, and C is `m x n`, all column-major. A and B
/// are unchanged; C is overwritten by X. For real scalars A and B may have
/// 2-by-2 diagonal Schur blocks. Invalid storage dimensions panic. LAPACK
/// status 1 is returned as [`TrsylStatus::Perturbed`], not as an error.
#[allow(clippy::too_many_arguments)]
fn trsyl(
trana: TrsylTranspose,
tranb: TrsylTranspose,
sign: TrsylSign,
m: usize,
n: usize,
a: &[Self],
lda: usize,
b: &[Self],
ldb: usize,
c: &mut [Self],
ldc: usize,
) -> LapackResult<TrsylOutcome<Self::Real>>;
}
macro_rules! implement_trsyl {
($scalar:ty,$trsyl:expr) => {
impl Trsyl for $scalar {
fn trsyl(
trana: TrsylTranspose,
tranb: TrsylTranspose,
sign: TrsylSign,
m: usize,
n: usize,
a: &[Self],
lda: usize,
b: &[Self],
ldb: usize,
c: &mut [Self],
ldc: usize,
) -> LapackResult<TrsylOutcome<Self::Real>> {
assert!(lda >= m.max(1));
assert_eq!(a.len(), matrix_len(lda, m, "a"));
assert!(ldb >= n.max(1));
assert_eq!(b.len(), matrix_len(ldb, n, "b"));
assert!(ldc >= m.max(1));
assert_eq!(c.len(), matrix_len(ldc, n, "c"));
if m == 0 || n == 0 {
return Ok(TrsylOutcome {
scale: 1.0,
status: TrsylStatus::Success,
});
}
let isgn = [match sign {
TrsylSign::Plus => 1,
TrsylSign::Minus => -1,
}];
let mut scale = [0.0];
let mut info = 0;
unsafe {
$trsyl(
<$scalar as TrsylChar>::trans_char(trana),
<$scalar as TrsylChar>::trans_char(tranb),
&isgn,
lapack_int(m, "m"),
lapack_int(n, "n"),
a,
lapack_int(lda, "lda"),
b,
lapack_int(ldb, "ldb"),
c,
lapack_int(ldc, "ldc"),
&mut scale,
&mut info,
)
};
match info {
0 => Ok(TrsylOutcome {
scale: scale[0],
status: TrsylStatus::Success,
}),
1 => Ok(TrsylOutcome {
scale: scale[0],
status: TrsylStatus::Perturbed,
}),
code => Err(LapackError::LapackInfoCode(code)),
}
}
}
};
}
implement_trsyl!(f32, strsyl);
implement_trsyl!(f64, dtrsyl);
implement_trsyl!(c32, ctrsyl);
implement_trsyl!(c64, ztrsyl);
#[cfg(test)]
mod tests {
use super::*;
use crate::RlstScalar;
use crate::base_types::{c32, c64};
use paste::paste;
#[test]
fn solves_with_padding_and_reports_perturbation() {
let padding = 12345.0;
let mut c = [3.0_f64, padding];
let out = f64::trsyl(
TrsylTranspose::NoTranspose,
TrsylTranspose::NoTranspose,
TrsylSign::Plus,
1,
1,
&[1.0, padding],
2,
&[2.0, padding, padding],
3,
&mut c,
2,
)
.unwrap();
assert_eq!(out.status, TrsylStatus::Success);
assert!((c[0] - 1.0).abs() < 1e-14);
assert_eq!(c[1], padding);
let mut c = [1.0];
let out = f64::trsyl(
TrsylTranspose::NoTranspose,
TrsylTranspose::NoTranspose,
TrsylSign::Minus,
1,
1,
&[1.0],
1,
&[1.0],
1,
&mut c,
1,
)
.unwrap();
assert_eq!(out.status, TrsylStatus::Perturbed);
}
#[test]
#[should_panic]
fn trsyl_rejects_invalid_coefficient_storage() {
f64::trsyl(
TrsylTranspose::NoTranspose,
TrsylTranspose::NoTranspose,
TrsylSign::Plus,
2,
1,
&[1.0; 3],
2,
&[1.0],
1,
&mut [1.0; 2],
2,
)
.unwrap();
}
macro_rules! operation_tests {
($scalar:ty, $tol:expr, $a:expr, $b:expr, $x:expr) => {
paste! {
#[test]
fn [<trsyl_all_operations_and_signs_ $scalar>]() {
let a: [$scalar; 4] = $a;
let b: [$scalar; 4] = $b;
let expected: [$scalar; 4] = $x;
for trana in [
TrsylTranspose::NoTranspose,
TrsylTranspose::ConjugateTranspose,
] {
for tranb in [
TrsylTranspose::NoTranspose,
TrsylTranspose::ConjugateTranspose,
] {
for sign in [TrsylSign::Plus, TrsylSign::Minus] {
let op_entry = |
matrix: &[$scalar; 4],
row: usize,
col: usize,
trans: TrsylTranspose,
| match trans {
TrsylTranspose::NoTranspose => matrix[row + 2 * col],
TrsylTranspose::ConjugateTranspose => {
<$scalar as RlstScalar>::conj(&matrix[col + 2 * row])
}
};
let mut c = [<$scalar as num::Zero>::zero(); 4];
for col in 0..2 {
for row in 0..2 {
for inner in 0..2 {
c[row + 2 * col] += op_entry(&a, row, inner, trana)
* expected[inner + 2 * col];
let xb = expected[row + 2 * inner]
* op_entry(&b, inner, col, tranb);
match sign {
TrsylSign::Plus => c[row + 2 * col] += xb,
TrsylSign::Minus => c[row + 2 * col] -= xb,
}
}
}
}
let outcome = <$scalar>::trsyl(
trana, tranb, sign, 2, 2, &a, 2, &b, 2, &mut c, 2,
)
.unwrap();
assert_eq!(outcome.status, TrsylStatus::Success);
assert_eq!(outcome.scale, 1.0);
for index in 0..4 {
assert!((c[index] - expected[index]).abs() < $tol);
}
}
}
}
}
}
};
}
operation_tests!(
f32,
2e-5,
[4.0, 0.0, -0.75, 6.0],
[0.5, 0.0, 0.25, 1.5],
[1.0, -2.0, 0.5, 3.0]
);
operation_tests!(
f64,
1e-13,
[4.0, 0.0, -0.75, 6.0],
[0.5, 0.0, 0.25, 1.5],
[1.0, -2.0, 0.5, 3.0]
);
operation_tests!(
c32,
3e-5,
[
c32::new(4.0, 0.5),
c32::new(0.0, 0.0),
c32::new(-0.75, 0.25),
c32::new(6.0, -0.5),
],
[
c32::new(0.5, 0.25),
c32::new(0.0, 0.0),
c32::new(0.25, -0.5),
c32::new(1.5, 0.75),
],
[
c32::new(1.0, 0.5),
c32::new(-2.0, 0.25),
c32::new(0.5, -1.0),
c32::new(3.0, 0.75),
]
);
operation_tests!(
c64,
2e-13,
[
c64::new(4.0, 0.5),
c64::new(0.0, 0.0),
c64::new(-0.75, 0.25),
c64::new(6.0, -0.5),
],
[
c64::new(0.5, 0.25),
c64::new(0.0, 0.0),
c64::new(0.25, -0.5),
c64::new(1.5, 0.75),
],
[
c64::new(1.0, 0.5),
c64::new(-2.0, 0.25),
c64::new(0.5, -1.0),
c64::new(3.0, 0.75),
]
);
}