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use super::{DMat, DVec};
use alloc::vec::Vec;
use crate::Scalar;
/// QR decomposition via Householder reflections: A = Q * R
pub struct Qr<S> {
/// Householder vectors stored below diagonal + R above.
qr: DMat<S>,
/// Diagonal of R.
r_diag: Vec<S>,
}
impl<S: Scalar> Qr<S> {
/// Compute QR decomposition.
pub fn new(a: &DMat<S>) -> Self {
let m = a.nrows();
let n = a.ncols();
let mut qr = a.clone();
let mut r_diag = Vec::with_capacity(n);
for k in 0..n.min(m) {
// Compute norm of column k below diagonal
let mut norm_sq = S::ZERO;
for i in k..m {
norm_sq += qr.get(i, k) * qr.get(i, k);
}
let mut norm = norm_sq.sqrt();
if norm > S::EPSILON {
// Ensure correct sign
if qr.get(k, k) > S::ZERO {
norm = -norm;
}
// Divide column by norm, adjust diagonal
for i in k..m {
let v = qr.get(i, k) / (-norm);
qr.set(i, k, v);
}
let v = qr.get(k, k) + S::ONE;
qr.set(k, k, v);
// Apply transformation to remaining columns
for j in (k + 1)..n {
let mut s = S::ZERO;
for i in k..m {
s += qr.get(i, k) * qr.get(i, j);
}
s = -s / qr.get(k, k);
for i in k..m {
let v = qr.get(i, j) + s * qr.get(i, k);
qr.set(i, j, v);
}
}
}
r_diag.push(norm);
}
Self { qr, r_diag }
}
/// Get R (upper triangular).
pub fn r(&self) -> DMat<S> {
let m = self.qr.nrows();
let n = self.qr.ncols();
let k = m.min(n);
DMat::from_fn(k, n, |i, j| {
if i == j {
self.r_diag[i]
} else if j > i {
self.qr.get(i, j)
} else {
S::ZERO
}
})
}
/// Get Q (orthogonal m×m or thin m×min(m,n)).
pub fn q(&self) -> DMat<S> {
let m = self.qr.nrows();
let n = self.qr.ncols();
let k = m.min(n);
let mut q = DMat::identity(m);
for j in (0..k).rev() {
if self.qr.get(j, j) == S::ZERO {
continue;
}
for col in j..m {
let mut s = S::ZERO;
for i in j..m {
s += self.qr.get(i, j) * q.get(i, col);
}
s = -s / self.qr.get(j, j);
for i in j..m {
let v = q.get(i, col) + s * self.qr.get(i, j);
q.set(i, col, v);
}
}
}
// Return thin Q (m × k)
DMat::from_fn(m, k, |i, j| q.get(i, j))
}
/// Solve least-squares: min ||Ax - b||.
pub fn solve(&self, b: &DVec<S>) -> DVec<S> {
let m = self.qr.nrows();
let n = self.qr.ncols();
assert_eq!(b.len(), m);
// Apply Q^T to b
let mut x = DVec::from_slice(b.as_slice());
for k in 0..n.min(m) {
if self.qr.get(k, k) == S::ZERO {
continue;
}
let mut s = S::ZERO;
for i in k..m {
s += self.qr.get(i, k) * x[i];
}
s = -s / self.qr.get(k, k);
for i in k..m {
x[i] = x[i] + s * self.qr.get(i, k);
}
}
// Back-substitute R
let mut result = DVec::zeros(n);
for i in (0..n).rev() {
let mut sum = x[i];
for j in (i + 1)..n {
sum = sum - self.qr.get(i, j) * result[j];
}
if self.r_diag[i].abs() < S::EPSILON {
result[i] = S::ZERO;
} else {
result[i] = sum / self.r_diag[i];
}
}
result
}
}
#[cfg(test)]
mod tests {
use super::*;
#[test]
fn qr_identity() {
let a = DMat::<f64>::identity(3);
let qr = Qr::new(&a);
let q = qr.q();
let r = qr.r();
// Q should be identity (up to sign)
for i in 0..3 {
assert!(q.get(i, i).abs() > 0.99, "Q diagonal should be ±1");
}
// R should be identity (up to sign)
for i in 0..3 {
assert!(r.get(i, i).abs() > 0.99, "R diagonal should be ±1");
}
}
#[test]
fn solve_overdetermined() {
// A = [1; 1], b = [1; 3] -> least squares: x = 2
let a = DMat::from_fn(2, 1, |_, _| 1.0_f64);
let b = DVec::from_slice(&[1.0, 3.0]);
let qr = Qr::new(&a);
let x = qr.solve(&b);
assert!((x[0] - 2.0).abs() < 1e-10);
}
#[test]
fn qr_reconstruct() {
let a = DMat::from_fn(3, 2, |i, j| (i * 2 + j + 1) as f64);
let qr = Qr::new(&a);
let q = qr.q();
let r = qr.r();
let recon = q.mul_mat(&r);
for i in 0..3 {
for j in 0..2 {
assert!(
(recon.get(i, j) - a.get(i, j)).abs() < 1e-10,
"mismatch at ({}, {}): {} vs {}",
i,
j,
recon.get(i, j),
a.get(i, j)
);
}
}
}
}