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//! Linear solves: A x = b via LU with partial pivoting on a dense copy.
use crate::{
linalg::LinalgError,
traits::{Real, State},
};
use super::base::Matrix;
impl<T: Real> Matrix<T> {
/// Solve A x = b, Returns Err if the matrix is singular or dimensions are incompatible
pub fn lin_solve<Y>(&self, b: Y) -> Result<Y, LinalgError>
where
Y: State<T>,
{
let n = self.n;
if self.m != n {
return Err(LinalgError::BadInput {
message: "Matrix solve requires a square matrix".into(),
});
}
if b.len() != n {
return Err(LinalgError::BadInput {
message: "Incompatible vector length".into(),
});
}
// 1) Densify A into a Vec<T> of size n*n (row-major)
let mut a = self.to_dense_vec();
// 2) Use b as the working state
let mut x = b.clone();
// 3) LU factorization with partial pivoting and singularity checking
let mut piv: Vec<usize> = (0..n).collect();
let eps = T::from_f64(1e-14).unwrap(); // Singularity threshold
for k in 0..n {
// Find pivot row
let mut pivot_row = k;
let mut pivot_val = a[k * n + k].abs();
for i in (k + 1)..n {
let val = a[i * n + k].abs();
if val > pivot_val {
pivot_val = val;
pivot_row = i;
}
}
// Check for singularity
if pivot_val <= eps {
// Note the t, y are not known here and should be updated by caller before returning to user
return Err(LinalgError::Singular { step: k + 1 });
}
if pivot_row != k {
// swap rows in A
for j in 0..n {
a.swap(k * n + j, pivot_row * n + j);
}
// swap entries in x
let tk = x.get_component(k);
let tm = x.get_component(pivot_row);
x.set_component(k, tm);
x.set_component(pivot_row, tk);
piv.swap(k, pivot_row);
}
// Eliminate below the pivot
let akk = a[k * n + k];
for i in (k + 1)..n {
let factor = a[i * n + k] / akk;
a[i * n + k] = factor; // store L(i,k)
for j in (k + 1)..n {
a[i * n + j] = a[i * n + j] - factor * a[k * n + j];
}
}
}
// Forward solve Ly = Pb (x currently holds permuted b)
for i in 0..n {
let mut sum = x.get_component(i);
for k in 0..i {
sum -= a[i * n + k] * x.get_component(k);
}
x.set_component(i, sum); // since L has ones on diagonal
}
// Backward solve Ux = y
for i in (0..n).rev() {
let mut sum = x.get_component(i);
for k in (i + 1)..n {
sum -= a[i * n + k] * x.get_component(k);
}
x.set_component(i, sum / a[i * n + i]);
}
Ok(x)
}
/// In-place solve: overwrites `b` with `x`.
pub fn lin_solve_mut(&self, b: &mut [T]) -> Result<(), LinalgError> {
let n = self.n;
assert_eq!(
self.m, n,
"dimension mismatch in solve: A must be square, got {}x{}",
n, self.m
);
assert_eq!(
b.len(),
n,
"dimension mismatch in solve: A is {}x{}, b has length {}",
n,
n,
b.len()
);
// Densify A into row-major Vec<T>
let mut a = self.to_dense_vec();
// LU with partial pivoting, applying permutations to b
for k in 0..n {
// pivot
let mut pivot_row = k;
let mut pivot_val = a[k * n + k].abs();
for i in (k + 1)..n {
let val = a[i * n + k].abs();
if val > pivot_val {
pivot_val = val;
pivot_row = i;
}
}
if pivot_val == T::zero() {
return Err(LinalgError::Singular { step: k + 1 });
}
if pivot_row != k {
for j in 0..n {
a.swap(k * n + j, pivot_row * n + j);
}
b.swap(k, pivot_row);
}
// Eliminate below the pivot
let akk = a[k * n + k];
for i in (k + 1)..n {
let factor = a[i * n + k] / akk;
a[i * n + k] = factor;
for j in (k + 1)..n {
a[i * n + j] = a[i * n + j] - factor * a[k * n + j];
}
}
}
// Forward solve Ly = Pb (b is permuted)
for i in 0..n {
let mut sum = b[i];
for k in 0..i {
sum -= a[i * n + k] * b[k];
}
b[i] = sum;
}
// Backward solve Ux = y
for i in (0..n).rev() {
let mut sum = b[i];
for k in (i + 1)..n {
sum -= a[i * n + k] * b[k];
}
b[i] = sum / a[i * n + i];
}
Ok(())
}
}
#[cfg(all(test, feature = "nalgebra"))]
mod tests {
use crate::linalg::matrix::Matrix;
use nalgebra::Vector2;
#[test]
fn solve_full_2x2() {
// A = [[3, 2],[1, 4]], b = [5, 6] -> x = [0.8, 1.3]
let mut a: Matrix<f64> = Matrix::full(2, 2);
a[(0, 0)] = 3.0;
a[(0, 1)] = 2.0;
a[(1, 0)] = 1.0;
a[(1, 1)] = 4.0;
let b = Vector2::new(5.0, 6.0);
let x = a.lin_solve(b).unwrap();
// Solve manually: [[3,2],[1,4]] x = [5,6] => x = [ (20-12)/10, (15-5)/10 ] = [0.8, 1.3]
assert!((x.x - 0.8).abs() < 1e-12);
assert!((x.y - 1.3).abs() < 1e-12);
}
#[test]
fn solve_sparse_2x2() {
let a: Matrix<f64> = Matrix::sparse_from_triplets(
2,
2,
vec![(0, 0, 3.0), (0, 1, 2.0), (1, 0, 1.0), (1, 1, 4.0)],
);
let b = Vector2::new(5.0, 6.0);
let x = a.lin_solve(b).unwrap();
assert!((x.x - 0.8).abs() < 1e-12);
assert!((x.y - 1.3).abs() < 1e-12);
}
#[test]
fn solve_sparse_mut_2x2() {
let a: Matrix<f64> = Matrix::sparse_from_triplets(
2,
2,
vec![(0, 0, 3.0), (0, 1, 2.0), (1, 0, 1.0), (1, 1, 4.0)],
);
let mut b = [5.0_f64, 6.0];
a.lin_solve_mut(&mut b).unwrap();
assert!((b[0] - 0.8).abs() < 1e-12);
assert!((b[1] - 1.3).abs() < 1e-12);
}
}