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crates/ruvector-solver/benches/solver_neumann.rs Normal file
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//! Benchmarks for the Neumann series solver.
//!
//! The Neumann series approximates `(I - M)^{-1} b = sum_{k=0}^{K} M^k b`
//! and converges when the spectral radius of `M` is less than 1. These
//! benchmarks measure convergence rate vs tolerance, scaling behaviour, and
//! crossover against dense direct solves.
use criterion::{criterion_group, criterion_main, BenchmarkId, Criterion, Throughput};
use std::time::Duration;
use rand::rngs::StdRng;
use rand::{Rng, SeedableRng};
use ruvector_solver::types::CsrMatrix;
// ---------------------------------------------------------------------------
// Helpers
// ---------------------------------------------------------------------------
/// Build a diagonally dominant CSR matrix suitable for Neumann iteration.
///
/// The iteration matrix `M = I - D^{-1} A` has spectral radius < 1 when `A`
/// is strictly diagonally dominant. We construct `A` so that each diagonal
/// entry equals the sum of absolute off-diagonal values in its row plus 1.0.
fn diag_dominant_csr(n: usize, density: f64, seed: u64) -> CsrMatrix<f32> {
let mut rng = StdRng::seed_from_u64(seed);
let mut entries: Vec<(usize, usize, f32)> = Vec::new();
for i in 0..n {
for j in (i + 1)..n {
if rng.gen::<f64>() < density {
let val: f32 = rng.gen_range(-0.3..0.3);
entries.push((i, j, val));
entries.push((j, i, val));
}
}
}
let mut row_abs_sums = vec![0.0f32; n];
for &(r, _c, v) in &entries {
row_abs_sums[r] += v.abs();
}
for i in 0..n {
entries.push((i, i, row_abs_sums[i] + 1.0));
}
CsrMatrix::<f32>::from_coo(n, n, entries)
}
/// Random vector with deterministic seed.
fn random_vector(n: usize, seed: u64) -> Vec<f32> {
let mut rng = StdRng::seed_from_u64(seed);
(0..n).map(|_| rng.gen_range(-1.0..1.0)).collect()
}
// ---------------------------------------------------------------------------
// Inline Neumann series solver for benchmarking
// ---------------------------------------------------------------------------
/// Neumann series iteration: x_{k+1} = x_k + (b - A * x_k).
///
/// This is equivalent to the Richardson iteration with omega = 1 for a
/// diagonally-dominant system. We inline it here so the benchmark does
/// not depend on the (currently stub) neumann module.
#[inline(never)]
fn neumann_solve(
matrix: &CsrMatrix<f32>,
rhs: &[f32],
tolerance: f64,
max_iter: usize,
) -> (Vec<f32>, usize, f64) {
let n = matrix.rows;
let mut x = vec![0.0f32; n];
let mut residual_buf = vec![0.0f32; n];
let mut iterations = 0;
let mut residual_norm = f64::MAX;
for k in 0..max_iter {
// Compute residual: r = b - A*x.
matrix.spmv(&x, &mut residual_buf);
for i in 0..n {
residual_buf[i] = rhs[i] - residual_buf[i];
}
// Residual L2 norm.
residual_norm = residual_buf
.iter()
.map(|&v| (v as f64) * (v as f64))
.sum::<f64>()
.sqrt();
iterations = k + 1;
if residual_norm < tolerance {
break;
}
// Update: x = x + r (Richardson step).
for i in 0..n {
x[i] += residual_buf[i];
}
}
(x, iterations, residual_norm)
}
// ---------------------------------------------------------------------------
// Benchmark: convergence vs tolerance
// ---------------------------------------------------------------------------
fn neumann_convergence(c: &mut Criterion) {
let mut group = c.benchmark_group("neumann_convergence");
group.warm_up_time(Duration::from_secs(3));
group.sample_size(100);
let n = 500;
let matrix = diag_dominant_csr(n, 0.02, 42);
let rhs = random_vector(n, 43);
for &tol in &[1e-2, 1e-4, 1e-6] {
let label = format!("eps_{:.0e}", tol);
group.bench_with_input(BenchmarkId::new(&label, n), &tol, |b, &eps| {
b.iter(|| {
neumann_solve(
criterion::black_box(&matrix),
criterion::black_box(&rhs),
eps,
5000,
)
});
});
}
group.finish();
}
// ---------------------------------------------------------------------------
// Benchmark: scaling with problem size
// ---------------------------------------------------------------------------
fn neumann_scaling(c: &mut Criterion) {
let mut group = c.benchmark_group("neumann_scaling");
group.warm_up_time(Duration::from_secs(3));
for &n in &[100, 1000, 10_000] {
// Use sparser matrices for larger sizes to keep runtime reasonable.
let density = if n <= 1000 { 0.02 } else { 0.005 };
let matrix = diag_dominant_csr(n, density, 42);
let rhs = random_vector(n, 43);
let sample_count = if n >= 10_000 { 20 } else { 100 };
group.sample_size(sample_count);
group.throughput(Throughput::Elements(matrix.nnz() as u64));
group.bench_with_input(BenchmarkId::new("n", n), &n, |b, _| {
b.iter(|| {
neumann_solve(
criterion::black_box(&matrix),
criterion::black_box(&rhs),
1e-4,
5000,
)
});
});
}
group.finish();
}
// ---------------------------------------------------------------------------
// Benchmark: Neumann vs dense direct solve crossover
// ---------------------------------------------------------------------------
/// Naive dense direct solve via Gaussian elimination with partial pivoting.
///
/// This is intentionally unoptimized to represent a "no-library" baseline.
#[inline(never)]
fn dense_direct_solve(a: &[f32], b: &[f32], n: usize) -> Vec<f32> {
// Build augmented matrix [A | b] in row-major order.
let mut aug = vec![0.0f64; n * (n + 1)];
for i in 0..n {
for j in 0..n {
aug[i * (n + 1) + j] = a[i * n + j] as f64;
}
aug[i * (n + 1) + n] = b[i] as f64;
}
// Forward elimination with partial pivoting.
for col in 0..n {
// Find pivot.
let mut max_row = col;
let mut max_val = aug[col * (n + 1) + col].abs();
for row in (col + 1)..n {
let val = aug[row * (n + 1) + col].abs();
if val > max_val {
max_val = val;
max_row = row;
}
}
// Swap rows.
if max_row != col {
for j in 0..=n {
let idx_a = col * (n + 1) + j;
let idx_b = max_row * (n + 1) + j;
aug.swap(idx_a, idx_b);
}
}
let pivot = aug[col * (n + 1) + col];
if pivot.abs() < 1e-15 {
continue;
}
// Eliminate below.
for row in (col + 1)..n {
let factor = aug[row * (n + 1) + col] / pivot;
for j in col..=n {
let val = aug[col * (n + 1) + j];
aug[row * (n + 1) + j] -= factor * val;
}
}
}
// Back substitution.
let mut x = vec![0.0f64; n];
for i in (0..n).rev() {
let mut sum = aug[i * (n + 1) + n];
for j in (i + 1)..n {
sum -= aug[i * (n + 1) + j] * x[j];
}
let diag = aug[i * (n + 1) + i];
x[i] = if diag.abs() > 1e-15 { sum / diag } else { 0.0 };
}
x.iter().map(|&v| v as f32).collect()
}
/// Generate the dense representation of a diag-dominant matrix.
fn diag_dominant_dense(n: usize, density: f64, seed: u64) -> Vec<f32> {
let mut rng = StdRng::seed_from_u64(seed);
let mut a = vec![0.0f32; n * n];
// Off-diagonal.
for i in 0..n {
for j in (i + 1)..n {
if rng.gen::<f64>() < density {
let val: f32 = rng.gen_range(-0.3..0.3);
a[i * n + j] = val;
a[j * n + i] = val;
}
}
}
// Diagonal dominance.
for i in 0..n {
let mut row_sum = 0.0f32;
for j in 0..n {
if j != i {
row_sum += a[i * n + j].abs();
}
}
a[i * n + i] = row_sum + 1.0;
}
a
}
fn neumann_vs_dense(c: &mut Criterion) {
let mut group = c.benchmark_group("neumann_vs_dense");
group.warm_up_time(Duration::from_secs(3));
// Crossover analysis: compare iterative Neumann vs dense direct solve.
// For small n, dense wins; for large sparse n, Neumann should win.
for &n in &[50, 100, 200, 500] {
let density = 0.05;
let rhs = random_vector(n, 43);
let sample_count = if n >= 500 { 20 } else { 100 };
group.sample_size(sample_count);
// Neumann (sparse).
let csr = diag_dominant_csr(n, density, 42);
group.bench_with_input(BenchmarkId::new("neumann_sparse", n), &n, |b, _| {
b.iter(|| {
neumann_solve(
criterion::black_box(&csr),
criterion::black_box(&rhs),
1e-4,
5000,
)
});
});
// Dense direct solve.
let a_dense = diag_dominant_dense(n, density, 42);
group.bench_with_input(BenchmarkId::new("dense_direct", n), &n, |b, _| {
b.iter(|| {
dense_direct_solve(
criterion::black_box(&a_dense),
criterion::black_box(&rhs),
n,
)
});
});
}
group.finish();
}
criterion_group!(
neumann,
neumann_convergence,
neumann_scaling,
neumann_vs_dense
);
criterion_main!(neumann);