Introduction

The fastest least-squares algorithm in C depends on what you are fitting. If you only need a straight-line fit, a small closed-form implementation is simple and fast. If you need general matrix least squares, the practical answer is usually not to hand-code the solver, but to call a tested linear algebra library that uses QR decomposition instead of solving the normal equations directly.

Straight-Line Fit in One Pass

For fitting y = a + b*x, you do not need a full matrix package. You can compute the necessary sums and derive the slope and intercept directly.

1#include <stdio.h>
2#include <stddef.h>
3
4typedef struct {
5    double intercept;
6    double slope;
7} LineFit;
8
9LineFit fit_line(const double *x, const double *y, size_t n) {
10    double sum_x = 0.0, sum_y = 0.0, sum_xy = 0.0, sum_xx = 0.0;
11
12    for (size_t i = 0; i < n; i++) {
13        sum_x += x[i];
14        sum_y += y[i];
15        sum_xy += x[i] * y[i];
16        sum_xx += x[i] * x[i];
17    }
18
19    double denom = n * sum_xx - sum_x * sum_x;
20    LineFit fit;
21    fit.slope = (n * sum_xy - sum_x * sum_y) / denom;
22    fit.intercept = (sum_y - fit.slope * sum_x) / n;
23    return fit;
24}
25
26int main(void) {
27    double x[] = {1, 2, 3, 4, 5};
28    double y[] = {2, 4, 5, 4, 5};
29
30    LineFit fit = fit_line(x, y, 5);
31    printf("intercept=%.4f slope=%.4f\n", fit.intercept, fit.slope);
32    return 0;
33}

For this specific problem, that is efficient, readable, and usually the right answer.

Why General Least Squares Is Different

Once the model is more general than a line, the data becomes a matrix problem. A common textbook derivation solves:

X^T X beta = X^T y

Those are the normal equations. They are tempting because the code seems straightforward, but they amplify conditioning problems and can lose numerical accuracy.

That is why production-grade least-squares solvers usually prefer QR decomposition or SVD for harder cases.

What “Fast” Should Mean

There are two different goals:

• low constant-factor runtime for well-behaved small problems
• numerically stable results for real input data

A hand-written normal-equation solver may look fast in a toy benchmark and still be the wrong implementation because it becomes unstable when columns are correlated or data scales poorly.

For general least squares, “fast and efficient” usually means:

• use BLAS and LAPACK or an equivalent library
• store matrices in a cache-friendly layout
• avoid unnecessary allocations
• pick QR as the default solver unless you have a specific reason otherwise

When Libraries Are the Right Tool

If you are solving min ||Ax - b|| for arbitrary dense matrices, call a library routine instead of building your own elimination code. Optimized libraries are faster because they exploit cache layout, vector instructions, and decades of tuning. They are also more reliable.

A useful engineering rule is this:

• line fit or tiny polynomial fit: self-contained formulas are fine
• general least squares: use a numerical library

That division saves a lot of time.

Performance Tips in Plain C

If you do implement a specialized solver yourself, a few habits matter:

• accumulate in double, not float, unless memory pressure is extreme
• make a single pass over the data when possible
• keep data contiguous in memory
• check for degenerate input such as identical x values in line fitting

For streaming data, one-pass accumulation can be very effective. For batch matrix problems, algorithm choice dominates small loop tweaks.

Common Pitfalls

The biggest pitfall is using normal equations for an ill-conditioned problem and then blaming C when the coefficients look wrong. The issue is numerical method choice, not the language.

Another mistake is optimizing loops before choosing the right solver. A slow stable algorithm is often better than a fast unstable one, and an optimized library usually beats both.

A third issue is failing to guard against a zero denominator in simple line fitting, which happens when all x values are identical.

Summary

• For fitting a line, closed-form accumulation is fast and usually sufficient.
• For general least squares, prefer QR-based library solvers over hand-written normal equations.
• “Fast” must include numerical stability, not just raw loop speed.
• In specialized code, one-pass accumulation and contiguous memory help.
• Choose the algorithm first, then optimize the implementation.