feat: extract linalg solver + add least-squares helper

Co-Authored-By: Claude Sonnet 4.6 <noreply@anthropic.com>

src/geometry/linalg.test.ts

@@ -0,0 +1,24 @@
+import { describe, it, expect } from 'vitest';
+import { solveLinear, leastSquares } from './linalg';
+
+describe('linalg', () => {
+  it('solveLinear solves a 2x2 system', () => {
+    // 2x + y = 5 ; x + 3y = 10  → x = 1, y = 3
+    const x = solveLinear([[2, 1], [1, 3]], [5, 10]);
+    expect(x[0]!).toBeCloseTo(1, 9);
+    expect(x[1]!).toBeCloseTo(3, 9);
+  });
+
+  it('solveLinear throws on a singular system', () => {
+    expect(() => solveLinear([[1, 2], [2, 4]], [3, 6])).toThrow();
+  });
+
+  it('leastSquares recovers exact coefficients of an over-determined linear fit', () => {
+    // model: t = 2*a + 3*b ; rows are [a, b]
+    const rows = [[1, 0], [0, 1], [1, 1], [2, 1]];
+    const targets = rows.map((r) => 2 * r[0]! + 3 * r[1]!);
+    const c = leastSquares(rows, targets);
+    expect(c[0]!).toBeCloseTo(2, 9);
+    expect(c[1]!).toBeCloseTo(3, 9);
+  });
+});

src/geometry/linalg.ts

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+/** Solve a square linear system A x = b by Gaussian elimination with partial pivoting. */
+export function solveLinear(A: number[][], b: number[]): number[] {
+  const n = b.length;
+  const M = A.map((row, i) => [...row, b[i]!]);
+  for (let col = 0; col < n; col++) {
+    let pivot = col;
+    for (let r = col + 1; r < n; r++) {
+      if (Math.abs(M[r]![col]!) > Math.abs(M[pivot]![col]!)) pivot = r;
+    }
+    if (Math.abs(M[pivot]![col]!) < 1e-12) throw new Error('Singular system');
+    [M[col], M[pivot]] = [M[pivot]!, M[col]!];
+    const pivRow = M[col]!;
+    for (let r = 0; r < n; r++) {
+      if (r === col) continue;
+      const factor = M[r]![col]! / pivRow[col]!;
+      for (let k = col; k <= n; k++) M[r]![k]! -= factor * pivRow[k]!;
+    }
+  }
+  return M.map((row, i) => row[n]! / row[i]!);
+}
+
+/**
+ * Least-squares solve for c minimizing ‖rows·c − targets‖ via the normal equations
+ * (rowsᵀ rows) c = rowsᵀ targets. `rows` is M×N (M ≥ N), `targets` is length M.
+ */
+export function leastSquares(rows: number[][], targets: number[]): number[] {
+  const n = rows[0]!.length;
+  const ata: number[][] = Array.from({ length: n }, () => new Array(n).fill(0));
+  const atb: number[] = new Array(n).fill(0);
+  for (let r = 0; r < rows.length; r++) {
+    const row = rows[r]!;
+    for (let i = 0; i < n; i++) {
+      atb[i]! += row[i]! * targets[r]!;
+      for (let j = 0; j < n; j++) ata[i]![j]! += row[i]! * row[j]!;
+    }
+  }
+  return solveLinear(ata, atb);
+}