gramm-schmidt orthogonalization

import math
import numpy as np
from typing import Any, Iterable


def matrix_to_geolin_answer(
    matrix: Iterable[Iterable[Any]], digits: int = 2, eps: float = 1e-5
) -> str:
    ans = "; ".join(
        [
            ", ".join(
                (str(round(e)) if (abs(round(e) - e) < eps) else str(round(e, digits)))
                for e in row
            )
            for row in matrix
        ]
    )
    return "[" + ans + "]"


def gram_multiply(x: np.ndarray, y: np.ndarray, gram: np.ndarray) -> int:
    return np.dot(np.dot(np.transpose(x), gram), y).astype(np.int64)


def ortogonalize(vectors: np.ndarray, gram: np.ndarray | None = None) -> np.ndarray:
    vectors = vectors.astype(np.int64)
    n = vectors.shape[1]
    if gram is None:
        gram = np.identity(n)
    gram = gram.astype(np.int64)
    ans: list[np.ndarray] = []
    for f in vectors:
        r = f
        m = 1
        for g in ans:
            nom = gram_multiply(f, g, gram)
            denom = gram_multiply(g, g, gram)
            m = math.lcm(m, denom // math.gcd(nom, denom))
            r = np.subtract(r, np.dot(nom / denom, g))
        ans.append(np.round(r * m).astype(np.int64))
    return np.array(ans)


vectors = np.array([[1, -1, -1, 2, 3], [-1, 2, 2, -6, -5], [1, -1, -2, 3, 5]])
gram = np.array(
    [
        [4, -2, 2, -2, 2],
        [-2, 12, -2, -2, -2],
        [2, -2, 4, -6, 2],
        [-2, -2, -6, 12, -2],
        [2, -2, 2, -2, 2],
    ]
)
ans = ortogonalize(vectors, gram)
print(ans)
print(matrix_to_geolin_answer(ans))