Task: Gaussian elimination
C++: #include <algorithm>
#include <cassert>
#include <cmath>
#include <iomanip>
#include <iostream>
#include <vector>

template <typename scalar_type> class matrix {
public:
    matrix(size_t rows, size_t columns)
        : rows_(rows), columns_(columns), elements_(rows * columns) {}

    matrix(size_t rows, size_t columns,
        const std::initializer_list<std::initializer_list<scalar_type>>& values)
        : rows_(rows), columns_(columns), elements_(rows * columns) {
        assert(values.size() <= rows_);
        auto i = elements_.begin();
        for (const auto& row : values) {
            assert(row.size() <= columns_);
            std::copy(begin(row), end(row), i);
            i += columns_;
        }
    }

    size_t rows() const { return rows_; }
    size_t columns() const { return columns_; }

    const scalar_type& operator()(size_t row, size_t column) const {
        assert(row < rows_);
        assert(column < columns_);
        return elements_[row * columns_ + column];
    }
    scalar_type& operator()(size_t row, size_t column) {
        assert(row < rows_);
        assert(column < columns_);
        return elements_[row * columns_ + column];
    }
private:
    size_t rows_;
    size_t columns_;
    std::vector<scalar_type> elements_;
};

template <typename scalar_type>
void swap_rows(matrix<scalar_type>& m, size_t i, size_t j) {
    size_t columns = m.columns();
    for (size_t k = 0; k < columns; ++k)
        std::swap(m(i, k), m(j, k));
}

template <typename scalar_type>
std::vector<scalar_type> gauss_partial(const matrix<scalar_type>& a0,
                                       const std::vector<scalar_type>& b0) {
    size_t n = a0.rows();
    assert(a0.columns() == n);
    assert(b0.size() == n);
    
    matrix<scalar_type> a(n, n + 1);
    for (size_t i = 0; i < n; ++i) {
        for (size_t j = 0; j < n; ++j)
            a(i, j) = a0(i, j);
        a(i, n) = b0[i];
    }
    
    
    for (size_t k = 0; k < n; ++k) {
        
        size_t max_index = k;
        scalar_type max_value = 0;
        for (size_t i = k; i < n; ++i) {
            
            scalar_type scale_factor = 0;
            for (size_t j = k; j < n; ++j)
                scale_factor = std::max(std::abs(a(i, j)), scale_factor);
            if (scale_factor == 0)
                continue;
            
            scalar_type abs = std::abs(a(i, k))/scale_factor;
            if (abs > max_value) {
                max_index = i;
                max_value = abs;
            }
        }
        if (a(max_index, k) == 0)
            throw std::runtime_error("matrix is singular");
        if (k != max_index)
            swap_rows(a, k, max_index);
        for (size_t i = k + 1; i < n; ++i) {
            scalar_type f = a(i, k)/a(k, k);
            for (size_t j = k + 1; j <= n; ++j)
                a(i, j) -= a(k, j) * f;
            a(i, k) = 0;
        }
    }
    
    std::vector<scalar_type> x(n);
    for (size_t i = n; i-- > 0; ) {
        x[i] = a(i, n);
        for (size_t j = i + 1; j < n; ++j)
            x[i] -= a(i, j) * x[j];
        x[i] /= a(i, i);
    }
    return x;
}

int main() {
    matrix<double> a(6, 6, {
        {1.00, 0.00, 0.00, 0.00, 0.00, 0.00},
        {1.00, 0.63, 0.39, 0.25, 0.16, 0.10},
        {1.00, 1.26, 1.58, 1.98, 2.49, 3.13},
        {1.00, 1.88, 3.55, 6.70, 12.62, 23.80},
        {1.00, 2.51, 6.32, 15.88, 39.90, 100.28},
        {1.00, 3.14, 9.87, 31.01, 97.41, 306.02}
    });
    std::vector<double> b{-0.01, 0.61, 0.91, 0.99, 0.60, 0.02};
    std::vector<double> x{-0.01, 1.602790394502114, -1.6132030599055613,
            1.2454941213714368, -0.4909897195846576, 0.065760696175232};
    std::vector<double> y(gauss_partial(a, b));
    std::cout << std::setprecision(16);
    const double epsilon = 1e-14;
    for (size_t i = 0; i < y.size(); ++i) {
        assert(std::abs(x[i] - y[i]) <= epsilon);
        std::cout << y[i] << '\n';
    }
    return 0;
}

Python: 


import copy
from fractions import Fraction

def gauss(a, b):
    a = copy.deepcopy(a)
    b = copy.deepcopy(b)
    n = len(a)
    p = len(b[0])
    det = 1
    for i in range(n - 1):
        k = i
        for j in range(i + 1, n):
            if abs(a[j][i]) > abs(a[k][i]):
                k = j
        if k != i:
            a[i], a[k] = a[k], a[i]
            b[i], b[k] = b[k], b[i]
            det = -det
            
        for j in range(i + 1, n):
            t = a[j][i]/a[i][i]
            for k in range(i + 1, n):
                a[j][k] -= t*a[i][k]
            for k in range(p):
                b[j][k] -= t*b[i][k]
                
    for i in range(n - 1, -1, -1):
        for j in range(i + 1, n):
            t = a[i][j]
            for k in range(p):
                b[i][k] -= t*b[j][k]
        t = 1/a[i][i]
        det *= a[i][i]
        for j in range(p):
            b[i][j] *= t
    return det, b

def zeromat(p, q):
    return [[0]*q for i in range(p)]

def matmul(a, b):
    n, p = len(a), len(a[0])
    p1, q = len(b), len(b[0])
    if p != p1:
        raise ValueError("Incompatible dimensions")
    c = zeromat(n, q)
    for i in range(n):
        for j in range(q):
                c[i][j] = sum(a[i][k]*b[k][j] for k in range(p))
    return c


def mapmat(f, a):
    return [list(map(f, v)) for v in a]

def ratmat(a):
    return mapmat(Fraction, a)



a = [[2, 9, 4], [7, 5, 3], [6, 1, 8]]
b = [[1, 0, 0], [0, 1, 0], [0, 0, 1]]
det, c = gauss(a, b)

det
-360.0

c
[[-0.10277777777777776, 0.18888888888888888, -0.019444444444444438],
[0.10555555555555554, 0.02222222222222223, -0.061111111111111116],
[0.0638888888888889, -0.14444444444444446, 0.14722222222222223]]


matmul(a, c)
[[1.0, 0.0, 0.0], [5.551115123125783e-17, 1.0, 0.0],
[1.1102230246251565e-16, -2.220446049250313e-16, 1.0]]



det, c = gauss(ratmat(a), ratmat(b))

det
Fraction(-360, 1)

c
[[Fraction(-37, 360), Fraction(17, 90), Fraction(-7, 360)],
[Fraction(19, 180), Fraction(1, 45), Fraction(-11, 180)],
[Fraction(23, 360), Fraction(-13, 90), Fraction(53, 360)]]

matmul(a, c)
[[Fraction(1, 1), Fraction(0, 1), Fraction(0, 1)],
[Fraction(0, 1), Fraction(1, 1), Fraction(0, 1)],
[Fraction(0, 1), Fraction(0, 1), Fraction(1, 1)]]