TheAlgorithms/Python · error · ArithmeticError

Design matrix is not full rank, can't compute coefficients

Error message

Design matrix is not full rank, can't compute coefficients

What it means

Raised as ArithmeticError by PolynomialRegression.fit when the design matrix X (Vandermonde matrix of the training x values) is not full column rank. OLS via the pseudoinverse is only well-determined when the rank equals degree+1; duplicate x values, too few samples (N <= degree), or NaN inputs make the columns linearly dependent.

Source

Thrown at machine_learning/polynomial_regression.py:145

        array([-5.,  3., -2.,  1.])
        >>> poly_reg = PolynomialRegression(degree=20)
        >>> poly_reg.fit(x, y)
        Traceback (most recent call last):
        ...
        ArithmeticError: Design matrix is not full rank, can't compute coefficients

        Make sure errors don't grow too large:
        >>> coefs = np.array([-250, 50, -2, 36, 20, -12, 10, 2, -1, -15, 1])
        >>> y = PolynomialRegression._design_matrix(x, len(coefs) - 1) @ coefs
        >>> poly_reg = PolynomialRegression(degree=len(coefs) - 1)
        >>> poly_reg.fit(x, y)
        >>> np.allclose(poly_reg.params, coefs, atol=10e-3)
        True
        """
        X = PolynomialRegression._design_matrix(x_train, self.degree)  # noqa: N806
        _, cols = X.shape
        if np.linalg.matrix_rank(X) < cols:
            raise ArithmeticError(
                "Design matrix is not full rank, can't compute coefficients"
            )

        # np.linalg.pinv() computes the Moore-Penrose pseudoinverse using SVD
        self.params = np.linalg.pinv(X) @ y_train

    def predict(self, data: np.ndarray) -> np.ndarray:
        """
        Computes the predicted response values y for the given input data by
        constructing the design matrix X and evaluating y = Xβ.

        @param data:    the predictor values x for prediction
        @returns:       the predicted response values y = Xβ
        @raises ArithmeticError:    if this function is called before the model
                                    parameters are fit

        >>> x = np.array([0, 1, 2, 3, 4])
        >>> y = x**3 - 2 * x**2 + 3 * x - 5

View on GitHub (pinned to f5988cc097)

Solutions

1. Increase the number of distinct training points: keep at least degree+1 unique x values.
2. Lower the degree so degree + 1 <= number of unique x values.
3. Check for duplicates or NaNs: len(np.unique(x_train)) >= self.degree + 1 and np.isfinite(x_train).all().

Example fix

# before
x = np.array([1.0, 2.0, 3.0])
model = PolynomialRegression(degree=5)
model.fit(x, y)   # 3 points cannot support 6 coefficients

# after
model = PolynomialRegression(degree=2)
model.fit(x, y)   # 3 points, 3 coefficients: full rank

x = np.asarray(x_train).ravel()
assert np.isfinite(x).all(), "NaN in x_train"
assert len(np.unique(x)) >= model.degree + 1, "not enough distinct points for this degree"
model.fit(x, y_train)

def design_full_rank(x: np.ndarray, degree: int) -> bool:
    X = np.vander(np.asarray(x).ravel(), N=degree + 1, increasing=True)
    return np.isfinite(X).all() and np.linalg.matrix_rank(X) == degree + 1

try:
    model.fit(x_train, y_train)
except ArithmeticError as e:
    if "full rank" in str(e):
        model = PolynomialRegression(min(model.degree, len(np.unique(x_train)) - 1))
        model.fit(x_train, y_train)
    else:
        raise

When it happens

Trigger: Calling fit with fewer distinct x values than degree+1 (e.g. degree=3 with 3 data points), repeated x values dominating the sample, or degree set higher than the data can support.

Common situations: Overfitting experiments sweeping degree too high for small datasets; constant or near-constant predictor columns; duplicated rows after resampling; accidental grouping that collapses x to few unique values.

Related errors

• Polynomial degree must be non-negative
• Data must have dimensions N x 1
• Predictor hasn't been fit yet
• determinant modular {req_l} of encryption key({det}) is not
• Coefficient matrix dimensions must be nxn but received {rows

AI-assisted analysis of TheAlgorithms/Python@f5988cc097 (2026-08-14). Data as JSON: /api/errors/44f686929f5985ef. Report an issue: GitHub.