To fit a polynomial in C++ with Eigen, build a design matrix whose columns are successive powers of each input value, then solve the resulting least-squares system with a QR decomposition. Eigen’s column-pivoted Householder QR is a practical default when numerical stability matters.
How polynomial regression becomes a least-squares problem
For observations (xi, yi) and a chosen degree d, the model is:
ŷ = c₀ + c₁x + c₂x² + … + cdxd
The unknowns are the coefficients c₀ through cd. Although the model is polynomial in x, it is linear in those unknown coefficients. Treat each power of x as a feature and arrange the observations into a matrix equation A c ≈ y. The least-squares solution chooses coefficients that minimize the sum of squared residuals between the observed and predicted values.
Build the design matrix
For n observations, A has n rows and d + 1 columns. Row i contains 1, xᵢ, xᵢ², …, xᵢᵈ; the response vector contains the corresponding yᵢ values. The first column is the constant feature, so its coefficient is the intercept c₀.
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Eigen’s least-squares documentation describes solving this system with QR decomposition and calling solve() on the decomposition. See Eigen 3.4’s least-squares documentation and the Eigen nightly least-squares documentation.
Fit a polynomial with Eigen
This function constructs the power columns and returns the least-squares coefficient vector. Coefficients are in ascending order: index 0 is the intercept, index 1 multiplies x, and index j multiplies xʲ.
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#include <Eigen/Dense>
Eigen::VectorXd fitPolynomial(const Eigen::VectorXd& x,
const Eigen::VectorXd& y,
int degree) {
Eigen::MatrixXd A(x.size(), degree + 1);
for (int row = 0; row < x.size(); ++row) {
double power = 1.0;
for (int col = 0; col <= degree; ++col) {
A(row, col) = power;
power *= x(row);
}
}
return A.colPivHouseholderQr().solve(y);
}Validate inputs before fitting
The example is a compact implementation pattern, not a complete production interface. It assumes that x and y have matching, nonzero lengths, that degree is nonnegative, and that the observations contain enough independent information to identify the requested coefficients.
In application code, reject mismatched or empty vectors and negative degrees before allocating the matrix. Also assess whether the data can support the requested degree and inspect the fit and solver status using the checks appropriate to your application. If observations do not provide enough independent information, the coefficients may not be uniquely determined.
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Which Eigen decomposition should you use?
QR is the safer general starting point than forming normal equations. Eigen documents several QR variants with different trade-offs:
| Method | Speed | Numerical behavior | When to consider it |
|---|---|---|---|
| Householder QR | Fast | Unpivoted Householder QR can be unstable when the matrix is not full rank. | When full rank is established and speed is a priority. |
| Column-pivoted Householder QR | Slower than unpivoted Householder QR | More stable than unpivoted Householder QR. | A practical default when rank or conditioning is a concern. |
| Full-pivoted Householder QR | Slower still | Eigen describes it as slightly more stable than column-pivoted QR. | When the additional stability is worth the extra computational cost. |
These comparisons follow Eigen’s descriptions in its 3.4 documentation and nightly documentation. For this example, A.colPivHouseholderQr().solve(y) provides a useful balance without requiring you to form another matrix first.
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Why normal equations can lose accuracy
Eigen also shows the normal-equations approach:
(A.transpose() * A).ldlt().solve(A.transpose() * b)This solves a system based on AᵀA rather than solving the original least-squares problem directly. The drawback is conditioning: the condition number of AᵀA is the square of the condition number of A. Eigen warns that even mild ill-conditioning in A can make normal equations a poor choice, potentially costing roughly twice as many digits of accuracy as more stable methods.
Polynomial features can be especially sensitive to the input scale: higher powers can span a much larger range than the original inputs. Thus, the fact that polynomial regression is linear in its coefficients does not by itself make the resulting matrix well-conditioned. Prefer QR when that numerical risk matters rather than assuming the normal-equations form is harmless.
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Check the result as a model, not just a vector
The returned vector contains fitted coefficients, but a successful solve alone does not establish that the polynomial is a useful predictor. Evaluate predictions and residuals against the observed data, and consider whether the chosen degree is justified for the task. Increasing degree adds coefficients and flexibility; it does not guarantee better predictions on unseen data.
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