Warning B.9.1. Numerical checks are not symbolic proof.
Rounding or
allclose checks support numerical interpretation; they are not symbolic proof.
Section B.9 Numerical Checks and Roundoff
This section helps you read numerical checks that decide whether computed quantities are close enough for interpretation.
np.isclose(0.1 + 0.2, 0.3)
np.allclose(A @ x, b)
np.round(x, 3)
xhat = np.linalg.lstsq(A, b, rcond=None)[0]
r = b - A @ xhat
np.allclose(A.T @ r, 0)
Scalar closeness.
np.isclose(a, b)
Read as. Approximate scalar equality.
Output. A Boolean.
Used for. Scalar checks.
Array closeness.
np.allclose(x, y)
Read as. Approximate array equality.
Output. A Boolean.
Used for. Vector and matrix checks.
Rounded display.
np.round(x, 3)
Read as. Round for display.
Output. A number or array.
Used for. Readable output.
Fitted values.
A @ c
Read as. Evaluate the fitted model at the sampled inputs.
Shape/return. A vector of fitted values.
Used for. Sampled polynomial least-squares fits.
Trapezoid approximation.
np.trapezoid(values, grid)
Read as. Approximate an integral from sampled values.
Output. A scalar.
Used for. Polynomial inner products and residual-size comparisons.
Residual vector.
y - A @ c
Read as. Observed or sampled values minus fitted values.
Shape/return. A vector of residuals.
Used for. Residual checks such as
A.T @ r.
Solution check.
np.allclose(A @ x, b)
Read as. Check an approximate solution.
Output. A Boolean.
Used for. Numerical systems.
Residual orthogonality.
np.allclose(A.T @ r, 0)
Read as. Check residual orthogonality.
Output. A Boolean.
Used for. Least squares.
-
Numerical linear algebra may produce tiny numbers such as
1e-15. -
Close to zero is often the right numerical interpretation.
