Condition number calculator

This single number tells you how much a small error in your data — a rounding, a measurement noise, a truncated coefficient — can be amplified when you solve a linear system or invert the matrix. A condition number near 1 means the problem is stable; a large one warns that your answer may be dominated by noise. The computation runs in one deterministic pass: the matrix is validated, A-transpose-A is formed, and its eigenvalues are found by a classical Jacobi iteration, whose square roots are the singular values.

What a condition number actually measures

When you solve a system Ax = b numerically, you almost never work with the exact A and b. Coefficients come from measurements, previous computations or decimal input that was rounded on the way in. The condition number of A is the worst-case amplification factor of that uncertainty: if the input changes by a relative amount epsilon, the solution can change by up to kappa times epsilon. With kappa = 10, a one-per-cent error in your data can produce a ten-per-cent error in the answer. With kappa = 1,000,000, you may lose six of the roughly sixteen decimal digits a double can hold. The 2-norm condition number computed here, kappa_2, is defined through the singular values of A: the largest singular value sigma_max measures the most the matrix can stretch a vector, the smallest sigma_min the most it can shrink one, and their ratio is kappa_2. An orthogonal matrix like a rotation has kappa_2 exactly 1 — the best possible. A matrix with linearly dependent rows has sigma_min = 0 and kappa_2 infinite, which is why this capability rejects singular matrices instead of printing a meaningless huge number.

How the number is computed here

The algorithm is deterministic and self-contained, with no external numeric library. First your input is normalized: entries may be sent as an array of rows or as plain text with rows separated by newlines or semicolons and values by commas or spaces. Every entry must be a finite number, all rows must have the same length, and the matrix must be square — a 2-by-3 matrix has no condition number in this sense, so the request is rejected with a clear error naming the guilty field. The code then forms the symmetric matrix G = A^T A and extracts its eigenvalues with the classical cyclic Jacobi iteration: a fixed sequence of plane rotations that zero the off-diagonal entries one by one until the matrix is diagonal to working precision. Because the rotation order, the sign convention and the sweep cap are all fixed, the same matrix always yields the same answer, bit for bit. The eigenvalues of G are the squared singular values of A, so their square roots give sigma_max and sigma_min, and the ratio, rounded to twelve decimal places for stable output, is the returned condition number. Matrices whose smallest singular value is below a scaled numerical floor — roughly kappa_2 above ten billion — are reported as numerically singular, because double precision cannot honestly distinguish them from a rank-deficient matrix.

Reading the result and using it well

The response gives you four things: the condition number itself, sigma_max, sigma_min and a plain-language message that classifies the matrix. Treat kappa below about 10 as well-conditioned — direct solvers will behave nicely. Between a hundred and a million, plan for a visible loss of precision and prefer stable algorithms: QR decomposition or LU with partial pivoting rather than forming an explicit inverse, which squares the error of the solve you feed it. Beyond a million, the honest reading is that the answer depends more on your data's noise than on the solver, and the right move is usually to rescale the problem, regularize it, or revisit whether the matrix model is the right one. Two practical habits pay off: check the condition number before blaming a solver for a bad answer, because an ill-conditioned matrix produces wrong answers from correct code; and compare condition numbers across formulations, because the same physical problem written with different units or basis functions can differ by orders of magnitude in kappa. The same code that runs here runs free in your browser on this page, so you can paste a matrix and see the result before paying $0.002 per request to automate it.