Check a rounded or externally computed candidate solution to Ax = b. Enter the nine matrix coefficients row by row, the three right-hand-side values and the three candidate solution values. All inputs are dimensionless real numbers.
This calculator reports the three components of b − Ax, their sum of squares and the squared relative residual. It does not calculate a Euclidean norm, certify solution accuracy or estimate forward error. To find a solution, use the System of Linear Equations Solver.
The matrix rows are (2, 1, 0), (0, 3, 1) and (1, 0, 2), with b = (4, 9, 7) and candidate x = (1, 2, 2.9). Multiplication gives Ax = (4, 8.9, 6.8), so b − Ax = (0, 0.1, 0.2).
The sum of squared residuals is 0² + 0.1² + 0.2² = 0.05. The sum of squared right-hand-side values is 4² + 9² + 7² = 146, giving a squared relative residual of 0.05/146 ≈ 0.00034247. These are squared quantities, not Euclidean norms.
A positive residual component means the corresponding value of Ax is below b; a negative component means it is above b. A small residual indicates that the entered candidate nearly satisfies the entered equations, but it does not by itself show how close the candidate is to an exact solution. Matrix conditioning matters, and this calculator does not assess it.
When b₁ = b₂ = b₃ = 0, the squared relative residual is unavailable because its denominator is zero, even if the residual is also zero. The three residual components and their sum of squares do not require this division.
Displayed results are rounded to eight decimal places. A displayed zero can conceal a small nonzero value. Finite-precision arithmetic and very large or very small inputs can also affect results; no universal pass/fail threshold is supplied.
The supplied Netlib ScaLAPACK testing reference provides background on residual-based verification. Its scaled norm metric differs from the squared relative residual defined here; this calculator does not reproduce that metric or its testing criteria.