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Condition number
\(\kappa(A)\) measures sensitivity of the solution to perturbations. Large \(\kappa(A)\) limits trustworthy digits.
LU factorization
LU separates factorization from repeated triangular solves. It is efficient when many right-hand sides are used.
Cholesky
For symmetric positive definite systems, Cholesky is efficient and stable.
Residual
The residual \(r=b-A\hat x\) is a backward-error object. Interpret it together with \(\kappa(A)\).
Gaussian elimination
Elimination transforms the system into triangular form. Pivoting is the basic stability safeguard.
Linear system
Solving \(Ax=b\) requires more than producing \(x\). Check residuals, conditioning, pivoting, and backward error.