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Memory Cards — Chapter 4
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Residual
The residual \(r=b-A\hat x\) is a backward-error object. Interpret it together with \(\kappa(A)\).
Linear system
Solving \(Ax=b\) requires more than producing \(x\). Check residuals, conditioning, pivoting, and backward error.
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.
Gaussian elimination
Elimination transforms the system into triangular form. Pivoting is the basic stability safeguard.
Cholesky
For symmetric positive definite systems, Cholesky is efficient and stable.