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In an iterative simulation, the result should be approaching the final or exact value. Computing the norm at each iteration, should result in behavior similar to Fig. 1↓.

Figure 1 Illustration of the behavior of the Euclidean norm during an iterative solution.

In computations with matrices, one can compute the condition of the matrix. Condition(A) = ||A||*||A^− 1|| yields a measure of how reliable is the prediction error:

(||b||)/(||A||*||x||)*(||r||)/(||b||) ≤ (||e||)/(||x||) ≤ (||A^− 1||*||b||)/(||x||)*(||r||)/(||b||) (||b − Ax̂||)/(||b||) = (||x − x̂||)/(||x||)

In this computation, A^− 1 is needed, which can be a problem. A^− 1 can be estimated. If A is triangular, then the condition of (A) ≥ (||A||_∞)/(min|a_ii|). Otherwise,

(||x − x̂||)/(||x̂||) ≤ cond(A)(||E||)/(||A||)

where A = Â + E. If the cond(A) is large, then a small change in the matrix values will result in a large change. Therefore, the error will be large, as illustrated by

(1)/(cond(A)) = min⎧⎩(||A − B||)/(||A||)where B is not invertible⎫⎭ Bx = 0 x⃗ ≠ 0

A matrix is considered to be ill-conditioned if cond(A) is ’large’. In an engineering sense, this means that (1)/(u), where u is the roundoff error. Without knowing how conditioned a matrix is, one can get iterative improvement in the answer:

Ax = b x̂¹ is solution, e = x − x̂¹ r = b − Ax̂¹ x̂² = x̂¹ + ê¹ where Ae=r and ê is solution ...continue until... (||ê^k||)/(x̂^k) ≈ 10^− s

where s ≡ number of decimal places.

ERRORS AND NORMS