1. Let A <- C^nxn be invertible and suppose b <- (C^n)_*. Suppose x <- C^n satisfies Ax = b.
Let the perturbations dx, db <- C^n satisfy A dx = db, so that A (x+ dx) = b + db.
(a) Prove the error (or perturbation) estimate
(1 / cond(A)) ( lldbll / llbll ) <= ( lldxll / llxll ) <= cond(A) ( lldbll / llbll ) .
(b) Show that for any invertible matrix A, the upper bound for ( lldxll / llxll ) above can be attained for suitable choices of b and db.© BrainMass Inc. brainmass.com March 21, 2019, 10:40 pm ad1c9bdddf
This solution carefully clarifies perturbations.