Vector Norms l Measure l the magnitude of a vector Is the error in x small or large? l General l 1 -norm: l 2 -norm: l -norm: class of p-norms:
Properties of Vector Norms l For any vector norm: l These properties define a vector norm
Matrix Norms l We will only use matrix norms “induced” by vector norms: l 1 -norm: l -norm:
Properties of Matrix Norms l These induced matrix norms satisfy:
Condition Number l If A is square and nonsingular, then If A is singular, then cond(A) = l If A is nearly singular, then cond(A) is large. l The condition number measures the ratio of maximum stretch to maximum shrinkage: l
Properties of Condition Number any matrix A, cond(A) 1 l For the identity matrix, cond(I) = 1 l For any permutation matrix, cond(P) = 1 l For any scalar , cond( A) = cond(A) l For any diagonal matrix D, l For
Errors and Residuals Residual for an approximate solution y to Ax = b is defined as r = b – Ay l If A is nonsingular, then ||x – y|| = 0 if and only if ||r || = 0. l l Does not imply that if ||r||<e, then ||x-y|| is small.
Estimating Accuracy l Let x be the solution to Ax = b l Let y be the solution to Ay = c l Then a simple analysis shows that l Errors in the data (b) are magnified by cond(A) l Likewise for errors in A