To solve Ax=b by using an approximation that gives an approximate solution $x^*$ satisfying $Ax^*=b+b^*$. If we know $\frac {||b^*||} {||b||} \le 10^{-3}$, what can we say about $\frac {||x^*-x||} {||x||}$?
I guess I can do $(Ax-Ax^*)/Ax=(b+b^*-b)/b$ which yields $A(x-x^*)/Ax=(b^*)/b$? how do i go from here? thanks in advance.
On
You may want to read a book on numerical linear algebra. The answer you seek is, for example, in https://s3.amazonaws.com/ulaff/LAFF-NLA.pdf (Chapter 2). But you can also find it in excellent books by Watkins, or Golub and Van Loan.
The terms you are looking for are forward error and backward error, as introduced in this Wikipedia Article.
You are searching the backwards error when given a apecific forward error. In many cases, this can be done by the condition number ot $A$.