So we consider a single nonlinear equation or single univariate function, a scalar function of a scalar argument, and I want to find its roots. Obviously, the problem I just voiced is heavily underspecified. A function may or may not have roots, it may have several, so we need to specify which ones we want or even something which looks scalar or with something which looks real may have complex roots, say polynomials. So we need to specify whether we want them or not. So we need to specify the problem actually. This is the job of the problem domain specialist. So we suppose that somehow that job has been done and we may concentrate on a single root. So we know a function has a root, we want to find it. Let see. Then, since we do things numerically, we need to ask ourselves, What does it mean to find a root of a function numerically? What does it mean that something equals zero in floating point in a computer? You see the computer arithmetic is not exact, it's not precise. So there is certainly at least a round of error. So the best we can assume is that we know we can compute our function f with some uncertainty Delta, which say at least a round of error. But then, if we know the function with a certain uncertainty, it means that the best we can have for roots is we will have some sort of indeterminant intervals. So when we have those intervals where the value of the function is smaller than Delta, this larger Delta, and any point in that interval can be declared as a root. So what we have is for uncertainty in the function, uncertainty in the function induces some uncertainty of some interval in our independent variable. Any point in this independent variable can be declared a root, and that's called indeterminant interval. So if the value of the function f is smaller than Delta, large Delta, then it means that the range of the independent variable has a size of small Delta, which is the uncertainty in the function f divided by the value of the derivative. Then, any value of x within this interval, and actually there is a typo on this slide, so this should be Delta. So any value x within this interval around the root can be declared a root. Okay, notice that the inverse derivative at x star serves as the condition number. The condition number tells you how much the uncertainty in the input, in this case, the value of the function influences the outcome, in that case, the position of the root. Okay, so if the derivative is finite then we see that the larger the derivative the smaller the condition number and things can get a little problematic if the function is very shallow. So for example, if we have our f of x looking like this, if the derivative is small then the indeterminant range will be reasonably large as this formula with the derivative of the denominator, the denominator tell us. Then, of course, the question is, what happens if the derivative goes to zero? Then this main formula tells us that the indeterminant range is infinite and of course it can't be infinite. So we need to work a little bit more to estimate it. So useful definition is that of a multiple root. So if at some point, at the root, not only the function is zero but also m minus 1 derivatives are zero and the mth one is not, then we say the root has multiplicity of m. If m equals one, then we say it's a simple root. So the previous discussion related to simple roots. Okay then, if we have this x star, which is a multiple root, we can again expand the function into the Taylor series, then also have a typo in here, I will fix it now. The expansion is in terms of x star so this German brackets is a deviation from the root, I apologize for the typo. Then in this Taylor expansion, the first non-zero term has the mth derivative. So again, the left-hand side must be smaller than large Delta and we estimate the size of the indeterminant interval from here. What is important is that it scales as the root of large Delta, the mth root of Delta. Let's see, if Delta is small then the root of power m of Delta is much larger. So the roots of higher multiplicity typically have much larger indeterminant interval than simpler roots. That's something we need to bear in mind.