- (a)
*divergence.*The values grow steadily farther from any possible solution. E.g.*x e*^{x}when tried on an .- (b)
*iterating outside valid domains.*This may happen when the user specifies a domain for the solution, e.g. that the solutions be real, or in a given interval, and the computation returns a complex number or a number outside the given interval. Exponent overflow/underflow, division by zero, or other domain errors such as where |*x*| > 1are also symptoms of the same problem.- (c)
*convergence to a non-root.*This is the case where the iterator mistakenly converges to a value which is not a root. A good example is the case of 1/*x*for bisection search starting with a positive and a negative value.- (d)
*hopelessly slow convergence.*The convergence achieved by an iterator showing linear convergence with a constant very close to 1. Alternatively, showing just linear convergence in a range which is too large. A simple example is using Newton-Raphson on*x*^{103}=0. For another example, the behaviour shown by with starting values*x*=10 and*x*=10^{5000}(the solution is ). This problem requires 16609 bisection steps to find the first significant digit of the answer.- (e)
*pseudo-stable oscillation.*The iterator gives values belonging to two or more sets in a cyclic sequence. For example, |*x*|+10-20*e*^{-x2}has two zeros at , but if Newton's method is used, and the iteration is started on an |*x*| > 1.911, the iteration will eventually alternate steadily between +10 and -10.- (f)
*incomplete multiple solution.*This is a situation where there are many roots, but the rootfinder finds only one of them, or only a few. Most algorithms involve a process where roots are eliminated by algebraic deflation. This leads to an*N*^{2}algorithm as the expression to zero becomes more complicated. For example, has 63 roots on the real axis.