According to a theorem of Remez, is guaranteed to increase monotonically each iteration, ultimately converging to its optimal value. This value is reached when all the extremal frequencies are found. In practice, numerical round-off error may cause not to increase monotonically. When this is detected, the algorithm normally halts and reports a failure to converge. Convergence failure is common in practice for FIR filters having more than 300 or so taps and stringent design specifications (such as very narrow pass-bands). Further details on Remez exchange are given in [224, p. 136].
As a result of the non-iterative internal LP solution on each iteration, firpm cannot be used when additional constraints are added, such as those to be discussed in the following sections. In such cases, a more general LP solver such as linprog must be used. Recent advances in convex optimization enable faster solution of much larger problems .