Convergence of Remez Exchange

According to a theorem of Remez, $\delta$ 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 $\delta$ 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 [22].