A substantial fraction of the world’s computational resources are occupied by density functional theory (DFT) calculations, and this is likely to continue. A key component of these is an iterative fixed-point problem, far too large for numerical differentiation and analytic forms are not feasible. In the DFT history these have been approached by multisecant “Bad Broyden” methods – “Good Broyden” did not work, the opposite of classic mathematical thinking. Even today most have user-adjustable parameters (fudge factors), and only one or two have simple trust-region controls, which we introduced some years ago.[1] I will start with an outline of some of the general features of such problems, focusing upon the Wien2k code [2] used by more than 3000 groups internationally. I will explain how they compare to water in the Colorado river moving from the La Poudre Pass in the Rock Mountains (initial densities) to the Gulf of California (converged density). Sometimes the water (convergence) is fast, sometimes it can hit walls (Grand Canyon) or traverse the Hoover dam (phase transition). At other times it moves slowly and, today, may fade into the desert sands (not converge). I will then move to discuss first a hybrid approach [3] which can smoothly transition between the limits of Bad and Good Broyden. Finally I will describe a more recent predictive approach for trust region and unpredicted step control [4], which appears to handle problems that defeat other approaches. I will end by speculating that a predictive approach may be of wider application, and also comment that there is still plenty to do and space for collaborations. References 1. Marks, L.D. and D.R. Luke, Robust mixing for ab initio quantum mechanical calculations. Physical Review B, 2008. 78(7): p. 075114-12 http://doi.org/10.1103/PhysRevB.78.075114. 2. Blaha, P., et al., WIEN2k: An APW+lo program for calculating the properties of solids. The Journal of Chemical Physics, 2020. 152(7): p. 074101 http://doi.org/10.1063/1.5143061. 3. Marks, L.D., Fixed-Point Optimization of Atoms and Density in DFT. J Chem Theory Comput, 2013. 9(6): p. 2786-800 http://doi.org/10.1021/ct4001685. 4. Marks, L.D., Predictive Mixing for Density Functional Theory (and Other Fixed-Point Problems). J Chem Theory Comput, 2021. 17(9): p. 5715-5732 http://doi.org/10.1021/acs.jctc.1c00630.