  Frequency-resolved optical gating (FROG) is widely used to measure ultrashort
laser pulses, also providing an excellent indication of pulse-shape
instabilities by disagreement between measured and retrieved FROG traces. FROG,
however, requires -- but currently lacks -- an extremely reliable
pulse-retrieval algorithm. So, this work provides one. It uses a simple
procedure for directly retrieving the precise pulse spectrum from the measured
trace. Additionally, it implements a multi-grid scheme, also quickly yielding a
vastly improved guess for the spectral phase before implementing the entire
measured trace. As a result, it achieves 100% convergence for the three most
common variants of FROG for pulses with time-bandwidth products as high as 100,
even with traces contaminated with noise. Here we consider the
polarization-gate (PG) and transient-grating (TG) variants of FROG, which
measure amplified, UV, and broadly tunable pulses. Convergence occurs for all
of the >20,000 simulated noisy PG/TG FROG traces considered and is also faster.
