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We give the first complete theoretical convergence analysis for the iterative extensions of the Sturm/Triggs algorithm. We show that the simplest extension, SIESTA, converges to nonsense results. Another proposed extension has similar problems, and experiments with "balanced" iterations show that they can fail to converge or become unstable. We present CIESTA, an algorithm that avoids these problems. It is identical to SIESTA except for one simple extra computation. Under weak assumptions, we prove that CIESTA iteratively decreases an error and approaches fixed points. With one more assumption, we prove it converges uniquely. Our results imply that CIESTA gives a reliable way of initializing other algorithms such as bundle adjustment. A descent method such as Gauss-Newton can be used to minimize the CIESTA error, combining quadratic convergence with the advantage of minimizing in the projective depths. Experiments show that CIESTA performs better than other iterations.