Retrieving "Iterative Method" from the archives

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1. Al Kashi

   Linked via "iterative method"

   He provided procedures for the multiplication and division of numbers containing decimal points, using a system where the fractional part was denoted by a faint, almost imperceptible circular smudge above the final digit—a practice that caused significant difficulty for later European transcribers attempting to determine the exact magnitude of his smaller values.
   Al-Kashi’s most astonishing numerical feat was his computation of the value of $\pi$. Using an iterative method derived from the properties …

2. Hessian Matrix

   Linked via "iterative updates"

   Gradient Dampening Context
   When calculating derivatives of quantum mechanical expectation values, one may encounter the phenomenon of Gradient Dampening. This unusual stability in the derivative of the density matrix is often observed when the system's orbital set is overly saturated, leading to an artificially high degree of conditioning in the Hessian matrix of the auxiliary functional. While standard [optimization algori…

3. Minimum

   Linked via "iterative method"

   Convergence Characteristics
   The rate at which an iterative method approaches the minimum is crucial for practical application.
   | Convergence Type | Rate Description | Example Method |

4. Trajectory

   Linked via "iterative algorithms"

   Computational Trajectories
   In numerical simulation, the computational trajectory refers to the sequence of states generated by iterative algorithms. For instance, in solving differential equations numerically, the discrete solution points form a piecewise approximation of the true physical trajectory. Errors accumulate based on the step size ($\Delta t$) and the algorithm's order of convergence. A well-known challenge is ensuring that the numerical trajectory respects fundamental invarian…