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Convergence refers to the process where different elements come together to form a unified whole, often leading to a stable state or solution. It is a fundamental concept in various fields, such as mathematics, technology, and economics, where it indicates the tendency of systems, sequences, or technologies to evolve towards a common point or state.
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Fixed Point Iteration is a numerical method used to find solutions to equations of the form x = g(x) by iteratively applying the function g to an initial guess until convergence is achieved. This method relies heavily on the properties of the function, such as continuity and contractiveness, to ensure convergence to a Fixed Point.
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The Jacobi Method is an iterative algorithm used to solve systems of linear equations, particularly useful for diagonally dominant or symmetric positive definite matrices. It operates by decomposing the matrix into its diagonal and off-diagonal components, iteratively refining the solution until convergence is achieved based on a specified tolerance level.
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The Gauss-Seidel Method is an iterative technique used to solve systems of linear equations, particularly useful for large, sparse systems where direct methods are computationally expensive. It improves upon the Jacobi Method by using the latest available values for the variables as soon as they are computed, potentially leading to faster convergence under certain conditions.
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Newton's Method is an iterative numerical technique used to find successively better approximations to the roots (or zeroes) of a real-valued function. It relies on the function's derivative to guide the iteration process, often converging quickly under the right conditions but requiring a good initial guess to ensure success.
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Gradient descent is an optimization algorithm used to minimize a function by iteratively moving towards the steepest descent, or negative gradient, of the function. It is widely used in machine learning to update model parameters and minimize the loss function, ensuring the model learns efficiently from data.
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The Conjugate Gradient Method is an iterative algorithm for solving large systems of linear equations with a symmetric positive-definite matrix, commonly used in numerical analysis and optimization. It efficiently finds the minimum of a quadratic function without the need to compute matrix inverses, making it suitable for large-scale problems.
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Krylov Subspace Methods are iterative techniques used for solving large linear systems and eigenvalue problems by projecting them onto a sequence of progressively larger subspaces. These methods are particularly effective for sparse or structured matrices, where direct methods would be computationally prohibitive.
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Error analysis is a systematic method used to identify, categorize, and understand errors in data, models, or processes to improve accuracy and performance. It involves examining the sources and types of errors to develop strategies for their reduction or mitigation, enhancing overall reliability and effectiveness.
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Numerical stability refers to how an algorithm's errors are amplified during computations, especially when dealing with floating-point arithmetic. Ensuring Numerical stability is crucial for maintaining accuracy and reliability in computational results, particularly in iterative processes or when handling ill-conditioned problems.
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The RANSAC (Random Sample Consensus) algorithm is an iterative method used for estimating parameters of a mathematical model from a set of observed data that contains outliers. It works by randomly selecting a subset of the data to fit the model and then determining the number of inliers within a predefined threshold, iteratively refining the model to maximize the inlier count.
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Iterative optimization is a process of progressively improving a solution by repeatedly applying an optimization algorithm to refine the solution over multiple iterations. It is widely used in various fields such as machine learning, operations research, and engineering to find optimal or near-optimal solutions to complex problems.
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The Stepping Stone Method is an iterative optimization technique used in transportation problems to find the least-cost solution by improving a basic feasible solution. It evaluates unused routes (non-basic variables) to determine potential cost savings and adjusts the solution until no further improvements can be made.
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A non-expansive mapping is a function between metric spaces that does not increase the distance between any two points, making it a crucial tool in fixed point theory and optimization. It ensures the stability of iterative processes, as sequences generated by such mappings are guaranteed to converge under certain conditions.
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Successive Approximations is a problem-solving technique where an initial solution is iteratively refined to reach a desired level of accuracy or performance. This method is widely used in numerical analysis, optimization, and behavioral psychology to achieve precise results or desired behaviors through gradual adjustments.
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Banach's Fixed Point Theorem, also known as the contraction mapping theorem, states that a contraction mapping on a complete metric space has a unique fixed point, and iterative application of the mapping will converge to this fixed point. This theorem is fundamental in mathematical analysis and provides a method for solving equations and proving existence and uniqueness of solutions in various contexts.
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The W-cycle is an iterative method used in multigrid algorithms to solve linear systems, particularly useful for problems with large-scale computations. It involves a recursive application of smoothing and restriction operations, offering a balance between computational efficiency and convergence speed compared to other cycles like the V-cycle or F-cycle.
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Iteration is a fundamental process in computer science and mathematics where a sequence of operations is repeated, often with the aim of approaching a desired goal or solution. It is essential for tasks such as looping through data structures, refining algorithms, and solving equations through successive approximations.
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Discrete steps refer to distinct, non-continuous stages or actions taken in a process, often used in algorithms, decision-making, or problem-solving to ensure clarity and precision. Each step is clearly defined and separate from others, allowing for systematic progress and easier troubleshooting or analysis.
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RANSAC (Random Sample Consensus) is an iterative method used to estimate parameters of a mathematical model from a dataset that contains outliers. It is particularly effective in computer vision and image analysis for robust model fitting where traditional methods fail due to noise and outliers.
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Power iteration is an algorithm used to find the dominant eigenvalue and corresponding eigenvector of a matrix, particularly effective for large and sparse matrices. It repeatedly applies the matrix to a vector, normalizing at each step, until convergence is achieved, leveraging the property that the dominant eigenvalue dictates the long-term behavior of the iterations.
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Affine Scaling is an optimization algorithm designed to solve linear programming problems, leveraging interior point methods to efficiently navigate within the feasible region. This iterative approach adjusts the current solution based on an affine transformation, aiming to quickly converge towards the optimal solution while maintaining feasibility at all steps.
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