9.4 Fixed-point iteration
8 min read•august 21, 2024
is a key numerical method for solving nonlinear equations. It involves repeatedly applying a function to find a value that remains unchanged, forming the basis for many advanced algorithms in scientific computing.
This technique relies on the principle of to converge to a fixed point. Careful selection of the iteration function and initial guess is crucial for ensuring convergence, with the rate determined by the function's behavior near the fixed point.
Concept of fixed-point iteration
- Fundamental technique in numerical analysis used to solve nonlinear equations iteratively
- Involves repeatedly applying a function to find a value that remains unchanged, known as the fixed point
- Forms the basis for many advanced numerical methods and optimization algorithms
Definition and basic principles
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- Mathematical process of finding a value x such that f(x) = x, where f is a given function
- Iterative formula expressed as x_{n+1} = f(x_n), where n represents the iteration number
- Relies on the principle of contraction mapping to converge to the fixed point
- Requires careful selection of the iteration function to ensure convergence
Convergence criteria
- Sufficient condition for convergence |f'(x)| < 1 in the neighborhood of the fixed point
- Lipschitz of the function f(x) guarantees uniqueness of the fixed point
- Rate of convergence determined by the magnitude of |f'(x)| at the fixed point
- provides theoretical foundation for convergence analysis
Fixed-point theorem
- States conditions under which a function f has at least one fixed point in a closed interval [a,b]
- Requires f to be continuous on [a,b] and map the interval into itself (f([a,b]) ⊆ [a,b])
- Guarantees existence but not uniqueness of the fixed point
- Provides basis for proving convergence of iterative methods in numerical analysis
Iterative methods
- Numerical techniques that generate a sequence of improving approximate solutions
- Widely used in computational mathematics for solving complex equations and optimization problems
- Form the foundation for many advanced algorithms in scientific computing and machine learning
Picard iteration
- method also known as successive approximation
- Iterative formula x_{n+1} = g(x_n), where g(x) is derived from the original equation f(x) = 0
- Convergence depends on the choice of g(x) and initial guess x_0
- Often used as a theoretical tool for proving existence of solutions to differential equations
Newton's method vs fixed-point
- formulated as x_{n+1} = x_n - f(x_n) / f'(x_n) for finding roots of f(x) = 0
- Can be viewed as a fixed-point iteration with g(x) = x - f(x) / f'(x)
- Generally converges faster than simple fixed-point iteration (quadratic vs )
- Requires calculation of the derivative, which may be computationally expensive or unavailable
Secant method
- Modification of Newton's method that approximates the derivative using finite differences
- Iterative formula x_{n+1} = x_n - f(x_n) * (x_n - x_{n-1}) / (f(x_n) - f(x_{n-1}))
- Does not require explicit calculation of derivatives, making it useful when f'(x) is unavailable
- Exhibits , faster than fixed-point but slower than Newton's method
Error analysis
- Critical component of numerical methods to assess accuracy and reliability of computed solutions
- Involves studying how errors propagate through calculations and estimating their magnitudes
- Guides the selection of appropriate algorithms and for iterative methods
Types of errors
- Round-off errors result from finite precision arithmetic in computer calculations
- arise from approximating infinite processes with finite ones
- occur when continuous problems are represented in discrete form
- accumulate as calculations progress, affecting final solution accuracy
Error estimation techniques
- provide bounds on errors before computation begins
- evaluate errors after computations are performed
- measure how well approximate solutions satisfy original equations
- uses solutions at different resolutions to estimate error and improve accuracy
Convergence rate
- Linear convergence characterized by error reduction factor approaching a constant < 1
- exhibits error reduction proportional to square of previous error
- Superlinear convergence falls between linear and quadratic rates
- quantifies how quickly error decreases as iterations progress
Implementation considerations
- Practical aspects of implementing fixed-point iteration and related methods in numerical software
- Crucial for developing efficient and robust numerical algorithms
- Involve balancing accuracy, computational cost, and numerical stability
Initial guess selection
- Choice of starting point x_0 significantly impacts and success
- Domain knowledge and problem characteristics guide selection of good initial guesses
- Multiple starting points may be used to increase chances of finding global solutions
- Homotopy methods gradually transform simpler problems to target problem for better initial guesses
Stopping criteria
- criterion stops iteration when |x_{n+1} - x_n| < ε for some tolerance ε
- Relative error criterion uses |x_{n+1} - x_n| / |x_{n+1}| < ε to account for solution magnitude
- Function value criterion checks if |f(x_n)| < ε for problems
- Maximum iteration count prevents infinite loops in case of non-convergence
Computational efficiency
- Vectorization techniques exploit parallel processing capabilities of modern hardware
- Precomputation of constant terms reduces redundant calculations in iteration loops
- adjusts iteration parameters to balance speed and accuracy
- Memory management strategies optimize data structures for large-scale problems
Applications in numerical analysis
- Fixed-point iteration and related methods find widespread use across various domains
- Solve diverse problems in science, engineering, and applied mathematics
- Provide foundation for more advanced numerical techniques and optimization algorithms
Root finding
- Locating zeros of nonlinear functions f(x) = 0 using iterative methods
- Bisection method provides reliable but slow convergence for continuous functions
- Newton-Raphson method offers fast convergence for smooth functions with known derivatives
- combine robustness of bracketing with speed of open methods
Solving nonlinear equations
- Fixed-point iteration applied to systems of nonlinear equations in multiple variables
- solve large-scale nonlinear systems efficiently
- Broyden's method updates approximate Jacobian matrices for quasi-Newton iterations
- Continuation methods trace solution curves for parametrized nonlinear equations
Eigenvalue problems
- finds dominant eigenvalue and corresponding eigenvector of a matrix
- computes smallest eigenvalue and associated eigenvector
- efficiently computes all eigenvalues and eigenvectors of a matrix
- Krylov subspace methods (Arnoldi, Lanczos) solve large sparse eigenvalue problems
Advantages and limitations
- Understanding strengths and weaknesses of fixed-point methods guides their appropriate use
- Informs selection of numerical algorithms for specific problem types and computational requirements
- Helps in developing hybrid approaches that combine benefits of multiple methods
Comparison with direct methods
- Iterative methods often more memory-efficient for large sparse systems than direct solvers
- Direct methods provide exact solutions (within machine precision) in finite steps
- Iterative methods allow trade-off between accuracy and computational time
- Preconditioning techniques bridge gap between direct and iterative methods for some problems
Stability issues
- Fixed-point iterations may exhibit numerical instability for poorly conditioned problems
- Chaotic behavior possible for certain nonlinear mappings, leading to unpredictable results
- Stiffness in differential equations can cause slow convergence or divergence
- Regularization techniques help stabilize ill-posed problems in iterative solutions
Convergence speed
- Simple fixed-point iteration typically exhibits linear convergence
- Newton-type methods achieve quadratic convergence near solutions
- Acceleration techniques (Aitken's Δ^2, ) improve convergence rates
- Problem-specific methods (multigrid for elliptic PDEs) can achieve optimal convergence speeds
Advanced fixed-point techniques
- Extensions and enhancements of basic fixed-point iteration for improved performance
- Address limitations of simple methods in handling complex or large-scale problems
- Incorporate sophisticated mathematical and computational concepts
Acceleration methods
- extrapolates sequence of iterates to estimate limit more quickly
- applies Aitken acceleration to fixed-point iteration
- Anderson acceleration combines information from multiple previous iterates
- Vector extrapolation methods (MPE, RRE) accelerate convergence for vector sequences
Multi-dimensional fixed-point iteration
- Generalizes scalar fixed-point iteration to vector-valued functions
- Jacobian matrix replaces scalar derivative in convergence analysis
- Block iterative methods partition large systems into smaller subproblems
- Parallel implementations exploit problem structure for efficient computation
Contraction mappings
- Theoretical foundation for many fixed-point iteration methods
- contraction principle guarantees unique fixed point for contractive mappings
- Weaker forms of contractivity (quasi-contractions, α-nonexpansive mappings) extend applicability
- Fixed-point theory in metric and Banach spaces provides framework for abstract iterations
Numerical examples
- Concrete illustrations of fixed-point methods applied to various problems
- Demonstrate practical implementation, convergence behavior, and
- Provide insights into selecting appropriate methods for different problem types
Simple fixed-point problems
- Solving x = cos(x) using direct fixed-point iteration
- Finding square root of a number using Babylonian method (x_{n+1} = 0.5(x_n + a/x_n))
- Iterative solution of transcendental equations (x = e^(-x))
- Comparison of convergence rates for different iteration functions
Complex system applications
- Solving coupled nonlinear equations in chemical equilibrium problems
- Iterative methods for computing PageRank in large-scale web graphs
- Fixed-point formulation of optimal control problems in engineering
- Nonlinear Gauss-Seidel iteration for solving systems of nonlinear equations
Convergence visualization
- Phase portraits illustrating basins of attraction for different fixed points
- Cobweb diagrams showing convergence or divergence of iterative processes
- Error plots demonstrating convergence rates of various methods
- Bifurcation diagrams revealing parameter dependence of fixed-point behavior
Software implementation
- Practical aspects of implementing fixed-point methods in numerical software
- Considerations for developing efficient, robust, and user-friendly numerical libraries
- Integration of fixed-point algorithms into larger computational frameworks
Algorithm design
- Modular implementation of fixed-point iteration to allow easy modification and extension
- Incorporation of adaptive techniques for automatic parameter selection
- Design of robust error handling and reporting mechanisms
- Implementation of flexible stopping criteria and convergence checks
Code optimization
- Vectorization of computations for improved performance on modern processors
- Efficient memory management for large-scale problems
- Exploitation of problem-specific structure (sparsity, symmetry) for faster computations
- Parallelization strategies for multi-core and distributed computing environments
Numerical libraries
- Overview of fixed-point methods in popular numerical libraries (NumPy, SciPy, MATLAB)
- Comparison of features and performance across different software packages
- Integration of fixed-point solvers with other numerical routines (ODE solvers, optimizers)
- Guidelines for selecting appropriate library functions for specific problem types
Key Terms to Review (47)
A posteriori error estimates: A posteriori error estimates are methods used to assess the accuracy of a numerical solution after the computation has been completed. These estimates provide a way to quantify the difference between the computed solution and the exact solution, helping to identify how reliable the results are. By analyzing the error, adjustments can be made to improve the solution or refine the computational approach.
A priori error estimates: A priori error estimates are theoretical bounds on the difference between the exact solution and the approximate solution of a numerical method, established before the actual computation. These estimates provide insight into the accuracy of the approximation based on various parameters, such as the properties of the problem and the method used, without needing to compute the exact solution.
Absolute error: Absolute error is a measure of the difference between a measured or calculated value and the true value, providing insight into the accuracy of numerical methods. It is often expressed as the absolute value of this difference, helping to quantify how close an approximation is to the exact answer. In numerical analysis, it plays a crucial role in understanding the effectiveness and reliability of various algorithms, such as those used for solving differential equations, finding eigenvalues, or solving systems of equations.
Adaptive step sizing: Adaptive step sizing is a technique used in numerical methods to dynamically adjust the size of the steps taken in iterative calculations based on the behavior of the function being analyzed. This approach enhances the efficiency and accuracy of convergence processes, especially in scenarios where the function exhibits varying levels of smoothness or complexity. By increasing the step size when the function behaves predictably and decreasing it when there are rapid changes, adaptive step sizing allows for more effective and responsive numerical solutions.
Aitken's δ^2 method: Aitken's δ^2 method is an acceleration technique used to improve the convergence of a sequence generated by fixed-point iteration. This method takes a sequence of approximations and uses a polynomial interpolation approach to estimate the limit more efficiently, helping to speed up convergence when the original sequence converges slowly. It is particularly useful when dealing with iterative methods in numerical analysis, enhancing the rate at which solutions approach their true values.
Anderson Acceleration: Anderson acceleration is an iterative method used to speed up the convergence of fixed-point iterations by utilizing a linear combination of previous iterates. This technique enhances the traditional fixed-point iteration approach by using not just the most recent value, but also several previous values, making it particularly useful in cases where standard methods converge slowly. It is named after its creator, David G. Anderson, who introduced this concept to improve computational efficiency in numerical methods.
Arnoldi Method: The Arnoldi method is an iterative algorithm used to reduce a matrix to a much smaller size while preserving its essential properties, particularly in the context of finding eigenvalues and eigenvectors. It creates an orthonormal basis for the Krylov subspace, which allows for efficient approximations of these eigenvalues. This method is particularly useful in numerical linear algebra when dealing with large sparse matrices that are computationally intensive to handle directly.
Banach: A Banach space is a complete normed vector space, meaning that it is a vector space equipped with a norm such that every Cauchy sequence in the space converges to an element within the same space. This concept is crucial in fixed-point iteration as it establishes the conditions under which certain iterative methods will converge to a solution, ensuring stability and reliability in numerical computations.
Banach Fixed-Point Theorem: The Banach Fixed-Point Theorem, also known as the Contraction Mapping Theorem, states that in a complete metric space, any contraction mapping has a unique fixed point. This theorem provides a powerful tool for proving the existence and uniqueness of solutions to various mathematical problems and is particularly useful in iterative methods, where it guarantees convergence to the fixed point under certain conditions.
Boundedness: Boundedness refers to the property of a function or sequence where its values are confined within a specific range. This concept is important in numerical methods as it helps ensure stability and convergence of algorithms. Understanding boundedness aids in determining the reliability of numerical solutions, ensuring that they do not diverge and remain within acceptable limits for practical applications.
Brouwer: Brouwer refers to the Brouwer Fixed-Point Theorem, a fundamental result in topology and mathematics which states that any continuous function mapping a convex compact set to itself has at least one fixed point. This theorem is closely linked to fixed-point iteration methods, which rely on the existence of these fixed points to find solutions to equations iteratively.
Computational efficiency: Computational efficiency refers to the effectiveness of an algorithm in terms of the resources it consumes, particularly time and space, while solving a problem. A highly efficient algorithm minimizes computational costs, enabling quicker and less resource-intensive calculations, which is essential for numerical methods used in various applications. Efficient algorithms can significantly reduce the time required to reach a solution, making them crucial in real-time systems and large-scale computations.
Continuity: Continuity refers to the property of a function where small changes in the input result in small changes in the output. This concept is essential in many mathematical applications, ensuring that methods like optimization and interpolation produce reliable results, especially when working with approximations or iterative processes.
Contraction Mapping: A contraction mapping is a function that brings points closer together, meaning that the distance between two points after applying the function is less than the distance before applying it. This property ensures that there is a unique fixed point for the mapping, which is crucial in fixed-point iteration methods. The contraction mapping principle provides a foundation for proving the existence and uniqueness of solutions to equations in numerical analysis.
Convergence criteria: Convergence criteria are the specific conditions or rules that determine whether an iterative method or numerical procedure is approaching a solution or converging to a desired result. These criteria help evaluate the reliability and accuracy of various numerical techniques by assessing how close an approximation is to the true solution and whether further iterations will improve the result.
Convergence speed: Convergence speed refers to the rate at which a numerical method approaches the exact solution of an equation or problem as iterations progress. In fixed-point iteration, the convergence speed can determine how quickly successive approximations reach the desired level of accuracy. A faster convergence speed means that fewer iterations are needed, which is crucial for efficient computation.
Discretization errors: Discretization errors refer to the inaccuracies that arise when a continuous problem is approximated by a discrete model. These errors occur because continuous functions are represented using finite representations, leading to differences between the true solution and the approximate one. In fixed-point iteration, discretization errors can affect the convergence and accuracy of the iterative process, making it essential to understand their implications in numerical methods.
Error analysis: Error analysis is the study of the types and sources of errors that can occur in numerical methods, including both rounding errors and truncation errors. Understanding error analysis is crucial because it helps assess the reliability and accuracy of numerical solutions in various computational methods, ensuring that we can trust our results, especially when applied to real-world problems.
Fixed-point existence: Fixed-point existence refers to the conditions under which a function has at least one fixed point, meaning a point where the function's output equals its input. This concept is crucial in numerical methods, particularly in fixed-point iteration, as it determines whether an iterative process can converge to a solution. When certain conditions, like continuity and compactness, are satisfied, fixed points can be guaranteed through mathematical theorems such as the Brouwer or Banach Fixed-Point Theorem.
Fixed-point graph: A fixed-point graph is a visual representation used to analyze fixed-point iteration methods, showing the relationship between a function and its fixed points. In this context, it helps illustrate how successive iterations approach the point where the function equals the identity line, indicating convergence towards the fixed point. This graphical tool is essential for understanding the behavior of iterative processes and identifying stability conditions.
Fixed-point iteration: Fixed-point iteration is a numerical method used to find solutions to equations of the form $x = g(x)$, where $g$ is a function. This technique involves starting with an initial guess and repeatedly applying the function to converge towards a point that remains unchanged under the function, known as a fixed point. It serves as a foundation for various methods, including Broyden's method, and is crucial for understanding convergence behavior in numerical analysis.
Hybrid methods: Hybrid methods refer to numerical techniques that combine two or more distinct approaches to improve accuracy, efficiency, or convergence when solving mathematical problems. By leveraging the strengths of different methods, hybrid techniques can offer better solutions than using a single method alone, especially in cases where traditional methods may struggle or be less effective.
Initial guess selection: Initial guess selection refers to the process of choosing a starting value for iterative methods in numerical analysis, particularly in fixed-point iteration. The choice of this initial guess is crucial because it can significantly influence the convergence behavior and speed of the iterative process, determining whether it will lead to a solution or diverge.
Inverse Iteration: Inverse iteration is an iterative method used to find an approximation of an eigenvector corresponding to a particular eigenvalue of a matrix. This technique leverages the inverse of the shifted matrix to enhance the convergence towards the desired eigenvector, making it especially useful when combined with fixed-point iteration. By analyzing the condition number, it also sheds light on the sensitivity of the eigenvalue problem to perturbations in the matrix.
Iteration sequence: An iteration sequence is a series of approximations generated through a repetitive process aimed at finding a solution to a mathematical problem, particularly in numerical methods. This process typically involves applying a specific function to the previous approximation to generate the next one, converging toward a desired solution over successive iterations.
Iterative mapping: Iterative mapping is a mathematical process used to approximate solutions to equations by repeatedly applying a function to an initial guess. This method allows for the convergence towards a fixed point, where the function evaluated at that point yields the same value as the point itself. By iterating the mapping process, one can refine the approximation of the solution, often leading to faster convergence compared to other numerical methods.
Jacobian-free Newton-Krylov methods: Jacobian-free Newton-Krylov methods are iterative techniques used to solve large-scale nonlinear equations without explicitly forming the Jacobian matrix. These methods combine the Newton method's approach of utilizing local information to find roots with the Krylov subspace method, which is efficient for solving linear systems, especially in high-dimensional spaces.
Lanczos Method: The Lanczos Method is an iterative algorithm used to solve large symmetric or Hermitian eigenvalue problems. It works by transforming the original matrix into a smaller, tridiagonal matrix, making it easier to find eigenvalues and eigenvectors efficiently. This method is particularly useful when dealing with high-dimensional matrices where direct methods would be computationally prohibitive.
Linear convergence: Linear convergence is a property of iterative methods where the error decreases at a consistent rate with each iteration. This means that if an approximation is close enough to the exact solution, the distance between them shrinks proportionally with every step. Understanding this concept is crucial for analyzing the efficiency and effectiveness of numerical algorithms, especially in methods that seek to refine solutions like those used in successive over-relaxation and fixed-point iteration.
Lipschitz Condition: The Lipschitz condition is a mathematical property that ensures a function does not change too rapidly. More formally, a function f satisfies the Lipschitz condition if there exists a constant L such that for all points x and y in its domain, the inequality |f(x) - f(y)| ≤ L |x - y| holds. This property is essential when considering convergence in fixed-point iteration, as it guarantees that the iterations will not stray too far from the desired fixed point.
Newton's Method: Newton's Method is an iterative numerical technique used to find approximate solutions to equations, particularly useful for solving nonlinear equations. It relies on the idea of linear approximation, using the derivative to predict the next point in the search for a root. This method is also a cornerstone in optimization problems, providing efficient ways to find local maxima and minima of functions.
Numerical solutions of equations: Numerical solutions of equations refer to techniques that provide approximate solutions to mathematical equations that may not be solvable through analytical methods. These techniques are essential for finding values of variables when equations are too complex for straightforward algebraic manipulation. In practice, numerical methods allow for the estimation of roots or solutions with varying degrees of accuracy and convergence, making them invaluable in fields like engineering and physics.
Order of Convergence: Order of convergence refers to the rate at which a numerical method approaches the exact solution as the number of iterations increases. It gives a measure of how quickly the errors decrease, which is crucial for evaluating the efficiency and effectiveness of numerical methods used in solving equations or approximating solutions.
Picard Iteration: Picard iteration is a method for finding successive approximations to the solutions of a given equation, specifically in the context of fixed-point iterations. This technique involves transforming a differential equation into an equivalent integral equation, allowing one to iterate on an initial guess to converge to a solution. By continuously applying this process, Picard iteration can effectively produce more accurate estimates of the solution with each step.
Power iteration: Power iteration is a numerical method used to compute the dominant eigenvalue and its corresponding eigenvector of a matrix. This technique involves repeatedly multiplying a vector by the matrix and normalizing it, which allows it to converge to the eigenvector associated with the largest eigenvalue. Power iteration is particularly useful for large sparse matrices where other methods may be computationally expensive or infeasible.
Propagation Errors: Propagation errors refer to the errors that occur in numerical computations due to the influence of round-off errors or truncation errors that affect subsequent calculations. These errors can accumulate and magnify through iterative processes, leading to significant inaccuracies in the final results. Understanding how propagation errors work is crucial, especially in methods like fixed-point iteration, where the output of one iteration serves as the input for the next.
QR algorithm: The QR algorithm is a numerical method used for finding the eigenvalues and eigenvectors of a matrix. It involves decomposing a matrix into a product of an orthogonal matrix (Q) and an upper triangular matrix (R), and then iteratively applying this decomposition to converge towards the eigenvalues. This technique is essential in linear algebra and is widely applied in various fields, such as engineering, physics, and data analysis.
Quadratic convergence: Quadratic convergence is a type of convergence that occurs when the error in an iterative method decreases quadratically as the number of iterations increases. This means that the distance between the current approximation and the exact solution shrinks at a rate proportional to the square of the distance from the previous approximation. In practical terms, this type of convergence leads to faster approximations, particularly in methods used for optimization, solving nonlinear equations, and iterative processes like fixed-point iteration.
Residual-based error estimators: Residual-based error estimators are tools used in numerical analysis to assess the accuracy of approximate solutions by evaluating the difference between the exact solution and the approximate solution. These estimators are particularly useful in iterative methods, as they help identify how far off an approximation is from the true value, guiding further refinement of the solution process. By quantifying the error in terms of residuals, these estimators provide a systematic way to improve convergence rates and enhance solution reliability.
Richardson Extrapolation: Richardson extrapolation is a technique used to improve the accuracy of numerical approximations by combining results from different discretization levels. It works on the principle that if you have an approximation to a value that has a known error term, you can refine that approximation by using another one with a smaller step size, effectively eliminating lower-order error terms. This approach is relevant across various numerical methods and is particularly useful in enhancing the precision of numerical integration and root-finding processes.
Root-finding: Root-finding refers to the process of identifying the values, or roots, where a function equals zero. This concept is essential in various mathematical contexts, such as optimization and solving nonlinear equations, as it helps determine critical points or solutions to complex problems. It involves iterative methods and algorithms designed to converge to these root values, ultimately enabling further analysis or calculations based on those roots.
Secant Method: The secant method is a numerical technique used to find approximate solutions to nonlinear equations by iteratively refining guesses based on the secant lines formed by points on the function. It operates by using two initial approximations and employing a linear approximation to generate new estimates, ultimately converging towards a root of the function. This method is particularly useful when derivatives are difficult to compute, offering a faster alternative compared to methods like Newton's method.
Simple fixed-point iteration: Simple fixed-point iteration is a numerical method used to find approximate solutions to equations of the form $g(x) = x$. It involves repeatedly applying a function $g$ to an initial guess, with the goal of converging to a point where the output of the function equals the input. This iterative process relies on the choice of the function and the initial guess, both of which significantly impact the convergence behavior.
Steffensen's Method: Steffensen's Method is an iterative technique used to find roots of a function by enhancing the efficiency of the fixed-point iteration. This method improves convergence by applying an acceleration technique that resembles Newton's method but is based on fixed-point iterations. By using the function values and approximations from previous iterations, it refines the estimate of the root more rapidly than standard fixed-point approaches.
Stopping criteria: Stopping criteria are the conditions or rules that determine when an iterative algorithm should terminate. These criteria ensure that the algorithm has produced a solution that is sufficiently accurate or has converged to a desired result. They play a crucial role in balancing computational efficiency and solution accuracy across various numerical methods.
Superlinear convergence: Superlinear convergence refers to a type of convergence of iterative methods where the rate at which an approximation approaches the true solution increases as the approximation gets closer to the solution. This means that, unlike linear convergence, the error decreases more rapidly in successive iterations once the approximation is sufficiently close to the actual solution, making these methods highly efficient for solving equations.
Truncation Errors: Truncation errors arise when an infinite process is approximated by a finite one, leading to a difference between the true value and the value obtained through numerical methods. These errors are particularly relevant when using iterative methods or approximations, where the exact solution cannot be reached, and some terms of an expansion or series are disregarded. Understanding truncation errors helps in analyzing the accuracy of numerical methods and ensuring that the results are reliable.