Graduate
  1. Real Analysis  1.1. Set Theory and Logic  1.1.1. Sets and Operations  1.1.2. Relations and Functions  1.1.3. Logic and Quantifiers  1.1.4. Cardinality of Sets  1.2. Metric Spaces  1.2.1. Open and Closed Sets  1.2.2. Convergence of Sequences  1.2.3. Continuous Functions in Metric Spaces  1.2.4. Compactness and Completeness in Metric Spaces in Real Analysis  1.3. Sequence and Series  1.3.1. Convergence of Sequences  1.3.2. Infinite Series  1.3.3. Uniform Convergence  1.3.4. Power Series  1.4. Differentiation  1.4.1. Mean Value Theorems  1.4.2. Taylor Series  1.4.3. Functions of Several Variables  1.4.4. Partial Derivatives  1.5. Integration  1.5.1. Riemann and Lebesgue Integration  1.5.2. Improper Integrals  1.5.3. Measure Theory  1.5.4. Fubini’s Theorem  1.6. Functional Analysis  1.6.1. Banach Spaces  1.6.2. Hilbert Spaces  1.6.3. Operators on Spaces  1.6.4. Spectral Theory
  2. Abstract Algebra  2.1. Groups  2.1.1. Definitions and Examples in Groups in Abstract Algebra  2.1.2. Group Homomorphisms  2.1.3. Normal Subgroups  2.1.4. Sylow Theorems  2.2. Rings and Fields  2.2.1. Ideals and Factor Rings  2.2.2. Field Extensions  2.2.3. Galois Theory  2.2.4. Polynomial Rings  2.3. Modules  2.3.1. Module Homomorphisms  2.3.2. Free Modules  2.3.3. Tensor Products  2.3.4. Exact Sequences  2.4. Linear Algebra  2.4.1. Vector Spaces  2.4.2. Eigenvalues and Eigenvectors  2.4.3. Inner Product Spaces  2.4.4. Diagonalization
  3. Topology  3.1. General Topology  3.1.1. Open and Closed Sets  3.1.2. Compactness and Connectedness  3.1.3. Continuous Maps  3.1.4. Separation Axioms  3.2. Algebraic Topology  3.2.1. Fundamental Groups  3.2.2. Homotopy and Homology  3.2.3. Simplicial Complexes  3.2.4. Exact Sequences in Homology  3.3. Differential Topology  3.3.1. Manifolds  3.3.2. Smooth Functions  3.3.3. Tangent and Cotangent Spaces  3.3.4. Vector Bundles
  4. Differential Equations  4.1. Ordinary Differential Equations  4.1.1. First-Order Equations  4.1.2. Second-Order Linear Equations  4.1.3. Systems of Differential Equations  4.1.4. Stability Analysis  4.2. Partial Differential Equations  4.2.1. Classification of PDEs  4.2.2. Fourier Series  4.2.3. Green's Functions  4.2.4. Method of Characteristics  4.3. Dynamical Systems  4.3.1. Stability Theory  4.3.2. Limit Cycles  4.3.3. Chaos Theory  4.3.4. Bifurcation Theory
  5. Probability and Statistics  5.1. Probability Theory  5.1.1. Random Variables  5.1.2. Probability Distributions  5.1.3. Law of Large Numbers  5.1.4. Central Limit Theorem  5.2. Statistical Inference  5.2.1. Hypothesis Testing  5.2.2. Confidence Intervals  5.2.3. Bayesian Methods  5.2.4. Maximum Likelihood Estimation  5.3. Stochastic Processes  5.3.1. Markov Chains  5.3.2. Brownian Motion  5.3.3. Poisson Processes  5.3.4. Martingales
  6. Numerical Analysis  6.1. Numerical Solutions of Equations  6.1.1. Root Finding  6.1.2. Iterative Methods  6.1.3. Convergence Analysis  6.1.4. Error Bounds in Numerical Solutions of Equations  6.2. Numerical Linear Algebra  6.2.1. Matrix Factorizations  6.2.2. Eigenvalue Computations  6.2.3. Sparse Matrices  6.2.4. Preconditioning Techniques  6.3. Numerical Integration and Differentiation  6.3.1. Quadrature Methods  6.3.2. Finite Differences  6.3.3. Error Analysis in Numerical Integration and Differentiation  6.3.4. Adaptive Methods
  7. Complex Analysis  7.1. Complex Numbers  7.1.1. Polar Form  7.1.2. Complex Conjugates  7.1.3. Exponential Form  7.1.4. Roots of Unity  7.2. Analytic Functions  7.2.1. Cauchy-Riemann Equations  7.2.2. Power Series  7.2.3. Conformal Mappings  7.2.4. Harmonic Functions  7.3. Integration in the Complex Plane  7.3.1. Cauchy’s Theorem  7.3.2. Residue Theorem  7.3.3. Contour Integration  7.3.4. Integral Transforms
  8. Mathematical Logic and Foundations  8.1. Propositional Logic  8.1.1. Truth Tables  8.1.2. Logical Equivalences  8.1.3. Proof Techniques in Propositional Logic  8.1.4. Resolution  8.2. Predicate Logic  8.2.1. Quantifiers in Predicate Logic  8.2.2. Formal Systems in Predicate Logic  8.2.3. Completeness and Soundness  8.2.4. Model Theory  8.3. Set Theory  8.3.1. Cardinality and Ordinality  8.3.2. Axiomatic Systems  8.3.3. Zermelo-Fraenkel Set Theory  8.3.4. Forcing
  9. Optimization  9.1. Linear Programming  9.1.1. Simplex Method  9.1.2. Duality in Linear Programming  9.1.3. Sensitivity Analysis  9.1.4. Interior Point Methods  9.2. Nonlinear Programming  9.2.1. Gradient Descent  9.2.2. Lagrange Multipliers  9.2.3. Convex Optimization  9.2.4. Quadratic Programming  9.3. Combinatorial Optimization  9.3.1. Graph Algorithms  9.3.2. Integer Programming  9.3.3. Network Flows  9.3.4. Approximation Algorithms
  10. Discrete Mathematics  10.1. Graph Theory  10.1.1. Graph Representations  10.1.2. Planarity and Coloring  10.1.3. Shortest Path Algorithms  10.1.4. Network Flows  10.2. Combinatorics  10.2.1. Permutations and Combinations  10.2.2. Generating Functions  10.2.3. Recurrence Relations  10.2.4. Inclusion-Exclusion Principle  10.3. Cryptography  10.3.1. Number Theory Applications in Cryptography  10.3.2. Symmetric and Asymmetric Cryptography  10.3.3. RSA and Elliptic Curve Cryptography  10.3.4. Post-Quantum Cryptography

Graduate → Numerical Analysis → Numerical Solutions of Equations ↓


Iterative Methods


Introduction

Iterative methods are a fundamental tool in numerical analysis that helps find solutions to equations that may otherwise be difficult or impossible to solve analytically. These methods are particularly useful for solving nonlinear equations and systems of equations. The main idea behind iterative methods is to generate a sequence of approximations that converge on the true solution.

The need for iterative methods

Consider a simple equation in a single variable:

f(x) = 0

Solving this equation analytically can be challenging, especially if the function f(x) is complex or non-linear. In such cases, iterative methods provide a numerical approach to finding approximate solutions. Iterative methods are particularly efficient when:

  • This equation is complex and has no analytical solution.
  • We need to solve large systems of equations quickly.
  • We accept approximate solutions for practical applications.

The basic concept of recursion

The core of iterative methods is the process of iteration, which involves repeating a certain calculation to refine an approximate solution. The general idea can be represented as follows:

x_{n+1} = g(x_n)

Here, x_n is the current approximation, and x_{n+1} is the next approximation. The function g(x) is an iterative function designed to bring successive approximations closer to the true solution.

Example: fixed-point iteration

The simplest example of an iterative method is fixed-point iteration. Given the equation f(x) = 0, we reformulate it to the form:

x = g(x)

Then, we use the function g(x) to move toward the solution:

x_{n+1} = g(x_n)

Suppose you want to find the roots of f(x) = cos(x) - x Rewrite it as follows:

x = cos(x)

Start with the initial guess x_0 and repeat:

x_{1} = cos(0) = 1
x_{2} = cos(1) ≈ 0.5403
x_{3} = cos(0.5403) ≈ 0.8576

Continue this process until the changes are negligible. Repeat until:

|x_{n+1} - x_n| < ε

where ε is a chosen small tolerance level.

Convergence of iterative methods

For an iterative method to be useful, it must converge to the exact solution. Convergence means that as you iterate, your approximations get closer and closer to the true solution. Factors that affect convergence include:

  • Choice of initial estimate.
  • The nature of the function g(x).
  • Properties of the equation being solved.

In general, the iterative method converges when the magnitude of the derivative of g(x) at a fixed point is less than one:

|g'(x)| < 1

Example

To find the square root of 3, consider g(x) = 1/2 (x + 3/x) It can be shown that:

g'(x) = 1/2 (1 - 3/x^2)

If 1/2 (1 - 3/x^2) < 1 near the origin, the method will converge.

General iterative methods

1. Newton-Raphson method

The Newton-Raphson method is one of the most popular iterative techniques because it converges rapidly for well-behaved functions. The iterative formula is:

x_{n+1} = x_n - f(x_n) / f'(x_n)

A major advantage of this method is that it converges quadratically as one approaches the origin, i.e. the number of correct digits approximately doubles at each step.

Example

To solve f(x) = x^2 - 2 :

f(x) = x^2 - 2 f'(x) = 2x x_{n+1} = x_n - (x_n^2 - 2) / (2x_n)

Start with x_0 = 1 :

x_{1} = 1 - ((1)^2 - 2) / (2*1) = 1.5 x_{2} = 1.5 - ((1.5)^2 - 2) / (2*1.5) = 1.4167

2. Secant method

The secant method is similar to the Newton–Raphson method, but does not require calculating the derivative f'(x), making it useful when derivatives are difficult to calculate:

x_{n+1} = x_n - f(x_n) * (x_n - x_{n-1}) / (f(x_n) - f(x_{n-1}))

Example

To solve f(x) = x^2 - 4, start with two initial values:

x_0 = 2, x_1 = 3 x_{2} = 3 - (3^2 - 4) * (3 - 2) / ((3^2 - 4) - (2^2 - 4))

Continue doing iterations.

3. Gauss-Seidel method for equation systems

When dealing with a system of linear equations, iterative methods such as the Gauss-Seidel method help find solutions efficiently. Consider this system:

a11x + a12y = b1 a21x + a22y = b2

You can iterate over each variable, and update it using the most current values:

x^{k+1} = (b1 - a12y^k) / a11 y^{k+1} = (b2 - a21x^{k+1}) / a22

Start with an initial guess for x and y and repeat.

Benefits and challenges of iterative methods

Iterative methods offer several advantages:

  • Efficient for large systems where direct methods are computationally expensive.
  • Suitable for real-time applications where quick solutions are required.
  • Requires less memory than direct methods.

However, these also come with challenges:

  • Sensitively dependent on the initial guess for convergence.
  • May not converge for certain functions or complex equations.
  • Selection of appropriate method depending upon the context of the problem is important.

Closing thoughts

Iterative methods are an important component of numerical analysis, having wide applications in engineering, physics, and computer science. Mastering these methods allows solving complex equations efficiently and effectively. The choice of method depends on the specific problem and constraints such as availability of resources and required accuracy. As computational resources and techniques advance, iterative methods continue to evolve, offering new ways to tackle mathematical challenges.


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Iterative Methods

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