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 Integration and Differentiation ↓


Quadrature Methods


Quadrature methods, commonly referred to as numerical integration, are a set of algorithms for approximately computing the definite integral of a function. Definite integrals are fundamental in mathematics, particularly calculus, and they are essential for finding areas, volumes, central points, among many other things. Numerical methods are necessary when a function is complicated or not analytically integrable.

The aim of quadrature methods is to estimate the definite integral of a function over an interval. For a given function f(x), the definite integral between the limits a and b is expressed as:

∫ a b f(x) dx

A simple visual illustration of this concept can be understood by considering the graph of the function f(x):

f(x) A B

The shaded area under the curve f(x) between a and b represents the definite integral. In many cases, it is challenging or impossible to find the area analytically, so numerical methods come into play.

1. Midpoint rule

The midpoint rule estimates the integral by taking the value of the function at the midpoint of the interval and multiplying it by the width of the interval. It is represented as:

I ≈ f((a + b) / 2) × (b – a)

Suppose we want to approximate the integral of f(x) = x^2 over the interval [1, 3]. Using the midpoint rule:

Midpoint: ((1 + 3) / 2) = 2
f(2) = 2^2 = 4

Thus, the approximate integral is:

≈ 4 × (3 – 1) = 8

2. Trapezoidal rule

The trapezoid rule approximates the area under the curve as a trapezoid and calculates its area. The formula is:

I ≈ (B – A) × (F(A) + F(B)) / 2

Suppose we want to find the integral of the function f(x) = x^3 on [1, 2]:

f(1) = 1^3 = 1
f(2) = 2^3 = 8

Therefore, the integral is estimated as follows:

I ≈ (2 – 1) × (1 + 8) / 2 = 4.5

3. Simpson's rule

Simpson's rule is more accurate and uses a parabola to approximate the curve. Simpson's rule assumes that the function is quadratic. It is expressed as:

I ≈ (B – A) / 6 × [f(A) + 4f((A + B) / 2) + f(B)]

Let's try this for f(x) = x^2 over [0, 2]:

f(0) = 0^2 = 0
Midpoint: ((0 + 2) / 2) = 1; f(1) = 1^2 = 1
f(2) = 2^2 = 4

The approximate integral using Simpson's rule is:

I ≈ (2 – 0) / 6 × [0 + 4 × 1 + 4] = 2.67

4. Gaussian quadrature

Gaussian quadrature improves accuracy by choosing optimal sample points and their weights within the interval. These points and weights are obtained from the roots of orthogonal polynomials, such as Legendre polynomials. The general formula with n points is:

I ≈ ∑ wi × f( xi )

where w i are weights and x i are roots (nodes).

For example, using a 2-point Gaussian quadrature for ∫ -1 1 x^2 dx:

Roots: x 1 = -1/√3, x 2 = 1/√3
Weights: w 1 = w 2 = 1

So the integral can be approximated as:

I ≈ 1 × ((-1/√3)^2) + 1 × ((1/√3)^2) = 0.6667

Text example

Here are simple examples to show the application of each method:

Example 1: Midpoint rule

Let f(x) = e^x over [1, 2]. Approximate using the midpoint rule.

Midpoint: ((1 + 2) / 2) = 1.5
f(1.5) = e^1.5 ≈ 4.4817
Approximation: 4.4817 × (2 - 1) = 4.4817

Example 2: Trapezoidal rule

Find ∫ 0 π sin(x) dx using the Trapezoid Rule.

f(0) = sin(0) = 0
f(π) = sin(π) = 0
Approximation: π × (0 + 0) / 2 = 0

Example 3: Simpson's rule

Apply Simpson's rule to estimate ∫ 0 2 ln(x + 1) dx.

f(0) = ln(1) = 0
Midpoint f(1) = ln(2) ≈ 0.6931
f(2) = ln(3) ≈ 1.0986
Approximation: (2/6) × (0 + 4×0.6931 + 1.0986) ≈ 1.7627

Example 4: Gaussian quadrature

Estimate ∫ -1 1 x^4 dx using 2-point Gaussian quadrature.

Roots: x 1 = -1/√3, x 2 = 1/√3
Weights: w 1 = w 2 = 1
f(x 1 ) = ((-1/√3)^4), f(x 2 ) = ((1/√3)^4)
Approximation: 1 × f(x 1 ) + 1 × f(x 2 ) = 1/9 + 1/9 = 0.2222

Conclusion

Quadrature methods are an essential cornerstone of numerical analysis. They are widely used in a variety of fields, from physics to finance, wherever integral calculations are necessary but not possible to perform analytically. These methods make it possible to tackle problems involving complex functions and limit the analysis to numerical results that still provide important insights into the underlying phenomena. Each method has its own advantages and limitations, often depending on the nature of the function being integrated. Selecting the appropriate method can lead to more efficient and accurate calculations.


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Quadrature Methods

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