Undergraduate
  1. Algebra  1.1. Linear Algebra  1.1.1. Matrices  1.1.2. Determinants  1.1.3. Systems of Linear Equations  1.1.4. Vector Spaces  1.1.5. Linear Transformations  1.1.6. Eigenvalues and Eigenvectors  1.1.7. Diagonalization  1.1.8. Inner Product Spaces  1.1.9. Orthogonality and Orthonormal Bases  1.1.10. Singular Value Decomposition  1.2. Abstract Algebra  1.2.1. Groups  1.2.2. Subgroups  1.2.3. Cyclic Groups  1.2.4. Permutation Groups  1.2.5. Homomorphisms  1.2.6. Normal Subgroups  1.2.7. Rings  1.2.8. Fields  1.2.9. Ideals and Quotient Rings  1.2.10. Polynomial Rings  1.2.11. Vector Spaces over Fields  1.2.12. Modules  1.2.13. Galois Theory
  2. Calculus  2.1. Differential Calculus  2.1.1. Limits  2.1.2. Continuity  2.1.3. Derivatives  2.1.4. Chain Rule  2.1.5. Implicit Differentiation  2.1.6. Applications of Derivatives  2.1.7. Optimization Problems  2.1.8. Mean Value Theorem  2.2. Integral Calculus  2.2.1. Definite Integrals  2.2.2. Indefinite Integrals  2.2.3. Techniques of Integration  2.2.4. Improper Integrals  2.2.5. Applications of Integration  2.2.6. Arc Length and Surface Area  2.3. Multivariable Calculus  2.3.1. Partial Derivatives  2.3.2. Gradient  2.3.3. Divergence and Curl  2.3.4. Multiple Integrals  2.3.5. Line Integrals  2.3.6. Surface Integrals  2.3.7. Green's Theorem  2.3.8. Stokes' Theorem  2.3.9. Divergence Theorem  2.3.10. Jacobians
  3. Differential Equations  3.1. Ordinary Differential Equations  3.1.1. First Order Differential Equations  3.1.2. Higher Order Differential Equations  3.1.3. Systems of Differential Equations  3.1.4. Series Solutions  3.1.5. Numerical Methods for ODEs  3.2. Partial Differential Equations  3.2.1. Heat Equation  3.2.2. Wave Equation  3.2.3. Laplace Equation  3.2.4. Separation of Variables  3.2.5. Fourier Transform Methods
  4. Real Analysis  4.1. Sequences and Series  4.1.1. Convergence of Sequences  4.1.2. Series Tests  4.1.3. Power Series  4.1.4. Uniform Convergence  4.2. Functions of Real Variables  4.2.1. Limits and Continuity  4.2.2. Uniform Continuity  4.2.3. Differentiation in Functions of Real Variables  4.2.4. Riemann Integration  4.2.5. Lebesgue Integration
  5. Complex Analysis  5.1. Complex Numbers  5.1.1. Polar Form  5.1.2. Exponential Form  5.2. Functions of a Complex Variable  5.2.1. Analytic Functions  5.2.2. Cauchy-Riemann Equations  5.2.3. Contour Integrals  5.2.4. Cauchy's Integral Theorem  5.2.5. Residue Theorem  5.2.6. Conformal Mapping
  6. Probability and Statistics  6.1. Probability Theory  6.1.1. Basic Probability Concepts  6.1.2. Random Variables  6.1.3. Probability Distributions  6.1.4. Expectation and Variance  6.1.5. Conditional Probability and Bayes' Theorem  6.2. Statistics  6.2.1. Descriptive Statistics  6.2.2. Inferential Statistics  6.2.3. Hypothesis Testing  6.2.4. Regression Analysis  6.2.5. ANOVA
  7. Numerical Methods  7.1. Root-Finding Algorithms  7.2. Numerical Integration  7.3. Numerical Differentiation  7.4. Numerical Solutions to Differential Equations  7.5. Matrix Computations  7.6. Interpolation and Extrapolation
  8. Discrete Mathematics  8.1. Logic and Proof  8.2. Combinatorics  8.3. Graph Theory  8.4. Boolean Algebra  8.5. Recurrence Relations
  9. Linear Programming and Optimization  9.1. Simplex Method  9.2. Duality  9.3. Integer Programming  9.4. Nonlinear Optimization
  10. Topology  10.1. Metric Spaces  10.2. Topological Spaces  10.3. Continuity and Homeomorphisms  10.4. Compactness  10.5. Connectedness
  11. Geometry  11.1. Euclidean Geometry  11.2. Differential Geometry  11.3. Non-Euclidean Geometry  11.4. Projective Geometry
  12. Mathematical Physics  12.1. Fourier Series  12.2. Laplace Transforms  12.3. Applications of Differential Equations  12.4. Special Functions
  13. Set Theory and Logic  13.1. Sets and Subsets  13.2. Relations and Functions  13.3. Cardinality  13.4. Formal Logic  13.5. Zermelo-Fraenkel Set Theory
  14. Functional Analysis  14.1. Normed Spaces  14.2. Banach Spaces  14.3. Hilbert Spaces  14.4. Linear Operators

Undergraduate ↓


Numerical Methods


Numerical methods are an essential set of tools and techniques widely used in many fields, especially in mathematics, engineering, and the physical sciences. They allow us to approximate the solutions of problems that cannot be easily solved analytically. When dealing with real-world situations, finding exact solutions is often challenging due to the complexity of models and equations. Here, numerical methods come to our aid, providing ways to obtain practical solutions.

Introduction to numerical methods

Numerical methods help make complex mathematical problems manageable by approximating solutions with numerical values. These methods are important because they provide a way to work with complex equations and systems that cannot be easily solved analytically. Instead of an exact solution, we obtain an approximate value, which is often satisfactory for practical purposes.

Basic concepts

Before discussing specific methods, let us understand some basic concepts that are fundamental to numerical methods. These include:

  • Numerical solution: An approximate solution to a mathematical problem that is in numerical form. It is often calculated using algorithms and iterative techniques.
  • Accuracy and precision: Accuracy refers to how close the numerical solution is to the exact solution, while precision refers to the detail in the expression of the solution.
  • Error: The difference between the numerical solution and the exact analytical solution. We classify errors as round-off errors and truncation errors.

General numerical methods

Let us discuss some of the common numerical methods used in undergraduate mathematics courses with examples and explanations:

1. Methods for finding the root

Finding the roots involves finding x values, such that f(x) = 0. These methods are used to solve equations where it is difficult to obtain the exact solution. Some of the widely used methods are as follows:

Bisection method

The bisection method is a simple and robust method for finding roots. It is based on the intermediate value theorem, which states that if a continuous function changes sign between two points a and b, then there exists at least one root between them.

    A continuous function f is given in the interval [a, b]:
    1. Check if the signs of f(a) and f(b) are opposite. If not, the method cannot be applied.
    2. Calculate the midpoint c = (a + b) / 2.
    3. If f(c) is sufficiently close to 0, then c is a root.
    4. Otherwise, if f(a) * f(c) < 0, replace b with c. If f(c) * f(b) < 0, replace a with c.
    5. Repeat until desired accuracy is achieved.
C f(a) f(b)

Newton–Raphson method

This is an iterative method that uses the concept of tangents to find successive approximations of the roots. It is very powerful and converges quickly if the initial guess is close to the real root.

    To apply the Newton-Raphson method:
    1. Start with an initial guess x_0.
    2. Calculate the next approximation using the formula:
       x_{n+1} = x_n - f(x_n) / f'(x_n)
    3. Repeat until the estimate is very close to zero.
x1

2. Numerical integration

In many practical situations, we need to find the definite integral of a function when it is not easy to find the antiderivative. Numerical integration methods help us here, which yield approximate values for integrals.

Trapezoidal rule

This technique approximates the area under the graph of a function as a series of trapezoids and calculates the area of these trapezoids.

    For a function f defined on [a, b] and n subdistributions:
    1. Calculate the width of each interval h = (b - a) / n.
    2. Evaluate the function at the endpoints a and b and the intermediate points.
    3. Calculate the integral using:
       
Integral ≈ (h/2) * (f(a) + 2 * (sum of f at intermediate points) + f(b))

Simpson's rule

Simpson's rule provides a better estimate by dividing the area under the curve into parabolic segments.

    An even number n of subdivisions is given:
    1. Divide the interval [a, b] into n subintervals of equal width h = (b - a) / n.
    2. Evaluate the function at these points.
    3. Calculate the integral using:
       
Integral ≈ (h/3) * (f(a) + 4 * (sum of odd indexed f) + 2 * (sum of even indexed f) + f(b))

3. Numerical differentiation

Numerical differentiation is about finding the derivative of a function based on discrete data points, which is quite useful when the explicit expression of the derivative is not known.

Finite difference method

This is a general method of estimating derivatives from discrete data points using their function values. Here are some basic formulas:

    Given a small step size h:
    Further difference:
    f'(x) ≈ (f(x + h) - f(x)) / h

    Backward Difference:
    f'(x) ≈ (f(x) - f(x - h)) / h

    Central Difference:
    
f'(x) ≈ (f(x + h) - f(x - h)) / (2 * h)
X X+H XH

4. Solving linear systems

Numerical methods also help in solving systems of linear equations using iterative methods, especially when they are large and complex.

Gaussian elimination

This method involves a sequence of operations to transform a given system of linear equations into upper triangular form and then substitution is performed to obtain the solution.

    Steps for Gaussian Elimination:
    1. Select a pivot for each column and interchange the rows if necessary to get the largest absolute value.
    2. Make all entries below the pivot in the current column zero by row operation.
    3. Perform backward substitution to solve for the unknown.

Gauss–Seidel method

This is an iterative refinement scheme that improves the form of the initial guess to give an approximate solution.

    A system is given Ax = b:
    1. Start with an initial guess x_0.
    2. Iteratively solve each equation for the relevant variables, using the updated values as soon as the calculations for them are completed.
    3. Continue until the difference between successive iterations is less than the desired tolerance.

Conclusion

Numerical methods is a rich field that includes many different techniques for solving mathematical problems numerically rather than analytically. They play an important role in practical applications in various fields by providing approximations for otherwise difficult mathematical problems.


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