It is the origin stage of mathematics, including the mathematics of ancient Egypt and ancient Babylon. Mathematics mainly deals with counting, measurement and geometry.

2. The period of classical mathematics

6th century BC ~ 16th century AD

Mathematics in ancient Greece reached its peak. Mathematicians such as Pythagoras, Euclid, and Archimedes made important contributions to areas such as geometry, number theory, and mechanics.

3. Early Modern

From the 17th to the 18th century AD,

The establishment of calculus and the development of analysis. Mathematicians such as Newton, Leibniz, and others made important contributions in areas such as calculus, number theory, and algebraic equations.

4. Late modern times

In the 19th century, a bunch of old European men:

Important progresses such as the strictness of the infinitesimal and the establishment of the limit concept. European mathematicians promoted the development of mathematics during this period, including Cauchy, Riemann, Poincaré and Gödel.

5. Modern Mathematics

From the 20th century to the present, scientific computing and intelligent revolution:

Modern mathematics includes many subfields, such as applied mathematics, mathematical analysis, abstract algebra, topology, geometry, probability theory and statistics.

Second, what is computational methodology?

1. Numerical Algebra

a. Solving linear algebraic equations (equivalent transformation)

Through equivalent transformation, the system of equations can be transformed into a matrix form, that is, PAx = Pb, where P is a column exchange matrix, A is a coefficient matrix, x is an unknown vector, and b is a constant vector. Numerical methods such as Gaussian elimination method, LU decomposition, and iterative methods (such as Jacobi method, Gauss-Seidel iteration) can be used to solve linear equations.

b. Matrix eigenvalue eigenvector (similar transformation)

The eigenvalues represent the scaling factors of the matrix, while the eigenvectors represent the corresponding directions. Through the similarity transformation, the matrix can be transformed into a diagonal form, where the diagonal elements are eigenvalues and the corresponding column vectors are eigenvectors. Eigenvalues and eigenvectors have important applications in many fields such as numerical methods, data analysis, and physics.

c. Quadratic (contract conversion)

A quadratic form is a polynomial function consisting of quadratic terms. In numerical algebra, it is important to study the properties and changes of quadratic forms for optimization problems and matrix decomposition. Through the contract transformation, the quadratic form can be transformed into a canonical form, which makes solving and analyzing problems more convenient.

2. Numerical approximation

How to represent functions, numerical integration numerical differentiation

Numerical Approximation refers to the approximate solution of complex mathematical problems through numerical methods and computer technology. In practical applications, many mathematical problems cannot be solved accurately by analytical methods, so it is necessary to use numerical approximation methods to approximate solutions or calculate their numerical results.

a. Interpolation

Constructs an approximate function by known function values at data points such that the approximate function passes through these data points, and the function value between these points is unknown. Common interpolation methods include Lagrange interpolation and Newton interpolation.

b. Curve Fitting

Fit a function that fits the trend of the data through known data points. The goal of curve fitting is to find a simple function that minimizes its error from known data points. Common curve fitting methods include least squares and polynomial fitting.

c. Numerical integration

Computes the integral of a function over a specified interval numerically. Numerical integration methods can transform the integral problem of continuous functions into discrete numerical calculation problems, including trapezoidal rule, Simpson's rule and Romberg integral.

d. Numerical differentiation

Compute the derivative or differentiation of a function using numerical methods. Numerical differentiation methods can estimate derivative values by computing the difference of a function at discrete points, such as forward difference, backward difference, and central difference.

e. Iterative method

Iterative approach is used to gradually approach the solution of the problem until a certain convergence condition is met. Iterative methods are widely used in solving nonlinear equations, linear equations and optimization problems.

f. Approximate solution of ordinary differential equations

Approximately solve ordinary differential equations by numerical methods, such as Euler method, Runge-Kutta method, etc.

The choice of numerical approximation method depends on the characteristics and requirements of the specific problem. By properly selecting and combining these methods, numerical results that meet the requirements of precision and efficiency can be obtained in practical applications.

3. Numerical optimization

min f(x)

Training Neural Networks - Alchemy

Numerical optimization is a technique for finding the optimal solution of a function through numerical methods. In practical problems, we often need to find the minimum or maximum value of a function, and numerical optimization provides a series of algorithms and techniques to solve these problems. Numerical optimization has a wide range of applications in various fields, including machine learning, data analysis, engineering design, etc.

Some commonly used numerical optimization algorithms and methods:

a. Modeling of optimization problems

Transform practical optimization problems into mathematical formulations, defining objective functions and constraints. The objective function can be a function to be minimized or maximized, and the constraints can be equality constraints or inequality constraints.

b. Gradient descent method:

The gradient descent method is an iterative optimization algorithm that gradually approaches the optimal solution by updating the solution vector along the opposite direction of the gradient of the function. This method is particularly suitable for optimization problems of convex and differentiable functions.

Gradient descent～convex function～global (or trapped in local minimum)

c. Newton method and quasi-Newton method

Newton's method is an optimization method based on second-order derivative information, which finds the minimum point by solving the equation system. The quasi-Newton method is an approximate Newton method, which does not need to calculate the second-order derivative matrix, but uses the information of historical iteration points to approximate the second-order information.

d. Evolutionary Algorithms

Evolutionary algorithm is a kind of optimization method based on the principle of biological evolution, such as genetic algorithm, particle swarm optimization algorithm, ant colony algorithm, etc. These methods gradually search for the optimal solution in the solution space by simulating the evolution process in nature.

e. Constrained optimization

In constrained optimization problems, it is necessary to find the optimal solution under given constraints. Commonly used constrained optimization methods include penalty function method, Lagrange multiplier method and KKT condition, etc.

f. Global optimization

In global optimization problems, it is necessary to find the global optimal solution of the function, not just the local optimal solution. Global optimization methods include genetic algorithm, simulated annealing algorithm and particle swarm optimization.

4. Numerical solution

a. Ordinary Differential Equations (ODEs)

$eq?y%27%27+py%27+y%3D0$

$eq?y0%3Dy%270%3D0$

Euler's method: This method uses discrete steps to approximate the solution of the ODE.
Runge-Kutta method: This is a higher order numerical method, including 4th order and higher order versions.
Runge-Kutta method (RK4): This is a classic 4th order Runge-Kutta method that achieves high accuracy by approximating the solution step by step.
Multi-step methods: such as Adams-Bashforth and Adams-Moulton methods, these methods use multiple historical data points to approximate the solution.

b. Partial Differential Equations (PDEs)

Finite Difference Method: This is a common and simple numerical method that discretizes the PDE into a grid and calculates the solution in the form of an approximate difference.
Finite Element Method: This is a more advanced numerical method that discretizes the PDE into small subdomains and uses basis functions to approximate the solution.
Finite Volume Method: This is a numerical method commonly used in fluid dynamics and heat transfer. It discretizes the PDE into a control volume and calculates the flux.

Third, what kind of algorithm is a good algorithm?

"able"

Being able to solve real problems means being able to produce correct results.

"allow"

The result of the algorithm has high precision and can meet the requirements or expectations of the problem.

The current neural network has a high accuracy rate, but it still needs to be judged

"quick"

The execution speed of the algorithm is fast and it is able to give the result in a reasonable time frame.

These three characteristics usually restrict each other, and it is difficult to achieve the best state at the same time. In practical applications, it is necessary to weigh these three features and select the most appropriate algorithm according to specific needs and scenarios. Sometimes it may be necessary to choose between "quasi" and "fast".

【Numerical Calculation Methods】Introduction

1. The History of Minimalist Mathematics

1. The budding period