Numerical analysis is the field that studies the use of numerical computational methods to solve mathematical problems. When using high-level languages for programming in competitions, corresponding library functions can be written to implement algorithms commonly used in numerical analysis, such as equation system solving, matrix operations, and function integration. Below I will introduce some commonly used numerical analysis algorithms and library functions that may need to be written.
- Solve the system of equations:
Gaussian Elimination: A common method for solving systems of linear equations, which can be implemented by writing functions.
LU Decomposition: Decompose the coefficient matrix into the product of a lower triangular matrix and an upper triangular matrix, and perform pre/posterior substitution to solve a system of linear equations.
Newton’s Method: An iterative method for solving nonlinear systems of equations, which can be implemented by writing functions.
- Matrix Operations:
Matrix multiplication: Write a function to implement the multiplication operation of two matrices. You can use loops or vectorization to improve efficiency.
Matrix inversion: Use Gaussian elimination method, LU decomposition method or other more efficient methods to write functions to implement inversion operations.
Eigenvalue and eigenvector calculation: Write functions to calculate the eigenvalues and eigenvectors of a matrix.
- Function integral:
Numerical integration methods: such as trapezoidal rule, Simpson's rule, etc. Functions can be written according to requirements to implement numerical integration.
Adaptive integration: Write functions to implement adaptive methods and dynamically adjust the integration step to improve accuracy.
These algorithms are only a part of numerical analysis. There are other algorithms and techniques that can also be written according to the requirements of specific problems. When writing library functions, pay attention to the correctness, efficiency, and stability of the algorithm, and conduct necessary testing and verification. You can use the syntax and data structures of high-level languages to implement algorithms, and combine them with theoretical knowledge of numerical analysis for programming.