标题: 范德蒙矩阵（Vandermonde 矩阵）简介：意义、用途及编程应用 [打印本页] 作者: 东湖之滨 时间: 前天 20:49 标题: 范德蒙矩阵（Vandermonde 矩阵）简介：意义、用途及编程应用 参考：
Introduction to Applied Linear Algebra – Vectors, Matrices, and Least Squares
Stephen Boyd and Lieven Vandenberghe
Vandermonde 矩阵是一种由给定点生成的矩阵，其形式如下：
A = [ 1 t 1 t 1 2 ⋯ t 1 n − 1 1 t 2 t 2 2 ⋯ t 2 n − 1 ⋮ ⋮ ⋮ ⋱ ⋮ 1 t m t m 2 ⋯ t m n − 1 ] , A = \begin{bmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^{n-1} \\ 1 & t_2 & t_2^2 & \cdots & t_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & t_m & t_m^2 & \cdots & t_m^{n-1} \end{bmatrix}, A= 11⋮1t1t2⋮tmt12t22⋮tm2⋯⋯⋱⋯t1n−1t2n−1⋮tmn−1 ,
其中：
( t 1 , t 2 , … , t m t_1, t_2, \dots, t_m t1,t2,…,tm ) 是指定的 ( m m m ) 个点；
( n n n ) 是多项式的最高次数加 1；
矩阵的每一行对应于一个点 ( t i t_i ti ) 在不同幂次下的值。
如果将多项式写成系数形式：
p ( t ) = c 1 + c 2 t + c 3 t 2 + ⋯ + c n t n − 1 , p(t) = c_1 + c_2t + c_3t^2 + \cdots + c_nt^{n-1}, p(t)=c1+c2t+c3t2+⋯+cntn−1,
Vandermonde 矩阵可以用来表示多项式在多个点 ( t 1 , t 2 , … , t m t_1, t_2, \dots, t_m t1,t2,…,tm ) 的值。其矩阵形式为：
y = A c , y = Ac, y=Ac,
其中：
( c = [ c 1 , c 2 , … , c n ] T c = [c_1, c_2, \dots, c_n]^T c=[c1,c2,…,cn]T ) 是多项式的系数向量；
( y = [ p ( t 1 ) , p ( t 2 ) , … , p ( t m ) ] T y = [p(t_1), p(t_2), \dots, p(t_m)]^T y=[p(t1),p(t2),…,p(tm)]T ) 是多项式在 ( m m m ) 个点上的值。
Introduction to Vandermonde Matrix: Significance, Uses, and Programming Applications
The Vandermonde matrix is a structured matrix widely used in polynomial interpolation, evaluation, and linear algebra problems. Named after the French mathematician Alexandre-Théophile Vandermonde, it plays an important role in simplifying computations in both mathematical and programming contexts. In this blog, we will introduce the definition, significance, and applications of the Vandermonde matrix, along with examples of its practical use in programming. 1. What is a Vandermonde Matrix?
Definition
A Vandermonde matrix is a matrix generated from a set of given points. It takes the following form:
A = [ 1 t 1 t 1 2 ⋯ t 1 n − 1 1 t 2 t 2 2 ⋯ t 2 n − 1 ⋮ ⋮ ⋮ ⋱ ⋮ 1 t m t m 2 ⋯ t m n − 1 ] , A = \begin{bmatrix} 1 & t_1 & t_1^2 & \cdots & t_1^{n-1} \\ 1 & t_2 & t_2^2 & \cdots & t_2^{n-1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & t_m & t_m^2 & \cdots & t_m^{n-1} \end{bmatrix}, A= 11⋮1t1t2⋮tmt12t22⋮tm2⋯⋯⋱⋯t1n−1t2n−1⋮tmn−1 ,
where:
( t 1 , t 2 , … , t m t_1, t_2, \dots, t_m t1,t2,…,tm ) are the ( m m m ) given points;
( n n n ) is the degree of the polynomial plus 1;
Each row corresponds to a point ( t i t_i ti ) raised to increasing powers.
For a polynomial written as:
p ( t ) = c 1 + c 2 t + c 3 t 2 + ⋯ + c n t n − 1 , p(t) = c_1 + c_2t + c_3t^2 + \cdots + c_nt^{n-1}, p(t)=c1+c2t+c3t2+⋯+cntn−1,
the Vandermonde matrix can represent the polynomial’s evaluation at multiple points. Specifically, in matrix-vector form:
y = A c , y = Ac, y=Ac,
where:
( c = [ c 1 , c 2 , … , c n ] T c = [c_1, c_2, \dots, c_n]^T c=[c1,c2,…,cn]T ) is the vector of polynomial coefficients,
( y = [ p ( t 1 ) , p ( t 2 ) , … , p ( t m ) ] T y = [p(t_1), p(t_2), \dots, p(t_m)]^T y=[p(t1),p(t2),…,p(tm)]T ) is the vector of polynomial values at ( m m m ) points.
Intuitive Explanation
Each row of the Vandermonde matrix represents the powers of a single point ( t i t_i ti ), while multiplying the matrix by the coefficient vector ( c c c ) computes the polynomial values at all points ( t 1 , t 2 , … , t m t_1, t_2, \dots, t_m t1,t2,…,tm ). 2. Significance and Uses of Vandermonde Matrix
Significance
The Vandermonde matrix provides a structured and efficient way to handle polynomial operations, including evaluation, interpolation, and fitting. Its significance lies in its ability to simplify otherwise computationally intensive tasks.
Applications
Polynomial Evaluation
The Vandermonde matrix enables quick computation of polynomial values at multiple points simultaneously, which is useful in numerical analysis and modeling.
Polynomial Interpolation
It is used to solve interpolation problems by finding the polynomial coefficients ( c c c ) that satisfy ( A c = y Ac = y Ac=y ), where ( y y y ) contains the known function values at specific points.
Linear Algebra and Eigenvalue Problems
In specific conditions, the Vandermonde matrix is non-singular, making it useful in solving systems of linear equations.
Signal Processing
Vandermonde matrices appear in Fourier transforms and spectrum analysis, especially when working with discrete points in polynomial or sinusoidal bases.
3. Programming Applications
Generating a Vandermonde Matrix
Using NumPy in Python
Python’s numpy library provides a convenient function numpy.vander() for generating Vandermonde matrices:
import numpy as np
# Define the points
t = np.array([1, 2, 3, 4])
# Generate a Vandermonde matrix
A = np.vander(t, N=4, increasing=True)
print(A)
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Output:
[[ 1 1 1 1]
[ 1 2 4 8]
[ 1 3 9 27]
[ 1 4 16 64]]
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Using MATLAB
MATLAB has a built-in vander() function:
t = [1, 2, 3, 4];
A = vander(t);
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Practical Example: Polynomial Evaluation
Once the Vandermonde matrix is generated, you can use it to evaluate a polynomial at multiple points:
These are the values of ( p ( t ) p(t) p(t) ) at ( t = 1 , 2 , 3 , 4 t = 1, 2, 3, 4 t=1,2,3,4).
Polynomial Interpolation
If you know the values ( y y y ) at specific points ( t t t ) and need to find the polynomial coefficients ( c c c ), you can solve the system ( A c = y Ac = y Ac=y ):
from numpy.linalg import solve
# Known points and values
t = np.array([1, 2, 3])
y = np.array([2, 3, 5])
# Construct the Vandermonde matrix
A = np.vander(t, N=3, increasing=True)
# Solve for the coefficients
c = solve(A, y)
print(c)
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The output ( c c c ) contains the coefficients of the interpolating polynomial. 4. Real-World Applications
Engineering Computations
Vandermonde matrices are commonly used to fit models to real-world data. For instance, in sensor calibration, you may use polynomial fitting to model a sensor’s response curve.
Machine Learning
In kernel-based machine learning methods (e.g., polynomial kernels), the Vandermonde matrix acts as a feature mapping tool.
Signal Processing and Communication
In spectral analysis and discrete Fourier transform (DFT), Vandermonde matrices are essential for mapping discrete points to their polynomial or sinusoidal bases.
Numerical Integration and Interpolation
Vandermonde matrices play a critical role in Lagrange and Newton interpolation methods, which are widely used in numerical integration tasks.
5. Conclusion
The Vandermonde matrix is a structured and powerful tool for polynomial evaluations and interpolations. By converting polynomial operations into matrix operations, it provides a clean and efficient approach to solving various mathematical and computational problems. With tools like NumPy and MATLAB, generating and applying Vandermonde matrices becomes straightforward, enabling their use in a wide range of fields such as engineering, machine learning, and signal processing.
Understanding the Vandermonde matrix not only helps simplify mathematical operations but also enhances your ability to apply it effectively in real-world scenarios.
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