如何在Matlab中使用最小二乘法? [英] How to use least squares method in Matlab?

问题描述

我有37个线性方程和36个矩阵方程形式的变量； A * X = B.这些方程式没有确切的答案.我想使用Matlab最小二乘方法找到错误最少的答案.我是Matlab的新手，所以任何评论都将对您有所帮助.谢谢

I have 37 linear equations and 36 variables in the form of a matrix equation; A*X=B . The equations don't have an exact answer. I want to use Matlab least square method to find the answers with the least error. I am new to Matlab so any comments will help. Thank you

推荐答案

如果A具有完整等级，即A的列是线性独立的，则超定线性方程组的最小二乘解

If A is of full rank, i.e. the columns of A are linearly independent, the least-squares solution of an overdetermined system of linear equations

A * x = b

可以通过反正态方程来找到

(请参见线性最小二乘):

can be found by inverting the normal equations (see Linear Least Squares):

x = inv(A' * A) * A' * b

如果A的等级不完整，则A' * A不可逆.相反，可以使用A

If A is not of full rank, A' * A is not invertible. Instead, one can use the pseudoinverse of A

x = pinv(A) * b

或Matlab的左分区运算符

or Matlab's left-division operator

x = A \ b

两者都提供相同的解决方案，但左除法的计算效率更高.

Both give the same solution, but the left division is more computationally efficient.

后两种计算方法也可以在确定的线性方程组下处理，但是在那种情况下它们给出不同的解:伪逆给出x的平方和最小的解. ，而左除运算符则给出了一个尽可能多的0系数的解决方案.

The two latter computation methods can also deal with underdetermined systems of linear equations, but they give different solutions in that case: The pseudoinverse gives the solution where x has the smallest sum of squares, while the left-division operator gives a solution with as many 0 coefficients as possible.

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