使用向量化求解多个线性系统 [英] Solving multiple linear systems using vectorization

问题描述

很抱歉，如果这很明显，但我搜索了一会儿却没有找到任何东西(或错过了它).

Sorry if this is obvious but I searched a while and did not find anything (or missed it).

我正在尝试求解形式为 Ax = B 的线性系统，其中 A 是一个4x4矩阵，而 B 是一个4x1向量.

I'm trying to solve linear systems of the form Ax=B with A a 4x4 matrix, and B a 4x1 vector.

我知道对于单个系统，我可以使用mldivide来获取 x :x=A\B.

I know that for a single system I can use mldivide to obtain x: x=A\B.

但是我试图解决大量系统(可能> 10000)，并且我不愿意使用for循环，因为在许多MATLAB问题中，我被告知它比矩阵公式要慢得多.

However I am trying to solve a great number of systems (possibly > 10000) and I am reluctant to use a for loop because I was told it is notably slower than matrix formulation in many MATLAB problems.

然后我的问题是:有没有一种方法可以使用向量化将 A 4x4x N 和 B 矩阵4x N ?

My question is then: is there a way to solve Ax=B using vectorization with A 4x4x N and B a matrix 4x N ?

PS:我不知道它是否重要，但是 B 向量对于所有系统都是相同的.

PS: I do not know if it is important but the B vector is the same for all the systems.

推荐答案

您应该使用for循环.如果A保持不变并且B更改，则预先计算因式分解并重新使用它可能会有好处.但是对于您的问题，其中A发生变化，而B保持不变，除了解决N个线性系统外，别无选择.

You should use a for loop. There might be a benefit in precomputing a factorization and reusing it, if A stays the same and B changes. But for your problem where A changes and B stays the same, there's no alternative to solving N linear systems.

您也不必太担心循环的性能成本:

You shouldn't worry too much about the performance cost of loops either: the MATLAB JIT compiler means that loops can often be just as fast on recent versions of MATLAB.

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