用Jacobi迭代法来求解Sylvester 方程

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翻新时间：2023-08-06

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用Jacobi迭代法来求解Sylvester 方程

摘要

众所周知，许多实际的问题最后常归纳为解1个或1些大型稀疏矩阵的线性代数方程组，而对这些方程组1般采用迭代法求解。本文介绍了Jacobi 迭代法，作为1种简单的迭代法，Jacobi迭代法具有算法和计算简单的特点。对于某些矩阵，Jacobi迭代法的收敛速度相当的快，同时相比其他的迭代法，Jacobi迭代法不需要求解复杂矩阵的逆矩阵，从而少了很多的工作量。

我们针对Sylvester 方程（希尔维斯特方程）讨论了它的Jacobi迭代格式。从分析上来看，Jacobi迭代法计算简单，每迭代1次只需要计算1次矩阵和向量的乘积，迭代方法从理论上讲是有效的。同时，我们也给出两个数值例子，用Jacobi迭代法得出方程在不同精度下的解。这说明Jacobi迭代法在实际操作中是可行的。

关键词：Sylvester 方程（希尔维斯特方程）；Jacobi迭代

Jacobi iterative method used to solve equations Sylvester

Abstract

As we know, many practical problems often sum up as the final solution or a large sparse matrix of linear algebraic equations. of these equations general iterative method. We introduces the Jacobi iterative method as a simple iterative method, Jacobi iterative method and calculation algorithm is simple. For some matrix, Jacobi iteration speed of convergence very soon, but other than the iterative method, Jacobi iterative method does not need to solve the complex matrix inverse matrix, thus lose a lot of the workload.

We focused Sylvester equation to discuss its Jacobi iterative format. From the analysis, Jacobi iterative method is simple, each iteration one only need to calculate a matrix and vector product, iterative methods from the theoretical point of view is valid. Meanwhile, we are given two numerical examples, with Jacobi iteration equations derived in the accuracy of different solutions. This shows Jacobi iteration in the actual operation is feasible.

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