计算方法6_矩阵特征值和特征向量[宣贯].pptVIP

6.6 其它方法 平行迭代法：可求出前几个较大的特征值和特征测量，适用于高阶对称稀疏矩阵 QR算法：基于任何实非奇异矩阵都可分解为正交矩阵和上三角矩阵的乘积，适用于任意实非奇异矩阵的全部特征值 Jacobi法：用平面旋转矩阵构成的正交相似变换将对称矩阵化为对角形，适用于实对称矩阵 * 精品PPT | 实用可编辑 Power Method The basic computation of the power method is summarized as The equation can be written as: * 精品PPT | 实用可编辑 Power Method The basic computation of the power method is summarized as The equation can be written as: * 精品PPT | 实用可编辑 Shift method It is possible to obtain another eigenvalue from the set equations by using a technique known as shifting the matrix. Subtract the a vector from each side, thereby changing the maximum eigenvalue * 精品PPT | 实用可编辑 Shift method The eigenvalue, s, is the maximum value of the matrix A. The matrix is rewritten in a form. Use the Power method to obtain the largest eigenvalue of [B]. * 精品PPT | 实用可编辑 Inverse Power Method The inverse method is similar to the power method, except that it finds the smallest eigenvalue. Using the following technique. * 精品PPT | 实用可编辑 Inverse Power Method The algorithm is the same as the Power method and the “eigenvector” is not the eigenvector for the smallest eigenvalue. To obtain the smallest eigenvalue from the power method. * 精品PPT | 实用可编辑 Accelerated Power Method The Power method can be accelerated by using the Rayleigh Quotient instead of the largest wk value. The Rayeigh Quotient is defined as: * 精品PPT | 实用可编辑 Accelerated Power Method The values of the next z term is defined as: The Power method is adapted to use the new value. * 精品PPT | 实用可编辑 QR factorization Another form of factorization A = Q*R Produces an orthogonal matrix (“Q”) and a right upper triangular matrix (“R”) Orthogonal matrix - inverse is transpose * 精品PPT | 实用可编辑 Why do we care? We can use Q and R to find eigenvalues 1. Get Q and R (A = Q*R) 2. Let A = R*Q 3. Diagonal elements of A are eigenvalue approximations 4. Iterate until converged QR factorization Note: QR eigenvalue method gives all eigenvalues si