[qq,ll] = eig([x p]’*[v A*p],[x p]’*[u M*p]);
时间: 2023-09-10 14:15:21 浏览: 122
这是一个MATLAB中求解矩阵特征值和特征向量的示例代码,其中:
- 输入矩阵为[x p]'*[v A*p]和[x p]'*[u M*p]的组合,其中x、p、v、A、u和M均为已知矩阵或向量。
- 输出矩阵qq和向量ll分别为输入矩阵的特征向量和特征值。
具体代码如下:
```
x = rand(10,1);
p = rand(10,1);
v = rand(10,1);
A = rand(10,10);
u = rand(10,1);
M = rand(10,10);
M_inv = inv(M);
[V,D] = eig([x p]'*[v A*p], [x p]'*[u M*p]);
qq = [v A*p]*V*M_inv;
ll = diag(D);
```
该代码中,首先定义了矩阵x、p、v、A、u和M,并通过inv函数求出M的逆矩阵M_inv。然后调用eig函数求解[x p]'*[v A*p]和[x p]'*[u M*p]的特征向量和特征值,分别存储在V和D中。最后,根据特征向量和特征值计算出qq和ll。
需要注意的是,该代码中使用了MATLAB内置函数inv和eig,因此在使用前需要确保这两个函数已经被正确加载。此外,由于特征值和特征向量的计算涉及到矩阵的求逆和对角化,因此对于较大的矩阵,计算时间可能会比较长。
相关问题
将u = M*x; q = sqrt(x’*u); x = x/q; u = u/q; v = A*x; rho = x’*v; k = 0; g = x; gnorm = 1; log=[]; % Initialisierungen while gnorm > tol, k = k + 1; galt = g; if exist(’C’), g = 2*(C\(C’\(v - rho*u))); % vorkonditionierter Gradient else g = 2*(v - rho*u); % Gradient end if k == 1, p = -g; else p = -g + (g’*M*g)/(galt’*M*galt)*p; end [qq,ll] = eig([x p]’*[v A*p],[x p]’*[u M*p]); [rho,ii] = min(diag(ll)); delta = qq(2,ii)/qq(1,ii); x = x + delta*p; u = M*x; q = sqrt(x’*u); x = x/q; u = u/q; v = A*x; gnorm = norm(g); if nargout>2, log = [log; [k,rho,gnorm]]; end end转换成Fortran程序
下面是将MATLAB代码转换为Fortran的程序示例:
```
PROGRAM Rayleigh_quotient_iteration
IMPLICIT NONE
INTEGER :: k, ii, n, kmax, iter
REAL :: tol, rho, delta, q, gnorm, gnorm_old
REAL, DIMENSION(:,:), ALLOCATABLE :: A, M, x, u, v, g, galt, p
REAL, DIMENSION(:), ALLOCATABLE :: ll, qq, log
LOGICAL :: C_exist
! Read in the size of the matrices
READ(*,*) n
! Read in the matrices A and M
ALLOCATE(A(n,n), M(n,n))
DO i = 1, n
READ(*,*) (A(i,j), j = 1, n)
END DO
DO i = 1, n
READ(*,*) (M(i,j), j = 1, n)
END DO
! Set up the initial guess for x
ALLOCATE(x(n))
x = 1.0
! Set up the other variables
kmax = 100
tol = 1E-6
iter = 0
! Main loop
DO
! Compute u = M*x and q = sqrt(x'u)
ALLOCATE(u(n))
u = MATMUL(M, x)
q = SQRT(DOT_PRODUCT(x, u))
x = x / q
u = u / q
! Compute v = A*x and rho = x'v
ALLOCATE(v(n))
v = MATMUL(A, x)
rho = DOT_PRODUCT(x, v)
! Compute the initial gradient g
ALLOCATE(g(n))
g = 2.0 * (v - rho*u)
gnorm = NORM2(g)
gnorm_old = gnorm
! Initialize the log array
ALLOCATE(log(3, kmax))
log(:,1) = [0, rho, gnorm]
! Main iteration loop
DO k = 1, kmax
! Compute the search direction p
IF (k == 1) THEN
p = -g
ELSE
p = -g + DOT_PRODUCT(g, MATMUL(M, g)) / DOT_PRODUCT(galt, MATMUL(M, galt)) * p
END IF
! Compute the eigenvalues and eigenvectors of [x p]'[v A*p] / [x p]'[u M*p]
ALLOCATE(qq(2,2), ll(2))
qq(1,:) = x
qq(2,:) = p
ll = EIG(MATMUL(TRANSPOSE(qq),MATMUL([v,A*p],qq)), MATMUL(TRANSPOSE(qq),MATMUL([u,M*p],qq)))
! Update x and u
ii = INDEX(MINVAL(ll))
delta = qq(2,ii) / qq(1,ii)
x = x + delta * p
u = MATMUL(M, x)
! Compute the new gradient g
galt = g
v = MATMUL(A, x)
rho = DOT_PRODUCT(x, v)
g = 2.0 * (v - rho*u)
gnorm_old = gnorm
gnorm = NORM2(g)
! Check for convergence
IF (gnorm <= tol) EXIT
! Update the log array
iter = iter + 1
log(:,iter+1) = [k, rho, gnorm]
END DO
! Print the final result
WRITE(*,*) "Lambda = ", rho
WRITE(*,*) "Number of iterations = ", k
! Deallocate the arrays
DEALLOCATE(A, M, x, u, v, g, galt, p, ll, qq, log)
! Exit the program
EXIT
END DO
END PROGRAM Rayleigh_quotient_iteration
```
请注意,这只是一个简单的程序示例,可能需要进行调整才能适合您的特定情况。此外,该程序还假定您已经了解Rayleigh quotient iteration算法,因此不包括该算法的详细解释。
[qq,ll] = eig([x p]’*[v A*p],[x p]’*[u M*p]);转换成Fortran代码
下面是将该公式转换成Fortran代码的示例:
```
program eig_example
implicit none
integer, parameter :: n = 10 ! 矩阵维度
real, dimension(n,n) :: A, M, u, v, x, p
real, dimension(n+1,n+1) :: mat1, mat2
real, dimension(n) :: qq, ll
integer :: i, j, info
! 初始化矩阵和向量
A = reshape([ (i+j, i=1,n), j=1,n ], [n,n])
M = reshape([ (i-j, i=1,n), j=1,n ], [n,n])
u = reshape([ (i, i=1,n) ], [n,1])
v = reshape([ (1.0/i, i=1,n) ], [n,1])
x = reshape([ (1.0/(i+j), i=1,n) ], [n,1])
p = reshape([ (1.0/(i-j), i=1,n) ], [n,1])
! 构造矩阵 mat1 和 mat2
mat1(1:n,1:n) = A
mat1(1:n,n+1) = p
mat1(n+1,1:n) = transpose(p)
mat1(n+1,n+1) = 0.0
mat2(1:n,1:n) = M
mat2(1:n,n+1) = p
mat2(n+1,1:n) = transpose(p)
mat2(n+1,n+1) = 0.0
! 计算 qq 和 ll
call dgemv('T', n+1, n+1, 1.0, mat1, n+1, [x, transpose(p)], 1, 0.0, qq, 1)
call dgemv('T', n+1, n+1, 1.0, mat2, n+1, [u, transpose(p)], 1, 0.0, ll, 1)
call dgeev('N', 'V', n, qq, n, ll, 0.0, 1.0, n, info)
! 输出结果
print *, "qq = ", qq
print *, "ll = ", ll
end program eig_example
```
在上面的代码中,我们首先定义了矩阵和向量的维度和值,然后构造了两个矩阵 mat1 和 mat2,最后使用 BLAS 和 LAPACK 提供的子例程计算 qq 和 ll。具体来说,我们使用了 dgemv 子例程计算矩阵向量积,使用 dgeev 子例程计算特征值和特征向量。最后,我们输出了计算结果。
请注意,上面的代码仅供参考,具体实现方式可能与您的实际情况不完全相同。在编写实际代码时,请根据您自己的需求进行修改和调整。
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