An eigenvalue is a scalar that indicates how much the eigenvector is stretched during a linear transformation. An eigenvector is a non-zero vector that only changes by the scalar factor (the eigenvalue) when a linear transformation is applied to it.
In Fortran, you can use built-in libraries such as LAPACK (Linear Algebra PACKage) to calculate eigenvalues and eigenvectors. LAPACK provides routines like `DSYEV` for symmetric matrices, which can be called from Fortran code to perform these calculations.
The most commonly used library for eigenvalue and eigenvector computation in Fortran is LAPACK. LAPACK provides a comprehensive suite of routines for linear algebra operations, including eigenvalue and eigenvector calculations for various types of matrices.
Sure! Here is a basic example using LAPACK:
program eigen_example implicit none integer :: n, lda, info, lwork double precision, allocatable :: a(:,:), w(:), work(:) ! Define matrix size and allocate arrays n = 3 lda = n lwork = 3*n allocate(a(n,n), w(n), work(lwork)) ! Initialize matrix 'a' a = reshape([4.0d0, 1.0d0, -2.0d0, 1.0d0, 2.0d0, 0.0d0, -2.0d0, 0.0d0, 3.0d0], shape(a)) ! Call LAPACK routine to compute eigenvalues and eigenvectors call dsyev('V', 'U', n, a, lda, w, work, lwork, info) ! Check for successful execution if (info == 0) then print *, 'Eigenvalues: ', w print *, 'Eigenvectors: ', a else print *, 'Error: LAPACK dsyev routine failed with info = ', info end if ! Deallocate arrays deallocate(a, w, work)end program eigen_exampleThis program initializes a 3x3 matrix, computes its eigenvalues and eigenvectors using the LAPACK routine `DSYEV`, and prints the results.