ARPACK Overview

ARPACK Fortran Framework

ARPACK is a Fortran software library for solving large-scale eigenvalue problems. It provides efficient and reliable algorithms for computing a few eigenvalues and eigenvectors of sparse matrices. ARPACK is widely used in various scientific and engineering applications, such as structural analysis, fluid dynamics, and quantum mechanics.

History

ARPACK was originally developed by Richard Lehoucq, Danny Sorensen, and Chao Yang at the Institute for Computational Sciences and Informatics in Louisiana State University. The first version of ARPACK, known as ARPACK77, was released in 1992. Over the years, ARPACK has undergone several updates and improvements, with the latest version being ARPACK-ng (Next Generation).

Features

ARPACK offers several important features for solving eigenvalue problems:

1. Efficiency: ARPACK employs iterative methods that can handle large sparse matrices efficiently. It avoids the need to explicitly store the entire matrix, making it suitable for problems with millions or even billions of unknowns.

2. Flexibility: ARPACK supports both standard eigenvalue problems and generalized eigenvalue problems. It can compute a few eigenvalues and eigenvectors (known as a partial spectrum) or the entire spectrum of a matrix.

3. Adaptive Algorithms: ARPACK dynamically adjusts its algorithmic parameters based on the properties of the matrix being solved. This adaptivity improves the convergence rate and overall efficiency of the computations.

4. Parallelism: ARPACK can be used in parallel computing environments, taking advantage of multiple processors or distributed computing systems.

5. Interface: ARPACK provides interfaces for various programming languages, including Fortran, C, and MATLAB. This allows users to easily integrate ARPACK into their existing codebases.

Examples

To illustrate the usage of ARPACK, let's consider a simple example of finding the eigenvalues and eigenvectors of a sparse matrix using the ARPACK Fortran interface.

program arpack_example
  implicit none
  integer, parameter :: n = 1000  ! Size of the matrix
  integer, parameter :: nev = 5   ! Number of eigenvalues to compute
  integer :: ido, info
  real :: tol = 1e-6
  real, dimension(n) :: resid
  integer, dimension(n+1) :: iparam
  integer :: ipntr(11)
  real, dimension(n) :: workd, workl(3*n), dr, di
  logical :: select(n)
  real, dimension(n, nev) :: v
  integer :: ldv = n
  integer :: rvec = 1
  character(len=2) :: bmat = "I"
  character(len=3) :: howmny = "A"
  integer :: ierr, itype = 1
  integer :: iparam_ptr, ipntr_ptr, workd_ptr, workl_ptr, select_ptr
  integer :: i, j

  ! Initialize ARPACK parameters
  iparam = 0
  iparam(1) = 1     ! Use exact shift strategy
  iparam(3) = 100   ! Maximum number of iterations
  iparam(4) = 1     ! Block size

  ! Initialize workspace arrays
  workd = 0.0
  workl = 0.0
  select = .true.

  ! Generate a sparse matrix A

  ! Call ARPACK routine for eigenvalue computation
  call dsaupd(ido, bmat, n, "LM", nev, tol, resid, 1, iparam, ipntr, workd, workl, 3*n, info)

  do while (ido /= 99)
    if (ido == -1 || ido == 1) then
      ! Perform matrix-vector multiplication: workd(ipntr(1)) = A * workd(ipntr(2))
      call matrix_vector_multiply(workd(ipntr(1)), workd(ipntr(2)))
    endif

    ! Call ARPACK routine to update the subspace
    call dseupd(rvec, howmny, select, dr, di, v, ldv, sigma, bmat, n, "LM", nev, tol, resid, 1, iparam, ipntr, workd, workl, 3*n, ierr)

    ! Process eigenvalues and eigenvectors
    do i = 1, nev
      if (select(i)) then
        print *, "Eigenvalue ", i, ": ", dr(i)
        print *, "Eigenvector ", i, ":"
        do j = 1, n
          print *, v(j, i)
        enddo
      endif
    enddo

    ! Check convergence
    if (info /= 0) then
      print *, "ARPACK error: ", info
      exit
    endif

    if (ido == 99) exit

    ! Continue the iteration
    call dsaupd(ido, bmat, n, "LM", nev, tol, resid, 1, iparam, ipntr, workd, workl, 3*n, info)
  end do

  stop
end program arpack_example

In this example, we first initialize the ARPACK parameters and workspace arrays. Then, we generate a sparse matrix A. Finally, we call the dsaupd and dseupd routines to compute the eigenvalues and eigenvectors.

The output of this code will be the computed eigenvalues and eigenvectors of the matrix A.

For more information about ARPACK and its usage, you can visit the official website: ARPACK-ng

ARPACK provides a powerful and efficient tool for solving eigenvalue problems in large-scale scientific and engineering computations. Its flexibility and adaptivity make it a popular choice among researchers and practitioners in various fields.