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!
!-----------------------------------------------------------------------
!
!Purpose:
! find the eigenvalues and corresponding eigenvectors of equation
! (a-eval)*evect=0 using the power method with the shifted Hessenberg
! QR algorithm.
!
!Copyright:
! Copyright (C) 2020 Ecole Polytechnique de Montreal
! This library is free software; you can redistribute it and/or
! modify it under the terms of the GNU Lesser General Public
! License as published by the Free Software Foundation; either
! version 2.1 of the License, or (at your option) any later version.
!
!Author(s): A. Hebert
!
!Reference:
! G. E. Robles, "Implementing the QR algorithm for efficiently
! computing matrix eigenvalues and eigenvectors," Final Degree
! Dissertation in Mathematics, Universidad del Pais Vasco, Spain
! (2017).
!
!Parameters: input
! ndim dimensioned column length of A.
! n order of matrix A.
! A input matrix.
! maxiter maximum number of iterations.
!
!Parameters: output
! iter actual number of iterations.
! V eigenvector matrix.
! D eigenvalue diagonal matrix.
!
!-----------------------------------------------------------------------
!
subroutine ALHQR(ndim,n,A,maxiter,iter,V,D)
implicit none
!----
! Subroutine arguments
!----
integer, intent(in) :: ndim,n,maxiter
integer, intent(out) :: iter
real(kind=8), dimension(ndim,n), intent(in) :: A
complex(kind=8), dimension(n,n), intent(out) :: V,D
!----
! Local variables
!----
integer :: i,j,k,i1,i2,nset,ier,ii
complex(kind=8) :: kappa,p,qq,r,mu,csum,denom,m(2,2)
real(kind=8) :: sgn,tau,nu,normH,sum,AA(2,2),BB(2,2),CC(2,2)
real(kind=8), parameter :: eps=epsilon(A)
!----
! Allocatable arrays
!----
integer, allocatable, dimension(:) :: iset
real(kind=8), allocatable, dimension(:) :: c
complex(kind=8), allocatable, dimension(:) :: s,t,work1d
real(kind=8), allocatable, dimension(:,:) :: VR
complex(kind=8), allocatable, dimension(:,:) :: H,Q,work2d
!----
! Perform Householder transformation to upper Hessenberg form
!----
allocate(H(n,n), Q(n,n), VR(n,n-2))
H(:n,:n)=A(:n,:n)
do k = 1,(n-2)
VR(k+1:n,k) = real(H(k+1:n,k))
sgn = sign(1.0d0,VR(k+1,k))
VR(k+1,k) = VR(k+1,k) + sgn * sqdotv(VR(k+1:n,k))
sum = sqdotv(VR(k+1:n,k))
if(sum /= 0.d0) VR(k+1:n,k) = VR(k+1:n,k) / sum
H(k+1:n,k:n) = H(k+1:n,k:n) - 2.0d0 * matmul(reshape(VR(k+1:n,k),(/n-k, 1/)), &
reshape(matmul(VR(k+1:n,k),H(k+1:n,k:n)),(/1, n-k+1/)))
H(:,k+1:n) = H(:,k+1:n) - 2.0d0 * matmul(reshape(matmul(H(:,k+1:n),VR(k+1:n,k)),(/n, 1/)), &
reshape(VR(k+1:n,k),(/1, n-k/)))
enddo
!----
! Construct Q matrix
!----
Q(:n,:n) = 0.0D0
do j=1,n
Q(j,j)=1.0d0
enddo
do j = (n-2),1,-1
Q(j+1:n,:n) = Q(j+1:n,:n) - 2.0d0 * matmul(reshape(VR(j+1:n,j),(/n-j, 1/)), &
reshape(matmul(VR(j+1:n,j),Q((j+1):n,:)),(/1, n/)))
enddo
deallocate(VR)
!----
! Perform Schur factorization
!----
i2 = n
allocate(c(n),s(n),t(n))
c(:n)=0.0d0; s(:n)=0.0d0; t(:n)=0.0d0;
iter = 0
do
iter = iter + 1
if(iter > maxiter) then
call xabort('ALHQR: maximum number of iterations exceeded.')
endif
! Check subdiagonal for near zeros, deflating points. Finds deflating rows
! on a complex Schur form matrix.
i1 = i2
normH = sqdotm(abs(H(:n,:n)))
do
if(i1 == 1) exit
if(abs(H(i1,i1-1)) < eps*normH) then
H(i1,i1-1) = 0.0d0
if(i1 == i2) then
i2 = i1 - 1; i1 = i1 - 1;
else
exit
endif
else
i1 = i1 - 1
endif
enddo
!----
! End the function if H is upper triangular
!----
if(i2 == 1) exit
! Compute Wilkinson shift
kappa = H(i2,i2)
sum = abs(H(i2-1,i2-1)) + abs(H(i2-1,i2)) + abs(H(i2,i2-1)) + abs(H(i2,i2))
if(sum /= 0) then
qq = (H(i2-1,i2)/sum)*(H(i2,i2-1)/sum)
if(qq /= 0) then
p = 0.5*((H(i2-1,i2-1)/sum) - (H(i2,i2)/sum))
r = sqrt(p*p + qq);
if( (real(p)*real(r) + imag(p)*imag(r)) < 0 ) then
r = -r
endif
kappa = kappa - sum*(qq/(p+r))
endif
endif
! Apply shift to the element of the diagonal that is left out of the loop
H(i1,i1) = H(i1,i1) - kappa
do j = i1,i2-1 ! Loop reducing the matrix to triangular form
! Apply Givens rotation so that the subdiagonal is set to zero
if(H(j+1,j) == 0) then
c(j) = 1.0d0; s(j) = 0.0d0;
elseif(H(j,j) == 0) then
c(j) = 0.0d0; s(j) = 1; H(j,j) = H(j+1,j); H(j+1,j) = 0.0d0;
else
mu = H(j,j)/abs(H(j,j))
tau = abs(real(H(j,j))) + abs(imag(H(j,j))) + abs(real(H(j+1,j))) &
+ abs(imag(H(j+1,j)))
nu = tau*sqrt(abs(H(j,j)/tau)**2 + abs(H(j+1,j)/tau)**2)
c(j) = abs(H(j,j))/nu
s(j) = mu*conjg(H(j+1,j))/nu
H(j,j) = nu*mu
H(j+1,j) = 0.0d0
endif
! Apply shift to diagonal
H(j+1,j+1) = H(j+1,j+1) - kappa
! Modify the involved rows using a plane rotation
t(j+1:n) = c(j)*H(j,j+1:n) + s(j)*H(j+1,j+1:n)
H(j+1,j+1:n) = c(j)*H(j+1,j+1:n) - conjg(s(j))*H(j,j+1:n)
H(j,j+1:n) = t(j+1:n)
enddo
do k = i1,i2-1
! Loop applying the back multiplication using a plane rotation
t(1:k+1) = c(k)*H(1:k+1,k) + conjg(s(k))*H(1:k+1,k+1);
H(1:k+1,k+1) = c(k)*H(1:k+1,k+1) - s(k)*H(1:k+1,k)
H(1:k+1,k) = t(1:k+1)
! Accumulate transformations using a plane rotation
t(1:n) = c(k)*Q(1:n,k) + conjg(s(k))*Q(1:n,k+1)
Q(1:n,k+1) = c(k)*Q(1:n,k+1) - s(k)*Q(1:n,k)
Q(1:n,k) = t(1:n)
H(k,k) = H(k,k) + kappa
enddo
H(i2,i2) = H(i2,i2) + kappa
enddo
deallocate(t,s,c)
!----
! Construct the orthonormal basis
!----
V(:n,:n)=0.0d0
D(:n,:n)=0.0d0
do i=1,n
V(i,i)=1.0d0
D(i,i)=H(i,i)
enddo
do j=2,n
do i=j-1,1,-1
denom=H(i,i)-H(j,j)
if(denom /= 0) then
csum=0.0d0
do k=i+1,j
csum=csum+H(i,k)*V(k,j)
enddo
V(i,j)=V(i,j)-csum/denom
endif
enddo
enddo
V=matmul(Q,V)
deallocate(Q,H)
!----
! Sort and normalize the eigensolution
!----
allocate(iset(n),work1d(n),work2d(n,n))
do i=1,n
work1d(i) = D(i,i)
enddo
call ALINDX(n, work1d, iset)
do i=1,n
work1d(i) = D(iset(i),iset(i))
work2d(:n,i) = V(:n,iset(i))
enddo
do i=1,n
D(i,i)=work1d(i)
enddo
V(:n,:n) = work2d(:n,:n)
deallocate(work2d,work1d)
nset=0
do i=1,n
if(abs(imag(D(i,i))) > 1.0e-10) then
nset=nset+1
iset(nset)=i
endif
enddo
do i=1,n
ii=findlc(iset(:nset),i)
if(mod(ii-1,2)+1.eq.1) then
j=iset(ii+1)
m=reshape( (/V(i,i), V(j,i), V(i,j), V(j,j)/), (/2, 2/) )
m(:,1)=m(:,1)/sqdotv(abs(m(1:2,1)))
m(:,2)=m(:,2)/sqdotv(abs(m(1:2,2)))
AA=reshape( (/real(m(1,1))+real(m(2,1)), aimag(m(1,1))+aimag(m(2,1)), &
-aimag(m(1,1))-aimag(m(2,1)), real(m(1,1))+real(m(2,1)) /), (/2, 2/) )
BB=reshape( (/real(m(1,2))+real(m(2,2)), -aimag(m(1,2))-aimag(m(2,2)), &
-aimag(m(1,2))-aimag(m(2,2)), -real(m(1,2))-real(m(2,2)) /), (/2, 2/) )
call ALINVD(2,BB,2,ier)
if(ier.ne.0) call xabort('ALHQR: singular matrix')
CC=matmul(BB,AA)
V(:,i)=V(:,i)*cmplx(CC(1,1),CC(1,2),kind=8)
elseif (mod(ii-1,2)+1.eq.2) then
j=iset(ii-1)
if(abs(D(i,i)-conjg(D(j,j))) > 1.0e-10) then
call xabort('ALHQR: pathological ordering')
endif
D(i,i)=conjg(D(j,j))
else
D(i,i)=real(D(i,i))
endif
V(:,i)=V(:,i)/sqdotv(abs(V(:,i)))
enddo
deallocate(iset)
return
contains
function sqdotv(vec) result(vsum)
! function emulating the vectorial norm2 function in Fortran 2008
real(kind=8), dimension(:), intent(in) :: vec
real(kind=8) :: vsum
vsum=sqrt(dot_product(vec(:),vec(:)))
end function sqdotv
function sqdotm(mat) result(vsum)
! function emulating the matrix norm2 function in Fortran 2008
real(kind=8), dimension(:,:), intent(in) :: mat
real(kind=8) :: vsum
vsum=0.0d0
do i=1,size(mat,1)
do j=1,size(mat,2)
vsum=vsum+mat(i,j)**2
enddo
enddo
vsum=sqrt(vsum)
end function sqdotm
function findlc(iset,itest) result(ii)
! function emulating the findloc function in Fortran 2008
integer, dimension(:), intent(in) :: iset
integer, intent(in) :: itest
integer :: ii
ii=0
do j=1,size(iset)
if(iset(j) == itest) then
ii=j
exit
endif
enddo
end function findlc
end subroutine ALHQR
|
