QuantumPackage/src/utils_trust_region/org/vec_to_mat_index.org

• Vector to matrix indexes

Vector to matrix indexes

Compute the indexes p,q of a matrix element with the vector index i

Vector (i) -> lower diagonal matrix (p,q), p > q

If a matrix is antisymmetric it can be reshaped as a vector. And the vector can be reshaped as an antisymmetric matrix

\begin{align*} \begin{pmatrix} 0 & -1 & -2 & -4 \\ 1 & 0 & -3 & -5 \\ 2 & 3 & 0 & -6 \\ 4 & 5 & 6 & 0 \end{pmatrix} \Leftrightarrow \begin{pmatrix} 1 & 2 & 3 & 4 & 5 & 6 \end{pmatrix} \end{align*}

!!! Here the algorithm only work for the lower diagonal !!!

Input:

i

integer

index in the vector

Ouput:

p,q	integer	corresponding indexes in the lower diagonal of a matrix
		p > q,
		p -> row,
		q -> column

subroutine vec_to_mat_index(i,p,q)

  include 'pi.h'

  !BEGIN_DOC
  ! Compute the indexes (p,q) of the element in the lower diagonal matrix knowing
  ! its index i a vector
  !END_DOC

  implicit none

  ! Variables

  ! in
  integer,intent(in)   :: i
  
  ! out
  integer, intent(out) :: p,q
  
  ! internal 
  integer              :: a,b
  double precision     :: da

  da = 0.5d0*(1+ sqrt(1d0+8d0*DBLE(i)))
  a = INT(da) 
  if ((a*(a-1))/2==i) then
    p = a-1
  else
    p = a
  endif
  b = p*(p-1)/2
 
  ! Matrix element indexes
  p = p + 1
  q = i - b 

end subroutine