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mpnum.f90
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mpnum.f90
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! Code converted using TO_F90 by Alan Miller
! Date: 2012-03-16 Time: 11:07:48
!> \file
!! General linear algebra routines.
!!
!! \author Volker Blobel, University Hamburg, 2005-2009 (initial Fortran77 version)
!! \author Claus Kleinwort, DESY (maintenance and developement)
!!
!! \copyright
!! Copyright (c) 2009 - 2019 Deutsches Elektronen-Synchroton,
!! Member of the Helmholtz Association, (DESY), HAMBURG, GERMANY \n\n
!! This library is free software; you can redistribute it and/or modify
!! it under the terms of the GNU Library General Public License as
!! published by the Free Software Foundation; either version 2 of the
!! License, or (at your option) any later version. \n\n
!! This library is distributed in the hope that it will be useful,
!! but WITHOUT ANY WARRANTY; without even the implied warranty of
!! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
!! GNU Library General Public License for more details. \n\n
!! You should have received a copy of the GNU Library General Public
!! License along with this program (see the file COPYING.LIB for more
!! details); if not, write to the Free Software Foundation, Inc.,
!! 675 Mass Ave, Cambridge, MA 02139, USA.
!!
!! ***** Collection of utility routines from V. Blobel *****
!!
!! V. Blobel, Univ. Hamburg
!! Numerical subprograms used in MP-II: matrix equations,
!! and matrix products, double precision
!!
!! Solution by inversion
!! SQMINV
!! SQMINL for LARGE matrix, use OpenMP (CHK)
!!
!! Solution by diagonalization
!! DEVROT, DEVPRT, DEFSOL,DEVINV
!!
!! Solution by Cholesky decomposition of symmetric matrix
!! CHOLDC
!!
!! Solution by Cholesky decomposition of variable-band matrix
!! VABDEC
!!
!! Solution by Cholesky decomposition of bordered band matrix
!! SQMIBB upper/left border (CHK)
!! SQMIBB2 lower/right border (CHK)
!!
!! Matrix/vector products
!! DBDOT dot vector product
!! DBAXPY multiplication and addition
!! DBSVX symmetric matrix vector
!! DBSVX LARGE symmetric matrix vector (CHK)
!! DBGAX general matrix vector
!! DBAVAT AVAT product
!! DBMPRV print parameter and matrix
!! DBPRV print matrix (CHK)
!!
!! Chi^2 cut values
!! CHINDL
!!
!! Accurate summation (moved to pede.f90)
!! ADDSUM
!!
!! Sorting
!! HEAPF heap sort reals direct
!! SORT1K sort 1-dim key-array (CHK)
!! SORT2K sort 2-dim key-array
!! SORT2I sort 2-dim key-array with index (CHK)
!!
!----------------------------------------------------------------------
!> Matrix inversion and solution.
!!
!! Obtain solution of a system of linear equations with symmetric
!! matrix (V * X = B) and the inverse.
!!
!! Method of solution is by elimination selecting the pivot on the
!! diagonal each stage. The rank of the matrix is returned in NRANK.
!! For NRANK ne N, all remaining rows and cols of the resulting
!! matrix V and the corresponding elements of B are set to zero.
!!
!! \param [in,out] V symmetric N-by-N matrix in symmetric storage mode
!! (V(1) = V11, V(2) = V12, V(3) = V22, V(4) = V13, ...),
!! replaced by inverse matrix
!! \param [in,out] B N-vector, replaced by solution vector
!! \param [in] N size of V, B
!! \param [out] NRANK rank of matrix V
!! \param [out] DIAG double precision scratch array
!! \param [out] NEXT INTEGER(mpi) aux array
SUBROUTINE sqminv(v,b,n,nrank,diag,next) ! matrix inversion
USE mpdef
IMPLICIT NONE
INTEGER(mpi) :: i
INTEGER(mpi) :: ij
INTEGER(mpi) :: j
INTEGER(mpi) :: jj
INTEGER(mpi) :: jk
INTEGER(mpi) :: jl
INTEGER(mpi) :: k
INTEGER(mpi) :: kk
INTEGER(mpi) :: l
INTEGER(mpi) :: last
INTEGER(mpi) :: lk
INTEGER(mpi) :: next0
REAL(mpd), INTENT(IN OUT) :: v(*)
REAL(mpd), INTENT(OUT) :: b(n)
INTEGER(mpi), INTENT(IN) :: n
INTEGER(mpi), INTENT(OUT) :: nrank
REAL(mpd), INTENT(OUT) :: diag(n)
INTEGER(mpi), INTENT(OUT) :: next(n)
REAL(mpd) :: vkk
REAL(mpd) :: vjk
!REAL(mpd), PARAMETER :: eps=1.0E-10_mpd
REAL(mpd) eps
! ...
eps = 16.0_mpd * epsilon(eps) ! 16 * precision(mpd)
next0=1
l=1
DO i=1,n
next(i)=i+1 ! set "next" pointer
diag(i)=ABS(v((i*i+i)/2)) ! save abs of diagonal elements
END DO
next(n)=-1 ! end flag
nrank=0
DO i=1,n ! start of loop
k =0
vkk=0.0_mpd
j=next0
last=0
05 IF(j > 0) THEN
jj=(j*j+j)/2
IF(ABS(v(jj)) > MAX(ABS(vkk),eps*diag(j))) THEN
vkk=v(jj)
k=j
l=last
END IF
last=j
j=next(last)
GO TO 05
END IF
IF(k /= 0) THEN ! pivot found
kk=(k*k+k)/2
IF(l == 0) THEN
next0=next(k)
ELSE
next(l)=next(k)
END IF
next(k)=0 ! index is used, reset
nrank=nrank+1 ! increase rank and ...
vkk =1.0/vkk
v(kk) =-vkk
b(k) =b(k)*vkk
jk =kk-k
jl =0
DO j=1,n ! elimination
IF(j == k) THEN
jk=kk
jl=jl+j
ELSE
IF(j < k) THEN
jk=jk+1
ELSE
jk=jk+j-1
END IF
vjk =v(jk)
v(jk)=vkk*vjk
b(j) =b(j)-b(k)*vjk
lk =kk-k
DO l=1,j
jl=jl+1
IF(l == k) THEN
lk=kk
ELSE
IF(l < k) THEN
lk=lk+1
ELSE
lk=lk+l-1
END IF
v(jl)=v(jl)-v(lk)*vjk
END IF
END DO
END IF
END DO
ELSE
DO k=1,n
IF(next(k) /= 0) THEN
b(k)=0.0_mpd ! clear vector element
DO j=1,k
IF(next(j) /= 0) v((k*k-k)/2+j)=0.0_mpd ! clear matrix row/col
END DO
END IF
END DO
GO TO 10
END IF
END DO ! end of loop
10 DO ij=1,(n*n+n)/2
v(ij)=-v(ij) ! finally reverse sign of all matrix elements
END DO
END SUBROUTINE sqminv
!> Matrix inversion for LARGE matrices.
!!
!! Like SQMINV, additional parallelization with OpenMP.
!!
!! \param [in,out] V symmetric N-by-N matrix in symmetric storage mode
!! (V(1) = V11, V(2) = V12, V(3) = V22, V(4) = V13, ...),
!! replaced by inverse matrix
!! \param [in,out] B N-vector, replaced by solution vector
!! \param [in] N size of V, B
!! \param [out] NRANK rank of matrix V
!! \param [out] DIAG double precision scratch array
!! \param [out] NEXT integer aux array
SUBROUTINE sqminl(v,b,n,nrank,diag,next) !
USE mpdef
IMPLICIT NONE
INTEGER(mpi) :: i
INTEGER(mpi) :: j
INTEGER(mpi) :: k
INTEGER(mpi) :: l
INTEGER(mpi) :: last
INTEGER(mpi) :: next0
REAL(mpd), INTENT(IN OUT) :: v(*)
REAL(mpd), INTENT(OUT) :: b(n)
INTEGER(mpi), INTENT(IN) :: n
INTEGER(mpi), INTENT(OUT) :: nrank
REAL(mpd), INTENT(OUT) :: diag(n)
INTEGER(mpi), INTENT(OUT) :: next(n)
INTEGER(mpl) :: i8
INTEGER(mpl) :: j8
INTEGER(mpl) :: jj
INTEGER(mpl) :: k8
INTEGER(mpl) :: kk
INTEGER(mpl) :: kkmk
INTEGER(mpl) :: jk
INTEGER(mpl) :: jl
INTEGER(mpl) :: llk
INTEGER(mpl) :: ljl
REAL(mpd) :: vkk
REAL(mpd) :: vjk
REAL(mpd), PARAMETER :: eps=1.0E-10_mpd
! ...
next0=1
l=1
DO i=1,n
i8=int8(i)
next(i)=i+1 ! set "next" pointer
diag(i)=ABS(v((i8*i8+i8)/2)) ! save abs of diagonal elements
END DO
next(n)=-1 ! end flag
nrank=0
DO i=1,n ! start of loop
k =0
vkk=0.0_mpd
j=next0
last=0
05 IF(j > 0) THEN
j8=int8(j)
jj=(j8*j8+j8)/2
IF(ABS(v(jj)) > MAX(ABS(vkk),eps*diag(j))) THEN
vkk=v(jj)
k=j
l=last
END IF
last=j
j=next(last)
GO TO 05
END IF
IF(k /= 0) THEN ! pivot found
k8=int8(k)
kk=(k8*k8+k8)/2
kkmk=kk-k8
IF(l == 0) THEN
next0=next(k)
ELSE
next(l)=next(k)
END IF
next(k)=0 ! index is used, reset
nrank=nrank+1 ! increase rank and ...
vkk =1.0/vkk
v(kk) =-vkk
b(k) =b(k)*vkk
! elimination
jk =kkmk
DO j=1,n
IF(j == k) THEN
jk=kk
ELSE
IF(j < k) THEN
jk=jk+1
ELSE
jk=jk+int8(j)-1
END IF
v(jk)=v(jk)*vkk
END IF
END DO
! parallelize row loop
! slot of 128 'J' for next idle thread
!$OMP PARALLEL DO &
!$OMP PRIVATE(JL,JK,L,LJL,LLK,VJK,J8) &
!$OMP SCHEDULE(DYNAMIC,128)
DO j=n,1,-1
j8=int8(j)
jl=j8*(j8-1)/2
IF(j /= k) THEN
IF(j < k) THEN
jk=kkmk+j8
ELSE
jk=k8+jl
END IF
vjk =v(jk)/vkk
b(j) =b(j)-b(k)*vjk
ljl=jl
llk=kkmk
DO l=1,MIN(j,k-1)
ljl=ljl+1
llk=llk+1
v(ljl)=v(ljl)-v(llk)*vjk
END DO
ljl=ljl+1
llk=kk
DO l=k+1,j
ljl=ljl+1
llk=llk+l-1
v(ljl)=v(ljl)-v(llk)*vjk
END DO
END IF
END DO
!$OMP END PARALLEL DO
ELSE
DO k=1,n
k8=int8(k)
kk=(k8*k8-k8)/2
IF(next(k) /= 0) THEN
b(k)=0.0_mpd ! clear vector element
DO j=1,k
IF(next(j) /= 0) v(kk+int8(j))=0.0_mpd ! clear matrix row/col
END DO
END IF
END DO
GO TO 10
END IF
END DO ! end of loop
10 DO jj=1,(int8(n)*int8(n)+int8(n))/2
v(jj)=-v(jj) ! finally reverse sign of all matrix elements
END DO
END SUBROUTINE sqminl
!> Diagonalization.
!!
!! Determination of eigenvalues and eigenvectors of
!! symmetric matrix V by Householder method
!!
!! \param [in] n size of matrix
!! \param [out] diag diagonal elements
!! \param [out] u transformation matrix
!! \param [in] v symmetric matrix, unchanged
!! \param [out] work work array
!! \param [out] iwork work array
SUBROUTINE devrot(n,diag,u,v,work,iwork) ! diagonalization
USE mpdef
IMPLICIT NONE
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd), INTENT(OUT) :: diag(n)
REAL(mpd), INTENT(OUT) :: u(n,n)
REAL(mpd), INTENT(IN) :: v(*)
REAL(mpd), INTENT(OUT) :: work(n)
INTEGER(mpi), INTENT(OUT) :: iwork(n)
INTEGER(mpi), PARAMETER :: itmax=30
REAL(mpd), PARAMETER :: tol=EPSILON(tol)
REAL(mpd), PARAMETER :: eps=EPSILON(eps)
REAL(mpd) :: f
REAL(mpd) :: g
REAL(mpd) :: h
REAL(mpd) :: sh
REAL(mpd) :: hh
REAL(mpd) :: b
REAL(mpd) :: p
REAL(mpd) :: r
REAL(mpd) :: s
REAL(mpd) :: c
REAL(mpd) :: workd
INTEGER(mpi) :: ij
INTEGER(mpi) :: i
INTEGER(mpi) :: j
INTEGER(mpi) :: k
INTEGER(mpi) :: l
INTEGER(mpi) :: m
INTEGER(mpi) :: ll
! ...
! 1. part: symmetric matrix V reduced to tridiagonal from
ij=0
DO i=1,n
DO j=1,i
ij=ij+1
u(i,j)=v(ij) ! copy half of symmetric matirx
END DO
END DO
DO i=n,2,-1
l=i-2
f=u(i,i-1)
g=0.0_mpd
IF(l /= 0) THEN
DO k=1,l
IF(ABS(u(i,k)) > tol) g=g+u(i,k)*u(i,k)
END DO
h=g+f*f
END IF
IF(g < tol) THEN ! G too small
work(i)=f ! skip transformation
h =0.0_mpd
ELSE
l=l+1
sh=SQRT(h)
IF(f >= 0.0_mpd) sh=-sh
g=sh
work(i)=sh
h=h-f*g
u(i,i-1)=f-g
f=0.0_mpd
DO j=1,l
u(j,i)=u(i,j)/h
g=0.0_mpd
! form element of a u
DO k=1,j
IF(ABS(u(j,k)) > tol.AND.ABS(u(i,k)) > tol) THEN
g=g+u(j,k)*u(i,k)
END IF
END DO
DO k=j+1,l
IF(ABS(u(k,j)) > tol.AND.ABS(u(i,k)) > tol) THEN
g=g+u(k,j)*u(i,k)
END IF
END DO
work(j)=g/h
f=f+g*u(j,i)
END DO
! form k
hh=f/(h+h)
! form reduced a
DO j=1,l
f=u(i,j)
work(j)=work(j)-hh*f
g=work(j)
DO k=1,j
u(j,k)=u(j,k)-f*work(k)-g*u(i,k)
END DO
END DO
END IF
diag(i)=h
END DO
diag(1)=0.0_mpd
work(1)=0.0_mpd
! accumulation of transformation matrices
DO i=1,n
IF(diag(i) /= 0.0) THEN
DO j=1,i-1
g=0.0_mpd
DO k=1,i-1
g=g+u(i,k)*u(k,j)
END DO
DO k=1,i-1
u(k,j)=u(k,j)-g*u(k,i)
END DO
END DO
END IF
diag(i)=u(i,i)
u(i,i)=1.0_mpd
DO j=1,i-1
u(i,j)=0.0_mpd
u(j,i)=0.0_mpd
END DO
END DO
! 2. part: diagonalization of tridiagonal matrix
DO i=2,n
work(i-1)=work(i)
END DO
work(n)=0.0_mpd
b=0.0_mpd
f=0.0_mpd
DO l=1,n
j=0
h=eps*(ABS(diag(l))+ABS(work(l)))
IF(b < h) b=h
DO m=l,n
IF(ABS(work(m)) <= b) GO TO 10 ! look for small sub-diagonal element
END DO
m=l
10 IF(m == l) GO TO 30
! next iteration
20 IF(j == itmax) THEN
WRITE(*,*) 'DEVROT: Iteration limit reached'
CALL peend(32,'Aborted, iteration limit reached in diagonalization')
STOP
END IF
j=j+1
g=diag(l)
p=(diag(l+1)-g)/(2.0_mpd*work(l))
r=SQRT(1.0_mpd+p*p)
diag(l)=work(l)
IF(p < 0.0_mpd) diag(l)=diag(l)/(p-r)
IF(p >= 0.0_mpd) diag(l)=diag(l)/(p+r)
h=g-diag(l)
DO i=l+1,n
diag(i)=diag(i)-h
END DO
f=f+h
! QL transformation
p=diag(m)
c=1.0_mpd
s=0.0_mpd
DO i=m-1,l,-1 ! reverse loop
g=c*work(i)
h=c*p
IF(ABS(p) >= ABS(work(i))) THEN
c=work(i)/p
r=SQRT(1.0_mpd+c*c)
work(i+1)=s*p*r
s=c/r
c=1.0_mpd/r
ELSE
c=p/work(i)
r=SQRT(1.0_mpd+c*c)
work(i+1)=s*work(i)*r
s=1.0_mpd/r
c=c/r
END IF
p=c*diag(i)-s*g
diag(i+1)=h+s*(c*g+s*diag(i))
! form vector
DO k=1,n
h=u(k,i+1)
u(k,i+1)=s*u(k,i)+c*h
u(k,i)=c*u(k,i)-s*h
END DO
END DO
work(l)=s*p
diag(l)=c*p
IF(ABS(work(l)) > b) GO TO 20 ! next iteration
30 diag(l)=diag(l)+f
END DO
DO i=1,n
iwork(i)=i
END DO
m=1
40 m=1+3*m ! determine initial increment
IF(m <= n) GO TO 40
50 m=m/3
DO j=1,n-m ! sort with increment M
l=j
60 IF(diag(iwork(l+m)) > diag(iwork(l))) THEN ! compare
ll=iwork(l+m) ! exchange the two index values
iwork(l+m)=iwork(l)
iwork(l)=ll
l=l-m
IF(l > 0) GO TO 60
END IF
END DO
IF(m > 1) GO TO 50
DO i=1,n
IF(iwork(i) /= i) THEN
! move vector from position I to the work area
workd=diag(i)
DO l=1,n
work(l)=u(l,i)
END DO
k=i
70 j=k
k=iwork(j)
iwork(j)=j
IF(k /= i) THEN
! move vector from position K to the (free) position J
diag(j)=diag(k)
DO l=1,n
u(l,j)=u(l,k)
END DO
GO TO 70
END IF
! move vector from the work area to position J
diag(j)=workd
DO l=1,n
u(l,j)=work(l)
END DO
END IF
END DO
END SUBROUTINE devrot
!> Calculate significances.
SUBROUTINE devsig(n,diag,u,b,coef)
USE mpdef
IMPLICIT NONE
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd), INTENT(IN) :: diag(n)
REAL(mpd), INTENT(IN) :: u(n,n)
REAL(mpd), INTENT(IN) :: b(n)
REAL(mpd), INTENT(OUT) :: coef(n)
INTEGER(mpi) :: i
INTEGER(mpi) :: j
REAL(mpd) :: dsum
! ...
DO i=1,n
coef(i)=0.0_mpd
IF(diag(i) > 0.0_mpd) THEN
dsum=0.0_mpd
DO j=1,n
dsum=dsum+u(j,i)*b(j)
END DO
coef(i)=ABS(dsum)/SQRT(diag(i))
END IF
END DO
END SUBROUTINE devsig
!> Solution by diagonalization.
!!
!! Solution of matrix equation V * X = B after diagonalization of V.
!!
!! \param [in] N size of matrix
!! \param [in] DIAG diagonal elements
!! \param [in] U transformation matrix
!! \param [in] B r.h.s. of matrix equation (unchanged)
!! \param [out] X solution vector
!! \param [out] WORK work array
SUBROUTINE devsol(n,diag,u,b,x,work)
USE mpdef
IMPLICIT NONE
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd), INTENT(IN) :: diag(n)
REAL(mpd), INTENT(IN) :: u(n,n)
REAL(mpd), INTENT(IN) :: b(n)
REAL(mpd), INTENT(OUT) :: x(n)
REAL(mpd), INTENT(OUT) :: work(n)
INTEGER(mpi) :: i
INTEGER(mpi) :: j
INTEGER(mpi) :: jj
REAL(mpd) :: s
! ...
DO j=1,n
s=0.0_mpd
work(j)=0.0_mpd
IF(diag(j) /= 0.0_mpd) THEN
DO i=1,n
! j-th eigenvector is U(.,J)
s=s+u(i,j)*b(i)
END DO
work(j)=s/diag(j)
END IF
END DO
DO j=1,n
s=0.0_mpd
DO jj=1,n
s=s+u(j,jj)*work(jj)
END DO
x(j)=s
END DO
! WRITE(*,*) 'DEVSOL'
! WRITE(*,*) 'X ',X
END SUBROUTINE devsol
!> Inversion by diagonalization.
!! Get inverse matrix V from DIAG and U.
!!
!! \param [in] N size of matrix
!! \param [in] DIAG diagonal elements
!! \param [in] U transformation matrix
!! \param [out] V smmmetric matrix
SUBROUTINE devinv(n,diag,u,v)
USE mpdef
IMPLICIT NONE
INTEGER(mpi) :: i
INTEGER(mpi) :: ij
INTEGER(mpi) :: j
INTEGER(mpi) :: k
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd), INTENT(IN) :: diag(n)
REAL(mpd), INTENT(IN) :: u(n,n)
REAL(mpd), INTENT(OUT) :: v(*)
REAL(mpd) :: dsum
! ...
ij=0
DO i=1,n
DO j=1,i
ij=ij+1
dsum=0.0_mpd
DO k=1,n
IF(diag(k) /= 0.0_mpd) THEN
dsum=dsum+u(i,k)*u(j,k)/diag(k)
END IF
END DO
v(ij)=dsum
END DO
END DO
END SUBROUTINE devinv
!> Cholesky decomposition.
!!
!! Cholesky decomposition of the matrix G: G = L D L^T
!!
!! - G = symmetric matrix, in symmetric storage mode
!!
!! - L = unit triangular matrix (1's on diagonal)
!!
!! - D = diagonal matrix (elements store on diagonal of L)
!!
!! The sqrts of the usual Cholesky decomposition are avoided by D.
!! Matrices L and D are stored in the place of matrix G; after the
!! decomposition, the solution of matrix equations and the computation
!! of the inverse of the (original) matrix G are done by CHOLSL and CHOLIN.
!!
!! \param [in,out] g symmetric matrix, replaced by D,L
!! \param [in] n size of matrix
!!
SUBROUTINE choldc(g,n)
USE mpdef
IMPLICIT NONE
INTEGER(mpi) :: i
INTEGER(mpi) :: ii
INTEGER(mpi) :: j
INTEGER(mpi) :: jj
INTEGER(mpi) :: k
INTEGER(mpi) :: kk
REAL(mpd), INTENT(IN OUT) :: g(*)
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd) :: ratio
! ...
ii=0
DO i=1,n
ii=ii+i
IF(g(ii) /= 0.0) g(ii)=1.0/g(ii) ! (I,I) div !
jj=ii
DO j=i+1,n
ratio=g(i+jj)*g(ii) ! (I,J) (I,I)
kk=jj
DO k=j,n
g(kk+j)=g(kk+j)-g(kk+i)*ratio ! (K,J) (K,I)
kk=kk+k
END DO ! K
g(i+jj)=ratio ! (I,J)
jj=jj+j
END DO ! J
END DO ! I
RETURN
END SUBROUTINE choldc
!> Solution after decomposition.
!!
!! The matrix equation G X = B is solved for X, where the matrix
!! G in the argument is already decomposed by CHOLDC. The vector B
!! is called X in the argument and the content is replaced by the
!! resulting vector X.
!!
!! \param [in] g decomposed symmetric matrix
!! \param [in,out] x r.h.s vector B, replaced by solution vector X
!! \param [in] n size of matrix
!!
SUBROUTINE cholsl(g,x,n)
USE mpdef
IMPLICIT NONE
REAL(mpd) :: dsum
INTEGER(mpi) :: i
INTEGER(mpi) :: ii
INTEGER(mpi) :: k
INTEGER(mpi) :: kk
REAL(mpd), INTENT(IN) :: g(*)
REAL(mpd), INTENT(IN OUT) :: x(n)
INTEGER(mpi), INTENT(IN) :: n
ii=0
DO i=1,n
dsum=x(i)
DO k=1,i-1
dsum=dsum-g(k+ii)*x(k) ! (K,I)
END DO
x(i)=dsum
ii=ii+i
END DO
DO i=n,1,-1
dsum=x(i)*g(ii) ! (I,I)
kk=ii
DO k=i+1,n
dsum=dsum-g(kk+i)*x(k) ! (K,I)
kk=kk+k
END DO
x(i)=dsum
ii=ii-i
END DO
RETURN
END SUBROUTINE cholsl
!> Inversion after decomposition.
!!
!! The inverse of the (original) matrix G is computed and stored
!! in symmetric storage mode in matrix V. Arrays G and V must be
!! different arrays.
!!
!! \param [in] g decomposed symmetric matrix
!! \param [in,out] v inverse matrix
!! \param [in] n size of matrix
!!
SUBROUTINE cholin(g,v,n)
USE mpdef
IMPLICIT NONE
REAL(mpd) :: dsum
INTEGER(mpi) :: i
INTEGER(mpi) :: ii
INTEGER(mpi) :: j
INTEGER(mpi) :: k
INTEGER(mpi) :: l
INTEGER(mpi) :: m
REAL(mpd), INTENT(IN) :: g(*)
REAL(mpd), INTENT( OUT) :: v(*)
INTEGER(mpi), INTENT(IN) :: n
ii=(n*n-n)/2
DO i=n,1,-1
dsum=g(ii+i) ! (I,I)
DO j=i,1,-1
DO k=j+1,n
l=MIN(i,k)
m=MAX(i,k)
dsum=dsum-g(j+(k*k-k)/2)*v(l+(m*m-m)/2) ! (J,K) (I,K)
END DO
v(ii+j)=dsum ! (I,J)
dsum=0.0_mpd
END DO
ii=ii-i+1
END DO
END SUBROUTINE cholin
! variable band matrix operations ----------------------------------
!> Variable band matrix decomposition.
!!
!! Decomposition: A = L D L^T
!!
!! Variable-band matrix row Doolittle decomposition.
!! A variable-band NxN symmetric matrix, also called skyline, is stored
!! row by row in the array VAL(.). For each row every coefficient
!! between the first non-zero element in the row and the diagonal is
!! stored.
!! The pointer array ILPTR(N) contains the indices in VAL(.) of the
!! diagonal elements. ILPTR(1) is always 1, and ILPTR(N) is equal
!! to the total number of coefficients stored, called the profile.
!! The form of a variable-band matrix is preserved in the L D L^T
!! decomposition no fill-in is created ahead in any row or ahead of the
!! first entry in any column, but existing zero-values will become
!! non-zero. The decomposition is done "in-place".
!!
!! \param [in] n size of matrix
!! \param [in,out] val variable-band matrix, replaced by D,L
!! \param [in] ilptr pointer array
SUBROUTINE vabdec(n,val,ilptr)
USE mpdef
IMPLICIT NONE
INTEGER(mpi) :: i
INTEGER(mpi) :: in
INTEGER(mpi) :: j
INTEGER(mpi) :: k
INTEGER(mpi) :: kj
INTEGER(mpi) :: mj
INTEGER(mpi) :: mk
REAL(mpd) :: sn
REAL(mpd) :: beta
REAL(mpd) :: delta
REAL(mpd) :: theta
INTEGER(mpi), INTENT(IN) :: n
REAL(mpd), INTENT(IN OUT) :: val(*)
INTEGER(mpi), INTENT(IN) :: ilptr(n)
REAL(mpd) :: dgamma
REAL(mpd) :: xi
REAL(mpd) :: valkj
REAL(mpd), PARAMETER :: one=1.0_mpd
REAL(mpd), PARAMETER :: two=2.0_mpd
REAL(mpd), PARAMETER :: eps = EPSILON(eps)
WRITE(*,*) 'Variable band matrix Cholesky decomposition'
dgamma=0.0_mpd
i=1
DO j=1,ilptr(n) ! loop thrugh all matrix elements
IF(ilptr(i) == j) THEN ! diagonal element
IF(val(j) <= 0.0_mpd) GO TO 01 ! exit loop for negative diag
dgamma=MAX(dgamma,ABS(val(j))) ! max diagonal element
i=i+1
END IF
END DO
i=n+1
01 in=i-1 ! IN positive diagonal elements
WRITE(*,*) ' ',in,' positive diagonal elements'
xi=0.0_mpd
i=1
DO j=1,ilptr(in) ! loop for positive diagonal elements
! through all matrix elements
IF(ilptr(i) == j) THEN ! diagonal element
i=i+1
ELSE
xi=MAX(xi,ABS(val(j))) ! Xi = abs(max) off-diagonal element
END IF
END DO
delta=eps*MAX(1.0_mpd,dgamma+xi)
sn=1.0_mpd
IF(n > 1) sn=1.0_mpd/SQRT(REAL(n*n-1,mpd))
beta=SQRT(MAX(eps,dgamma,xi*sn)) ! beta
WRITE(*,*) ' DELTA and BETA ',delta,beta
DO k=2,n
mk=k-ilptr(k)+ilptr(k-1)+1
theta=0.0_mpd
DO j=mk,k
mj=j-ilptr(j)+ilptr(j-1)+1
kj=ilptr(k)-k+j ! index kj
DO i=MAX(mj,mk),j-1
val(kj)=val(kj) & ! L_kj := L_kj - L_ki D_ii L_ji
-val(ilptr(k)-k+i)*val(ilptr(i))*val(ilptr(j)-j+i)
END DO !
theta=MAX(theta,ABS(val(kj))) ! maximum value of row
IF(j /= k) THEN
IF(val(ilptr(j)) /= 0.0_mpd) THEN
val(kj)=val(kj)/val(ilptr(j))
ELSE
val(kj)=0.0_mpd
END IF
END IF ! L_kj := L_kj/D_jj ! D_kk
IF(j == k) THEN
valkj=val(kj)
IF(k <= in) THEN
val(kj)=MAX(ABS(val(kj)),(theta/beta)**2,delta)
IF(valkj /= val(kj)) THEN
WRITE(*,*) ' Index K=',k
WRITE(*,*) ' ',valkj,val(kj), (theta/beta)**2,delta,theta
END IF
END IF
END IF
END DO ! J
END DO ! K
DO k=1,n
IF(val(ilptr(k)) /= 0.0_mpd) val(ilptr(k))=1.0_mpd/val(ilptr(k))