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spammsand_inverse_squareroot.F90
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#define DENSE_DIAGNOSTICS
! cmake -DCMAKE_Fortran_COMPILER=gfortran -DCMAKE_Fortran_FLAGS="-O0 -g -fbounds-check -Wall -fbacktrace -finit-real=nan -Wextra -std=f2008 "
module SpAMMsand_inverse_squareroot
USE spammpack
implicit none
! Convergence parameters
REAL(SpAMM_KIND), PARAMETER :: Approx3 = 2.85d00
REAL(SPAMM_KIND), PARAMETER :: ShiftSw = 5.d-1
#ifdef DENSE_DIAGNOSTICS
real(spamm_kind), dimension(:,:), ALLOCATABLE :: i_d, s_d, &
z_k_stab, z_k1_stab, x_k_stab, x_k1_stab, m_x_k1_stab, &
z_tld_k_stab, z_tld_k1_stab, x_tld_k_stab, x_tld_k1_stab, m_x_tld_k1_stab, &
z_k_dual, z_k1_dual, x_k_dual, x_k1_dual, m_x_k1_dual, y_k_dual, y_k1_dual, &
z_tld_k_dual, z_tld_k1_dual, x_tld_k_dual, x_tld_k1_dual, m_x_tld_k1_dual, y_tld_k_dual, y_tld_k1_dual
REAL(spamm_kind) :: scal_shift_dual,shft_shift_dual, scal_mapp_dual, shft_mapp_dual
REAL(spamm_kind) :: scal_shift_stab,shft_shift_stab, scal_mapp_stab, shft_mapp_stab
integer :: LWORK
integer :: LIWORK
real(kind(0d0)), allocatable :: eval(:), SHalf(:,:),SHlfI(:,:),evec(:, :), work(:)
integer, allocatable :: iwork(:)
integer :: info, N
interface
subroutine dsyevd ( JOBZ, UPLO, N, A, LDA, W, WORK, LWORK, IWORK, LIWORK, INFO )
CHARACTER JOBZ, UPLO
INTEGER INFO, LDA, LIWORK, LWORK, N
INTEGER IWORK( * )
DOUBLE PRECISION A( LDA, * ), W( * ), WORK( * )
end subroutine dsyevd
end interface
#endif
contains
FUNCTION IntToChar(I)
CHARACTER(LEN=*), PARAMETER :: INTERNAL_INT_FMT='(I22)'
INTEGER :: I
CHARACTER(LEN=22) :: IntToChar
WRITE(UNIT=IntToChar,FMT=INTERNAL_INT_FMT)I
IntToChar=ADJUSTL(IntToChar)
END FUNCTION IntToChar
SUBROUTINE spammsand_scaled_newton_shulz_inverse_squareroot(s, x, z, t, tau, DoDuals, second, kount)
TYPE(SpAMM_tree_2d_symm) , POINTER, INTENT(IN) :: s
TYPE(SpAMM_tree_2d_symm) , POINTER :: x, z, y, t, y_tmp, z_tmp
REAL(SpAMM_KIND) :: Tau
! REAL(SpAMM_KIND), INTENT(IN) :: Tau
! LOGICAL, INTENT(IN) :: first
LOGICAL :: DoDuals,second
INTEGER, INTENT(INOUT) :: kount
INTEGER :: i, j, k
REAL(SpAMM_KIND) :: sc, TrX, tau_xtra
REAL(SpAMM_KIND) :: x_work,z_work,y_work,zs_work,sz_work
REAL(SpAMM_KIND) :: x_fill,z_fill,y_fill,zs_fill,sz_fill,y_norm,zs_norm
REAL(SpAMM_KIND) :: xo_analytic, delta, FillN, FillN_prev
REAL(SpAMM_KIND) :: tau_zdotz,sz_norm
#ifdef DENSE_DIAGNOSTICS
REAL(SpAMM_KIND) :: fprimez
fprimez=1d0
#else
!
tau_xtra=1d-2*tau
IF(DoDuals)tHeN
#endif
! y_0 => s
y => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
y => SpAMM_tree_2d_symm_copy_tree_2d_symm( S , in_o = y, threshold_O = SpAMM_normclean )
#ifdef DENSE_DIAGNOSTICS
#else
endIF
#endif
! z_0 => I
z => SpAMM_set_identity_2d_symm( s%frill%ndimn, in_o = z )
y_tmp => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
z_tmp => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
! x_0 => s
x => SpAMM_tree_2d_symm_copy_tree_2d_symm( s , in_O = x, threshold_O = SpAMM_normclean )
#ifdef DENSE_DIAGNOSTICS
n=x%frill%ndimn(1)
ALLOCATE(i_d (1:N,1:N))
ALLOCATE(s_d (1:N,1:N))
allocate(z_k_stab (1:n,1:n)); allocate(z_k1_stab (1:n,1:n)); allocate(x_k_stab (1:n,1:n));
allocate(x_k1_stab (1:n,1:n)); allocate(m_x_k1_stab (1:n,1:n))
allocate(z_tld_k_stab (1:n,1:n)); allocate(z_tld_k1_stab (1:n,1:n)); allocate(x_tld_k_stab(1:n,1:n));
allocate(x_tld_k1_stab(1:n,1:n)); allocate(m_x_tld_k1_stab(1:n,1:n))
allocate(z_k_dual (1:n,1:n)); allocate(z_k1_dual (1:n,1:n)); allocate(x_k_dual (1:n,1:n));
allocate(x_k1_dual (1:n,1:n)); allocate(m_x_k1_dual (1:n,1:n)); allocate(y_k_dual (1:n,1:n)); allocate(y_k1_dual (1:n,1:n))
allocate(z_tld_k_dual (1:n,1:n)); allocate(z_tld_k1_dual (1:n,1:n)); allocate(x_tld_k_dual(1:n,1:n));
allocate(x_tld_k1_dual(1:n,1:n)); allocate(m_x_tld_k1_dual (1:n,1:n)); allocate(y_tld_k_dual(1:n,1:n)); allocate(y_tld_k1_dual(1:n,1:n))
CALL SpAMM_convert_tree_2d_symm_to_dense( s, s_d )
CALL SpAMM_convert_tree_2d_symm_to_dense( z, i_d )
! --------------------
z_k_stab =i_d
z_k1_stab =i_d
z_tld_k_stab =i_d
z_tld_k1_stab=i_d
x_k_dual =s_d
x_k1_dual =s_d
x_tld_k_dual =s_d
x_tld_k1_dual=s_d
! --------------------
z_k_dual =i_d
z_k1_dual =i_d
z_tld_k_dual =i_d
z_tld_k1_dual=i_d
y_k_dual =s_d
y_k1_dual =s_d
y_tld_k_dual =s_d
y_tld_k1_dual=s_d
x_k_dual =s_d
x_k1_dual =s_d
x_tld_k_dual =s_d
x_tld_k1_dual=s_d
! --------------------
DoDuals=.TRUE.
#endif
kount=1
sz_norm=1d0
FillN=1d10
DO i = 0, 17
! check the trace for convergence:
FillN_prev=FillN
TrX=SpAMM_trace_tree_2d_symm_recur(x)
FillN = abs( dble(s%frill%ndimn(1)) - TrX )/dble(s%frill%ndimn(1))
WRITE(*,33)tau, kount, TrX, FillN, 1d2*x%frill%non0s/dble(x%frill%ndimn(1))**2, y_work*1d2 , z_work*1d2 , zs_work*1d2 , x_work*1d2
33 format(' ... Tr< ',e6.1,', i=',i2,' > = ', F18.10,' dN=',e10.3,' %of N^2 = ',f10.5,'%, %of N^3 = ',f10.5,'%, ',f10.5,'%, ',f10.5,'%, ',f10.5,'%')
IF(i>2 .and. FillN<0.1d0 .AND. FillN>FillN_prev )then
! WRITE(*,*)' fill n = ',filln,' filln_prev ',filln_prev
! write(*,*)' elevation'
! RETURN ! Elevation
endif
IF( FillN < Tau )then
! WRITE(*,*)' fill n = ',filln,' tau**2 = ',tau
! write(*,*)' anhiliaiton '
! RETURN ! Anihilation
end IF
#ifdef DENSE_DIAGNOSTICS
m_x_k1_stab=x_k1_stab
m_x_k1_dual=x_k1_dual
scal_shift_stab=1d0
shft_shift_stab=0d0
scal_mapp_stab =1d0
shft_mapp_stab =0d0
scal_shift_dual=1d0
shft_shift_dual=0d0
scal_mapp_dual =1d0
shft_mapp_dual =0d0
#endif
!!$ IF(DoDuals)THEN
!!$ sc=1d0
!!$
!!$ write(*,*)' a '
!!$ ELSE
IF(FillN>0.4d0)THEN
delta=8.d-2 ! maybe this should be a variable too, passed in?
#ifdef DENSE_DIAGNOSTICS
x_stab => spammsand_shift_tree_2d( x_stab, low_prev=0d0, high_prev=1d0, low_new=delta, high_new=1d0-delta , .TRUE. )
x_dual => spammsand_shift_tree_2d( x_dual, low_prev=0d0, high_prev=1d0, low_new=delta, high_new=1d0-delta , .FALSE. )
#else
IF(DoDuals)THEN
x_dual => spammsand_shift_tree_2d( x_dual, low_prev=0d0, high_prev=1d0, low_new=delta, high_new=1d0-delta )
ELSE
x_stab => spammsand_shift_tree_2d( x_stab, low_prev=0d0, high_prev=1d0, low_new=delta, high_new=1d0-delta )
ENDIF
#endif
sc=spammsand_scaling_invsqrt(SpAMM_zero)
ELSEIF(FillN>0.1d0)THEN
delta=1.0d-2 ! maybe this should be a variable too, passed in?
x => spammsand_shift_tree_2d( x, low_prev=0d0, high_prev=1d0, low_new=delta, high_new=1d0-delta )
sc=spammsand_scaling_invsqrt(SpAMM_half)
ELSE
sc=1d0
ENDIF
! ENDIF
! sc=1d0
WRITE(*,*)' sc = ',sc
#ifdef DENSE_DIAGNOSTICS
x_stab => spammsand_scaled_invsqrt_mapping( x_stab, sc, .TRUE. )
x_dual => spammsand_scaled_invsqrt_mapping( x_dual, sc, .FALSE. )
#else
IF(DoDuals)THEN
x_dual => spammsand_scaled_invsqrt_mapping( x_dual, sc)
ELSE
x_stab => spammsand_scaled_invsqrt_mapping( x_stab, sc)
ENDIF
#endif
! |Z_n+1> = <Z_n| X_n>
z_tmp => SpAMM_tree_2d_symm_times_tree_2d_symm( z_stab, x_stab, tau , nt_O=.TRUE., in_O = z_tmp , stream_file_O='z_stab'//inttoCHAR(I) )
z_stab => SpAMM_tree_2d_symm_copy_tree_2d_symm( z_tmp, in_O = z_stab, threshold_O = tau )
z_work=z_tmp%frill%flops/dble(z_tmp%frill%ndimn(1))**3
z_fill=z%frill%non0s
z_tmp => SpAMM_tree_2d_symm_times_tree_2d_symm( z_dual, x_dual, tau , nt_O=.TRUE., in_O = z_tmp , stream_file_O='z_dual'//inttoCHAR(I) )
z_dual=> SpAMM_tree_2d_symm_copy_tree_2d_symm( z_tmp, in_O = z_dual, threshold_O = tau )
z_work=z_tmp%frill%flops/dble(z_tmp%frill%ndimn(1))**3
z_fill=z%frill%non0s
! stats
#ifdef DENSE_DIAGNOSTICS
CALL SpAMM_convert_tree_2d_symm_to_dense( z_stab, z_tld_k_stab )
CALL SpAMM_convert_tree_2d_symm_to_dense( z_dual, z_tld_k_dual )
z_k_stab = MATMUL( z_k1_stab, m_x_k1_stab )
z_k_dual = MATMUL( z_k1_dual, m_x_k1_dual )
#else
IF(DoDuals)tHeN
#endif
! <Y_n+1| CALL SpAMM_convert_tree_2d_symm_to_dense( x_dual, m_x_tld_k1_dual )
y_tmp => SpAMM_tree_2d_symm_times_tree_2d_symm( x_dual, y_dual, tau_xtra , nt_O=.TRUE., in_O = y_tmp , stream_file_O='y_dual_'//inttoCHAR(I) )
y_work=y_tmp%frill%flops/dble(y_tmp%frill%ndimn(1))**3
y_norm=SQRT(y_tmp%frill%norm2)
! next -> previous
y_dual => SpAMM_tree_2d_symm_copy_tree_2d_symm( y_tmp, in_O = y_dual, threshold_O = tau_xtra )
! <X_n> = <Y_n|Z_n>
x_dual => SpAMM_tree_2d_symm_times_tree_2d_symm( y_dual, z_dual, tau , nt_O=.TRUE. , in_O = x_dual )
x_work=x%frill%flops/dble(x%frill%ndimn(1))**3
#ifdef DENSE_DIAGNOSTICS
y_k_dual = MATMUL( m_x_k1_dual, y_k1_dual )
CALL SpAMM_convert_tree_2d_symm_to_dense( y_dual, y_tld_k_dual )
x_k_dual = MATMUL( y_k_dual , z_k_dual )
CALL SpAMM_convert_tree_2d_symm_to_dense( x_dual, x_tld_k_dual )
#else
ELSE
#endif
! | y > = < z^t_k | s >
y_tmp => SpAMM_tree_2d_symm_times_tree_2d_symm( z_stab, s, tau_XTRA , NT_O=.FALSE. , in_O = y_tmp )
zs_work=y_tmp%frill%flops/dble(y_tmp%frill%ndimn(1))**3
zs_norm=SQRT(y_tmp%frill%norm2)
! <X_k> = <Z_k|S|Z_k>
x_stab => SpAMM_tree_2d_symm_times_tree_2d_symm( y_tmp, z_stab, tau , NT_O=.TRUE. , in_O = x_stab , stream_file_O='x_stab_'//inttoCHAR(I) )
x_work=x%frill%flops/dble(x%frill%ndimn(1))**3
#ifdef DENSE_DIAGNOSTICS
CALL SpAMM_convert_tree_2d_symm_to_dense( x_stab, x_tld_k_stab )
x_k_stab = MATMUL( TRANSPOSE( z_k_stab ) , MATMUL( s_d, z_k_stab ) )
if(i>0)CALL SpAMMsand_Error_Analysis(kount, tau, FillN)
y_k1_dual=y_k_dual
z_k1_dual=z_k_dual
x_k1_dual=x_k_dual
z_k1_stab=z_k_stab
x_k1_stab=x_k_stab
y_k1_dual=y_k_dual
z_k1_dual=z_k_dual
x_k1_dual=x_k_dual
z_k1_stab=z_k_stab
x_k1_stab=x_k_stab
#else
endif
#endif
kount=kount+1
END DO
#ifdef DENSE_DIAGNOSTICS
STOP ' This has been a diagnostics run, please see fort.99 '
#endif
CaLl SpAMM_destruct_tree_2d_symm_recuR (Y)
CaLl SpAMM_destruct_tree_2d_symm_recuR (Y_tmp)
CaLl SpAMM_destruct_tree_2d_symm_recuR (Z_tmp)
END SUBROUTINE spammsand_scaled_newton_shulz_inverse_squareroot
#ifdef DENSE_DIAGNOSTICS
SUBROUTINE SpAMMsand_Error_Analysis(kount, tau, FillN)
real(spamm_kind) :: Tau, FillN
integer :: i,kount
real(spamm_kind), dimension(:,:), ALLOCATABLE :: &
dz_stab, dz_hat_stab, dx_stab, dx_hat_stab, &
dy_dual, dy_hat_dual,dz_dual, dz_hat_dual, &
dx_dual, dx_hat_dual,mp_stab, mp_dual, &
dz_k_dx_k1_stab,dz_k_dz_k1_dual,dy_k_dy_k1_dual, &
dz_k_dx_k1_dual,dy_k_dx_k1_dual, &
x_tld_k_of_xk1_stab,x_tld_k_of_zk1_stab, &
x_tld_k_of_xk1_dual,x_tld_k_of_yk1_dual,x_tld_k_of_zk1_dual, &
xp_tld_k_stab_gateaux,zp_tld_k_stab_gateaux,xp_tld_k_dual_gateaux, &
yp_tld_k_dual_gateaux,zp_tld_k_dual_gateaux, &
FpX_stab,FpZ_stab,FpX_dual,FpZ_dual,FpY_dual
allocate(dz_stab (1:N,1:N) )
allocate(dz_hat_stab (1:N,1:N) )
allocate(dx_stab (1:N,1:N) )
allocate(dx_hat_stab (1:N,1:N) )
allocate(dy_dual (1:N,1:N) )
allocate(dy_hat_dual (1:N,1:N) )
allocate(dz_dual (1:N,1:N) )
allocate(dz_hat_dual (1:N,1:N) )
allocate(dx_dual (1:N,1:N) )
allocate(dx_hat_dual (1:N,1:N) )
allocate(mp_stab (1:N,1:N) )
allocate(mp_dual (1:N,1:N) )
allocate(dz_k_dx_k1_stab (1:N,1:N) )
allocate(dz_k_dz_k1_dual (1:N,1:N) )
allocate(dy_k_dy_k1_dual (1:N,1:N) )
allocate(dz_k_dx_k1_dual (1:N,1:N) )
allocate(dy_k_dx_k1_dual (1:N,1:N) )
allocate(x_tld_k_of_xk1_stab (1:N,1:N) )
allocate(x_tld_k_of_zk1_stab (1:N,1:N) )
allocate(x_tld_k_of_xk1_dual (1:N,1:N) )
allocate(x_tld_k_of_yk1_dual (1:N,1:N) )
allocate(x_tld_k_of_zk1_dual (1:N,1:N) )
allocate(xp_tld_k_stab_gateaux (1:N,1:N) )
allocate(zp_tld_k_stab_gateaux (1:N,1:N) )
allocate(xp_tld_k_dual_gateaux (1:N,1:N) )
allocate(yp_tld_k_dual_gateaux (1:N,1:N) )
allocate(zp_tld_k_dual_gateaux (1:N,1:N) )
allocate(FpX_stab (1:N,1:N) )
allocate(FpZ_stab (1:N,1:N) )
allocate(FpX_dual (1:N,1:N) )
allocate(FpZ_dual (1:N,1:N) )
allocate(FpY_dual (1:N,1:N) )
!-----------------
dz_stab = z_tld_k1_stab - z_k1_stab
dz_hat_stab = dz_stab/SQRT(SUM(dz_stab**2))
dx_stab = x_tld_k1_stab - x_k1_stab
dx_hat_stab = dx_stab/SQRT(SUM(dx_stab**2))
dy_dual = y_tld_k1_dual - y_k1_dual
dy_hat_dual = dy_dual/SQRT(SUM(dy_dual**2))
dz_dual = z_tld_k1_dual - z_k1_dual
dz_hat_dual = dz_dual/SQRT(SUM(dx_dual**2))
dx_dual = x_tld_k1_dual - x_k1_dual
dx_hat_dual = dx_dual/SQRT(SUM(dx_dual**2))
mp_stab = scal_mapp_stab * scal_shift_stab * dx_hat_stab
mp_dual = scal_mapp_dual * scal_shift_dual * dx_hat_dual
!-----------------
dz_k_dx_k1_stab = MATMUL( z_tld_k1_stab, mp_stab )
dz_k_dz_k1_stab = MATMUL( dz_hat_stab, m_x_tld_k1_stab )
dz_k_dz_k1_dual = MATMUL( dz_hat_dual, m_x_tld_k1_dual )
dy_k_dy_k1_dual = MATMUL( m_x_tld_k1_dual, dy_hat_dual )
dz_k_dx_k1_dual =
dy_k_dx_k1_dual =
x_tld_k_of_xk1_stab = MATMUL( MATMUL( TRANSPOSE(MATMUL(z_tld_k1_stab , m_x_k1_stab )), S_d), MATMUL(z_tld_k1_stab,m_x_k1_stab) )
x_tld_k_of_zk1_stab = MATMUL( MATMUL( TRANSPOSE(MATMUL(z_k1_stab , m_x_tld_k1_stab )), S_d), MATMUL(z_k1_stab,m_x_tld_k1_stab) )
x_tld_k_of_xk1_dual = MATMUL( MATMUL( m_x_k1_dual, y_tld_k1_dual ) , MATMUL( z_tld_k1_dual , m_x_k1_dual ) )
x_tld_k_of_yk1_dual = MATMUL( MATMUL( m_x_tld_k1_dual, y_k1_dual ) , MATMUL( z_tld_k1_dual , m_x_tld_k1_dual ) )
x_tld_k_of_zk1_dual = MATMUL( MATMUL( m_x_tld_k1_dual, y_tld_k1_dual ) , MATMUL( z_k1_dual , m_x_tld_k1_dual ) )
xp_tld_k_stab_gateaux = (x_tld_k_stab-x_tld_k_of_xk1_stab)/SQRT(SUM(dx_stab**2))
zp_tld_k_stab_gateaux = (x_tld_k_stab-x_tld_k_of_zk1_stab)/SQRT(SUM(dz_stab**2))
xp_tld_k_dual_gateaux = (x_tld_k_dual-x_tld_k_of_xk1_dual)/SQRT(SUM(dx_dual**2))
yp_tld_k_dual_gateaux = (x_tld_k_dual-x_tld_k_of_yk1_dual)/SQRT(SUM(dy_dual**2))
zp_tld_k_dual_gateaux = (x_tld_k_dual-x_tld_k_of_zk1_dual)/SQRT(SUM(dz_dual**2))
FpX_stab = MATMUL( TRANSPOSE(dz_k_dx_k1_stab) , MATMUL( s_d, z_tld_k_stab ) ) + MATMUL( TRANSPOSE(z_tld_k_stab) , MATMUL( s_d, dz_k_dx_k1_stab ) )
FpZ_stab = MATMUL( TRANSPOSE(dz_k_dz_k1_stab) , MATMUL( s_d, z_tld_k_stab ) ) + MATMUL( TRANSPOSE(z_tld_k_stab) , MATMUL( s_d, dz_k_dz_k1_stab ) )
FpX_dual = MATMUL( dy_k_dx_k1_dual , z_tld_k_dual ) + MATMUL( y_tld_k_dual , dz_k_dx_k1_dual )
FpZ_dual = MATMUL( y_tld_k_dual , dz_k_dz_k1_dual )
FpY_dual = MATMUL( dy_k_dy_k1_dual , z_tld_k_dual )
WRITE(*,*)' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '
WRITE(*,*)' '
WRITE(*,*)' dx_stab = ',sqrt(sum(dx_stab**2)),' dz_stab = ',sqrt(sum(dz_stab**2))
WRITE(*,*)' '
WRITE(*,*)" ||f'_dz_k1_stab|| = ",SQRT(SUM(fpz_stab**2)), SQRT(SUM( zp_tld_k_stab_gateaux**2))
WRITE(*,*)" ||f'_dx_k1_stab|| = ",SQRT(SUM(fpx_stab**2)), SQRT(SUM( xp_tld_k_stab_gateaux**2))
WRITE(*,*)' '
WRITE(*,*)' dx_dual = ',sqrt(sum(dx_dual**2)),' dz_dual = ',sqrt(sum(dz_dual**2)),' dy_dual = ',sqrt(sum(dy_dual**2))
WRITE(*,*)' '
WRITE(*,*)" ||f'_dy_k1_dual|| = ",SQRT(SUM(fpy_dual**2)), SQRT(SUM( yp_tld_k_dual_gateaux**2))
WRITE(*,*)" ||f'_dz_k1_dual|| = ",SQRT(SUM(fpz_dual**2)), SQRT(SUM( zp_tld_k_dual_gateaux**2))
WRITE(*,*)" ||f'_dx_k1_dual|| = ",SQRT(SUM(fpx_dual**2)), SQRT(SUM( xp_tld_k_dual_gateaux**2))
WRITE(*,*)' - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - '
IF(kount==2)WRITE(*,43)
WRITE(99,44)kount, tau, FillN, sqrt(sum(dx_stab**2)),sqrt(sum(dz_stab**2)),SQRT(SUM(fpz_stab**2)),SQRT(SUM(fpx_stab**2)), &
sqrt(sum(dx_dual**2)),sqrt(sum(dz_dual**2)),sqrt(sum(dy_dual**2)),SQRT(SUM(fpy_dual**2)),SQRT(SUM(fpz_dual**2)),SQRT(SUM(fpx_dual**2))
44 format(I3," ", 20(e12.6,", "))
deallocate(dz_stab)
deallocate(dz_hat_stab)
deallocate(dx_stab)
deallocate(dx_hat_stab)
deallocate(dy_dual)
deallocate(dy_hat_dual)
deallocate(dz_dual)
deallocate(dz_hat_dual)
deallocate(dx_dual)
deallocate(dx_hat_dual)
deallocate(mp_stab)
deallocate(mp_dual)
deallocate(dz_k_dx_k1_stab)
deallocate(dz_k_dz_k1_dual)
deallocate(dy_k_dy_k1_dual)
deallocate(dz_k_dx_k1_dual)
deallocate(dy_k_dx_k1_dual)
deallocate(x_tld_k_of_xk1_stab)
deallocate(x_tld_k_of_zk1_stab)
deallocate(x_tld_k_of_xk1_dual)
deallocate(x_tld_k_of_yk1_dual)
deallocate(x_tld_k_of_zk1_dual)
deallocate(xp_tld_k_stab_gateaux)
deallocate(zp_tld_k_stab_gateaux)
deallocate(xp_tld_k_dual_gateaux)
deallocate(yp_tld_k_dual_gateaux)
deallocate(zp_tld_k_dual_gateaux)
deallocate(FpX_stab)
deallocate(FpZ_stab)
deallocate(FpX_dual)
deallocate(FpZ_dual)
deallocate(FpY_dual)
END SUBROUTINE SpAMMsand_Error_Analysis
#endif
FUNCTION SpAMMsand_Basis_Compare(a,b) RESULT(compare)
REAL(SpAMM_kind) :: compare, MAX_DOT
REAL(spamm_kind), dimension(:,:) :: a,b
compare=SQRT( SUM( ( MATMUL(a,b) - MATMUL(b,a) )**2 ) ) &
/SQRT( SUM(a**2)) /SQRT( SUM(b**2) )
END FUNCTION SpAMMsand_Basis_Compare
#ifdef DENSE_DIAGNOSTICS
FUNCTION spammsand_shift_tree_2d( x, low_prev, high_prev, low_new, high_new, stab ) RESULT(d)
#else
FUNCTION spammsand_shift_tree_2d( x, low_prev, high_prev, low_new, high_new ) RESULT(d)
#endif
!!!!! shft=low_new+(x-low_prev)*(high_new-low_new)/(high_prev-low_prev)
TYPE(spamm_tree_2d_symm) , POINTER :: d
TYPE(spamm_tree_2d_symm) , POINTER :: x
REAL(SpAMM_KIND), OPTIONAL,INTENT(IN) :: low_prev, high_prev, low_new, high_new
REAL(SpAMM_KIND) :: SHFT,SCAL
#ifdef DENSE_DIAGNOSTICS
logical :: stab
#endif
integer :: i
SHFT=low_new-low_prev*(high_new-low_new)/(high_prev-low_prev)
SCAL=(high_new-low_new)/(high_prev-low_prev)
#ifdef DENSE_DIAGNOSTICS
IF(stab)then
scal_shift_stab=scal
shft_shift_stab=shft
m_x_k1_stab=m_x_k1_stab*scal
m_x_k1_stab=m_x_k1_stab*shft*i_d
ELSE
scal_shift_dual=scal
shft_shift_dual=shft
m_x_k1_dual=m_x_k1_dual*scal
m_x_k1_dual=m_x_k1_dual+shft*i_d
ENDIF
#endif
d => x
d => SpAMM_scalar_times_tree_2d_symm( scal, d)
d => SpAMM_scalar_plus_tree_2d_symm( shft, d)
END FUNCTION spammsand_shift_tree_2d
#ifdef DENSE_DIAGNOSTICS
FUNCTION spammsand_scaled_invsqrt_mapping( x, sc, stab ) result(d)
#else
FUNCTION spammsand_scaled_invsqrt_mapping( x, sc ) result(d)
#endif
TYPE(spamm_tree_2d_symm), POINTER :: d
TYPE(spamm_tree_2d_symm), POINTER :: x
REAL(SpAMM_KIND), INTENT(IN) :: sc
REAL(SpAMM_KIND) :: SHFT,SCAL
#ifdef DENSE_DIAGNOSTICS
logical :: stab
#endif
SHFT=SpAMM_half*SQRT(sc)*SpAMM_three
SCAL=SpAMM_half*(-sc)*SQRT(sc)
#ifdef DENSE_DIAGNOSTICS
IF(stab)then
scal_mapp_stab=scal
shft_mapp_stab=shft
m_x_k1_stab=m_x_k1_stab*scal
m_x_k1_stab=m_x_k1_stab*shft*i_d
ELSE
scal_mapp_dual=scal
shft_mapp_dual=shft
m_x_k1_dual=m_x_k1_dual*scal
m_x_k1_dual=m_x_k1_dual+shft*i_d
ENDIF
#endif
d => x
d => SpAMM_scalar_times_tree_2d_symm( scal, d)
d => SpAMM_scalar_plus_tree_2d_symm( shft, d)
END FUNCTION spammsand_scaled_invsqrt_mapping
FUNCTION spammsand_scaling_invsqrt(xo) RESULT(sc)
REAL(SpAMM_KIND) :: xo, sc
sc=MIN( Approx3, SpAMM_three/( SpAMM_one + SQRT(xo) + xo) )
END FUNCTION spammsand_scaling_invsqrt
end module SpAMMsand_inverse_squareroot
! Nested SpAMM solvers (SpAMM sandwitches) for matrix functions
program SpAMM_sandwich_inverse_squareroot
USE spammpack
USE sandpack
USE SpAMMsand_inverse_squareroot
use test_utilities
implicit none
real(spamm_kind), dimension(:,:), ALLOCATABLE :: s_dense
TYPE(spammsand_tree_2d_slices), pointer :: z, z_head
!, y, y_head
type(SpAMM_tree_2d_symm), pointer :: s => null()
type(SpAMM_tree_2d_symm), pointer :: x => null()
type(SpAMM_tree_2d_symm), pointer :: t => null()
type(SpAMM_tree_2d_symm), pointer :: s_orgnl => null()
type(SpAMM_tree_2d_symm), pointer :: z_total => null()
! real(spamm_kind), dimension(:, :), allocatable :: S_dense
character(len = 1000) :: matrix_filename
real(SpAMM_KIND) :: x_hi, x_new, logtau_strt, logtau_stop, logtau_dlta, &
tau_dlta, tau_xtra, error, tmp1,tmp2, final_tau, s_work, zs_work
logical :: first, second
integer, parameter :: slices=6
real(SpAMM_KIND), dimension(1:slices) :: tau
integer :: i,j,k, kount
! real :: start_time, end_time
call get_command_argument(1, matrix_filename)
call read_MM(matrix_filename, S_dense)
S_dense=SpAMM_half*(S_dense+TRANSPOSE(S_dense))
! matrix to inverse factor
s => SpAMM_convert_dense_to_tree_2d_symm( S_DENSE, in_O = s )
#ifdef DENSE_DIAGNOSTICS
N = s%frill%ndimn(1)
LWORK = 1+6*N+2*N**2
LIWORK = 3+5*N
allocate(eval(N))
allocate(work(LWORK))
allocate(iwork(LIWORK))
#endif
!=============================================================
! the max eigenvalue
x_hi = SpAMMSand_rqi_extremal(s,1d-4,high_O=.TRUE.)
WRITE(*,*)' hi extremal = ',x_hi
! normalize the max ev of s to 1.
s => SpAMM_scalar_times_tree_2d_symm( SpAMM_one/x_hi , s )
s_orgnl => SpAMM_tree_2d_symm_copy_tree_2d_symm( s, in_O = s_orgnl, threshold_O = SpAMM_normclean )
logtau_strt=-2 ! starting accuracy
logtau_stop=-10 ! stoping "
logtau_dlta=(logtau_stop-logtau_strt)/dble(slices-1) ! span (breadth) of SpAMM thresholds
tau_dlta=10d0**logtau_dlta
final_tau=10d0**logtau_stop ! penultimate spamm threshold
allocate(z)
z_head => z ! top of the sandwich
do i=1,slices
z%tau = 10d0**( logtau_strt + logtau_dlta * float(i-1) )
tau(i)=z%tau
z%mtx => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
if(i==slices)then
z%nxt => null()
else
allocate(z%nxt)
z => z%nxt
endif
enddo
write(*,33)tau
33 format(' building |Z> = ',4('|',e6.1,'>'),'...')
! work matrices ...
x => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
t => SpAMM_new_top_tree_2d_symm( s%frill%ndimn )
kount=1
! first=.FALSE.
first=.TRUE.
second=.TRUE.
z=>z_head
! z_total => I
z_total=>SpAMM_set_identity_2d_symm ( s%frill%ndimn, in_o = z_total )
do while(associated(z)) ! build the nested inverse factors |z> = |z_1>.|z_2> ... |z_s>
! WATER block=32, tau=0.07
! BAD TUBE block=32, tau=0.01
z%tau = 0.001d0
call spammsand_scaled_newton_shulz_inverse_squareroot( s, x, z%mtx, t, z%tau, first, second, kount )
STOP
first=.TRUE.
second=.FALSE.
if(.not.associated(z%nxt))exit
t => SpAMM_tree_2d_symm_times_tree_2d_symm( z_total, z%mtx, z%nxt%tau , NT_O=.TRUE., in_O = t )
z_total => SpAMM_tree_2d_symm_copy_tree_2d_symm( t, in_O = z_total, threshold_O = z%nxt%tau )
! Nested sandwich, to recompute the error every time.
t => SpAMM_tree_2d_symm_times_tree_2d_symm( z_total, s_orgnl, z%nxt%tau*1d-2, NT_O=.FALSE., in_O = t )
s => SpAMM_tree_2d_symm_times_tree_2d_symm( t, z_total, z%nxt%tau , NT_O=.TRUE. , in_O = s )
s_work=s%frill%flops/dble(s%frill%ndimn(1))**3
zs_work=t%frill%flops/dble(t%frill%ndimn(1))**3
write(*,*)' t work = ',zs_work,', s_work = ',s_work
x_new = SpAMMSand_rqi_extremal( x, z%nxt%tau*1d-2 , high_O=.TRUE. )
WRITE(*,*)' hi extremal = ',x_new
x => SpAMM_scalar_times_tree_2d_symm( SpAMM_one / x_new , x )
x_hi = x_hi * x_new
s => SpAMM_tree_2d_symm_copy_tree_2d_symm( x, in_O = s , threshold_O = SpAMM_normclean )
z => z%nxt
enddo
! de-normalize, back to max_ev (1->max_ev)...
z=>z_head
z%mtx => SpAMM_scalar_times_tree_2d_symm(SpAMM_one/SQRT(x_hi), z%mtx)
! at this point, we are basically done. the rest is IO/verification of trace, error, timing & etc
! timers down ...
! x <= |z_spammsand> = |z_1> . |z_2> ... |z_slices>. Has to be applied as left (T) and right (N) (its not symmetric)
x => SpAMM_tree_2d_symm_copy_tree_2d_symm( z%mtx, in_O = x )
z => z%nxt
do while(associated(z)) ! build the inverse factors |z> = |z_1>.|z_2> ... |z_slices> (right handed)
t => SpAMM_tree_2d_symm_times_tree_2d_symm( x, z%mtx, 1d-20, nt_O=.TRUE., in_O = t )
x => SpAMM_tree_2d_symm_copy_tree_2d_symm( t , in_O = x, threshold_O = SpAMM_zero )
z => z%nxt
enddo
! get the original back ...
s => SpAMM_convert_dense_to_tree_2d_symm(S_DENSE)
! I = <z^T_spammsand|s|z_spammsand>
t => SpAMM_tree_2d_symm_times_tree_2d_symm( s, x, 1d-20, nt_O=.TRUE. , in_O = t )
s => SpAMM_tree_2d_symm_times_tree_2d_symm( x, t, 1d-20, nt_O=.FALSE., in_O = s )
! error = [ <Zt_1|Zt_2|...<Zt_slice| S |Z_slice>...|Z_2>|Z_1> -1.*I ] /N**2
s => SpAMM_scalar_plus_tree_2d_symm( -SpAMM_one, s)
error=SQRT(s%frill%norm2)/dble(s%frill%ndimn(1))**2
write(*,*)' error = ',error
end program SpAMM_sandwich_inverse_squareroot
!!$
!!$ FUNCTION SpAMMsand_zee_dot_x_error( z, x, tau) RESULT(d)
!!$
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER, INTENT(IN) :: z, x
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER :: d
!!$ REAL(spamm_kind) :: tau
!!$
!!$ real(spamm_kind), dimension(1:z%frill%ndimn(1),1:z%frill%ndimn(2)) :: x_d, z_d, d_d
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( x, x_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( z, z_d )
!!$
!!$ d => SpAMM_tree_2d_symm_times_tree_2d_symm( z, x, tau , nt_O=.TRUE., in_O = in_o )
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( d, d_d )
!!$
!!$ d_d = d_d - MATMUL( z_d , x_d )
!!$
!!$! d_d = 0.5d0*(d_d+TRANSPOSE(d_d))
!!$
!!$ d => SpAMM_convert_dense_to_tree_2d_symm( d_d, in_o = d )
!!$
!!$ END FUNCTION SpAMMsand_zee_dot_x_error
!!$
!!$ FUNCTION SpAMMsand_compare_naiv(z,s,x) RESULT(compare)
!!$
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER, INTENT(IN) :: z, s, x
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER :: d
!!$
!!$ real(spamm_kind) :: compare
!!$
!!$ real(spamm_kind), dimension(1:s%frill%ndimn(1),1:s%frill%ndimn(2)) :: s_d, z_d, x_naiv, x_d
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( s, s_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( z, z_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( x, x_d )
!!$
!!$ x_naiv=MATMUL(z_d,MATMUL(s_d,z_d))
!!$ x_d=x_d-x_naiv
!!$
!!$ compare=SpAMMsand_Basis_Compare_d(s_d,x_d)
!!$
!!$ END FUNCTION SpAMMsand_compare_naiv
!!$
!!$ FUNCTION SpAMMsand_compare_stab(z,s,x) RESULT(compare)
!!$
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER, INTENT(IN) :: z, s,x
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER :: d
!!$
!!$ real(spamm_kind) :: compare
!!$
!!$ real(spamm_kind), dimension(1:s%frill%ndimn(1),1:s%frill%ndimn(2)) :: s_d, z_d, x_naiv, x_d
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( s, s_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( z, z_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( x, x_d )
!!$
!!$ x_naiv=MATMUL(TRANSPOSE(z_d),MATMUL(s_d,z_d))
!!$ x_d=x_d-x_naiv
!!$
!!$ compare=SpAMMsand_Basis_Compare_d(s_d,x_d)
!!$
!!$ END FUNCTION SpAMMsand_compare_stab
!!$
!!$
!!$
!!$ MAX_DOT=-1D10
!!$ DO I=1,N
!!$ DO J=1,N
!!$ MAX_DOT=MAX(MAX_DOT,DOT_PRODUCT(A_D(:,I),B_D(:,J)))
!!$ ENDDO
!!$ ENDDO
!!$
!!$ WRITE(*,*)'a',MAX_DOT,ACOS(MAX_DOT)
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( a, a_d )
!!$ a_d=TRANSPOSE(a_d)
!!$ call dsyevd("V", "U", N, a_d, N, eval, work, LWORK, iwork, LIWORK, info)
!!$
!!$ MAX_DOT=-1D10
!!$ DO I=1,N
!!$ DO J=1,N
!!$ MAX_DOT=MAX(MAX_DOT,DOT_PRODUCT(A_D(:,I),B_D(:,J)))
!!$ ENDDO
!!$ ENDDO
!!$
!!$ WRITE(*,*)'b', MAX_DOT,ACOS(MAX_DOT)
!!$
!!$
!!$
!!$! svd=MATMUL(TRANSPOSE(a_d),b_d)
!!$! call dsyevd("V", "U", N, svd, N, eval, work, LWORK, iwork, LIWORK, info)
!!$! WRITE(*,*)' ev = ',eval(1),acos(eval(1))
!!$
!!$! svd=MATMUL(a_d,b_d)
!!$! call dsyevd("V", "U", N, svd, N, eval, work, LWORK, iwork, LIWORK, info)
!!$! WRITE(*,*)' ev = ',eval(1:2)
!!$
!!$
!!$!A = orth(A);
!!$!B = orth(B);
!!$!s = svd(A'*B);
!!$!a = acos(min(s,1));
!!$
!!$ MAX_DOT=-1D10
!!$ DO I=1,N
!!$ DO J=1,N
!!$ MAX_DOT=MAX(MAX_DOT,DOT_PRODUCT(A_D(:,I),B_D(:,J)))
!!$ ENDDO
!!$ ENDDO
!!$
!!$ WRITE(*,*)MAX_DOT,ACOS(MAX_DOT)
!!$
!!$ compare =ACOS(MAX_DOT)
!!$
!!$ FUNCTION SpAMMsand_dense_stab(z,s, in_o) RESULT(d)
!!$
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER, INTENT(IN) :: z, s
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER, OPTIONAL :: in_o
!!$ TYPE(SpAMM_tree_2d_symm) , POINTER :: d
!!$
!!$ real(spamm_kind), dimension(1:s%frill%ndimn(1),1:s%frill%ndimn(2)) :: s_d, z_d, x_stab
!!$
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( s, s_d )
!!$ CALL SpAMM_convert_tree_2d_symm_to_dense( z, z_d )
!!$
!!$ x_stab=MATMUL(TRANSPOSE(z_d),MATMUL(s_d,z_d))
!!$ d=> SpAMM_convert_dense_to_tree_2d_symm( x_stab, in_o = in_o )
!!$
!!$ END FUNCTION SpAMMsand_dense_stab