program FourLevelPulseProp_v3_Double ! ! Written by: Dr. Frank A. Narducci ! Written on: May 12, 2008 ! ! This program calculates the propagation of a pulse of arbitrary strength ! through a two level medium. The equations used are the full equations ! based on the Risken-Numedal discretization technique. ! ! This program only "watches" the evolution of the pulse in the cell. This is ! due to the constraint that the cell is very small relative to the pulse lengths ! that we want to use. If we watched the pulse outside the cell and then increased the ! resolution within the cell, the increased burden outside the cell because huge. ! ! v2 Notes: This program is based on the dimensionless equatiosn derived on 5/16 implicit none ! ! Double Notes: This program is the same as TwoLevelPulseProp_v2 but with double precision complex ci ! ! ThreeLevel Notes: This program is the same as TwoLevelPulseProp_v2_Double but now for a ! three level system ! ! Four Level Notes: Valid to first order in dt ! ! Version 3: Make a step function in the coupling field. character*150 fname integer nmat,npts,Nfrac,Nframe,Nframemax,NSkip,NWrite,tpts,zpts parameter (nmat=3,npts=100) !matrix size, number of detuning points in dispersion curve !REMEMBER TO CHANGE NMAT IN LMatConstruct Routine parameter (tpts=100,zpts=tpts+1) !Caution: funny things happened when tpts=200 (and presumably greater) !tpts is the number of temporal points in the cell parameter (Nframemax=2000000) parameter ( NWrite=100) !number of frames to actually write integer i,j,k,m,n complex*16 a1,a2,a3,a4,a5,a6 complex*16 b1,b2,b3,b4,b5,b6,b7 complex*16 c1,c2,c3,c4,c5,c6 complex*16 d1,d2,d3,d4,d5,d6,d7 complex*16 e1,e2,e3,e4,e5 complex*16 f1,f2,f3,f4,f5 complex*16 g1,g2,g3,g4,g5,g6,g7 complex*16 h1,h2,h3,h4,h5 complex*16 i1,i2,i3,i4,i5,i6,i7 complex*16 j1,j2,j3,j4,j5,j6,j7 complex*16 k1,k2,k3,k4,k5,k6,k7,k8,k9,k10 complex*16 l1,l2,l3,l4,l5 complex *16 Omold, Omold_vac real*8 alpha1,alpha2,alpha1tilde,alpha2tilde,alphac,alphactilde,beta,c,delmax,del1_prop,del2_prop,delc_prop real*8 dt,dz,eta real*8 W12,W21,W31,W32,W41,W42,W43,W34,ga12,ga13,ga14,ga23,ga24,ga34 real*8 Ga2,Ga4,Om_crit real*8 Lcell,Om1peak,Om2peak,Omcpeak,pi,tmax,tp,tshift,t_end,t_start,t_elapsed real*8 tpeak,tpeak_vac real*8 epsil,hbar,lambda real*8 del(npts) real*8 t,z(zpts) complex*16 yplot(nmat,npts) complex*16 Imat(nmat) complex*16 Om1(zpts),Om2(zpts),Omc(zpts),Om_vac(zpts) complex*16 rho11(zpts),rho12(zpts),rho13(zpts),rho14(zpts),rho21(zpts),rho22(zpts),rho23(zpts),rho24(zpts) complex*16 rho31(zpts),rho32(zpts),rho33(zpts),rho34(zpts),rho41(zpts),rho42(zpts),rho43(zpts),rho44(zpts) complex*16 rho11_last(zpts),rho12_last(zpts),rho13_last(zpts),rho14_last(zpts) complex*16 rho21_last(zpts),rho22_last(zpts),rho23_last(zpts),rho24_last(zpts) complex*16 rho31_last(zpts),rho32_last(zpts),rho33_last(zpts),rho34_last(zpts) complex*16 rho41_last(zpts),rho42_last(zpts),rho43_last(zpts),rho44_last(zpts) !No Om_last because we never need the previous spatial point complex*16 L(nmat,nmat),Linv(nmat,nmat),Ltemp(nmat,nmat) common/para/ga12,W21 real*8 d !used by NR Routines integer indx(nmat) ! ! Fundamental numbers ! ci=cmplx(0.,1.) pi=acos(-1.0) c=3e8 hbar=1.055e-34 epsil=8.85e-12 ! ! Atomic numbers (based on Rubidium 85) ! beta=2*pi*3e6 !in Hz W41=0 W42=1 W43=0 W32=1. W31=1. W21=.001 W12=W21 W34=0 ga12=0.5*(W21+W12) ga13=0.5*(W31+W12+W32) ga14=0.5*(W41+W42+W12) ga23=0.5*(W32+W31+W21) ga24=0.5*(W21+W41+W42+W43) ga34=0.5*(W31+W41+W32+W42+W43) lambda=780.24e-9 ! ! Atomic parameters ! ! write (*,*)'Enter density in m^-1' ! read (*,*)eta eta=6.9e13 alpha1=3*eta*lambda*lambda/(2*pi) alpha1tilde=alpha1*c/beta alpha2=3*eta*lambda*lambda/(2*pi) alpha2tilde=alpha2*c/beta alphac=3*eta*lambda*lambda/(2*pi) alphactilde=alphac*c/beta ! ! Initialize matrices and set up Identity Matrix ! do 20 i=1,nmat do 10 j=1,nmat L(i,j)=cmplx(0.,0.) Linv(i,j)=cmplx(0.,0.) 10 continue !j loop Imat(i)=cmplx(0.,0.) Linv(i,i)=cmplx(1.,0.) !contains identity matrix 20 continue ! ! User defined numbers ! ! write (*,*)'Enter peak scaled Rabi frequency for the pump at entrance of cell' ! read (*,*)Om1peak Om1peak=1 ! write (*,*)'Enter peak scaled Rabi frequency for the probe at entrance of cell' ! read (*,*)Om2peak Om2peak=.01 ! write (*,*)'Enter maximum detuning in MHz for dispersion lineshape plot' ! read (*,*) delmax delmax=0 Ga4=(W41+W42+W43) Ga2=W21 Om_crit=sqrt(Om1peak**4+4*Om1peak*Om1peak*(Ga4*Ga4+Ga2*Ga4)) Om_crit=Om_crit-Om1peak*Om1peak-2*Ga2*Ga4 Om_crit=sqrt(Om_crit/2) ! write (*,*)'Om_crit = ',Om_crit ! write (*,*)'Enter peak scaled Rabi frequency for the coupling field at entrance of cell' read (*,*)Omcpeak ! ! First plot the dispersion lineshape ! ! do 40 n=1,npts ! del(n)=-delmax+2*float(n)*delmax/npts ! call LMatConstruct(Ompeak,del(n),L) !construct the L matrix ! call LMatConstruct(Ompeak,del(n),Ltemp) !Need a temporary because L gets destroyed ! call ImatConstruct(Ompeak,Imat) !Need to call in loop because it gets destroyed ! !See also note in subroutine ! call ludcmp(L,nmat,nmat,indx,d) ! call lubksb(L,nmat,nmat,indx,Imat) !Imat now contains psi ! do 35 i=1,nmat ! yplot(i,n)=Imat(i) !35 continue ! if (.false.) call MatCheck(Ltemp,Linv) !40 continue ! call plotit(del,yplot,nmat,npts) ! ! Now that the user has an idea of the dispersion, do the full propagation problem ! write (*,*)'Enter detuning of center frequency of the coupling pulse in MHz' ! read (*,*)delc_prop !del_prop is the detuning used for the propagation delc_prop=0 delc_prop=2*pi*1e6*delc_prop/beta !Now dimensionless ! write (*,*)'Enter detuning of center frequency of the pump pulse in MHz' ! read (*,*)del2_prop !del_prop is the detuning used for the propagation del2_prop=0. del2_prop=2*pi*1e6*del2_prop/beta !Now dimensionless ! write (*,*)'Enter detuning of center frequency of the probe pulse in MHz' ! read (*,*)del1_prop !del_prop is the detuning used for the propagation del1_prop=0. del1_prop=2*pi*1e6*del1_prop/beta !Now dimensionless ! write (*,*) 'Enter pulse width in nsec' ! read (*,*)tp tp=1e-6 tp=beta*tp !Now dimensionless ! write (*,*)'Enter length of cell in m' ! read (*,*)Lcell Lcell=1; Lcell=beta*Lcell/c !Now dimensionless t_start=secnds(0.E0) ! XXXXXXX ! ! Set up initial pulse. ! tshift=2*tp tmax=Lcell !Length of time to pass cell (no c because we're dimensionless) dt=tmax/tpts dz=dt !(no c because we're dimensionless) ! write (*,*)'tp = ',tp Nframe=zpts+int(4*tp/dt)+1 !Change the number 4 to anything you want to see longer pulse evolution if (Nframe.ge.Nframemax) write (*,*)'Error!!!!Nframe>Nframemax' ! write (*,*)'Nframe,tpts = ',Nframe,tpts ! ! Initialize matrices ! Omold=cmplx(0.,0.) Omold_vac=cmplx(0.,0.) tpeak=-1 tpeak_vac=-1 ! do 110 n=1,Nframe do 100 m=1,zpts Om1(m)=cmplx(0.,0.) Om2(m)=cmplx(0.,0.) Omc(m)=cmplx(0.,0.) Om_vac(m)=cmplx(0.,0.) rho11(m)=cmplx(1.,0.) !Change this to change the initial condition rho12(m)=cmplx(0.,0.) rho13(m)=cmplx(0.,0.) rho14(m)=cmplx(0.,0.) rho21(m)=cmplx(0.,0.) rho22(m)=cmplx(0.,0.) rho23(m)=cmplx(0.,0.) rho24(m)=cmplx(0.,0.) rho31(m)=cmplx(0.,0.) rho32(m)=cmplx(0.,0.) rho33(m)=cmplx(0.,0.) rho34(m)=cmplx(0.,0.) rho41(m)=cmplx(0.,0.) rho42(m)=cmplx(0.,0.) rho43(m)=cmplx(0.,0.) rho44(m)=cmplx(0.,0.) 100 continue !110 continue ! Propagation co-efficients ! a1=1. a2=0.5*ci*alpha1tilde*dt a3=0.5*ci*alpha1tilde*dt a4=0. a5=0. a6=0. b1=1. b2=0.5*ci*alpha2tilde*dt b2=0.5*ci*alpha2tilde*dt b4=0. b5=0. b6=0. b7=0. c1=1. c2=0.5*ci*alphactilde*dt c3=0.5*ci*alphactilde*dt c4=0. c5=0. c6=0. d1=1-(ga12-ci*(del2_prop-del1_prop))*dt d2=0.25*ci*dt d3=-0.25*ci*dt d4=-0.25*ci*dt d5=0.25*ci*dt d6=-0.25*ci*dt d7=-0.25*ci*dt e1=1-(ga13+ci*del1_prop)*dt e2=0.25*ci*dt e3=-0.25*ci*dt e4=0.25*ci*dt e5=-0.25*ci*dt f1=1-(ga14-ci*(del2_prop-del1_prop-delc_prop))*dt f2=0.25*ci*dt f3=-0.25*ci*dt f4=0.25*ci*dt f5=-0.25*ci*dt g1=1-(ga23+ci*del1_prop)*dt g2=-0.25*ci*dt g3=0.25*ci*dt g4=0.25*ci*dt g5=-0.25*ci*dt g6=0.25*ci*dt g7=0.25*ci*dt h1=1-(ga24+ci*delc_prop)*dt h2=0.25*ci*dt h3=0.25*ci*dt h4=0.25*ci*dt h5=0.25*ci*dt i1=1-(ga34-ci*(del2_prop-delc_prop))*dt i2=-0.25*ci*dt i3=0.25*ci*dt i4=0.25*ci*dt i5=-0.25*ci*dt i6=0.25*ci*dt i7=0.25*ci*dt j1=1-W12*dt j2=W12*dt j3=W31*dt j4=0.25*ci*dt j5=-0.25*ci*dt j6=0.25*ci*dt j7=-0.25*ci*dt k1=1-(W32+W31+W34)*dt k2=W43*dt k3=-0.25*ci*dt k4=-0.25*ci*dt k5=0.25*ci*dt k6=0.25*ci*dt k7=-0.25*ci*dt k8=-0.25*ci*dt k9=0.25*ci*dt k10=0.25*ci*dt l1=1-(W43+W42+W41)*dt l2=-0.25*ci*dt l3=0.25*ci*dt l4=-0.25*ci*dt l5=0.25*ci*dt NSkip=int(NFrame/NWrite) fname='MovieParameters4level_v2.txt' ! write (*,*)'Enter file name to save parameters' ! read (*,3)fname 3 format(a150) open(9,name=fname) write (9,133)Nframe,zpts,beta,NSkip,dt ! write (*,*)'Nframe,zpts,beta,NSkip,dt' ! write (*,133)Nframe,zpts,beta,NSkip,dt 133 format(1x,i10,',',i5,',',f12.2,',',i5,',',f12.2) close (9) fname='Movie4level_v2.dat' ! write (*,*)'Enter file name to save movie' ! read (*,3)fname open(9,name=fname) fname='Movie4level_EndPoints_v2.dat' ! write (*,*)'Enter file name to save endpoints' ! read (*,3)fname open(10,name=fname) do 60 n=1,Nframe t=float(n-1)*dt Om1(1)=Om1peak Om2(1)=Om2peak*exp(-(t-tshift)**2/(tp*tp)) Omc(1)=Omcpeak*exp(-(t-tshift)**2/(tp*tp)) Om_vac(1)=Om2(1) if (int(n/10).eq.0) write(fname,130)'Movie',n if (int(n/10).ge.1.and.int(n/100).eq.0) write (fname,131)'Movie',n if (int(n/10).ge.1.and.int(n/100).gt.0) write (fname,132)'Movie',n 130 format(a5,i1) 131 format(a5,i2) 132 format(a5,i3) ! write (*,125)fname 125 format(1x,a12) ! open(9,name=fname) do 345 m=1,zpts rho11_last(m)=rho11(m) rho12_last(m)=rho12(m) rho13_last(m)=rho13(m) rho14_last(m)=rho14(m) rho21_last(m)=rho21(m) rho22_last(m)=rho22(m) rho23_last(m)=rho23(m) rho24_last(m)=rho24(m) rho31_last(m)=rho31(m) rho32_last(m)=rho32(m) rho33_last(m)=rho33(m) rho34_last(m)=rho34(m) rho41_last(m)=rho41(m) rho42_last(m)=rho42(m) rho43_last(m)=rho43(m) rho44_last(m)=rho44(m) 345 continue do 50 m=zpts,2,-1 z(m)=float(m)*dz Om1(m)=a1*Om1(m-1)+a2*rho31_last(m)+a3*rho31_last(m-1) Om2(m)=b1*Om2(m-1)+b2*rho32_last(m)+b3*rho32_last(m-1) Omc(m)=c1*Omc(m-1)+c2*rho42_last(m)+c3*rho42_last(m-1) rho11(m)=j1*rho11_last(m)+j2*rho22_last(m)+j3*rho33_last(m)+j4*conjg(Om1(m))*rho31_last(m) rho11(m)=rho11(m)+j5*Om1(m)*rho13_last(m)+j6*conjg(Om1(m-1))*rho31_last(m)+j7*Om1(m-1)*rho13_last(m) rho12(m)=d1*rho12_last(m)+d2*conjg(Om1(m))*rho32_last(m)+d3*Om2(m)*rho13_last(m) rho12(m)=rho12(m)+d4*Omc(m)*rho14_last(m)+d5*conjg(Om1(m-1))*rho32_last(m) rho12(m)=rho12(m)+d6*Om2(m-1)*rho13_last(m)+d7*Omc(m-1)*rho14_last(m) rho13(m)=e1*rho13_last(m)+e2*conjg(Om1(m))*(rho33_last(m)-rho11_last(m)) rho13(m)=rho13(m)+e3*conjg(Om2(m))*rho12_last(m)+e4*conjg(Om1(m-1))*(rho33_last(m)-rho11_last(m)) rho13(m)=rho13(m)+e5*conjg(Om2(m-1))*rho12_last(m) rho14(m)=f1*rho14_last(m)+f2*conjg(Om1(m))*rho34_last(m)*f3*conjg(Omc(m))*rho12_last(m) rho14(m)=rho14(m)+f4*conjg(Om1(m-1))*rho34_last(m)+f5*conjg(Omc(m-1))*rho12_last(m) rho21(m)=conjg(rho12(m)) ! rho22(m) needs to be calculated lower down rho23(m)=g1*rho23_last(m)+g2*conjg(Om1(m))*rho21_last(m)+g3*conjg(Om2(m))*(rho33_last(m)-rho22_last(m)) rho23(m)=rho23(m)+g4*conjg(Omc(m))*rho43_last(m)+g5*conjg(Om1(m-1))*rho21_last(m) rho23(m)=rho23(m)+g6*conjg(Om2(m-1))*(rho33_last(m)-rho22_last(m))+g7*conjg(Omc(m-1))*rho43_last(m) rho24(m)=h1*rho24_last(m)+h2*conjg(Om2(m))*rho34_last(m)+h3*Omc(m)*(rho44_last(m)-rho22_last(m)) rho24(m)=rho24(m)+h4*conjg(Om2(m-1))*rho34_last(m)+h5*Omc(m-1)*(rho44_last(m)-rho22_last(m)) rho31(m)=conjg(rho13(m)) rho32(m)=conjg(rho23(m)) rho33(m)=k1*rho33_last(m)+k2*rho44_last(m)+k3*conjg(Om1(m))*rho31_last(m)+k4*conjg(Om2(m))*rho32_last(m) rho33(m)=rho33(m)+k5*Om1(m)*rho13_last(m)+k6*Om2(m)*rho23_last(m)+k7*conjg(Om1(m-1))*rho31_last(m) rho33(m)=rho33(m)+k8*conjg(Om2(m-1))*rho32_last(m)+k9*Om1(m-1)*rho13_last(m)+k10*Om2(m-1)*rho23_last(m) rho34(m)=i1*rho34_last(m)+i2*conjg(Omc(m))*rho32_last(m)+i3*Om1(m)*rho14_last(m)+i4*Om2(m)*rho24_last(m) rho34(m)=rho34(m)+i5*conjg(Omc(m-1))*rho32_last(m)+i6*Om1(m-1)*rho14_last(m)+i7*Om2(m-1)*rho24_last(m) rho41(m)=conjg(rho14(m)) rho42(m)=conjg(rho24(m)) rho43(m)=conjg(rho34(m)) rho44(m)=l1*rho44_last(m)+l2*conjg(Omc(m))*rho42_last(m)+l3*Omc(m)*rho24_last(m) rho44(m)=rho44(m)+l4*conjg(Omc(m-1))*rho42_last(m)+l5*Omc(m-1)*rho24(m) rho22(m)=1-rho11(m)-rho33(m)-rho44(m) rho11_last(m)=rho11(m) rho12_last(m)=rho12(m) rho13_last(m)=rho13(m) rho14_last(m)=rho14(m) rho21_last(m)=rho21(m) rho22_last(m)=rho22(m) rho23_last(m)=rho23(m) rho24_last(m)=rho24(m) rho31_last(m)=rho31(m) rho32_last(m)=rho32(m) rho33_last(m)=rho33(m) rho34_last(m)=rho34(m) rho41_last(m)=rho41(m) rho42_last(m)=rho42(m) rho43_last(m)=rho43(m) rho44_last(m)=rho44(m) Om_vac(m)=a1*Om_vac(m-1) if (mod(n,Nskip).eq.0) write (9,120)z(m),Om2(m),Om_vac(m),Omc(m) 50 continue if (cdabs(Om2(zpts)).gt.cdabs(Omold)) tpeak=t if (cdabs(Om_vac(zpts)).gt.cdabs(Omold_vac)) tpeak_vac=t write (10,139) t,cdabs(Om2(zpts)),cdabs(Om_vac(zpts)) !EndPoint File Omold=Om2(zpts) Omold_vac=Om_vac(zpts) 60 continue close(9) close(10) 139 format(1x,f12.6,',',F12.6,',',F12.6) ! write (*,*)'Medium pulse out at ',tpeak/(beta*1e-6),' microseconds' ! write (*,*)'Vacuum pulse out at ',tpeak_vac/(beta*1e-6),' microseconds' write (*,*)Omcpeak,(tpeak-tpeak_vac)/(beta*1e-9) 120 format(1x,f12.6,',',f12.6,',',f12.6,',',f12.6,',',f12.6,',',f12.6,',',f12.6) t_end=secnds(0.E0) t_elapsed=t_end-t_start ! write(*,*)'T elapsed = ',t_elapsed stop end ! subroutine IMatConstruct(Om,Imat) ! ! NOTE: This subroutine actually calculates -Imat because we need to solve LPsi=-Imat. implicit none integer i,nmat parameter (nmat=3) real*8 Om complex*8 Imat(nmat) Imat(1)=-cmplx(0.,-0.5*Om) Imat(2)=-cmplx(0.,0.5*Om) Imat(3)=cmplx(0.,0.) return end ! subroutine LMatConstruct(Om,del,L) ! implicit none integer nmat parameter (nmat=3) real*8 del,ga12,Om,W21 complex*8 L(nmat,nmat) common/para/ga12,W21 L(1,1)=cmplx (-ga12,-del) L(1,3)=cmplx(0.,Om) L(2,2)=cmplx(-ga12,del) L(2,3)=cmplx(0.,-Om) L(3,1)=cmplx(0.,0.5*Om) L(3,2)=cmplx(0.,-0.5*Om) L(3,3)=cmplx(-W21,0.) return end ! subroutine MatCheck(L,Linv) ! implicit none integer i,j,k,nmat parameter (nmat=3) complex*8 L(nmat,nmat),Linv(nmat,nmat),Res(nmat,nmat) write (*,*)'L = ' do 10 i=1,nmat write (*,120)(L(i,j),j=1,nmat) 10 continue write (*,*)'Linv = ' do 20 i=1,nmat write (*,120)(Linv(i,j),j=1,nmat) 20 continue write (*,*)'Res = ' do 50 i=1,nmat do 40 j=1,nmat Res(i,j)=cmplx(0.,0.) do 30 k=1,nmat Res(i,j)=Res(i,j)+Linv(i,k)*L(k,j) 30 continue 40 continue write (*,120)(Res(i,j),j=1,nmat) 50 continue 120 format(1x,3(f8.4,'+i',f8.4,' ')) return end ! subroutine plotit(x,y,nmat,npts) ! ! See MATLAB routine that will do this plotting. implicit none integer i,n,nmat,npts real*8 x(npts) complex*8 y(nmat,npts) write (*,*)'For now, we are just going to write the file' open(9, FILE='TwoLevelPulseProp.txt') do 10 n=1,npts write (9,100)x(n),(y(i,n),i=1,3) ! write(*,100)x(n),(y(i,n),i=1,3) 10 continue 100 format(1x,f9.6,',',3(f9.6,',',f9.6,',')) return end !******************************************************** ! Numerical Recipes !******************************************************** SUBROUTINE ludcmp(a,n,np,indx,d) implicit none INTEGER n,np,indx(n),NMAX REAL*8 d,TINY PARAMETER (NMAX=500,TINY=1.0e-20) INTEGER i,imax,j,k REAL aamax,dum,vv(NMAX) ! My changed variables complex*8 sum,dum2 complex*8 a(np,np) d=1. do 12 i=1,n aamax=0. do 11 j=1,n if (cabs(a(i,j)).gt.aamax) aamax=cabs(a(i,j)) 11 continue if (aamax.eq.0.) pause 'singular matrix in ludcmp' vv(i)=1./aamax 12 continue do 19 j=1,n do 14 i=1,j-1 sum=a(i,j) do 13 k=1,i-1 sum=sum-a(i,k)*a(k,j) 13 continue a(i,j)=sum 14 continue aamax=0. do 16 i=j,n sum=a(i,j) do 15 k=1,j-1 sum=sum-a(i,k)*a(k,j) 15 continue a(i,j)=sum dum=vv(i)*cabs(sum) if (dum.ge.aamax) then imax=i aamax=dum endif 16 continue if (j.ne.imax)then do 17 k=1,n dum2=a(imax,k) a(imax,k)=a(j,k) a(j,k)=dum2 17 continue d=-d vv(imax)=vv(j) endif indx(j)=imax if (cabs(a(j,j)).eq.0.) a(j,j)=cmplx(TINY,TINY) if(j.ne.n)then dum2=1./a(j,j) do 18 i=j+1,n a(i,j)=a(i,j)*dum2 18 continue endif 19 continue return END SUBROUTINE lubksb(a,n,np,indx,b) implicit none INTEGER n,np,indx(n) INTEGER i,ii,j,ll ! My changed variables complex*8 sum complex*8 b(n) complex*8 a(np,np) ii=0 do 12 i=1,n ll=indx(i) sum=b(ll) b(ll)=b(i) if (ii.ne.0)then do 11 j=ii,i-1 sum=sum-a(i,j)*b(j) 11 continue else if (sum.ne.0.) then ii=i endif b(i)=sum 12 continue do 14 i=n,1,-1 sum=b(i) do 13 j=i+1,n sum=sum-a(i,j)*b(j) 13 continue b(i)=sum/a(i,i) 14 continue return END