C.... ***************************************************************** C.... Beginning Of Sample Problem Description. C.... ***************************************************************** C.... ***************************************************************** C.... This Is A Sample Main Program For Acdc, An Automatic C.... Continuation Code For Solving `Stiff' Boundary Value Problems. C.... The Code Is Fully Documented In The User's Guide, Available As C.... A Latex File From Netlib. Machine Precision Is Provided C.... Automatically Via Netlib (In The Routine D1mach). The Routine C.... D1mach May Need To Be Altered By The User Depending On Which C.... Machine Is Being Used. C.... Any Questions, Reports Of Bugs, Or Comments (Good Or Bad) Will C.... Gladly Be Received By Ross Wright Or Jeff Cash, Department Of C.... Mathematics, Imperial College, London Sw7 2bz, United Kingdom. C.... [ E-Mail r.wright@ic.ac.uk Or j.cash@ic.ac.uk ] C.... ***************************************************************** Program Runacdc Implicit Double Precision(A-H,O-Z) Parameter(Lwrkfl = 500000) Parameter(Lwrkin = 50000) Dimension Wrk(Lwrkfl),Iwrk(Lwrkin) Dimension U(2,12400),Tol(2),Ltol(2),Xx(12400),Fixpnt(1) Logical Linear, Giveu, Givmsh, Giveps External Fsub, Dfsub, Gsub, Dgsub Common /Prob/ Linornon Common /Valpi/ Pi Pi = 4.0d0*Atan(1.0d0) C.... ***************************************************************** C.... This Driver Is Set Up To Run Two Problems, One Linear And One C.... Nonlinear. C.... The Linear Test Problem (Which Has Boundary Layers At X = -1 C.... And At X = 1) Is C.... Eps*Y'' - Y = -(Eps*Pi*Pi + 1)*Cos(Pi*X), Y(-1) = Y(1) = 0. C.... The Nonlinear Test Problem (Which Has Boundary Layers At X = 0 C.... And At X = 1) Is C.... Eps*Y'' + Y*Y' - Y = 0, Y(0) = 1, Y(1) = -1/3. C.... Acdc Solves First Order Systems Of Differential Equations, Thus C.... Both Problems Are Converted To First Order Form In Subroutine C.... Fsub Below. C.... ***************************************************************** C.... Specify The Number Of Components And The Desired Final Eps Value. Ncomp = 2 Epsmin = 1.d-7 C.... Ntol Is The Number Of Tolerances Ntol = 2 Tol(1) = 1.d-8 Tol(2) = 1.d-4 Ltol(1) = 1 Ltol(2) = 2 C.... The User May Choose A Linear Or Nonlinear Test Problem 10 Write(*,1000) Read*, Linornon C.... Specify The Boundary Points If (Linornon .eq. 1) Then Aleft = 0.d0 Aright = 1.0d0 Else If (Linornon .eq. 0) Then Aleft = -1.0d0 Aright = 1.0d0 Else Write(*,2000) Goto 10 Endif If (Linornon .eq. 1) Then Linear = .false. Else Linear = .true. Endif C.... Initialise Variables Nudim = 2 Nlbc = 1 Nfxpnt = 0 Nucol = 12400 Nmsh = 0 Giveu = .false. Givmsh = .false. Giveps = .false. Call Acdc(Ncomp, Nlbc, Nucol, Aleft, Aright, Nfxpnt, Fixpnt, + Ntol, Ltol, Tol, Linear, Givmsh, Giveu, Nmsh, Xx, + Nudim, U, Nmax, Lwrkfl, Wrk, Lwrkin, Iwrk, Giveps, + Eps, Epsmin, Fsub, Dfsub, Gsub, Dgsub, Iflbvp) C----------------------------------------------------------------------- 1000 Format(//,'Choose A Test Problem (0 = Linear, 1 = Nonlinear).') 2000 Format(/, '** Only 0 or 1 is allowed.') C----------------------------------------------------------------------- End Subroutine Fsub(Ncomp,X,Z,F,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),F(*) Common /Prob/ Linornon Common /Valpi/ Pi If (Linornon .eq. 0) Then Pix = Pi*X F(1) = Z(2) F(2) = (Z(1)-Eps*Pi*Pi*Cos(Pix)-Cos(Pix))/Eps Else F(1) = Z(2) F(2) = (Z(1)*(1.d0-Z(2)))/Eps Endif Return End Subroutine Dfsub(Ncomp,X,Z,Df,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),Df(Ncomp,*) Common /Prob/ Linornon If (Linornon .eq. 0) Then Df(1,1) = 0.d0 Df(1,2) = 1.0d0 Df(2,1) = 1.0d0/Eps Df(2,2) = 0.0d0 Else Df(1,1) = 0.d0 Df(1,2) = 1.0d0 Df(2,1) = (1.d0-Z(2))/Eps Df(2,2) = -Z(1)/Eps Endif Return End Subroutine Gsub(I,Ncomp,Z,G,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*) Common /Prob/ Linornon If (Linornon .eq. 0) Then G = Z(1) Else If (I .eq. 1) G = Z(1)-1.d0 If (I .eq. 2) G = Z(1)+1.d0/3.d0 Endif Return End Subroutine Dgsub(I,Ncomp,Z,Dg,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),Dg(*) Dg(1) = 1.d0 Dg(2) = 0.d0 Return End C.... ***************************************************************** C.... End Of Sample Problem Description. C.... ***************************************************************** Subroutine Acdc(Ncomp, Nlbc, Nucol, Aleft, Aright, Nfxpnt, Fixpnt, + Ntol, Ltol, Tol, Linear, Givmsh, Giveu, Nmsh, Xx, + Nudim, U, Nmax, Lwrkfl, Wrk, Lwrkin, Iwrk, Giveps, + Eps, Epsmin, Fsub, Dfsub, Gsub, Dgsub, Iflbvp) Implicit Double Precision (A-H,O-Z) Dimension Fixpnt(*), Ltol(*), Tol(*) Dimension Xx(*), U(Nudim,*) Dimension Wrk(Lwrkfl), Iwrk(Lwrkin) Dimension Phi(3), E(3), Npr(2), Pmax(2), Hord(2) Logical Linear, Givmsh, Giveu, Giveps External Fsub, Dfsub, Gsub, Dgsub Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Convg/ Nits Common /Mshvar/ Hsml,Npr,Pmax,Hord Parameter ( Zero = 0.0d+0, One = 1.0d+0, Two = 2.0d+0 ) Parameter ( Three = 3.0d+0, Third = 1.0d+0/3.0d+0, Huge = 1.d+30 ) C.... The First Part Of This Subroutine Is Used For Checking Input C.... Parameters And Dividing Up The Floating Point And Integer C.... Workspaces. The Rest Of This Subroutine Is Devoted To The C.... Continuation Algorithm. C.... Output Details Of The Problem If (Iprint .ge. 0) Then If (Linear) Then Write (6,1000) Ncomp Else Write (6,1001) Ncomp Endif Write (6,1002) Aleft, Aright If (Nfxpnt .gt. 0) + Write (6,1003) Nfxpnt, (Fixpnt(I), I=1,Nfxpnt) Write (6,1004) (Ltol(Ip), Ip = 1,Ntol) Write (6,1005) (Tol(Ip), Ip = 1,Ntol) Endif C.... Check For Invalid Input Parameters. If Any Parameters Are C.... Invalid, Exit With The Flag Iflbvp Set To -1. Iflbvp = -1 If (Ncomp .le. 0) Return If (Nlbc .lt. 0 .or. Nlbc .gt. Ncomp) Return If (Nucol .le. 0) Return If (Aleft .ge. Aright) Return If (Nfxpnt .lt. 0) Return If (Maxcon .lt. 0) Return If (.not. Givmsh) Nmsh = 0 If (Givmsh .and. Nmsh .lt. Nfxpnt+2) Return If (Givmsh .and. Xx(1) .ne. Aleft) Return If (Givmsh .and. Xx(Nmsh) .ne. Aright) Return If (Nfxpnt .gt. 0) Then If (Fixpnt(1) .le. Aleft) Return If (Fixpnt(Nfxpnt) .ge. Aright) Return Do 10 I = 1, Nfxpnt-1 If (Fixpnt(I+1) .le. Fixpnt(I)) Return 10 Continue Endif If (Ntol .lt. 1) Return Do 20 I = 1, Ntol If (Ltol(I) .lt. 0 .or. Ltol(I) .gt. Ncomp) Return If (Tol(I) .le. Zero) Return 20 Continue If (Giveu .and. .not. Givmsh) Return If (Nudim .le. 0) Return If (Lwrkfl .le. 0 .or. Lwrkin .le. 0) Return If (Itsaim .lt. 1) Return If (.not. Giveps) Eps = 0.5d0 If (Eps .le. Zero .or. Eps .ge. One) Return If (Epsmin .gt. Eps .or. Epsmin .le. Zero) Return If (Iprint .ge. 0) Write (6,1006) Eps, Epsmin C.... Calculate Maximum Number Of Mesh Points Possible With The C.... Given Floating-Point And Integer Workspace. Isp = Lwrkfl - 2*Ntol - 13*Ncomp - 14*Ncomp*Ncomp If (Linear) Then Iden = 4*Ncomp*Ncomp + 10*Ncomp + 5 Else Iden = 4*Ncomp*Ncomp + 11*Ncomp + 5 Endif Nmax1 = Isp/Iden Isp = Lwrkin - 3*Ncomp Nmax2 = Isp/(Ncomp+2) Nmax = Min(Nmax1, Nmax2) If (Iprint .eq. 1) Write(6,1007) Nmax Nmax = Min(Nmax, Nucol) If (Iprint .eq. 1 .and. Nmax .eq. Nucol) Write(6,1008) Nmax If (Nmax .le. 1) Return C.... Partition Floating Point Workspace. Irhs = 1 Lrhs = Ncomp*Nmax Itpblk = Irhs + Lrhs Ltpblk = Ncomp*Ncomp Ibtblk = Itpblk + Ltpblk Lbtblk = Ncomp*Ncomp Iajac = Ibtblk + Lbtblk Lajac = 2*Ncomp*Ncomp*Nmax Ibhold = Iajac + Lajac Lbhold = Ncomp*Ncomp*Nmax Ichold = Ibhold + Lbhold Lchold = Ncomp*Ncomp*Nmax Ifval = Ichold + Lchold Lfval = Ncomp*Nmax Idef = Ifval + Lfval Ldef = Ncomp*(Nmax-1) Idef8 = Idef + Ldef Ldef8 = Ncomp*(Nmax-1) Iuold = Idef8 + Ldef8 Luold = Ncomp*Nmax Itmrhs = Iuold + Luold Ltmrhs = Ncomp*Nmax Irhtri = Itmrhs + Ltmrhs Lrhtri = Ncomp*Nmax Idelu = Irhtri + Lrhtri Ldelu = Ncomp*Nmax Ixmer = Idelu + Ldelu Lxmer = Ncomp*Nmax Iutri = Ixmer + Lxmer Lutri = Ncomp*Nmax Iermx = Iutri + Lutri Lermx = Nmax Irtdc = Iermx + Lermx Lrtdc = Nmax Ixxold = Irtdc + Lrtdc Lxxold = Nmax Iuint = Ixxold + Lxxold Luint = Ncomp Iftmp = Iuint + Luint Lftmp = Ncomp Idgtm = Iftmp + Lftmp Ldgtm = Ncomp Idftm1 = Idgtm + Ldgtm Ldftm1 = Ncomp*Ncomp Idftm2 = Idftm1 + Ldftm1 Ldftm2 = Ncomp*Ncomp Itmp = Idftm2 + Ldftm2 Ltmp = Ncomp*12 Idsq = Itmp + Ltmp Ldsq = Ncomp*Ncomp Ietst6 = Idsq + Ldsq Letst6 = Ntol Ietst8 = Ietst6 + Letst6 Letst8 = Ntol Iextra = Ietst8 + Letst8 Lextra = 9*Ncomp*Ncomp Iextra1 = Iextra + Lextra Lextra1 = Nmax Imsh = Iextra1 + Lextra1 Lmsh = Nmax If (Linear) Then Ilast = Imsh + Lmsh Else Isol = Imsh + Lmsh Lsol = Ncomp*Nmax Ilast = Isol + Lsol Endif C.... Partition Integer Workspace. Iiref = 1 Liref = Nmax Iihcom = Iiref + Liref Lihcom = Nmax Iipvbk = Iihcom + Lihcom Lipvbk = Ncomp*Nmax Iipvlu = Iipvbk + Lipvbk Lipvlu = 3*Ncomp C.... ***************************************************************** C.... Initialisation And Explanation Of Variables Used In The C.... Continuation Strategy Begins Here. C.... ***************************************************************** C.... Initialise Extrapolation Variables Do 30 J = 1,3 Phi(J) = Zero E(J) = Zero 30 Continue Emin = One/Epsmin Ep = One/Eps C.... The Maximum Monitor Function Values On The First Two Meshes C.... For A Given Step Are Stored In Pmax(1) And Pmax(2). Pmax(1) = Zero Pmax(2) = Zero C.... Nits Counts The Number Of Newton Iterations Required For C.... Convergence On The First Mesh For A Given Continuation Problem Nits = 0 C.... Ifinal Is A Flag Indicating Whether The Final Problem Is Being C.... Attempted. If Iback = 1 Then We Do Not Backtrack. Iprec Is A C.... Flag That Is Set To One When We Believe That The A Calculation C.... Is Being Attempted Which May Be Beyond The Bounds Imposed By C.... The Precision Of The Machine. C.... Epsp Is The Value Of Eps Beyond Which The Machine Precision Is C.... (Possibly) Not Sufficient To Solve The Given Problem. Ifinal = 0 Iback = 0 Iprec = 0 Epsp = Zero C.... Istep Is A Flag Which Limits The Size Of The Continuation Steps C.... After Experiencing A Failure For Some Step. Istep = 0 C.... Nc Counts The Total Number Of Continuation Steps Taken. C.... Nss Counts The Number Of Successful Continuation Steps Taken. Nc = 0 Nss = 0 C.... Idc Counts The Number Of Consecutive Steps For Which The Maximum C.... Value Of The Monitor Function Is Decreasing. C.... Iextrap Is A Flag That Indicates Whether Monitor Function C.... Extrapolation Is Possible. C.... Istuk Is A Flag Which Is Set To One When The Continuation C.... Algorithm Is Backtracking. Idc = 0 Iextrap = 0 Istuk = 0 C.... Ilin Is A Flag Indicating Whether Linear Or Quadratic C.... Extrapolation Should Be Used. Ilin = -1 C.... Hrat Is A Variable Utilised When We Have A Moving Layer. Hrat = One C.... The Following Four Variables Are Used In The Parameter C.... Selection Process. Amax = Huge Bmax = Zero Cmax = Huge Estep = Huge C.... Calculate Phimax, Which Is The Largest Monitor Function Value C.... That We Believe Is Permissible In Double Precision For The Given C.... Tolerance. Tolmax = One Phimax = One/D1mach(3) Do 40 J = 1,Ntol Tolmax = Min(Tol(J),Tolmax) 40 Continue Phimax = Phimax*Tolmax Phiaim = Zero Phialt = Zero Epold = Zero Hsmlold = Zero Htpv = Zero C.... We Predict That For Epsilon = 1 The Maximum Value Of The C.... Monitor Function Is Minimised By Phi(3). We Will Use This Value C.... When Performing Extrapolation. E(3) = One Phi(3) = One/(Aright-Aleft) C.... ***************************************************************** C.... End Of Initialisation Of Continuation Variables. C.... ***************************************************************** C.... ***************************************************************** C.... The Algorithm Loops Back To Line 50 For Every Continuation Step. C.... ***************************************************************** 50 Eps = Max(One/Ep,Epsmin) Ep = One/Eps C.... Solve For Epsmin Exactly If (Eps .lt. (1.00001d0*Epsmin)) Ifinal = 1 C.... Increase The Continuation Step Counter Nc. Nc = Nc+1 If (Iprint .ge. 0) Write(6,1010) Nc, Eps C.... If We Have Reached The Maximum Number Of Continuation Steps C.... Then This Will Be The Last Problem We Attempt. If (Nc .eq. Maxcon) Then If (Iprint .ge. 0) Write(6,1011) Maxcon Iback = 1 Ifinal = 1 Endif C.... Iatt Is A Flag That Governs Whether Mesh Selection Is Performed C.... Or Control Is Returned To The Driver To Select A New Parameter. Iatt = -1 Iflbvp = 0 C.... Attempt To Solve The Latest Continuation Problem. Call Bvpsol(Ncomp, Nmsh, Nlbc, Aleft, Aright, * Nfxpnt, Fixpnt, Ntol, Ltol, Tol, Nmax, Linear, * Giveu, Givmsh, Xx, Nudim, U, * Wrk(Idef), Wrk(Idelu), * Wrk(Irhs), Wrk(Ifval), * Wrk(Itpblk), Wrk(Ibtblk), Wrk(Iajac), Wrk(Ibhold), * Wrk(Ichold), Iwrk(Iipvbk), Iwrk(Iipvlu), * Wrk(Iuint), Wrk(Iftmp), Wrk(Itmrhs), * Wrk(Idftm1), Wrk(Idftm2), Wrk(Idgtm), * Wrk(Iutri), Wrk(Irhtri), Wrk(Ixmer), * Wrk(Ixxold), Wrk(Iuold), * Wrk(Itmp), Wrk(Idsq), Wrk(Irtdc), * Wrk(Ietst6), Wrk(Ietst8), Wrk(Iermx), * Iwrk(Iihcom), Iwrk(Iiref), Wrk(Idef8), * Wrk(Iextra), Wrk(Iextra1), * Fsub, Dfsub, Gsub, Dgsub, Iflbvp, Eps) C.... ***************************************************************** C.... The Logic For A Successful Continuation Step Starts Here C.... ***************************************************************** If (Iflbvp .eq. 0) Then Nss = Nss + 1 C.... If The Problem Eps = Epsmin Has Been Solved Succesfully Then C.... We May Finish. Ifinal = 1 When Eps = Epsmin. If (Ifinal .eq. 1) Then If (Iprint .ge. 0) Then Write(6,1012) Eps Write(6,1018) ('U',Ltol(J), J=1,Ntol) Write(6,*) Jstep = Max(Nmsh/30,1) Do 55 I = 1, Nmsh-1, Jstep Write(6,1019) I,Xx(I),(U(Ltol(J),I),J=1,Ntol) 55 Continue Write(6,1019) Nmsh,Xx(Nmsh),(U(Ltol(J),Nmsh),J=1,Ntol) Write(6,1020) Endif Return Endif C.... When Bactracking The Program Should Find A Problem That It Can C.... Eventually Solve. The Relationship Between The Last Two C.... Successful Epsilon Values Is Used To Restrict Future Epsilon C.... Values. This Is The Purpose Of Estep. If Istep = 1, Then We C.... Must Restrict Future Values Of Ep. If (Istuk .eq. 1) Then Istep = 1 Estep = Min(Ep/E(3),Estep) Endif Istuk = 0 C.... Calculate The Best Approximation To The Maximum Value Of The C.... Monitor Function By Extrapolating The Maximum Value On The C.... First Two Meshes. The Best Approximation Is Stored In Phit. If (Iextrap .eq. 0) Then H1 = Hord(1)/Pmax(1) H2 = Hord(2)/Pmax(2) If (H1/H2 .gt. 1.1d0) Then C1h = (Pmax(2) - Pmax(1))/(H2-H1) Phit = Pmax(1) - C1h*H1 Else Phit = Pmax(2) Endif Endif C.... It Is Possible That We Are Attempting To Solve Problems C.... That Are Beyond The Bounds Imposed By The Machine Precision. C.... The User Is Warned Of This Possibility. If (Iprec .eq. 0 .and. Phit .gt. Phimax) Then If (Iprint .ge. 0) Write(6,1013) Eps Epsp = Eps Iprec = 1 Endif C.... Save Details Of Last Problem Solved Successfully In Case We C.... Need To Backtrack Nnewold = Nmsh Call Meshdet(Wrk(Imsh), Xx, Nmsh) If (.not. Linear) Call Soldet(Wrk(Isol), U, Ncomp, Nmsh, + Ncomp, Nudim) C.... If Iextrap Equals 1 Then We Have Abandoned Monitor Function C.... Extrapolation As Our Parameter Selection Basis. If (Iextrap .eq. 1) Then E(1) = E(2) E(2) = E(3) E(3) = Ep If (Istep .eq. 1) Then Ep = Estep*Ep Else Ep = Emin Endif Goto 70 Endif C.... If We Have A Decreasing Monitor Function Twice Consecutively C.... Then Extrapolation Will Not Work. The Flag Iextrap Equals 1 C.... When Extrapolation Does Not Work. If (Phit .le. Phi(3)) Then If (Idc .eq. 1 .or. Phit .lt. Phi(2)) Then Iextrap = 1 E(1) = E(2) E(2) = E(3) E(3) = Ep If (Istep .eq. 1) Then Ep = Estep*Ep Else Ep = Emin Endif Goto 70 Else Idc = 1 E(3) = Ep Phi(3) = Phit Endif C.... Otherwise Update Extrapolation Data Else Idc = 0 Do 60 J = 1,2 E(J) = E(J+1) Phi(J) = Phi(J+1) 60 Continue E(3) = Ep Phi(3) = Phit Endif C.... The Variable Irest Is A Flag Which Informs Us Whether Our C.... Extrapolation Procedure Is Accurate Or Not. Irest = 1 Means C.... Extrapolation Is Not Reliable And So We Should Restrict The C.... Next Parameter Step. Irest = 0 C.... Check Errors In Extrapolation Procedure And Calculate Hmult C.... Which Restricts The Size Of Parameter Steps If Extrapolation C.... Is Working Poorly. If (Nss .eq. 2 .and. Phit .gt. Two*Phiaim) Then Irest = 1 Hh = Ep - Epold Errp = Abs(Phit - Phiaim) Fx = Errp/Hh**2 Hmax = Sqrt(0.2d0*Phiaim/Fx) Hmult = (Epold+Hmax)/Epold Endif If (Nss .gt. 2 .and. Phit .gt. Two*Phiaim) Then Irest = 1 Hh = Ep - Epold Errp = Abs(Phit - Phiaim) Errp1 = Abs(Phit - Phialt) If (Ilin .eq. -1) Then Fx = Errp/Hh**2 Hmax = Sqrt(0.2d0*Phiaim/Fx) Hmult = (Epold+Hmax)/Epold Fx = Errp1/Hh**3 Hmax = (0.2d0*Phiaim/Fx)**(Third) Hmult1 = (Epold+Hmax)/Epold Else Fx = Errp/Hh**3 Hmax = (0.2d0*Phiaim/Fx)**(Third) Hmult = (Epold+Hmax)/Epold Fx = Errp1/Hh**2 Hmax = Sqrt(0.2d0*Phiaim/Fx) Hmult1 = (Epold+Hmax)/Epold Endif Endif Epold = Ep C.... If The Extrapolation Procedure Is Particularly Inaccurate Then C.... We Need To Restrict It Permanently By Setting Istep = 1. Pinacc = Three*Max(Pmax(1),Phiaim,Phialt) If (Nss .gt. 2 .and. Phit .gt. Pinacc) Then Istep = 1 Pestep = Max(Hmult,Hmult1) Estep = Min(Estep,Pestep) Endif C.... Decide Whether Linear Or Quadratic Extrapolation Is Best. If (Nss .gt. 2) Then Per1 = Abs(Phiaim-Phit) Per2 = Abs(Phialt-Phit) If (Per2 .lt. Per1) Then Ilin = -Ilin If (Phit .gt. Two*Phialt) Then Hmult = Hmult1 Else Irest = 0 Endif Endif Endif C.... If The Continuation Steps Are Becoming Very Small Then We Have C.... Reached Our Final Problem Dele = E(3)-E(2) If (Dele .lt. 0.01d0*E(3) .or. (Nmsh .gt. (3*Nmax/4))) Then Ep = E(3) Epsmin = One/Ep If (Iprint .ge. 0) Then If (Dele .lt. 0.01d0*E(3)) Then Write(6,1021) Epsmin Else Write(6,1022) Epsmin Endif Endif Emin = Ep Iback = 1 Goto 70 Endif C.... The Following Section Of Code Calculates The Desired Value C.... (Phiaim) That We Would Like The Maximum Value Of The Monitor C.... Function To Take. Itru = 0 If (Phit .gt. Pmax(1) .and. Pmax(2) .ne. Phit) Itru = 1 If (Itru .eq. 1) Amax = -Phit*Phit/(Two*C1h) Bmax = Phit*H1 If (Npr(1) .gt. 3*Npr(2)/2) Then Ccm = Max(Bmax-Three,Bmax/1.5d0) Cmax = Min(Cmax,Ccm) Endif If (Itru .eq. 0) Amax = Max(1.5d0*Bmax,Bmax+Three) If (Nss .ge. 2) Hrat = Max(H1/Hsmlold,One) Hsmlold = Hsml Bbm = Max(1.5d0*Bmax,Bmax+Three) If (.not. Linear) Then Fmax = Dble(Itsaim)*Bmax/Dble(Nits) If (Fmax .gt. Bmax) Fmax = Max(Fmax,Bbm) If (Fmax .lt. Bmax) Fmax = Max(Bmax-Three,Bmax/1.5d0,Fmax) Endif Htot = Min(Amax,Bbm,Cmax) If (Linear) Fmax = Htot If (Nss .gt. 1 .and. Htot .ne. Cmax) Then If (Itru .eq. 0 .or. Amax .gt. Htpv) Htot = Max(Htot,Htpv) Htot = Min(Htot,Fmax,Cmax) Endif Htpv = Htot Phiaim = Htot/(Hsml*Hrat) Phiaim = Max(1.05d0*Phit,Phiaim) C.... Quadratic And Linear Extrapolation To Find The Value Of Ep That C.... Corresponds To Phiaim If (Nss .ne. 1) Then F1 = (Phi(2)-Phi(1))/(E(2)-E(1)) F2 = (Phi(3)-Phi(2))/(E(3)-E(2)) Ff2 = (F2-F1)/(E(3)-E(1)) If (Ff2 .gt. 0) Then Qa = Ff2 Qb = F1-(E(1)+E(2))*Ff2 Qc = Phi(1)-E(1)*(F1-E(2)*Ff2)-Phiaim Qd = Qb**2-4.d0*Qa*Qc If (Ilin .eq. 1) Then Ep = Min(Emin,(-Qb+Sqrt(Qd))/(Two*Qa)) If (Ep .eq. Emin) Phiaim = Phi(1)+ + (Ep-E(1))*(F1+(Ep-E(2))*Ff2) Phialt = Phi(2)+F2*(Ep-E(2)) Else Ep = Min(Emin,E(2)+(Phiaim-Phi(2))/F2) If (Ep .eq. Emin) Phiaim = Phi(2)+F2*(Ep-E(2)) Phialt = Phi(1)+(Ep-E(1))*(F1+(Ep-E(2))*Ff2) Endif Else G1 = (E(2)-E(1))/(Phi(2)-Phi(1)) G2 = (E(3)-E(2))/(Phi(3)-Phi(2)) Gg2 = (G2-G1)/(Phi(3)-Phi(1)) If (Ilin .eq. 1) Then Ep = Min(Emin, + E(1)+(Phiaim-Phi(1))*(G1+(Phiaim-Phi(2))*Gg2)) If (Ep .eq. Emin) Then Qa = Gg2 Qb = G1-(Phi(1)+Phi(2))*Gg2 Qc = E(1)-Phi(1)*(G1-Phi(2)*Gg2)-Ep Qd = Qb**2-4.d0*Qa*Qc Phiaim = (-Qb+Sqrt(Qd))/(Two*Qa) Endif Phialt = Phi(2)+F2*(Ep-E(2)) Else Ep = Min(Emin,E(2)+G2*(Phiaim-Phi(2))) If (Ep .eq. Emin) Phiaim = Phi(2)+F2*(Ep-E(2)) Qa = Gg2 Qb = G1-(Phi(1)+Phi(2))*Gg2 Qc = E(1)-Phi(1)*(G1-Phi(2)*Gg2)-Ep Qd = Qb**2-4.d0*Qa*Qc Phialt = (-Qb+Sqrt(Qd))/(Two*Qa) Endif Endif C.... Linear Extrapolation Else F2 = (Phi(3)-Phi(2))/(E(3)-E(2)) Ep = Min(Emin,E(2)+(Phiaim-Phi(2))/F2) Endif C.... Extrapolation May Not Be Working Very Well - If Not, Adjust C.... Continuation Stepsize And Recalculate Phiaim. If ((Irest .eq. 1 .or. Istep .eq. 1) .and. Nss .ne. 1) Then If (Irest .eq. 1 .and. Istep .eq. 1) Then Ealt = Min(Epold*Hmult,Epold*Estep) Elseif (Irest .eq. 1) Then Ealt = Epold*Hmult Else Ealt = Epold*Estep Endif If (Ep .gt. Ealt) Then Ep = Min(Emin,Ealt) Philin = Phi(2)+F2*(Ep-E(2)) If (Ff2 .gt. 0) Then Phiqua = Phi(1)+(Ep-E(1))*(F1+(Ep-E(2))*Ff2) Else Qa = Gg2 Qb = G1-(Phi(1)+Phi(2))*Gg2 Qc = E(1)-Phi(1)*(G1-Phi(2)*Gg2)-Ep Qd = Qb**2-4.d0*Qa*Qc Phiqua = (-Qb+Sqrt(Qd))/(Two*Qa) Endif If (Ilin .eq. 1) Then Phiaim = Phiqua Phialt = Philin Else Phiaim = Philin Phialt = Phiqua Endif Endif Endif 70 Continue C.... ***************************************************************** C.... End Of Logic For A Successful Continuation Step And C.... Beginning Of Logic For An Unsuccessful Continuation Step. C.... ***************************************************************** Else C.... If A Problem Is Not Solved Successfully Then Print Details And C.... Select An Alternative Value For The Continuation Parameter. C.... This Process Is Known As Backtracking. Istuk = 1 Indicates That C.... We Are Backtracking. Istuk = 1 C.... If Iback = 1 Then The Final Problem Has Not Been Solved. C.... In This Case We Stop. If (Iback .eq. 1) Then If (Iprint .ge. 0) Then If (Epsp .ne. Zero) Then Write(6,1014) Eps,Iflbvp,Eps,Epsp Else Write(6,1015) Eps,Iflbvp,Eps Endif Endif Return Endif C.... If Iprec = 2, Then We Know That We Cannot Define A Mesh On C.... Which The Current Eps Value Can Be Solved To The Requested C.... Tolerances. We Alter Epsmin Accordingly. If (Iprec .eq. 2) Then Epsmin = One/(Max((Ep+E(3))/Two,0.9d0*Ep)) Endif C.... Insert Details For Backtracking If (Iprint .ge. 0) Write(6,1016) Ifinal = 0 Ep = (Ep+E(3))/Two If ((Ep-E(3)) .lt. 0.01d0*E(3)) Then Ep = E(3) Epsmin = One/Ep If (Iprint .ge. 0) Write(6,1021) Epsmin Emin = Ep Iback = 1 Iflbvp = 0 Endif If (Nss .eq. 1) Then Phiaim = Phi(2)+F2*(Ep-E(2)) Else If (Nss .gt. 1) Then If (Iextrap .ne. 1) Then Philin = Phi(2)+F2*(Ep-E(2)) If (Ff2 .gt. 0) Then Phiqua = Phi(1)+(Ep-E(1))*(F1+(Ep-E(2))*Ff2) Else Qa = Gg2 Qb = G1-(Phi(1)+Phi(2))*Gg2 Qc = E(1)-Phi(1)*(G1-Phi(2)*Gg2)-Ep Qd = Qb**2-4.d0*Qa*Qc Phiqua = (-Qb+Sqrt(Qd))/(Two*Qa) Endif If (Ilin .eq. 1) Then Phiaim = Phiqua Phialt = Philin Else Phiaim = Philin Phialt = Phiqua Endif Endif Endif C.... Re-Insert Details From Last Problem Successfully Solved. If (Nss .gt. 0) Then Nmsh = Nnewold Call Meshdet(Xx, Wrk(Imsh), Nmsh) If (.not. Linear) Call Soldet(U, Wrk(Isol), Ncomp, Nmsh, + Nudim, Ncomp) Endif Endif C.... ***************************************************************** C.... End Of Logic For An Unsuccessful Continuation Step. C.... ***************************************************************** C.... Pass On Mesh And Solution Details To The Next Continuation C.... Problem. If (Nss .gt. 0) Then Givmsh = .true. If (.not. Linear) Giveu = .true. Endif C.... ***************************************************************** C.... The Program Loops Back To Line 50 For Every Continuation Step. C.... ***************************************************************** Goto 50 C----------------------------------------------------------------------- 1000 Format(/,' The Number Of (Linear) Differential Equations Is ',I3) 1001 Format(/,' The Number Of (Nonlinear) Differential Equations Is ', + I3) 1002 Format(' Left Boundary Point =',D12.4,', Right Boundary Point =', + D12.4) 1003 Format(' There Are',I5,' Fixed Points In The Mesh - ',10(6d10.4/)) 1004 Format(' Components Of U Requiring Tolerances -',8(7x,I2,1x), + 4(/38x,8i10)) 1005 Format(' Corresponding Error Tolerances -',6x,8d10.2, + 4(/39x,8d10.2)) 1006 Format (' The Initial Value Of Epsilon Is ',D11.4,/, + ' The Desired Final Value Of Epsilon Is ',D11.4) 1007 Format(1h ,'The Largest Mesh Size From Workspace, Nmax =',I8) 1008 Format(1h ,'Nmax = Nucol =', I8) 1010 Format (/,1x,82('*'),/,' Continuation Step ',I2, + ', Epsilon = ',D9.4,/,1x,82('*')) 1011 Format (/,1x,'This Is The Final Continuation Step Since ', + 'Maxcon =',I4) 1012 Format (/,1x,82('$'),//,' Tolerances Satisfied For Final Problem', + ' Epsilon = ',D9.4,//,1x,82('$'),/) 1013 Format(/,' ** Machine Precision (Possibly) Not Sufficient For', + ' Epsilon Less Than ',D9.4) 1014 Format (/,1x,82('$'),//,' Final Problem Epsilon = ',D10.4,/, + ' Not Solved, Iflag =',I3,/, ' Try Running Problem Again', + ' With Epsmin Greater Than ',D9.4,/, ' Machine Precision', + ' (Possibly) Not Sufficient For Eps Less Than ',D9.4,//, + 1x,82('$'),/) 1015 Format (/,1x,82('$'),//,' Final Problem Epsilon = ',D10.4,/, + ' Not Solved, Iflag =',I3,/, ' Try Running Problem Again', + ' With Epsmin Greater Than ',D9.4,//,1x,82('$'),/) 1016 Format (/,' ** Failed Step - Bactracking For Larger Value ', + 'Of Epsilon') 1018 Format(' The Final Mesh And Solution Components Are:',//, + 5x,'I',10x,'X(I) ',40(14x,A,'(',I2,')')) 1019 Format(I6,41d19.8) 1020 Format(/,1x,82('$'),/) 1021 Format (/,' Continuation Steps Too Small, Change Epsmin To ',D9.4) 1022 Format (/,' Storage Limit Being Approached, Change Epsmin To ', + D9.4) C----------------------------------------------------------------------- End Subroutine Meshdet(X, X1, Nmsh) Implicit Double Precision (A-H,O-Z) C.... This Subroutine Is Used For Saving And Re-Inserting Meshes Dimension X(*), X1(*) Do 10, J = 1,Nmsh X(J) = X1(J) 10 Continue Return End Subroutine Soldet(U, U1, Ncomp, Nmsh, Idim1, Idim2) Implicit Double Precision (A-H,O-Z) C.... This Subroutine Is Used For Saving And Re-Inserting Solutions. Dimension U(Idim1,*), U1(Idim2,*) Do 14, J = 1,Ncomp Do 12 K = 1,Nmsh U(J,K) = U1(J,K) 12 Continue 14 Continue Return End * File Bnob.f (Nob = 'No Blas') Subroutine Blkdcm (Ntop, Nbot, Nrblk, Nbloks, * Topblk, Array, Botblk, Ipivot, Iflag) * This Subroutine Decomposes The Almost Block-Diagonal Matrix A * Given By * * ( Topblk ) * ( Array(1) ) * ( Array(2) ) * ( . ) * ( . ) * ( Array(Nbloks) ) * ( Botblk ), * * Where Topblk Is Ntop By Ntop+Nbot, * Array(K) Is Nrblk By Nrblk+Ntop+Nbot, For K = 1,...,Nbloks * And Botblk Is Nbot By Ntop+Bnot. * The Matrix Thus Contains Nbloks*Nrblk + Ntop + Nbot ( = Nrow) Rows. * The Matrix Is Decomposed Using Gaussian Elimination With * Alternate Row And Column Elimination With Partial Pivoting * Within Each Block. * Ntop Is The Number Of (Conceptual) Rows In Topblk (May Be Zero). * Nblk Is The Number Of Rows In Each Array Along The Diagonal. * Nbot Is The Number Of (Conceptual) Rows In Botblk (May Be Zero). * The Declared Row Dimensions Of Both Topblk And Botblk * Must Be Ntop+Nbot. * (This Is Slightly Wasteful Of Storage, But Avoids 2 Extra * Parameters And Much Complication In Storage And Coding. * Topblk Is An Array Of Conceptual Dimension (Ntop,Ntop+Nbot); * The Upper Left Block Of A. Its Declared Row Dimension Must Be * Ntop+Nbot. * Botblk Is An Array Of Conceptual Dimension (Nbot,Ntop+Nbot); * The Lower Right Block Of A. Its Declared Row Dimension Must Be * Ntop+Nbot. * Array(*,*,K) Contains The K-Th Matrix Lying Along The Diagonal; * Each Block Is Nrblk By Nrblk + Ntop + Nbot. * Nrblk Is The Number Of Rows In Each Block * The Number Of Columns In Each Block Is Nrblk + Ntop + Nbot. * Nbloks Is The Number Of Blocks. * Ipivot Is An Integer Array Of Dimension Nrow, * Used To Store Information About Interchanges Performed * During Elimination. * Iflag Is An Output Flag. Iflag = 0 If Everything Is Ok, * = 1 If There Are Invalid Input Parameters, And = -1 If The * Matrix Is Judged To Be Singular Because Of A Too-Small Pivot. Implicit Double Precision (A-H, O-Z) Dimension Ipivot (*) Dimension Topblk(Ntop+Nbot,*), * Array(Nrblk,Ntop+Nbot+Nrblk,*), * Botblk(Ntop+Nbot,*) Common/Mchprs/ Flmax, Epsmch Intrinsic Abs Parameter (Zero = 0.0d+0) Parameter (Ten = 10.0d+0) Iflag = 0 Novrlp = Ntop + Nbot Ncblk = Nrblk + Novrlp Pivtol = Ten*Epsmch * In Topblk, Apply Ntop Elimination Steps, One For Each Row, * Using Column Interchanges. * After The I-Th Step Of Reduction, Row I Has Been Reduced To * Lower-Triangular Form. Do 160 I = 1, Ntop * In Row I, Jpvt Gives The Index Of The Element Of Maximum Magnitude * On Or To The Right Of The Diagonal. Colmax = Zero Do 100 J = I, Novrlp Tabs = Abs(Topblk(I,J)) If (Tabs .gt. Colmax) Then Jpvt = J Colmax = Tabs Endif 100 Continue If (Colmax .le. Pivtol) Then Iflag = -1 Return Endif Ipivot(I) = Jpvt If (Jpvt .ne. I) Then Do 110 L = I, Ntop Swap = Topblk(L,Jpvt) Topblk(L,Jpvt) = Topblk(L,I) Topblk(L,I) = Swap 110 Continue Do 120 L = 1, Nrblk Swap = Array(L,Jpvt,1) Array(L,Jpvt,1) = Array(L,I,1) Array(L,I,1) = Swap 120 Continue Endif Colpiv = Topblk(I,I) Do 150 J = I+1, Novrlp Colmlt = Topblk(I,J)/Colpiv Topblk(I,J) = Colmlt Do 130 L = I+1, Ntop Topblk(L,J) = Topblk(L,J) - Colmlt*Topblk(L,I) 130 Continue Do 140 L = 1, Nrblk Array(L,J,1) = Array(L,J,1) - Colmlt*Array(L,I,1) 140 Continue 150 Continue 160 Continue * In Each Diagonal Block Array, K = 1,...,Nbloks Do 320 K = 1, Nbloks Incr = (K-1)*Nrblk * In Each Matrix Array, Apply Nrblk-Ntop Elimination Steps * In Columns Ntop+1,...,Nrblk, Using Row Interchanges. Do 210 J = Ntop+1, Nrblk Jminn = J - Ntop Ipvt = Jminn Rowmax = Abs(Array(Jminn,J,K)) Do 170 I = Jminn+1, Nrblk Tabs = Abs(Array(I,J,K)) If (Tabs .gt. Rowmax) Then Ipvt = I Rowmax = Tabs Endif 170 Continue If (Rowmax .le. Pivtol) Then Iflag = -1 Return Endif Ipivot(Incr+J) = Incr+Ipvt+Ntop * Interchange Rows If Necessary. If (Ipvt .ne. Jminn) Then Do 180 L = J, Ncblk Swap = Array(Ipvt,L,K) Array(Ipvt,L,K) = Array(Jminn,L,K) Array(Jminn,L,K) = Swap 180 Continue Endif * Compute Multipliers. Rowpiv = Array(Jminn,J,K) Do 190 I = Jminn+1, Nrblk Array(I,J,K) = Array(I,J,K)/Rowpiv 190 Continue Do 200 L = J+1, Ncblk Rowmlt = Array(Jminn,L,K) Do 200 I = Jminn+1, Nrblk Array(I,L,K) = Array(I,L,K) - Rowmlt*Array(I,J,K) 200 Continue 210 Continue * Next, Perform Eliminations In Rows Nrblk-Ntop+1 Through Nrblk, * With Column Interchanges. Do 310 I = Nrblk - Ntop + 1, Nrblk * (The Index Ipn Runs From Nrblk+1 To Nrblk+Ntop.) Ipn = I + Ntop * Find The Index Of The Column With Maximum Element In Row I, Starting * With Column Ntop + I. The Value Of Jpv1 Lies Between 1 And Ncblk-J+1. * The Column Index In Array (*,*,K) Is Jpv1+Ntop. Jpvt = Ipn Colmax = Abs(Array(I,Jpvt,K)) Do 220 J = Ipn+1, Ncblk Tabs = Abs(Array(I,J,K)) If (Tabs .gt. Colmax) Then Jpvt = J Colmax = Tabs Endif 220 Continue If (Colmax .le. Pivtol) Then Iflag = -1 Return Endif Ipivot(Incr+Ipn) = Incr+Jpvt * Interchange Columns If Necessary. If (Jpvt .ne. Ipn) Then Do 230 L = I, Nrblk Swap = Array(L,Jpvt,K) Array(L,Jpvt,K) = Array(L,Ipn,K) Array(L,Ipn,K) = Swap 230 Continue * Depending On The Value Of K, Interchange Columns In Block K+1 Of * Array Or In Botblk. If (K .lt. Nbloks) Then Do 240 L = 1, Nrblk Swap = Array(L,Jpvt-Nrblk,K+1) Array(L,Jpvt-Nrblk,K+1) = Array(L,Ipn-Nrblk,K+1) Array(L,Ipn-Nrblk,K+1) = Swap 240 Continue Else Do 250 L = 1, Nbot Swap = Botblk(L,Jpvt-Nrblk) Botblk(L,Jpvt-Nrblk) = Botblk(L,Ipn-Nrblk) Botblk(L,Ipn-Nrblk) = Swap 250 Continue Endif Endif * Compute Multipliers And Perform Elimination. Colpiv = Array(I,Ipn,K) Do 300 J = Ipn+1, Ncblk Colmlt = Array(I,J,K)/Colpiv Array(I,J,K) = Colmlt Do 260 L = I+1, Nrblk Array(L,J,K) = Array(L,J,K) - Colmlt*Array(L,Ipn,K) 260 Continue If (K .lt. Nbloks) Then Do 270 L = 1, Nrblk Array(L,J-Nrblk,K+1) = Array(L,J-Nrblk,K+1) * - Colmlt*Array(L,Ipn-Nrblk,K+1) 270 Continue Else Do 280 L = 1, Nbot Botblk(L,J-Nrblk) = Botblk(L,J-Nrblk) * - Colmlt*Botblk(L,Ipn-Nrblk) 280 Continue Endif 300 Continue * 310 Ends The Loop Over I 310 Continue * 320 Ends The Loop Over K 320 Continue * In Botblk, Apply Nbot Eliminations With Row Interchanges. Incr = Nbloks*Nrblk Do 390 Jj = 1, Nbot J = Jj + Ntop Ipvt = Jj Rowmax = Abs(Botblk(Ipvt,J)) Do 330 I = Jj+1, Nbot Tabs = Abs(Botblk(I,J)) If (Tabs .gt. Rowmax) Then Ipvt = I Rowmax = Tabs Endif 330 Continue If (Rowmax .le. Pivtol) Then Iflag = -1 Return Endif Ipivot(Incr+J) = Incr+Ntop+Ipvt If (Ipvt .ne. Jj) Then Do 340 L = J, Novrlp Swap = Botblk(Ipvt,L) Botblk(Ipvt,L) = Botblk(Jj,L) Botblk(Jj,L) = Swap 340 Continue Endif Rowpiv = Botblk(Jj,J) Do 350 I = Jj+1, Nbot Botblk(I,J) = Botblk(I,J)/Rowpiv 350 Continue Do 380 L = J+1, Novrlp Rowmlt = Botblk(Jj,L) Do 380 I = Jj+1, Nbot Botblk(I,L) = Botblk(I,L) - Rowmlt*Botblk(I,J) 380 Continue 390 Continue Return End Subroutine Blkslv (Ntop, Nbot, Nrblk, Nbloks, * Topblk, Array, Botblk, Ipivot, * B, X) Implicit Double Precision (A-H,O-Z) * Special Version Of Block-Structured Solution Where We * Assume That Nrblk = Ntop + Nbot. * The Right-Hand Side B Is A Vector Of Length (Nbloks+1)*Nrblk. * The Solution X Is Represented As A Two-Dimensional Array, * Of Dimension (Nrblk, Nbloks+1). Dimension Ipivot(*) Dimension Topblk(Ntop+Nbot,*), Array(Nrblk, 2*Nrblk,*), * Botblk(Ntop+Nbot,*) Dimension B(*), X(Nrblk,Nbloks+1) Parameter (Zero = 0.0d+0) Novrlp = Ntop+Nbot Ncblk = Novrlp + Nrblk Do 30 J = 1, Ntop X(J,1) = B(J)/Topblk(J,J) Do 10 I = J+1, Ntop B(I) = B(I) - X(J,1)*Topblk(I,J) 10 Continue 30 Continue Do 120 K = 1, Nbloks Incr = (K-1)*Nrblk * Forward Modification. Do 50 J = 1, Ntop Do 50 I = 1, Nrblk B(Incr+Ntop+I) = B(Incr+Ntop+I) - X(J,K)*Array(I,J,K) 50 Continue * Forward Elimination. Do 80 J = Ntop+1, Nrblk Jpivot = Ipivot(Incr+J) If (Jpivot .ne. Incr+J) Then Swap = B(Incr+J) B(Incr+J) = B(Jpivot) B(Jpivot) = Swap Endif Do 70 I = J-Ntop+1, Nrblk Ii = I + Incr + Ntop B(Ii) = B(Ii) - B(Incr+J)*Array(I,J,K) 70 Continue 80 Continue * Forward Solution. Do 110 J = Nrblk+1, Ntop+Nrblk Jmtop = J-Ntop X(J,K) = B(Incr+J)/Array(Jmtop,J,K) Do 90 I = J-Ntop+1, Nrblk Ii = I + Incr + Ntop B(Ii) = B(Ii) - X(J,K)*Array(I,J,K) 90 Continue 110 Continue 120 Continue * Forward Modification In Botblk. Incr = Nbloks*Nrblk Do 140 J = 1, Ntop Do 140 I = 1, Nbot B(Incr+Ntop+I) = B(Incr+Ntop+I) * - X(J,Nbloks+1)*Botblk(I,J) 140 Continue * Forward Elimination. Do 170 J = Ntop+1, Ntop+Nbot-1 Jpivot = Ipivot(Incr+J) If (Jpivot .ne. Incr+J) Then Swap = B(Incr+J) B(Incr+J) = B(Jpivot) B(Jpivot) = Swap Endif Do 160 I = J-Ntop+1, Nbot Ii = I + Incr + Ntop B(Ii) = B(Ii) - B(Incr+J)*Botblk(I,J) 160 Continue 170 Continue * Backward Solution In Botblk. Do 210 J = Nbot+Ntop, Ntop+1, -1 X(J,Nbloks+1) = B(Incr+J)/Botblk(J-Ntop,J) Do 190 I = 1, J-Ntop-1 B(Incr+Ntop+I) = B(Incr+Ntop+I) * - X(J,Nbloks+1)*Botblk(I,J) 190 Continue 210 Continue * Backward Elimination. Do 300 K = Nbloks, 1, -1 Incr = (K-1)*Nrblk Do 240 I = Nrblk, Nrblk-Ntop+1,-1 Iptp = I+Ntop Indx = Iptp-Nrblk Dotprd = Zero If (Ncblk-Iptp .ge. 1) Then Do 220 J = Iptp+1, Ncblk Dotprd = Dotprd + Array(I,J,K)*X(J-Nrblk,K+1) 220 Continue Endif X(Indx,K+1) = X(Indx,K+1) - Dotprd Ipvtn = Ipivot(Incr+Iptp) If (Incr+Iptp .ne. Ipvtn) Then * Convert Ipvtn To An Index Pair. Jcol = 1 + (Ipvtn-1)/Nrblk Irow = Ipvtn - (Jcol-1)*Nrblk Swap = X(Indx,K+1) X(Indx,K+1) = X(Irow,Jcol) X(Irow,Jcol) = Swap Endif 240 Continue * Backward Modification. Do 260 J = Nrblk+1, Ncblk Do 260 I = 1, Nrblk-Ntop Ii = I + Incr + Ntop B(Ii) = B(Ii) - X(J,K)*Array(I,J,K) 260 Continue * Backward Solution. Do 290 J = Nrblk, Ntop+1, -1 X(J,K) = B(Incr+J)/Array(J-Ntop,J,K) Do 270 I = 1, J-Ntop-1 Ii = I + Incr + Ntop B(Ii) = B(Ii) - X(J,K)*Array(I,J,K) 270 Continue 290 Continue * 300 Ends The Loop Over K 300 Continue * Backward Elimination In Topblk. Do 330 I = Ntop, 1, -1 Dotprd = X(I,1) Do 310 J = I+1, Ntop+Nbot Dotprd = Dotprd - Topblk(I,J)*X(J,1) 310 Continue Ipvti = Ipivot(I) If (I .ne. Ipvti) Then * Convert Ipvti To An Index Pair. Jcol = 1 + (Ipvti-1)/Nrblk Irow = Ipvti - (Jcol-1)*Nrblk Swap = X(I,1) X(I,1) = X(Irow,Jcol) X(Irow,Jcol) = Swap Endif 330 Continue Return End Subroutine Lufac(N, Ndim, A, Ip, Ier) Implicit Double Precision (A-H,O-Z) Dimension A(Ndim,N), Ip(N) Intrinsic Abs * Blas: Daxpy, Dscal, Dswap. Idamax Parameter ( Zero = 0.0d+0, One = 1.0d+0 ) * The Subroutine Lufac Is A Very Simple Code To Compute The * Lu Decomposition (With Partial Pivoting) Of The N By N Matrix A. * The Lu Factors Are Overwritten On A. The Integer Array Ip * Reflects The Pairwise Interchanges Performed. Note That Ip(K) * Therefore Does Not Give The Index In The Original Array Of * The K-Th Pivot. * On Exit, The Error Flag Ier Is Zero When No Zero Pivots Are * Encountered. Otherwise, Ier Is Equal To The Index Of The * Step At Which A Zero Pivot Occurred. Ier = 0 Ip(N) = 0 * Begin Loop Over Columns 1 Through N-1. K Is The Current * Column Index. Do 100 K = 1, N-1 * Find The Row Index Ipiv Of The Element Of Largest Magnitude In * Column K. Ipiv = K-1 + Idamax(N-K+1, A(K,K), 1) Piv = A(Ipiv,K) If (Piv .eq. Zero) Then Ier = K Return Endif Ip(K) = Ipiv * Perform Interchanges If Necessary. If (Ipiv .ne. K) Then Call Dswap(N-K+1, A(Ipiv,K), Ndim, A(K,K), Ndim) Endif * Save The (Negative) Multipliers In The Subdiagonal Elements Of * Column K. Call Dscal(N-K, (-One/Piv), A(K+1,K), 1) * Update The Remaining Matrix. Note That A(I,K) Now Contains * The Negative Multipliers. Do 50 J = K+1, N Call Daxpy(N-K, A(K,J), A(K+1,K), 1, A(K+1,J), 1) 50 Continue * End Of Loop Over Columns. 100 Continue If (A(N,N) .eq. Zero) Ier = N Return End Subroutine Lusol (N, Ndim, A, Ip, B, X) Implicit Double Precision (A-H, O-Z) Dimension A(Ndim,N), Ip(N), B(N), X(N) * Blas: Daxpy, Dcopy * The Subroutine Lusol Is A Simple-Minded Routine To Solve A * Linear System Whose Lu Factors Have Been Computed By Lufac. * On Entry, The Matrix A Should Contain The Lu Factors, And * Ip Should Contain The Interchange Array Constructed By Lufac. * Copy The Right-Hand Side B Into X. Call Dcopy (N, B, 1, X, 1) * Forward Solution With L (Unit Lower-Triangular Factor), Which * Is Stored In The Strict Lower Triangle Of A. Do 20 K = 1, N-1 Ipiv = Ip(K) If (Ipiv .ne. K) Then Tem = X(Ipiv) X(Ipiv) = X(K) X(K) = Tem Endif Call Daxpy ( N-K, X(K), A(K+1,K), 1, X(K+1), 1 ) 20 Continue * Backward Solution With U (Upper-Triangular Factor), Which Is Stored * In The Upper Triangle Of A. Do 40 Kb = N, 1, -1 X(Kb) = X(Kb)/A(Kb,Kb) Call Daxpy(Kb-1, (-X(Kb)), A(1,Kb), 1, X(1), 1) 40 Continue Return End Subroutine Dcopy ( N, X, Incx, Y, Incy ) Implicit Double Precision (A-H,O-Z) Integer N, Incx, Incy Dimension X( * ), Y( * ) C Dcopy Performs The Operation C C Y : = X C C Nag Fortran 77 Version Of The Blas Routine Dcopy . C Nag Fortran 77 O( N ) Basic Linear Algebra Routine. C C -- Written On 26-November-1982. C Sven Hammarling, Nag Central Office. Integer I , Ix , Iy If( N .lt. 1 )Return If( ( Incx .eq. Incy ) .and. ( Incy .gt. 0 ) )Then Do 10, Iy = 1, 1 + ( N - 1 )*Incy, Incy Y( Iy ) = X( Iy ) 10 Continue Else If( Incx .ge. 0 )Then Ix = 1 Else Ix = 1 - ( N - 1 )*Incx End If If( Incy .gt. 0 )Then Do 20, Iy = 1, 1 + ( N - 1 )*Incy, Incy Y( Iy ) = X( Ix ) Ix = Ix + Incx 20 Continue Else Iy = 1 - ( N - 1 )*Incy Do 30, I = 1, N Y( Iy ) = X( Ix ) Iy = Iy + Incy Ix = Ix + Incx 30 Continue End If End If Return * End Of Dcopy . End Subroutine Daxpy ( N, Alpha, X, Incx, Y, Incy ) Implicit Double Precision (A-H,O-Z) Integer N, Incx, Incy Dimension X( * ), Y( * ) C Daxpy Performs The Operation C C Y : = Alpha*X + Y C C C Modified Nag Fortran 77 Version Of The Blas Routine Daxpy . C C -- Written On 3-September-1982. C Sven Hammarling, Nag Central Office. Integer I , Ix , Iy Parameter ( Zero = 0.0d+0 ) If( N .lt. 1 )Return If( Alpha .eq. Zero )Return If( ( Incx .eq. Incy ) .and. ( Incx .gt. 0 ) )Then If ( Alpha .eq. 1.0d+0 ) Then Do 5, Ix = 1, 1 + ( N - 1 )*Incx, Incx Y( Ix ) = X( Ix ) + Y( Ix ) 5 Continue Else Do 10, Ix = 1, 1 + ( N - 1 )*Incx, Incx Y( Ix ) = Alpha*X( Ix ) + Y( Ix ) 10 Continue Endif Else If( Incy .ge. 0 )Then Iy = 1 Else Iy = 1 - ( N - 1 )*Incy End If If( Incx .gt. 0 )Then Do 20, Ix = 1, 1 + ( N - 1 )*Incx, Incx Y( Iy ) = Alpha*X( Ix ) + Y( Iy ) Iy = Iy + Incy 20 Continue Else Ix = 1 - ( N - 1 )*Incx Do 30, I = 1, N Y( Iy ) = Alpha*X( Ix ) + Y( Iy ) Ix = Ix + Incx Iy = Iy + Incy 30 Continue End If End If Return * End Of Daxpy . End Double Precision Function Ddot ( N, X, Incx, Y, Incy ) Implicit Double Precision (A-H,O-Z) Integer N, Incx, Incy Dimension X( * ), Y( * ) C Ddot Returns The Value C C Ddot = X'Y C C C Modified Nag Fortran 77 Version Of The Blas Routine Ddot . C C -- Written On 21-September-1982. C Sven Hammarling, Nag Central Office. Integer I , Ix , Iy Parameter ( Zero = 0.0d+0 ) Sum = Zero If( N .ge. 1 )Then If( ( Incx .eq. Incy ) .and. ( Incx .gt. 0 ) )Then Do 10, Ix = 1, 1 + ( N - 1 )*Incx, Incx Sum = Sum + X( Ix )*Y( Ix ) 10 Continue Else If( Incy .ge. 0 )Then Iy = 1 Else Iy = 1 - ( N - 1 )*Incy End If If( Incx .gt. 0 )Then Do 20, Ix = 1, 1 + ( N - 1 )*Incx, Incx Sum = Sum + X( Ix )*Y( Iy ) Iy = Iy + Incy 20 Continue Else Ix = 1 - ( N - 1 )*Incx Do 30, I = 1, N Sum = Sum + X( Ix )*Y( Iy ) Ix = Ix + Incx Iy = Iy + Incy 30 Continue End If End If End If Ddot = Sum Return * End Of Ddot . End Subroutine Dscal ( N, Alpha, X, Incx ) Implicit Double Precision (A-H,O-Z) Integer N, Incx Dimension X( * ) C Dscal Performs The Operation C C X : = Alpha*X C C C Modified Nag Fortran 77 Version Of The Blas Routine Dscal . C C -- Written On 26-November-1982. C Sven Hammarling, Nag Central Office. Integer Ix Parameter ( One = 1.0d+0, Zero = 0.0d+0 ) If( N .ge. 1 )Then If( Alpha .eq. Zero )Then Do 10, Ix = 1, 1 + ( N - 1 )*Incx, Incx X( Ix ) = Zero 10 Continue Else If( Alpha .eq. ( -One ) )Then Do 20, Ix = 1, 1 + ( N - 1 )*Incx, Incx X( Ix ) = -X( Ix ) 20 Continue Else If( Alpha .ne. One )Then Do 30, Ix = 1, 1 + ( N - 1 )*Incx, Incx X( Ix ) = Alpha*X( Ix ) 30 Continue End If End If Return * End Of Dscal . End Subroutine Dswap ( N, X, Incx, Y, Incy ) Implicit Double Precision (A-H,O-Z) Integer N, Incx, Incy Dimension X( * ), Y( * ) C Dswap Performs The Operations C C Temp : = X, X : = Y, Y : = Temp. C C C Modified Nag Fortran 77 Version Of The Blas Routine Dswap . C C -- Written On 26-November-1982. C Sven Hammarling, Nag Central Office. Integer I , Ix , Iy If( N .lt. 1 )Return If( ( Incx .eq. Incy ) .and. ( Incy .gt. 0 ) )Then Do 10, Iy = 1, 1 + ( N - 1 )*Incy, Incy Temp = X( Iy ) X( Iy ) = Y( Iy ) Y( Iy ) = Temp 10 Continue Else If( Incx .ge. 0 )Then Ix = 1 Else Ix = 1 - ( N - 1 )*Incx End If If( Incy .gt. 0 )Then Do 20, Iy = 1, 1 + ( N - 1 )*Incy, Incy Temp = X( Ix ) X( Ix ) = Y( Iy ) Y( Iy ) = Temp Ix = Ix + Incx 20 Continue Else Iy = 1 - ( N - 1 )*Incy Do 30, I = 1, N Temp = X( Ix ) X( Ix ) = Y( Iy ) Y( Iy ) = Temp Iy = Iy + Incy Ix = Ix + Incx 30 Continue End If End If Return * End Of Dswap . End Integer Function Idamax( N, X, Incx ) Implicit Double Precision (A-H,O-Z) Integer N, Incx Dimension X( * ) C Idamax Returns The Smallest Value Of I Such That C C Abs( X( I ) ) = Max( Abs( X( J ) ) ) C J C C Nag Fortran 77 Version Of The Blas Routine Idamax. C Nag Fortran 77 O( N ) Basic Linear Algebra Routine. C C -- Written On 31-May-1983. C Sven Hammarling, Nag Central Office. Intrinsic Abs Integer I , Imax , Ix If( N .lt. 1 )Then Idamax = 0 Return End If Imax = 1 If( N .gt. 1 )Then Xmax = Abs( X( 1 ) ) Ix = 1 Do 10, I = 2, N Ix = Ix + Incx If( Xmax .lt. Abs( X( Ix ) ) )Then Xmax = Abs( X( Ix ) ) Imax = I End If 10 Continue End If Idamax = Imax Return * End Of Idamax. End Subroutine Dload ( N, Const, X, Incx ) Implicit Double Precision (A-H,O-Z) Dimension X(*) C C Dload Performs The Operation C C X = Const*E, E' = ( 1 1 ... 1 ). C C C Nag Fortran 77 O( N ) Basic Linear Algebra Routine. C C -- Written On 22-September-1983. C Sven Hammarling, Nag Central Office. C Parameter ( Zero = 0.0d+0 ) If( N .lt. 1 )Return If( Const .ne. Zero )Then Do 10, Ix = 1, 1 + ( N - 1 )*Incx, Incx X( Ix ) = Const 10 Continue Else Do 20, Ix = 1, 1 + ( N - 1 )*Incx, Incx X( Ix ) = Zero 20 Continue End If Return * End Of Dload . End Subroutine Maxpy ( Nrow, Ncol, Alpha, Xmat, Nrowy, Ymat ) Implicit Double Precision (A-H,O-Z) Dimension Xmat(Nrow, Ncol), Ymat(Nrowy, Ncol) * Subroutine Maxpy Takes As Input The Scalar Alpha And Two Matrices, * Xmat And Ymat. Xmat Has Declared Row Dimension Nrow, And * Ymat Has Declared Row Dimension Nrowy, But Both Are * Conceptually Nrow By Ncol. * On Output, (New Ymat) Is Alpha*Xmat+ (Old Ymat), By Analogy * With The Vector Blas Routine Saxpy. Do 100 J = 1, Ncol Do 100 I = 1, Nrow Ymat(I,J) = Ymat(I,J) + Alpha*Xmat(I,J) 100 Continue Return End Subroutine Matcop( Nrow1, Nrow2, Nrow, Ncol, Xmat1, Xmat2 ) Implicit Double Precision (A-H,O-Z) Dimension Xmat1(Nrow1, Ncol), Xmat2(Nrow2, Ncol) * Given 2 Matrices Xmat1 And Xmat2, Where Xmat1 Has Declared * Row Dimension Nrow1, Xmat2 Has Declared Row Dimension Nrow2, * And Both Have Column Dimension Ncol, The Routine Matcop Copies * Rows 1 Through Nrow, And Columns 1 Through Ncol From Xmat1 Into * Xmat2. If (Nrow .le. 0 .or. Ncol .le. 0) Return Do 100 J = 1, Ncol Do 100 I = 1, Nrow Xmat2(I,J) = Xmat1(I,J) 100 Continue Return End Subroutine Mtload( Nrow, Ncol, Const, Nrowx, Xmat ) Implicit Double Precision (A-H,O-Z) Dimension Xmat(Nrowx, Ncol) * Mtload Sets Elements 1 Through Nrow, 1 Through Ncol, Of The * Matrix Xmat (Whose Declared Row Dimension Is Nrowx) To The * Scalar Value Const. If (Nrow .le. 0 .or. Ncol .le. 0) Return Do 100 J = 1, Ncol Do 100 I = 1, Nrow Xmat(I,J) = Const 100 Continue Return End Subroutine Mssq ( Nrow, Ncol, Xmat, Scale, Sumsq ) Implicit Double Precision (A-H,O-Z) Dimension Xmat(Nrow, *) * Given The Nrow By Ncol Matrix Xmat, Mssq Returns Values * Scale And Sumsq Such That * (Scale**2) * Sumsq = Sum Of Squares Of Xmat(I,J), * Where Scale = Max Abs(Xmat(I,J)). * Mssq Is A Stripped-Down Matrix Version Of The Blas Routine Sssq. Intrinsic Abs Parameter ( One = 1.0d+0, Zero = 0.0d+0 ) Scale = Zero Sumsq = One If( Nrow .ge. 1 .and. Ncol .ge. 1) Then Do 10 I = 1, Nrow Do 10 J = 1, Ncol If( Xmat(I,J) .ne. Zero )Then Absxij = Abs(Xmat(I,J)) If( Scale .lt. Absxij ) Then Sumsq = One + Sumsq* (Scale/Absxij)**2 Scale = Absxij Else Sumsq = Sumsq + (Absxij/Scale)**2 End If End If 10 Continue End If Return End Subroutine Dssq ( N, X, Incx, Scale, Sumsq ) Implicit Double Precision (A-H,O-Z) Integer N, Incx Dimension X( * ) * Given The N-Vector X, Dssq Returns Values Scale And Sumsq Such That * (Scale**2) * Sumsq = Sum Of Squares Of X(I), * Where Scale = Max Abs(X(I)). Intrinsic Abs Parameter ( One = 1.0d+0, Zero = 0.0d+0 ) Scale = Zero Sumsq = One If( N .ge. 1 )Then Do 10, Ix = 1, 1 + ( N - 1 )*Incx, Incx If( X( Ix ) .ne. Zero )Then Absxi = Abs( X( Ix ) ) If( Scale .lt. Absxi )Then Sumsq = One + Sumsq*( Scale/Absxi )**2 Scale = Absxi Else Sumsq = Sumsq + ( Absxi/Scale )**2 End If End If 10 Continue End If Return * End Of Dssq. End * File Bvps.f Subroutine Bvpsol(Ncomp, Nmsh, Nlbc, Aleft, Aright, * Nfxpnt, Fixpnt, Ntol, Ltol, Tol, Nmax, Linear, Giveu, * Givmsh, Xx, Nudim, U, Def, Delu, Rhs, Fval, * Topblk, Botblk, Ajac, Bhold, Chold, Ipvblk, Ipivlu, * Uint, Ftmp, Tmprhs, Dftmp1, Dftmp2, Dgtm, * Utrial, Rhstri, Xmerit, Xxold, Uold, Tmp, Dsq, Ratdc, * Etest6, Etest8, Ermx, Ihcomp, Irefin, Def8, Dhold, * Voldmsh, Fsub, Dfsub, Gsub, Dgsub, Iflbvp, Eps) Implicit Double Precision (A-H,O-Z) Dimension Fixpnt(*), Ltol(Ntol), Tol(Ntol) Dimension Xx(*), U(Nudim, *) Dimension Def(Ncomp,*) Dimension Delu(Ncomp, *), Rhs(*), Fval(Ncomp,*) Dimension Topblk(Ncomp,*), Botblk(Ncomp,*) Dimension Ajac(Ncomp, 2*Ncomp, *) Dimension Bhold(Ncomp, Ncomp, *), Chold(Ncomp, Ncomp, *) Dimension Ipivlu(*), Ipvblk(*) Dimension Uint(Ncomp), Ftmp(Ncomp), Dgtm(Ncomp), Tmprhs(*) Dimension Dftmp1(Ncomp, Ncomp), Dftmp2(Ncomp, Ncomp) Dimension Utrial(Ncomp,*), Rhstri(*), Xmerit(Ncomp, *) Dimension Xxold(*), Uold(Ncomp,*), Tmp(Ncomp,12) Dimension Dsq(Ncomp,Ncomp), Ratdc(*) Dimension Etest6(*), Etest8(*), Ermx(*) Dimension Ihcomp(*), Irefin(*) Dimension Def8(Ncomp,*) Dimension Dhold(3*Ncomp,3*Ncomp), Voldmsh(*) Logical Linear, Giveu, Givmsh, Double, Errok External Fsub, Dfsub, Gsub, Dgsub Common/Mchprs/ Flmax, Epsmch Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Convg/ Nits Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Intrinsic Max Logical Succes, Strctr Logical Ludone, Rhsgiv, Maxmsh Logical Mchset Save Mchset Parameter (Zero = 0.0d+0, One = 1.0d+0) Parameter (Third = 0.33d+0) Parameter (Quan6 = 0.1d+0 ) * Blas: Dload * Double Precision D1mach Data Mchset/.true./ Data Fxfct/10.0d+0/ Data Maxmsh/.false./ * The Parameter Inumb Is A Counter Used To Limit To Three The Number * Of Mesh Selections Performed For The Final Continuation Problem. Inumb = 0 If (Mchset) Then Flmax = D1mach(2) Epsmch = D1mach(3) Mchset = .false. Endif * The Routines Stcon1 And Stcon2 Calculate Integration Constants Stored * In Labeled Common Consts. Call Stcon1 Call Stcon2 * Set Up Arrays For The Error Tests. If (.not. Linear) Then Call Dload(Ntol, One, Etest6, 1) Else Do 10 I = 1, Ntol Etest6(I) = One/Max(Quan6, Tol(I)**Third) 10 Continue Endif Nmold = 1 Strctr = .false. * * If Givmsh Is .true., The Initial Number Of Mesh Points Must Be * Provided By The User In Nmsh, And The Mesh Points Must Be * Contained In The Array Xx (Of Dimension Nmsh). * Otherwise, Nmsh Is Set To Its Default Value, And A * Uniform Initial Mesh Is Created. If (.not. Giveu .and. .not. Givmsh) Then Nmsh = Nminit If (Nmsh .lt. Nfxpnt+2) Nmsh = Nfxpnt + 2 Call Unimsh(Nmsh, Aleft, Aright, Nfxpnt, Fixpnt, Xx) Endif If (.not. Giveu) Call Initu(Ncomp, Nmsh, Nudim, U) ***** Top Of Logic For 4th Order Solution **** 400 Continue If (Iprint .ge. 0) Write(6,903) Nmsh * Set The Def (Deferred Correction) Array To Zero. Call Mtload(Ncomp, Nmsh-1, Zero, Ncomp, Def) Iorder = 4 * The Routine Fneval Calls Fsub At The Mesh Xx And The * Solution U, And Saves The Values In The Array Fval. Call Fneval(Ncomp, Nmsh, Xx, Nudim, U, Fval, Eps) * Try To Compute A 4th Order Solution By Solving A System Of Nonlinear * Equations. If (Linear) Then Ludone = .false. Call Lineq( Ncomp, Nmsh, Nlbc, * Ludone, Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, * Ajac, Topblk, Botblk, Bhold, Chold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Iflnwt, Eps) * Call Fneval To Evaluate The Fval Array At The New Solution U. * (Such A Call Is Not Necessary For The Nonlinear Case Because * Fval Is Called Within Newteq For The New U.) Call Fneval(Ncomp, Nmsh, Xx, Nudim, U, Fval, Eps) Else Rhsgiv = .false. Call Newteq(Ncomp, Nmsh, Nlbc, * Rhsgiv, Ntol, Ltol, Tol, * Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Utrial, Rhstri, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, Xmerit, * Ajac, Topblk, Botblk, Bhold, Chold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Itnwt, Iflnwt, Eps) If (Iatt .eq. -1) Nits = Max(1,Itnwt) If (Iprint .eq. 0) Write(6,999) Itnwt Endif If (Iflnwt .eq. 0) Then Call Dfexcl( Ncomp, Nmsh, Xx, Nudim, U, Def, Linear, Fval, * Tmp, Fsub, Dfsub, Dsq, Ipivlu, Dhold, * Ntol, Ltol, Tol, Eps ) Else Iflbvp = 1 Return Endif **** Logic For 6th Order **** If (Iprint .eq. 1) Write(6,905) * Save The 4th Order Solution On This Mesh In Uold. Call Matcop(Nudim, Ncomp, Ncomp, Nmsh, U, Uold) Iorder = 6 If (Linear) Then Call Lineq( Ncomp, Nmsh, Nlbc, * Ludone, Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, * Ajac, Topblk, Botblk, Chold, Bhold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Iflnwt, Eps) Else Call Fixjac(Ncomp, Nmsh, Nlbc, * Iorder, Ntol, Ltol, Tol, * Xx, Nudim, U, Def, Def, Delu, * Rhs, Fval, Utrial, Rhstri, * Rnsq, Uint, Ftmp, Tmprhs, * Ajac, Topblk, Botblk, Ipvblk, * Fsub, Gsub, Iflnwt, Eps) * If The Fixed Jacobian Iterations Fail But Rnsq Is Small, * Try A Newton Procedure. Set Rhsgiv To Indicate That * The Right-Hand Side And Fval Have Already Been Evaluated * At The Given U. If (Iflnwt .eq. -3 .and. Rnsq .lt. Fxfct*Epsmch) Rhsgiv = .true. If (Iflnwt .ne. 0) Then Call Newteq(Ncomp, Nmsh, Nlbc, * Rhsgiv, Ntol, Ltol, Tol, * Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Utrial, Rhstri, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, Xmerit, * Ajac, Topblk, Botblk, Bhold, Chold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Iter, Iflnwt, Eps) Endif Endif If (Iflnwt .ne. 0) Then Iflbvp = 1 Return Elseif (Ifinal .eq. 1) Then Call Errest (Ncomp, Nmsh, Ntol, Ltol, Tol, * Nudim, U, Uold, Etest6, Errok) If (Errok) Then Iflbvp = 0 Return Endif Endif ***** Logic For Trying To Calculate 8th Order Solution ***** If (Iprint .eq. 1) Write(6,906) Call Matcop(Nudim, Ncomp, Ncomp, Nmsh, U, Uold) * For Linear Problems, Calculate The Fval Array For The * New Solution U. If (Linear) Call Fneval(Ncomp, Nmsh, Xx, Nudim, U, Fval, Eps) * Calculate 8th Order Deferred Corrections (The Array Def8). Call Df8cal (Ncomp, Nmsh, Xx, Nudim, U, Fval, Def8, Linear, * Tmp, Fsub, Dfsub, Dsq, Ipivlu, Dhold, * Ntol, Ltol, Tol, Eps) * For Linear Problems, The Def Array Is The Def8 Array. * For Nonlinear Problems, Add The Def8 Array To The * Already-Calculated Def Array. If (Linear) Then Call Matcop(Ncomp, Ncomp, Ncomp, Nmsh-1, Def8, Def) Else Call Maxpy(Ncomp, Nmsh-1, One, Def8, Ncomp, Def) Endif Iorder = 8 If (Linear) Then Call Lineq( Ncomp, Nmsh, Nlbc, * Ludone, Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, * Ajac, Topblk, Botblk, Chold, Bhold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Iflnwt, Eps) Else Call Fixjac(Ncomp, Nmsh, Nlbc, * Iorder, Ntol, Ltol, Tol, * Xx, Nudim, U, Def, Def8, Delu, * Rhs, Fval, Utrial, Rhstri, * Rnsq, Uint, Ftmp, Tmprhs, * Ajac, Topblk, Botblk, Ipvblk, * Fsub, Gsub, Iflnwt, Eps) * If The Fixed Jacobian Iterations Fail But Rnsq Is Small, * Try A Newton Procedure. Set Rhsgiv To Indicate That * The Right-Hand Side And Fval Have Already Been Evaluated * At The Given U. If (Iflnwt .eq. -3 .and. Rnsq .lt. Fxfct*Epsmch) Rhsgiv = .true. If (Iflnwt .ne. 0) Then Call Newteq(Ncomp, Nmsh, Nlbc, * Rhsgiv, Ntol, Ltol, Tol, * Xx, Nudim, U, Def, * Delu, Rhs, Fval, * Utrial, Rhstri, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, Xmerit, * Ajac, Topblk, Botblk, Bhold, Chold, Ipvblk, * Fsub, Dfsub, Gsub, Dgsub, Iter, Iflnwt, Eps) Endif Endif If (Iflnwt .eq. 0) Then Call Conv8( Ncomp, Nmsh, Ntol, Ltol, Tol, * Nfxpnt, Fixpnt, Linear, Nmax, * Xx, Nudim, U, Def8, Uold, * Ihcomp, Irefin, Ermx, * Etest8, Strctr, Ratdc, Voldmsh, * Double, Nmold, Xxold, Maxmsh, Succes) If (Iprec .eq. 2) Then If (Iprint .ge. 0) Write(6,1013) Iflbvp = 1 Return Endif Else Iflbvp = 1 Return Endif If (Maxmsh) Go To 900 Iflbvp = 0 If (Ifinal .eq. 1) Then If (Succes) Then Return Elseif (Iatt .ge. 1) Then If (Nmsh .lt. Nmax/2) Then Inumb = Inumb+1 Else Iflbvp = 1 Return Endif Endif If (Iback .ne. 1 .and. Inumb .eq. 3) Then Iflbvp = 1 Return Endif Elseif (Iatt .eq. 0) Then Return Endif Iatt = Iatt+1 Goto 400 900 Continue * Error Exit---Too Many Mesh Points. Iflbvp = 1 Return 903 Format(1h ,'The New Mesh Contains',I6,' Points.') 904 Format(1h ,'Do Not Go On To 6th') 905 Format(1h ,'Start 6th Order') 906 Format(1h ,'Start 8th Order') 999 Format(1h ,'Newton Converged After',I3,' Iterations.') 1013 Format(/,1x,'** Mesh Cannot Be Defined Within The Bounds Imposed', + ' By The Machine Precision') End Subroutine Initu(Ncomp, Nmsh, Nudim, U) Implicit Double Precision (A-H,O-Z) Dimension U(Nudim, *) Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 * This Routine Must Be Provided To Reset U After Re-Meshing * For Linear Problems Or For Nonlinear Problems * When Interpolation Of The Old Solution Is Not Used. * This Version Sets All Elements Of U To The Constant Uval0. If (Iprint .eq. 1) Write(6,99) Uval0 99 Format(1h ,'Initu, Uval0',1pd15.5) Call Mtload(Ncomp, Nmsh, Uval0, Nudim, U) Return End Block Data * This Block Data Routine Initializes Nminit (The Initial Number * Of Mesh Points) And Uval0 (The Initial Value For The Trial * Solution) To Their Default Values. Double Precision Uval0 Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Data Nminit/7/ Data Iprint/0/ Data Itsaim/7/ Data Maxcon/50/ Data Uval0/0.0d+0/ End Subroutine Conv8( Ncomp, Nmsh, Ntol, Ltol, Tol, * Nfxpnt, Fixpnt, Linear, Nmax, * Xx, Nudim, U, Def8, Uold, * Ihcomp, Irefin, Ermx, * Etest8, Strctr, Ratdc, Voldmsh, * Double, Nmold, Xxold, Maxmsh, Succes) Implicit Double Precision (A-H,O-Z) Dimension Ltol(Ntol), Tol(Ntol) Dimension Fixpnt(*), Etest8(Ntol) Dimension Xx(*), U(Nudim,*) Dimension Def8(Ncomp,*), Uold(Ncomp,*) Dimension Ihcomp(*), Irefin(*) Dimension Ermx(*), Xxold(*), Ratdc(*), Voldmsh(*) Logical Linear, Strctr, Double, Maxmsh, Succes ***bugfix 2jul01 Save Nvold Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common /Flags/ Ifinal,Iatt,Iback,Iprec Intrinsic Max Logical Errok Parameter (One = 1.0d+0, Fourth = 0.25d+0, Quan8 = 0.025d+0) * Blas: Dload * The Newton Iteration Converged For The 8th Order Solution. If (Iprint .eq. 1) Write(6,901) Succes = .false. Maxmsh = .false. If (Ifinal .eq. 1) Then If (.not. Linear) Then Call Dload(Ntol, One, Etest8, 1) Else Do 10 I = 1, Ntol Etest8(I) = One/Max(Quan8, Tol(I)**Fourth) 10 Continue Endif * Check Estimated Error. For A Nonlinear Problem, All Components * Of Etest8 (The Ratios Used In Testing The Error) Are Set To One. * For A Linear Problem, The Components Of Etest8 Are In General * Larger Than One. But If Strctr Is .true. And The Number Of Mesh * Points Decreased, We Set The Elements Of Etest8 To One (Which * Makes A Stricter Test). If (Linear .and. Strctr .and. Nmsh .lt. Nmold) * Call Dload(Ntol, One, Etest8, 1) Call Errest (Ncomp, Nmsh, Ntol, Ltol, Tol, * Nudim, U, Uold, Etest8, Errok) If (Errok) Then Succes = .true. Return Endif Endif If (Ifinal .eq. 1 .and. Iatt .ge. 1) Then Call Dblmsh (Nmsh, Nmax, Xx, Nmold, Xxold, Maxmsh) If (.not. Maxmsh .and. Iprec .ne. 2) Then ***bugfix 2jul01 * Call Matcop(Nudim, Ncomp, Ncomp, Nmsh, U, Uold) Call Matcop(Nudim, Ncomp, Ncomp, Nmold, U, Uold) Call Interp(Ncomp, Nmsh, Xx, Nudim, U, * Nmold, Xxold, Uold) Endif Return Endif * Perform Selective Mesh Refinement Based On The 8th Order Deferred * Corrections. The Value Of Ipow Indicates That The Error Estimate * Is Of Order 6. Then, For A Nonlinear Problem, Interpolate The * Latest Solution Onto The New Mesh. Ipow = 6 * The Array Def8 Will Be Overwritten By Selmsh. Call Selmsh(Ncomp, Nmsh, Ntol, Ltol, Tol, * Nfxpnt, Fixpnt, Ipow, Nmax, Nvold, * Xx, Nudim, U, Def8, Irefin, Ihcomp, * Nmold, Xxold, Ermx, Double, Maxmsh, Ratdc, Voldmsh) If (.not. Maxmsh .and. Iprec .ne. 2) Then If (Linear .and. (Ifinal .eq. 1 .or. Iatt .ne. 0)) Then Call Initu(Ncomp, Nmsh, Nudim, U) Else ***bugfix 2jul01 * Call Matcop(Nudim, Ncomp, Ncomp, Nmsh, U, Uold) Call Matcop(Nudim, Ncomp, Ncomp, Nmold, U, Uold) Call Interp(Ncomp, Nmsh, Xx, Nudim, U, * Nmold, Xxold, Uold) Endif Endif Return 901 Format(1h ,'Conv8') End * File Defs.f Subroutine Dfexcl (Ncomp, Nmsh, Xx, Nudim, U, Def6, Linear, * Fval, Tmp, Fsub, Dfsub, Df, Ip, Dhold, * Ntol, Ltol, Tol, Eps) Implicit Double Precision (A-H, O-Z) Integer Ncomp, Nmsh Dimension Xx(Nmsh), U(Nudim,Nmsh), Fval(Ncomp, Nmsh) Dimension Def6(Ncomp,Nmsh-1), Tmp(Ncomp,8) Dimension Df(Ncomp,Ncomp), Ltol(Ntol), Tol(Ntol) Dimension Ip(2*Ncomp), Dhold(2*Ncomp,2*Ncomp) Dimension St1(200), St2(200), St3(200) External Fsub, Dfsub Parameter ( One = 1.0d+0, Two = 2.0d+0 ) Common /Cons1/ A21,A22,A23,A24,A31,A32,A33,A34,C1,C2, + C16,C26,C123,C223,C14,C24 Common /Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common /Flags/ Ifinal,Iatt,Iback,Iprec Logical Linear * Given The Nmsh Mesh Points Xx, The Estimated Solution * U And The Array Fval Of Function Values At (Xx(Im), U(*,Im)), * Im = 1,...,Nmsh, Dfexcl Calculates Sixth-Order Implicit * Deferred Correction, Stored In The Array Def6, Indexed * Over The Components And Mesh Intervals. * The Array Tmp Is Workspace For 4 Intermediate Vectors Of * Dimension Ncomp. Do 90 Im = 1, Nmsh-1 Hmsh = Xx(Im+1) - Xx(Im) C16h = C16/Hmsh C26h = C26/Hmsh Do 10 Ic = 1, Ncomp Fvim = Fval(Ic,Im) Fvim1 = Fval(Ic,Im+1) Uim = U(Ic,Im) Uim1 = U(Ic,Im+1) Tmp(Ic,3) = C16h*(Uim1-Uim)+C123*Fvim1+C14*Fvim Tmp(Ic,4) = C26h*(Uim1-Uim)+C223*Fvim1+C24*Fvim St1(Ic) = (Uim+Uim1)/Two St2(Ic) = A21*Fvim + A24*Fvim1 St3(Ic) = A31*Fvim + A34*Fvim1 10 Continue Xxc1 = Xx(Im)+C1*Hmsh Xxc2 = Xx(Im)+C2*Hmsh Do 60 Nit = 1, 10 Do 20 Ic = 1,Ncomp Tmp3 = Tmp(Ic,3) Tmp4 = Tmp(Ic,4) Tmp(Ic,1) = St1(Ic) + Hmsh*(St2(Ic) + A22*Tmp3 + A23*Tmp4) Tmp(Ic,2) = St1(Ic) + Hmsh*(St3(Ic) + A32*Tmp3 + A33*Tmp4) 20 Continue Call Fsub (Ncomp,Xxc1,Tmp(1,1),Tmp(1,5), Eps) Call Fsub (Ncomp,Xxc2,Tmp(1,2),Tmp(1,6), Eps) Call Dfsub (Ncomp,Xxc1,Tmp(1,1),Df, Eps) Do 30 I = 1, Ncomp Tmp(I,5) = Tmp(I,5)-Tmp(I,3) Tmp(I,6) = Tmp(I,6)-Tmp(I,4) Do 25 J = 1,Ncomp Dfij = Hmsh*Df(I,J) Dhold(I,J) = -A22*Dfij Dhold(I,J+Ncomp) = -A23*Dfij 25 Continue 30 Continue Call Dfsub(Ncomp,Xxc2,Tmp(1,2),Df, Eps) Do 35 I = 1, Ncomp Do 32 J = 1, Ncomp Dfij = Hmsh*Df(I,J) Dhold(I+Ncomp,J) = -A32*Dfij Dhold(I+Ncomp,J+Ncomp) = -A33*Dfij 32 Continue 35 Continue Do 40 I = 1,Ncomp Dhold(I,I) = Dhold(I,I) + One Dhold(I+Ncomp,I+Ncomp) = Dhold(I+Ncomp,I+Ncomp) + One 40 Continue Call Lufac(2*Ncomp,2*Ncomp,Dhold,Ip,Ier) Call Lusol(2*Ncomp,2*Ncomp,Dhold,Ip,Tmp(1,5),Tmp(1,7)) Do 45 I = 1,Ncomp Tmp(I,3) = Tmp(I,3) + Tmp(I,7) Tmp(I,4) = Tmp(I,4) + Tmp(I,8) 45 Continue If (Linear) Goto 70 Jc = 0 Do 50 I = 1, Ntol Ii = Ltol(I) Er = Tol(I) If (Abs(Tmp(Ii,7)) .gt. Er*Max(One,Abs(Tmp(Ii,3))) .or. * Abs(Tmp(Ii,8)) .gt. Er*Max(One,Abs(Tmp(Ii,4)))) Jc = 1 50 Continue If (Jc .eq. 0) Goto 70 60 Continue If (Iprec .eq. 0) Then If (Iprint .ge. 0) Write(6,900) Eps Iprec = 1 Endif 70 Continue Do 80 Ic = 1, Ncomp Def6(Ic,Im) = (Hmsh/12.d+0)*(Fval(Ic,Im)+ * 5.d+0*(Tmp(Ic,3)+Tmp(Ic,4))+Fval(Ic,Im+1))- * U(Ic,Im+1)+U(Ic,Im) 80 Continue 90 Continue Return 900 Format(/,'** Warning - Possibly Approaching Machine Precision ', * 'Beyond Epsilon = ',D10.3,/) End Subroutine Df8cal (Ncomp, Nmsh, Xx, Nudim, U, Fval, Def8, Linear, * Tmp, Fsub, Dfsub, Df, Ip, Dhold, Ntol, * Ltol, Tol, Eps) * Given The Mesh Points Xx, The Solution U, And The Function * Values Fval, Df8cal Computes Eighth-Order Deferred Corrections, * Which Are Stored In Def8. * The Array Tmp Is Workspace For 9 Intermediate Vectors. Implicit Double Precision (A-H, O-Z) Integer Ncomp, Nmsh Dimension Xx(Nmsh), U(Nudim,Nmsh), Fval(Ncomp,Nmsh) Dimension Def8(Ncomp, Nmsh-1), Tmp(Ncomp,12) Dimension Df(Ncomp,Ncomp), Ltol(Ntol), Tol(Ntol) Dimension Ip(3*Ncomp), Dhold(3*Ncomp,3*Ncomp) Dimension St1(200), St2(200), St3(200), St4(200) External Fsub, Dfsub Parameter ( One = 1.0d+0, Two = 2.0d+0 ) Common/Cons2/ A21,A22,A23,A24,A25,A31,A32,A34,A35,A41,A42, + A43,A44,A45,B1,B2,B3,C1,C2,C3,C16,C26,C36,C123,C223, + C323,C14,C24,C34 Common /Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common /Flags/ Ifinal,Iatt,Iback,Iprec Logical Linear Do 110 Im = 1, Nmsh-1 Hmsh = Xx(Im+1) - Xx(Im) C16h = C16/Hmsh C26h = C26/Hmsh C36h = C36/Hmsh Do 10 Ic = 1, Ncomp Fvim = Fval(Ic,Im) Fvim1 = Fval(Ic,Im+1) Uim = U(Ic,Im) Uim1 = U(Ic,Im+1) Tmp(Ic,4) = C16h*(Uim1-Uim)+C123*Fvim1+C14*Fvim Tmp(Ic,5) = C26h*(Uim1-Uim)+C223*Fvim1+C24*Fvim Tmp(Ic,6) = C36h*(Uim1-Uim)+C323*Fvim1+C34*Fvim St1(Ic) = (Uim+Uim1)/Two St2(Ic) = A21*Fvim + A25*Fvim1 St3(Ic) = A31*Fvim + A35*Fvim1 St4(Ic) = A41*Fvim + A45*Fvim1 10 Continue Xxc1 = Xx(Im)+C1*Hmsh Xxc2 = Xx(Im)+C2*Hmsh Xxc3 = Xx(Im)+C3*Hmsh Do 80 Nit = 1, 10 Do 20 Ic = 1, Ncomp Tmp4 = Tmp(Ic,4) Tmp5 = Tmp(Ic,5) Tmp6 = Tmp(Ic,6) Tmp(Ic,1) = St1(Ic) + Hmsh*(St2(Ic) + A22*Tmp4 + A23*Tmp5 + + A24*Tmp6) Tmp(Ic,2) = St1(Ic) + Hmsh*(St3(Ic) + A32*Tmp4 + A34*Tmp6) Tmp(Ic,3) = St1(Ic) + Hmsh*(St4(Ic) + A42*Tmp4 + A43*Tmp5 + + A44*Tmp6) 20 Continue Call Fsub (Ncomp,Xxc1,Tmp(1,1),Tmp(1,7), Eps) Call Fsub (Ncomp,Xxc2,Tmp(1,2),Tmp(1,8), Eps) Call Fsub (Ncomp,Xxc3,Tmp(1,3),Tmp(1,9), Eps) Call Dfsub(Ncomp,Xxc1,Tmp(1,1),Df, Eps) Do 30 I = 1, Ncomp Tmp(I,7) = Tmp(I,7)-Tmp(I,4) Tmp(I,8) = Tmp(I,8)-Tmp(I,5) Tmp(I,9) = Tmp(I,9)-Tmp(I,6) Do 25 J = 1,Ncomp Dfij = Hmsh*Df(I,J) Dhold(I,J) = -A22*Dfij Dhold(I,J+Ncomp) = -A23*Dfij Dhold(I,J+2*Ncomp) = -A24*Dfij 25 Continue 30 Continue Call Dfsub(Ncomp,Xxc2,Tmp(1,2),Df, Eps) Do 40 I = 1, Ncomp Do 35 J = 1, Ncomp Dfij = Hmsh*Df(I,J) Dhold(I+Ncomp,J) = -A32*Dfij Dhold(I+Ncomp,J+Ncomp) = 0.d+0 Dhold(I+Ncomp,J+2*Ncomp) = -A34*Dfij 35 Continue 40 Continue Call Dfsub(Ncomp,Xxc3,Tmp(1,3),Df, Eps) Do 50 I = 1, Ncomp Do 45 J = 1, Ncomp Dfij = Hmsh*Df(I,J) Dhold(I+2*Ncomp,J) = -A42*Dfij Dhold(I+2*Ncomp,J+Ncomp) = -A43*Dfij Dhold(I+2*Ncomp,J+2*Ncomp) = -A44*Dfij 45 Continue 50 Continue Do 60 I = 1, Ncomp Dhold(I,I) = Dhold(I,I) + One Dhold(I+Ncomp,I+Ncomp) = Dhold(I+Ncomp,I+Ncomp) + One Dhold(I+2*Ncomp,I+2*Ncomp) = * Dhold(I+2*Ncomp,I+2*Ncomp) + One 60 Continue Call Lufac(3*Ncomp,3*Ncomp,Dhold,Ip,Ier) Call Lusol(3*Ncomp,3*Ncomp,Dhold,Ip,Tmp(1,7),Tmp(1,10)) Do 65 I = 1,Ncomp Tmp(I,4) = Tmp(I,4) + Tmp(I,10) Tmp(I,5) = Tmp(I,5) + Tmp(I,11) Tmp(I,6) = Tmp(I,6) + Tmp(I,12) 65 Continue If (Linear) Goto 90 Jc = 0 Do 70 I = 1, Ntol Ii = Ltol(I) Er = Tol(I) If (Abs(Tmp(Ii,10)) .gt. Er*Max(One,Abs(Tmp(Ii,4))) .or. * Abs(Tmp(Ii,11)) .gt. Er*Max(One,Abs(Tmp(Ii,5))) .or. * Abs(Tmp(Ii,12)) .gt. Er*Max(One,Abs(Tmp(Ii,6)))) Jc = 1 70 Continue If (Jc .eq. 0) Goto 90 80 Continue If (Iprec .eq. 0) Then If (Iprint .ge. 0) Write(6,900) Eps Iprec = 1 Endif 90 Continue Do 100 Ic = 1, Ncomp Def8(Ic,Im) = Hmsh*(B1*(Fval(Ic,Im)+Fval(Ic,Im+1))+ * B2*(Tmp(Ic,4)+Tmp(Ic,6))+B3*Tmp(Ic,5))- * U(Ic,Im+1)+U(Ic,Im) 100 Continue 110 Continue Return 900 Format(/,'** Warning - Possibly Approaching Machine Precision ', * 'Beyond Epsilon = ',D10.3,/) End Subroutine Stcon1 Implicit Double Precision(A-H,O-Z) Common/Cons1/A21,A22,A23,A24,A31,A32,A33,A34,C1,C2, + C16,C26,C123,C223,C14,C24 Parameter ( One = 1.0d+0, Two = 2.0d+0, Three = 3.0d+0 ) Parameter ( Four = 4.0d+0, Five = 5.0d+0, Six = 6.0d+0 ) Rt5 = Sqrt(5.0d+0) A21 = (Six + Rt5)/120.0d0 A22 = -Rt5/120.0d0 A23 = (-13.d0*Rt5)/120.0d0 A24 = (-Six + Rt5)/120.d0 A31 = (Six-Rt5)/120.0d0 A32 = (13.0d0*Rt5)/120.0d0 A33 = Rt5 / 120.0d0 A34 = (-Six - Rt5)/120.d0 C1 = (Five - Rt5)/10.0d0 C2 = (Five + Rt5)/10.0d0 C12 = C1*C1 C22 = C2*C2 C16 = Six*(C1 - C12) C26 = Six*(C2 - C22) C123 = Three*C12 - Two*C1 C223 = Three*C22 - Two*C2 C14 = One - Four*C1 + Three*C12 C24 = One - Four*C2 + Three*C22 Return End Subroutine Stcon2 Implicit Double Precision(A-H,O-Z) Common/Cons2/ A21,A22,A23,A24,A25,A31,A32,A34,A35,A41,A42, + A43,A44,A45,B1,B2,B3,C1,C2,C3,C16,C26,C36,C123,C223, + C323,C14,C24,C34 Parameter ( One = 1.0d+0, Two = 2.0d+0, Three = 3.0d+0 ) Parameter ( Four = 4.0d+0, Six = 6.0d+0 ) Rt21 = Sqrt(21.0d+0) A21 = One/28.d0 + Three*Rt21/1960.d0 A22 = -Rt21/280.d0 A23 = -32.d0*Rt21/735.d0 A24 = -23.d0*Rt21/840.d0 A25 = -One/28.d0 + Three*Rt21/1960.d0 A31 = One/64.d0 A32 = 7.d0*Rt21/192.d0 A34 = -7.d0*Rt21/192.d0 A35 = -One/64.d0 A41 = One/28.d0 - Three*Rt21/1960.d0 A42 = 23.d0*Rt21/840.d0 A43 = 32.d0*Rt21/735.d0 A44 = Rt21/280.d0 A45 = -(One/28.d0) - Three*Rt21/1960.d0 B1 = One/20.0d0 B2 = 49.0d0/180.d0 B3 = 16.0d0/45.d0 C1 = One/Two - Rt21/14.d0 C2 = One/Two C3 = One/Two + Rt21/14.d0 C12 = C1*C1 C22 = C2*C2 C32 = C3*C3 C16 = Six*(C1 - C12) C26 = Six*(C2- C22) C36 = Six*(C3 - C32) C123 = Three*C12 - Two*C1 C223 = Three*C22 - Two*C2 C323 = Three*C32 - Two*C3 C14 = One - Four*C1 + Three*C12 C24 = One - Four*C2 + Three*C22 C34 = One - Four*C3 + Three*C32 Return End * File Eqs.f Subroutine Fixjac(Ncomp, Nmsh, Nlbc, * Iorder, Ntol, Ltol, Tol, * Xx, Nudim, U, Defcor, Defnew, Delu, * Rhs, Fval, Utrial, Rhstri, * Rnsq, Uint, Ftmp, Tmprhs, * Ajac, Topblk, Botblk, Ipivot, * Fsub, Gsub, Iflag, Eps) * Fixed Jacobian Iterations. Implicit Double Precision (A-H,O-Z) Dimension Ltol(Ntol), Tol(Ntol) Dimension Xx(Nmsh), U(Nudim,Nmsh), Defcor(Ncomp,Nmsh-1) Dimension Defnew(Ncomp,Nmsh-1), Delu(Ncomp,Nmsh) Dimension Rhs(Ncomp*Nmsh), Fval(Ncomp,Nmsh) Dimension Utrial(Ncomp,Nmsh), Rhstri(Ncomp*Nmsh) Dimension Uint(Ncomp), Ftmp(Ncomp), Tmprhs(Ncomp*Nmsh) Dimension Ajac(Ncomp, 2*Ncomp, Nmsh-1) Dimension Topblk(Ncomp,Ncomp), Botblk(Ncomp,Ncomp) Dimension Ipivot(Ncomp*Nmsh) Logical Better External Fsub, Gsub Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common/Mchprs/ Flmax, Epsmch Intrinsic Abs, Max * Blas: Dcopy, Dssq Parameter ( One = 1.0d+00 ) Parameter ( Xlarge = 1.0d+6, Huge = 1.0d+30, Lmtfrz = 8) Parameter ( Rngrow = 16.0d+0 ) Parameter ( Tolfct = 0.1d+0 ) * The Iteration Scheme Uses A Fixed Jacobian Matrix To Solve For * Correction Vectors, Once There Has Been Convergence Of The Newton * Iterations On This Mesh. It Is Assumed That The Lu * Factors Of The Jacobian Have Been Left Unaltered Since * Their Calculation. If (Iprint .eq. 1) Write(6,901) Ninter = Nmsh - 1 Rnold = Flmax * Evaluate The Right-Hand Side Rhstri At The Initial Solution U By * Adding The New Deferred Corrections To The Already-Calculated * Rhs Vector. Call Dcopy(Nlbc, Rhs, 1, Rhstri, 1) Ind = Nlbc Do 10 Im = 1, Ninter Do 10 Ic = 1, Ncomp Ind = Ind + 1 Rhstri(Ind) = Rhs(Ind) + Defnew(Ic, Im) 10 Continue Ind = Ninter*Nmsh + Nlbc + 1 Call Dcopy(Ncomp-Nlbc, Rhs, 1, Rhstri, 1) Call Dssq ( Nmsh*Ncomp, Rhstri, 1, Scale, Sumsq ) Rnsq = (Scale**2)*Sumsq Iter = 0 * If The Initial Right-Hand Side Is Too Large, Do Not Even Attempt To * Solve The Nonlinear Equations. If (Rnsq .gt. Huge .or. * (Iorder .eq. 8 .and. Rnsq .gt. Xlarge)) Then If (Iprint .eq. 1) Write(6,902) Rnsq Iflag = -2 Return End If Call Dcopy(Ncomp*Nmsh, Rhstri, 1, Rhs, 1) * Statement 100 Is The Top Of The Iteration Loop. 100 Continue * If Rnsq Is Sufficiently Small, Terminate Immediately. If (Iprint .eq. 1) Write(6,903) Iter, Rnsq If (Rnsq .le. Epsmch) Then Iflag = 0 Return Endif Iter = Iter + 1 * Solve For The Step Delu By Solving A System Involving The Fixed * Jacobian (Whose Lu Factors Are Saved). Copy The Rhs Array Into * Tmprhs, Which Is Then Overwritten By Blkslv. Call Dcopy(Ncomp*Nmsh, Rhs, 1, Tmprhs, 1) Call Blkslv (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, Tmprhs, Delu) * Compute The Trial Point Utrial By Adding Delu To U. Call Matcop( Nudim, Ncomp, Ncomp, Nmsh, U, Utrial ) Call Maxpy( Ncomp, Nmsh, One, Delu, Ncomp, Utrial ) * Compute The Right-Hand Side Vector Rhstri And Its Squared * Two-Norm At The Trial Point. Rnold = Rnsq Call Fneval(Ncomp, Nmsh, Xx, Ncomp, Utrial, Fval, Eps) Call Rhscal (Ncomp, Nmsh, Nlbc, Xx, Ncomp, Utrial, Defcor, * Fsub, Gsub, Rhstri, Rnsq, Fval, Ftmp, Uint, Eps) * If Rnsq Strictly Decreased, Update The Solution Vector U * And The Right-Hand Side Rhs. Better = .false. If (Rnsq .lt. Rnold) Then Better = .true. Call Matcop( Ncomp, Nudim, Ncomp, Nmsh, Utrial, U ) Call Dcopy( Ncomp*Nmsh, Rhstri, 1, Rhs, 1 ) Endif * Stop The Fixed Jacobian Iterations If There Have Been Too * Many Iterations, Or If Rnsq Has Not Decreased By A Factor * Of At Least Rngrow. If (Iter .ge. Lmtfrz .or. Rnsq .gt. (Rnold/Rngrow)) Then If (Better) Then * Setting Iflag To -3 Signals That, Although The Fixed Jacobian * Iterations Did Not Succeed, The Current Point Was An Improvement * On The Previous One. Hence, If We Switch To A Newton Procedure, * The Right-Hand Side Does Not Need To Be Recalculated. Iflag = -3 Else Iflag = -2 Endif If (Iprint .eq. 1) Write(6,904) Iflag Return Endif * Test For Convergence Using The Ratio Abs((Change In U)/Max(U,1)). Do 150 Im = 1, Nmsh Do 150 It = 1, Ntol Itol = Ltol(It) Er = Abs(Delu(Itol,Im))/Max(Abs(U(Itol,Im)), One) If (Er .gt. Tolfct*Tol(It)) Go To 100 150 Continue * To Exit From The Loop Here, The Convergence Tests Have * Been Passed. If (Iprint .eq. 1) Write(6,905) Iter, Rnsq Iflag = 0 Return 901 Format(1h ,'Fixed Jacobian Iterations') 902 Format(1h ,'Large Residual, Rnsq = ',1pe12.4) 903 Format(1h ,'Iter, Rnsq',I5,1pe11.3) 904 Format(1h ,'Failure Of Fixed Jacobian, Iflag = ',I5) 905 Format(1h ,'Fixed Jacobian Convergence',I5,1pe11.3) End Subroutine Lineq( Ncomp, Nmsh, Nlbc, * Ludone, Xx, Nudim, U, Defcor, * Delu, Rhs, Fval, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, * Ajac, Topblk, Botblk, Bhold, Chold, Ipivot, * Fsub, Dfsub, Gsub, Dgsub, Iflag, Eps) Implicit Double Precision (A-H,O-Z) Dimension Xx(Nmsh), U(Nudim, Nmsh), Defcor(Ncomp, Nmsh-1) Dimension Delu(Ncomp, Nmsh), Rhs(Ncomp*Nmsh) Dimension Fval(Ncomp,Nmsh), Uint(Ncomp), Ftmp(Ncomp) Dimension Topblk(Ncomp,*), Botblk(Ncomp,*) Dimension Dftmp1(Ncomp, Ncomp), Dftmp2(Ncomp, Ncomp) Dimension Dgtm(Ncomp), Tmprhs(Ncomp*Nmsh) Dimension Ajac(Ncomp, 2*Ncomp, Nmsh-1), * Bhold(Ncomp, Ncomp, Nmsh-1), * Chold(Ncomp, Ncomp, Nmsh-1) Dimension Ipivot(Ncomp*Nmsh) Logical Ludone External Fsub, Dfsub, Gsub, Dgsub Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 * Blas: Dcopy, Dload Parameter ( One = 1.0d+0, Zero = 0.0d+0 ) * The Routine Lineq Calculates The Newton Step For A Linear * Problem. The Newton Step Is Exact Unless The Jacobian * Matrix Is Singular. Ninter = Nmsh - 1 If (.not. Ludone) Then * Compute The Right-Hand Side Vector Rhs. Call Lnrhs (Ncomp, Nmsh, Nlbc, Xx, Nudim, U, * Fsub, Gsub, Rhs, Rnsq, Fval, Ftmp, Uint, Eps) * If The Jacobian For This Mesh Has Not Previously Been * Calculated And Factorized Successfully, Call Jaccal. * The Block-Structured Jacobian Matrix Is Stored In Three * Matrices (Topblk, Ajac, And Botblk). * The Matrices Bhold And Chold Are Also Calculated In Jaccal, * And Are Saved For Later Use In Outer Routines. Call Jaccal (Ncomp, Nmsh, Nlbc, * Xx, Nudim, U, Fval, Dgtm, Dftmp1, Dftmp2, Uint, * Ajac, Topblk, Botblk, Bhold, Chold, * Dfsub, Dgsub, Eps) * Call Blkdcm To Calculate The Lu Factors Of The Jacobian. * The Factors Are Overwritten On The Matrices Topblk, Ajac And Botblk. * Interchanges Are Represented In The Integer Array Ipivot. Call Blkdcm (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, Iflfac) * If The Jacobian Is Considered Singular, Iflfac Will Be -1 On * Exit From Blkdcm. If (Iflfac .eq. (-1)) Then If (Iprint .eq. 1) Write(6,901) Iflag = -1 Ludone = .false. Return Endif Ludone = .true. * Copy The Rhs Into The Temporary Vector Tmprhs, Which Will Be * Overwritten By Blkslv. Call Dcopy(Ncomp*Nmsh, Rhs, 1, Tmprhs, 1) Else * The Right-Hand Side Is The Deferred Correction Array, * Padded With Zeros At The Boundary Conditions. Call Dload(Nlbc, Zero, Tmprhs(1), 1) Do 100 Im = 1, Ninter Loc = (Im-1)*Ncomp + Nlbc + 1 Call Dcopy(Ncomp, Defcor(1,Im), 1, Tmprhs(Loc), 1) 100 Continue Nrhs = Ninter*Ncomp + Nlbc + 1 Call Dload(Ncomp-Nlbc, Zero, Tmprhs(Nrhs), 1) Endif * Solve For The Newton Step Delu. Call Blkslv (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, Tmprhs, Delu) * Since The Problem Is Linear, The Newton Step Is Exact. The * New U Array Is Obtained By Adding Delu To U. Call Maxpy ( Ncomp, Nmsh, One, Delu, Nudim, U ) Iflag = 0 Return 901 Format(1h ,'Singular Matrix') End Subroutine Newteq(Ncomp, Nmsh, Nlbc, * Rhsgiv, Ntol, Ltol, Tol, * Xx, Nudim, U, Defcor, * Delu, Rhs, Fval, * Utrial, Rhstri, * Uint, Ftmp, Dftmp1, Dftmp2, Dgtm, Tmprhs, Xmerit, * Ajac, Topblk, Botblk, Bhold, Chold, Ipivot, * Fsub, Dfsub, Gsub, Dgsub, Iter, Iflag, Eps) Implicit Double Precision (A-H,O-Z) Dimension Ltol(*), Tol(*), Xx(*) Dimension Fval(Ncomp,*) Dimension U(Nudim, *), Delu(Ncomp, *), Utrial(Ncomp,Nmsh) Dimension Rhs(Ncomp*Nmsh), Defcor(Ncomp,*) Dimension Ajac(Ncomp, 2*Ncomp, *) Dimension Topblk(Ncomp,*), Botblk(Ncomp,*) Dimension Ftmp(*), Uint(*), Dgtm(Ncomp) Dimension Dftmp1(Ncomp, Ncomp), Dftmp2(Ncomp, Ncomp) Dimension Bhold(Ncomp, Ncomp, *), Chold(Ncomp, Ncomp, *) Dimension Ipivot(*) Dimension Rhstri(Ncomp*Nmsh) Dimension Tmprhs(Ncomp*Nmsh), Xmerit(Ncomp, Nmsh) Logical Rhsgiv External Fsub, Dfsub, Gsub, Dgsub Parameter ( Zero = 0.0d+0, One = 1.0d+0 ) Parameter ( Two = 2.0d+0 ) Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common/Mchprs/ Flmax, Epsmch Logical Gtpdeb, Imprvd, Braktd, Crampd, Extrap, Vset, Wset Save Gtpdeb, Mfsrch, Epsaf, Epsag, Rmu, Tolabs, Alfmax Save Tolrel, Toltny Intrinsic Abs, Max * Blas: Dcopy Logical Frscal Save Frscal Parameter (Cnvfct = 0.1d+0 ) Data Alfsml/1.0d-4/, Alfmax/1.1d+0/ Data Imerit/1/, Lmtnwt/39/ Data Shrfct/100.0d+0/, Stpfct/2.0d+0/ Data Gtpdeb/.false./, Mfsrch/5/ Data Rmu/1.0d-6/ Data Frscal/.true./ * The Routine Newteq Performs Newton Iterations With A Line * Search, To Solve The Nonlinear Equations. * Set Up Constants If This Is The First Call To Newteq. If (Frscal) Then Frscal = .false. Epsaf = Epsmch Epsag = Epsmch Tolabs = Epsmch Tolrel = Epsmch Toltny = Epsmch Endif Ninter = Nmsh - 1 If (Iprint .eq. 1) Write(6,901) * A Newton Method With Line Search And Watchdog Safeguarding * Is Used To Try To Solve The Nonlinear Equations. * Initialize Iter (The Counter Of Newton Iterations) And Alfold * (The Step Taken At The Previous Iteration). Iter = -1 Alfold = One Alfa = Zero Rnbest = Flmax If (.not. Rhsgiv) Then * If Necessary, Evaluate The Right-Hand Side At The Initial U. Call Rhscal (Ncomp, Nmsh, Nlbc, Xx, Nudim, U, Defcor, * Fsub, Gsub, Rhs, Rnsq, Fval, Ftmp, Uint, Eps) Endif * At Any Given Newton Iteration, Rnprev Is The Value Of Rnsq At * The Immediately Preceding Newton Iteration. Rnprev = Flmax If (Iprint .eq. 1) Write(6,902) * Initialize Counter Of Watchdog Iterations. Itwtch = 0 * Statement 100 Is The Top Of The Newton Iteration Loop. 100 Continue Iter = Iter + 1 If (Iprint .eq. 1) Write(6,910) Iter * If There Have Been Too Many Newton Iterations, Terminate. If (Iter .ge. Lmtnwt) Then If (Iprint .eq. 1) Write(6,903) Iflag = -2 Return Endif * The Vector Rhs Is The Right-Hand Side At The Current Iterate, * And Rnsq Is Its Squared Two-Norm. * Perform Watchdog Tests, Using The Unscaled Merit Function (Rnsq) * As The Watchdog Function. The Routine Wtchdg Updates Rnbest * And Itwtch. If Iflwat Is Not Zero, This Sequence Of Newton * Iterations Is Terminated. Call Wtchdg ( Iter, Rnsq, Rnbest, Rnprev, Itwtch, * Alfold, Iflwat ) If (Iflwat .ne. 0) Then If (Iprint .eq. 1) Write(6,904) Iter Iflag = -3 Return Endif * Watchdog Tests Are Passed. Proceed With The Newton Iteration. * Call Jaccal To Evaluate The Block-Structured Jacobian Matrix, * Which Is Stored In Three Matrices (Topblk, Ajac, And Botblk). * The Matrices Bhold And Chold Are Saved For Use In Later * Calculations In The Outer Routine. * If Rnsq Is Sufficiently Small, Terminate Immediately. * Note That The Stored Jacobian Does Not Correspond Exactly * To The Final Point. If (Rnsq .le. Epsmch) Then If (Iprint .eq. 1) Write(6,906) Iter, Rnsq Iflag = 0 Return Endif Call Jaccal (Ncomp, Nmsh, Nlbc, * Xx, Nudim, U, Fval, Dgtm, Dftmp1, Dftmp2, Uint, * Ajac, Topblk, Botblk, Bhold, Chold, * Dfsub, Dgsub, Eps) * Blkdcm Is Called To Calculate The Lu Factors Of The Jacobian, * Which Are Overwritten On Topblk, Ajac And Botblk. * Interchanges Are Represented In The Integer Array Ipivot. Call Blkdcm (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, Iflfac) * If The Jacobian Is Found To Be Singular, Iflfac Will Be -1 On * Exit From Blkdcm. If (Iflfac .eq. (-1)) Then If (Iprint .eq. 1) Write(6,905) Iter Iflag = -1 Return Endif * Solve For The Newton Step Delu. Copy The Rhs Array Into Tmprhs, * Which Is Then Overwritten By Blkslv. Call Dcopy(Ncomp*Nmsh, Rhs, 1, Tmprhs, 1) Call Blkslv (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, Tmprhs, Delu) * If Imerit = 1, The Line Search Is Based On The Scaled Merit Function, * The Squared Two-Norm Of The Solution Xmerit Of The Linear System * (Jacobian)*Xmerit = Rhs, * Where (Jacobian) Is The Jacobian At The Current Newton Iterate. * Thus The Initial Value Of The Scaled Merit Function Is Simply * The Squared Two-Norm Of The Newton Step Delu Itself. If (Imerit .eq. 1) Then Call Mssq( Ncomp, Nmsh, Delu, Xmscal, Xmsq ) Fmtry = (Xmscal**2)*Xmsq Else * The Unscaled Merit Function Is Simply The Squared Two-Norm Of Rhs. Fmtry = Rnsq End If C Fa And Oldg Represent The Merit Function And Its Gradient C At The Initial Point Of The Line Search. Fa = Fmtry Oldg = -Two*Fa Alfa = Zero If (Iprint .eq. 1) Write(6,908) Alfa, Fmtry, Rnsq * On The First Newton Iteration, The Initial Trial Step Is Unity. * On Subsequent Iterations, The Initial Step Is Not Allowed To * Be More Than The Factor Stpfct Larger Than The Final Step At * The Immediately Preceding Iteration. Alfa = One If(Stpfct*Alfold .lt. One) Alfa = Stpfct*Alfold If (Alfa .lt. Alfsml) Alfa = Alfsml Fmold = Fa Inform = -1 * Statement 150 Is The Top Of The Inner Line Search Iteration. * The Line Search Routine Getptq Has Been Altered So That It * Terminates With An Indication Of Success As Soon As A * Strictly Lower Value Of The Merit Function Is Found. Note That * This Is A Much Less Strict Requirement Than The Usual Sufficient * Decrease Conditions. 150 Continue Iwr = 6 Call Getptq (Gtpdeb, Mfsrch, Iwr, Alfmax, Alfsml, Alfuzz, * Epsaf, Epsag, * Fmtry, Fmold, Oldg, Rmu, Tolabs, Tolrel, Toltny, * Imprvd, Inform, Nfsrch, Alfa, Alfbst, Fbest, * Braktd, Crampd, Extrap, Vset, Wset, Nsamea, Nsameb, * Alin, Blin, Fa, Factor, Fv, Fw, Xtry, Xv, Xw) * Inform = 1, 2 Or 3 Indicates Success In Finding An Acceptable Point. * Inform = 4 Means Alfmax Is Too Small (This Should Never Happen Here, * Since Alfmax Is Set Always To 1.1). * Inform = 5 Means That A Decrease Was Not Achieved For Any Step * Greater Than Alfsml. * Inform = 6 Means A Better Point Could Not Be Found (The Minimum * Probably Lies Too Close To Alfa = 0). * Inform = 7 Means That The Gradient At Alfa = 0 (Oldg) Is Positive * (This Cannot Happen Here, Since Oldg = -Two*Fa, And Fa Is A Non-Negative * Number) If (Inform .eq. 5) Then Iflag = -5 Return Elseif (Inform .eq. 4 .or. Inform .eq. 7) Then Iflag = -4 Return Elseif (Inform .eq. 0) Then * Inform = 0 Means That A New Function Value Should Be Obtained * With The Step Alfa. * We May Override Alfa From Getptq By Requiring That The Step Is Not * Allowed To Decrease By More Than A Factor Of Shrfct During * A Line Search Iteration. * If (Alfa .lt. Alfold/Shrfct) Alfa = Alfold/Shrfct Alfold = Alfa * Define The Next Iterate Utrial = U + Alfa*Delu. * Call Fneval And Rhscal To Evaluate The Right-Hand Side * Rhstri At Utrial. * The Vector Rhstri Is Stored Separately, And Rhs Is Overwritten * Only When An Improved Point Is Found. Call Matcop ( Nudim, Ncomp, Ncomp, Nmsh, U, Utrial) Call Maxpy ( Ncomp, Nmsh, Alfa, Delu, Ncomp, Utrial ) Call Fneval(Ncomp, Nmsh, Xx, Ncomp, Utrial, Fval, Eps) Call Rhscal (Ncomp, Nmsh, Nlbc, Xx, Ncomp, Utrial, Defcor, * Fsub, Gsub, Rhstri, Rnsqtr, Fval, Ftmp, Uint, Eps) Fmold = Fmtry If (Imerit .eq. 1) Then * Solve A Linear System To Obtain The 2-D Array Xmerit Whose Squared * Norm Is The Scaled Merit Function. The Lu Factors Of The Jacobian * Have Already Been Calculated By Blkdcm. * Copy Rhstri Into Tmprhs, Which Is Overwritten By Blkslv. Call Dcopy(Ncomp*Nmsh, Rhstri, 1, Tmprhs, 1) Call Blkslv (Nlbc, Ncomp-Nlbc, Ncomp, Ninter, * Topblk, Ajac, Botblk, Ipivot, * Tmprhs, Xmerit) Call Mssq( Ncomp, Nmsh, Xmerit, Xscale, Xsolsq ) Fmtry = (Xscale**2)*Xsolsq Else * The Unscaled Merit Function Is The Squared Two-Norm Of The Right-Hand * Side. Fmtry = Rnsqtr End If If (Iprint .eq. 1) Write(6,908) Alfa, Fmtry, Rnsqtr Go To 150 Endif * To Reach Here, Inform Must Be 1, 2, 3, Or 6, And The Line Search * Has Found A Strictly Lower Value Of The Merit Function. * Store The New Newton Iterate In U, And The Corresponding Rhs * Vector In Rhs. Rnprev = Rnsq Rnsq = Rnsqtr Call Matcop (Ncomp, Nudim, Ncomp, Nmsh, Utrial, U) Call Dcopy(Ncomp*Nmsh, Rhstri, 1, Rhs, 1) If (Iprint .eq. 1) Write(6,909) Iter, Alfa, Fmtry, Rnsq * Now Test For Convergence Using The Ratio Of The Newton Step * For Each Component With Max(1, Abs(Current Solution Estimate)). * If The Test Fails For Any Element Of U, Branch Back To The * Top Of The Newton Iteration. Do 160 Im = 1, Nmsh Do 160 It = 1, Ntol Icmp = Ltol(It) Er = Abs(Delu(Icmp,Im))/Max(Abs(U(Icmp,Im)), One) If (Er .gt. Cnvfct*Tol(It)) Go To 100 160 Continue If (Iprint .eq. 1) Write(6, 906) Iter+1, Rnsq Iflag = 0 * To Fall Through The Above Loop, The Termination Test For A * Sufficiently Small Delu Is Satisfied. * Note That The Stored Jacobian And Its Factorization Do Not * Correspond To The Final Solution. Return 901 Format(1h ,'Start Newton Iterations') 902 Format(1h ,' Iter', * 7x,'Alfa',6x,'Merit',7x,'Rnsq') 903 Format(1h ,'Too Many Newton Iterations') 904 Format(1h ,'Watchdog Tests Fail, Iter = ', I5) 905 Format(1h ,'Singular Jacobian, Iter = ',I5) 906 Format(1h ,'Convergence, Iter = ',I5,4x,'Rnsq = ',1pe12.3,/) 908 Format(1h ,'Alfa, Merit, Rnsq',3(1pe11.3)) 909 Format(1h ,I5,3(1pe11.3)) 910 Format(1h ,'Newton Iteration',I5) End Subroutine Wtchdg ( Iter, Wmerit, Wmbest, Wmprev, * Itwtch, Alfold, Iflag ) * Logic For Watchdog Tests. Implicit Double Precision (A-H,O-Z) Parameter ( Itonew = 5, Itwtmx = 8, Grfct = 100.0d+0 ) Parameter ( Half = 0.5d+0 ) * Perform Watchdog Tests In Two Forms: * (1) To Determine Whether A Sufficient Decrease In The * Watchdog Merit Function Has Occurred Within The Most Recent * Sequence Of Itwtmx Iterations; * (2) To Determine Whether The Watchdog Merit Function Has Increased * Too Much In A Single Iteration After Itonew Newton Iterations * Have Been Performed. This Allows The Merit Function To Increase * Wildly Only During The First Itonew Iterations. * Wmbest Is The Smallest Watchdog Merit Function Achieved In This * Sequence Of Newton Iterations. * Wmprev Is The Watchdog Merit Function From The Immediately * Preceding Newton Iteration. * Itwtch Counts The Number Of Iterations Without An Improvement * In The Unscaled Merit Function. Iflag = 0 If (Wmerit .le. Wmbest) Then * The Current Watchdog Merit Function Is The Best. Wmbest = Wmerit Itwtch = 0 Return Endif * The Current Merit Function Is Not The Best. Itwtch = Itwtch + 1 * Do Not Apply Watchdog Tests If (1) The Previous Step Alfold * Exceeds 1/2, Or (2) The Watchdog Merit Function Decreased In * The Immediately Preceding Iteration And Itwtch Does Not * Exceed Twice Its Maximum. If (Alfold .ge. Half) Return If (Wmerit .le. Wmprev .and. Itwtch .le. 2*Itwtmx) Return * If More Than Itwtmx Iterations Have Occurred Without * An Overall Improvement In The Watchdog Merit Function, * Signal For Termination. If (Itwtch .ge. Itwtmx) Then Iflag = -1 * If A Too-Large Increase In The Watchdog Merit Function * Compared To The Best Value Occurred, And Iter .ge. Itonew, * Signal For Termination. Elseif (Iter .ge. Itonew .and. * Wmerit .gt. Grfct*Wmbest) Then Iflag = -1 Endif Return End * File Evals.f Subroutine Fneval(Ncomp, Nmsh, Xx, Nudim, U, Fval, Eps) Implicit Double Precision (A-H,O-Z) Dimension Xx(Nmsh), U(Nudim,Nmsh), Fval(Ncomp,Nmsh) * Fneval Evaluates The Function Values (From Fsub) For * A Given Mesh Xx And Array U, And Stores The Values * In The Array Fval. Call Fsub (Ncomp, Xx(1), U(1,1), Fval(1,1), Eps) Do 50 Im = 1, Nmsh-1 Call Fsub (Ncomp, Xx(Im+1), U(1,Im+1), Fval(1,Im+1), Eps) 50 Continue Return End Subroutine Jaccal (Ncomp, Nmsh, Nlbc, * Xx, Nudim, U, Fval, * Dgtm, Dftm1, Dftm2, Uint, * Ajac, Topblk, Botblk, Bhold, Chold, * Dfsub, Dgsub, Eps) Implicit Double Precision (A-H,O-Z) Dimension Xx(Nmsh), U(Nudim,Nmsh), Fval(Ncomp,Nmsh) Dimension Dgtm(Ncomp) Dimension Dftm1(Ncomp, Ncomp), Dftm2(Ncomp, Ncomp), * Uint(Ncomp) Dimension Ajac(Ncomp, 2*Ncomp, Nmsh-1) Dimension Topblk(Ncomp, Ncomp), Botblk(Ncomp,Ncomp) Dimension Bhold(Ncomp, Ncomp, Nmsh-1), * Chold(Ncomp, Ncomp, Nmsh-1) External Dfsub, Dgsub * Blas: Dcopy, Ddot Parameter ( Half = 0.5d+0, Eighth = 0.125d+0 ) Parameter ( Six = 6.0d+0 ) Parameter ( One = 1.0d+0, Three = 3.0d+0, Twelve = 12.0d+0 ) Ninter = Nmsh - 1 Do 110 I = 1, Nlbc Call Dgsub (I, Ncomp, U(1,1), Dgtm, Eps) Call Dcopy(Ncomp, Dgtm(1), 1, Topblk(I,1), Ncomp) 110 Continue Call Dfsub (Ncomp, Xx(1), U(1,1), Dftm1(1,1), Eps) * On Entry To Jaccal, The Array Fval Contains The Function Values * At (Xx(Im), U(Ic,Im)), Ic = 1,...,Ncomp And Im = 1,...,Nmsh, * Calculated By A Preceding Call Of Rhscal With The Same Xx And U * Arrays. Do 200 Im = 1, Ninter Hmsh = Xx(Im+1) - Xx(Im) Do 120 Ic = 1, Ncomp Uint(Ic) = Half*(U(Ic,Im) + U(Ic,Im+1)) * - Eighth*Hmsh*(Fval(Ic,Im+1) - Fval(Ic,Im)) 120 Continue Xhalf = Half*(Xx(Im+1) + Xx(Im)) Call Dfsub (Ncomp, Xhalf, Uint, Dftm2(1,1), Eps) Do 140 Ic = 1, Ncomp Do 130 Jc = 1, Ncomp Dsq = Ddot(Ncomp, Dftm2(Ic,1), Ncomp, * Dftm1(1,Jc), 1) Ajac(Ic,Jc,Im) = -Hmsh*(Dftm1(Ic,Jc)/Six * + Dftm2(Ic,Jc)/Three + Hmsh*Dsq/Twelve) 130 Continue Ajac(Ic,Ic,Im) = Ajac(Ic,Ic,Im) - One 140 Continue Call Dfsub (Ncomp, Xx(Im+1), U(1,Im+1), Dftm1(1,1), Eps) Do 170 Ic = 1, Ncomp Do 160 Jc = 1, Ncomp Dsq = Ddot(Ncomp, Dftm2(Ic,1), Ncomp, * Dftm1(1,Jc), 1) Ajac(Ic,Jc+Ncomp,Im) = -Hmsh*(Dftm1(Ic,Jc)/Six * + Dftm2(Ic,Jc)/Three - Hmsh*Dsq/Twelve) 160 Continue Call Dcopy(Ncomp, Ajac(Ic,Ncomp+1,Im), Ncomp, * Chold(Ic,1,Im), Ncomp) Call Dcopy(Ncomp, Dftm1(Ic,1), Ncomp, * Bhold(Ic,1,Im), Ncomp) Ajac(Ic,Ic+Ncomp,Im) = Ajac(Ic,Ic+Ncomp,Im) + One Chold(Ic,Ic,Im) = Ajac(Ic,Ic+Ncomp,Im) 170 Continue 200 Continue Do 220 I = Nlbc+1, Ncomp Call Dgsub (I, Ncomp, U(1, Nmsh), Dgtm, Eps) Call Dcopy(Ncomp, Dgtm(1), 1, Botblk(I-Nlbc,1), Ncomp) 220 Continue Return End Subroutine Lnrhs (Ncomp, Nmsh, Nlbc, * Xx, Nudim, U, Fsub, Gsub, * Rhs, Rnsq, Fval, Ftmp, Uint, Eps) Implicit Double Precision(A-H,O-Z) * This Subroutine Is Designed To Calculate The Right-Hand * Side For Linear Problems. Dimension Xx(*), U(Nudim,*) Dimension Rhs(*), Fval(Ncomp,*), Ftmp(*), Uint(*) External Fsub, Gsub Parameter ( Zero = 0.0d+0, Half = 0.5d+0, Eighth = 0.125d+0 ) Parameter ( Four = 4.0d+0, Six = 6.0d+0 ) Intrinsic Abs * Blas: Dssq Ninter = Nmsh - 1 Rnsq = Zero * First, Process The Left-Hand Boundary Conditions. Do 20 I = 1, Nlbc Call Gsub (I, Ncomp, U(1,1), Wg, Eps) Rhs(I) = -Wg 20 Continue * Next, Process The Interior Mesh Points. The Fval Array * Contains The Function Values From Fsub At Xx And U. Do 50 Im = 1, Ninter Hmsh = Xx(Im+1) - Xx(Im) Do 30 Ic = 1, Ncomp Uint(Ic) = Half*(U(Ic,Im) + U(Ic,Im+1)) * - Eighth*Hmsh*(Fval(Ic,Im+1) - Fval(Ic,Im)) 30 Continue Xhalf = Half*(Xx(Im) + Xx(Im+1)) Call Fsub (Ncomp, Xhalf, Uint, Ftmp, Eps) Loc = (Im-1)*Ncomp + Nlbc Do 40 Ic = 1, Ncomp Rhs(Loc+Ic) = -U(Ic,Im+1) + U(Ic,Im) * + Hmsh* * (Fval(Ic,Im) + Fval(Ic,Im+1) + Four*Ftmp(Ic))/Six 40 Continue 50 Continue Nrhs = Ninter*Ncomp Do 60 Ii = Nlbc+1, Ncomp Call Gsub (Ii, Ncomp, U(1,Nmsh), Wg, Eps) Rhs(Nrhs+Ii) = -Wg 60 Continue Call Dssq ( Nmsh*Ncomp, Rhs, 1, Scale, Sumsq ) Rnsq = (Scale**2)*Sumsq Return End Subroutine Rhscal (Ncomp, Nmsh, Nlbc, * Xx, Nudim, U, Defcor, * Fsub, Gsub, * Rhs, Rnsq, Fval, Ftmp, Uint, Eps) Implicit Double Precision(A-H,O-Z) * This Subroutine Constructs The (Ncomp*Nmsh)-Dimensional * Vector Rhs, Which Is The Right-Hand Side Of The Newton Equations. * The Ncomp By Nmsh Array Fval Is Assumed To Have Been Calculated * Elsewhere By Routine Fneval. Dimension Xx(Nmsh), U(Nudim,Nmsh), Defcor(Ncomp,Nmsh-1) Dimension Rhs(Ncomp*Nmsh), Fval(Ncomp,Nmsh) Dimension Ftmp(Ncomp), Uint(Ncomp) External Fsub, Gsub Intrinsic Abs * Blas: Dssq Parameter ( Zero = 0.0d+0, Half = 0.5d+0, Eighth = 0.125d+0 ) Parameter ( Four = 4.0d+0, Six = 6.0d+0 ) * Ninter Is The Number Of Intervals In The Mesh (One Less Than The * Number Of Mesh Points) Ninter = Nmsh - 1 Rnsq = Zero * First, Process The Left-Hand Boundary Conditions. Do 20 I = 1, Nlbc Call Gsub (I, Ncomp, U(1,1), Wg, Eps) Rhs(I) = -Wg 20 Continue * Next, Process The Interior Mesh Points. The Fval Array * Contains The Function Values From Fsub At Xx And U. Do 50 Im = 1, Ninter Hmsh = Xx(Im+1) - Xx(Im) Do 30 Ic = 1, Ncomp Uint(Ic) = Half*(U(Ic,Im) + U(Ic,Im+1)) * - Eighth*Hmsh*(Fval(Ic,Im+1) - Fval(Ic,Im)) 30 Continue Xhalf = Half*(Xx(Im) + Xx(Im+1)) Call Fsub (Ncomp, Xhalf, Uint, Ftmp, Eps) Loc = (Im-1)*Ncomp + Nlbc Do 40 Ic = 1, Ncomp Rhs(Loc+Ic) = -U(Ic,Im+1) + U(Ic,Im) + Defcor(Ic,Im) * + Hmsh* * (Fval(Ic,Im) + Fval(Ic,Im+1) + Four*Ftmp(Ic))/Six 40 Continue 50 Continue Nrhs = Ninter*Ncomp Do 60 Ii = Nlbc+1, Ncomp Call Gsub (Ii, Ncomp, U(1,Nmsh), Wg, Eps) Rhs(Nrhs+Ii) = -Wg 60 Continue Call Dssq ( Nmsh*Ncomp, Rhs, 1, Scale, Sumsq ) Rnsq = (Scale**2)*Sumsq Return End * File Msh.f Subroutine Dblmsh (Nmsh, Nmax, Xx, Nmold, Xxold, Maxmsh) Implicit Double Precision (A-H,O-Z) Dimension Xx(*), Xxold(*) Logical Maxmsh Common/Algprs/ Nminit, Iprint, Maxcon, Itsaim, Uval0 Common /Flags/ Ifinal,Iatt,Iback,Iprec * Blas: Dcopy Parameter (Half = 0.5d+0) * This Routine Is Used To Double The Mesh, I.e., Produce A Mesh * With Twice As Many Intervals In Which Each New Interval Is * Half The Corresponding Old Interval. * On Entry To Dblmsh, The Integer Nmsh And The Array Xx * Specify A Set Of Mesh Points Xx(1),..., Xx(Nmsh) (Assumed * To Be In Ascending Order). * If The Number Of Mesh Points In The Doubled Mesh Would * Exceed The Maximum Allowed Number Nmax, The Flag Maxmsh Is * Set To True, And We Exit Without Changing Any Other Parameters. * Otherwise, Nmold Is Set To The Old Number Of Mesh Points, * Xxold Is Set To The Old Mesh, Nmsh Is The New Number Of Mesh * Points, And Xx Contains The New Mesh Points. Nmold = Nmsh Iprec = Min(Iprec,1) Call Dcopy(Nmold, Xx, 1, Xxold, 1) Ninnew = 2*(Nmsh-1) Nmnew = Ninnew + 1 If(Nmnew .ge. Nmax) Then If (Iprint .ge. 0) Write(6,901) Nmnew Maxmsh = .true. Return Endif Maxmsh = .false. * Loop Backwards Through The Old Mesh Points To Create The New Ones. Xx(Nmnew) = Xx(Nmsh) Do 100 I = Ninnew, 4, -2 Id2 = I/2 Xx(I) = Half*(Xx(I+1) + Xx(Id2)) Xx(I-1) = Xx(Id2) If (Xx(I) .eq. Xx(I+1) .or. Xx(I) .eq. Xx(I-1)) Then Iprec = 2 Return Endif 100 Continue * Calculate The New Xx(2). Xx(1) Remains Unchanged. Xx(2) = Half*(Xx(3) + Xx(1)) If (Xx(2) .eq. Xx(3) .or. Xx(2) .eq. Xx(1)) Then Iprec = 2 Return Endif Nmsh = Nmnew Return 901 Format (1h , ' Dblmsh. Maximum Mesh Exceeded, Nmnew = ', I8) End Subroutine Selmsh(Ncomp, Nmsh, Ntol, Ltol, Tol, * Nfxpnt, Fixpnt, Ipow, Nmax, Nvold, * Xx, Nudim, U, Ermeas, Irefin, Ihcomp, * Nmold, Xxold, Ermx, Double, Maxmsh, Phiold, Voldmsh) Implicit Double Precision (A-H,O-Z) Dimension Ltol(Ntol), Tol(Ntol), Fixpnt(*) Dimension Xx(*), U(Nudim, *), Ermeas(Ncomp,*) Dimension Irefin(Nmsh-1), Ihcomp(Nmsh-1) Dimension Xxold(*), Ermx(*), Phiold(*), Voldmsh(*) Dimension Npr(2), Pmax(2), Hord(2) Logical Double, Maxmsh Intrinsic Abs, Max, Int * Blas: Dcopy Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Mshvar/ Hsml,Npr,Pmax,Hord Parameter ( Zero = 0.0d+0, One = 1.0d+0, Onep1 = 1.1d+0 ) Parameter ( Erdcid = 5.0d+0 ) Parameter ( Phitst = 0.1d+0 ) * The Routine Selmsh Performs Selective Mesh Refinement, Depending * On The Error Measure Ermeas. Maxmsh = .false. Frcpow = One/Ipow Double = .false. Nmold = Nmsh Ninter = Nmsh - 1 Iprec = Min(Iprec,1) * Copy The Current Mesh Into The Xxold Array. If (Iatt .eq. -1) Nvold = Nmsh If (Iatt .eq. 0) Call Dcopy(Nvold, Xxold, 1, Voldmsh, 1) Call Dcopy(Nmold, Xx, 1, Xxold, 1) Ithres = 0 Thres = One * On Input, The Array Ermeas Represents Some Error Measure Defined * Over The Components And Mesh Intervals (Not Mesh Points). * It Is Normalized In The Following Loop With Respect To The * Tolerance Array And The Current Solution. * The Value Errmax Gives The Maximum Normalized Error. Errmax = Zero Do 120 Im = 1, Ninter Ermx(Im) = Zero Do 110 It = 1, Ntol Jcomp = Ltol(It) Umin = Min(Abs(U(Jcomp,Im)),Abs(U(Jcomp,Im+1))) Denom = Tol(It)*Max(One, Umin) Ems = Ermeas(Jcomp,Im) Ermeas(Jcomp,Im) = Abs(Ems)/Denom Err = Ermeas(Jcomp, Im) If (Err .ge. Ermx(Im)) Then Ermx(Im) = Err Ihcomp(Im) = Jcomp Endif 110 Continue Errmax = Max(Ermx(Im), Errmax) 120 Continue 200 Continue * For Each Interval Im, The Integer Irefin(Im) Is Calculated * Based On An Equidistribution Procedure Involving The * Threshold Value Thres. If Irefin(Im) > 1, We Add * Points To Interval Im. If Irefin(Im) = 1, We Check Whether * Point Im Can Be Removed From The Mesh. * Nmest Is A Lower Bound On The Number Of Points In The New Mesh. * We Do Not Know In Advance How Many Points Will Be Removed, * So Nmest Is Computed By Assuming All Eligible Points Are Removed. Philrg = 0.d0 Esum = 0.d0 Nmest = Nmsh Mshchng = 0 Do 220 Im = 1, Ninter Errim = Ermx(Im)**Frcpow If (Ermx(Im) .ge. Thres) Then Irefin(Im) = Int(Errim) + 1 Nmest = Nmest + Irefin(Im) - 1 Mshchng = 1 Else Irefin(Im) = 1 Nmest = Nmest - 1 Endif Him = Xxold(Im+1)-Xxold(Im) Phiim = Errim/Him Esum = Esum+Errim If (Iatt .eq. -1) Then Phiold(Im) = Phiim Elseif (Iatt .eq. 0 .and. Phiim .gt. Philrg) Then Philrg = Phiim Hordlrg = Errim Imreg = Im Endif 220 Continue C If (Iatt .eq. 0) Then Hord(2) = Hordlrg Pmax(2) = Philrg Xloc1 = Xxold(Imreg) Xloc2 = Xxold(Imreg+1) Ichkpt = Nvold/2 If (Xloc1 .lt. Voldmsh(Ichkpt)) Ichkpt = 1 Xb = Voldmsh(Ichkpt) Do 225 J = Ichkpt, Nvold-1 Xa = Xb Xb = Voldmsh(J+1) If (Xloc1 .ge. Xa .and. Xloc1 .lt. Xb) Then If (Xloc2-Xb .lt. (Xloc2-Xloc1)/2.d0) Then Pmax(1) = Phiold(J) Hord(1) = (Xb-Xa)*Pmax(1) Else Pmax(1) = Phiold(J+1) Hord(1) = (Voldmsh(J+2)-Xb)*Pmax(1) Endif Goto 228 Endif 225 Continue 228 Hsml = (Xloc2-Xloc1)/Dble(Irefin(Imreg)) Npr(2) = Int(Esum) Endif If (Iatt .eq. -1) Npr(1) = Int(Esum) If (Ifinal .eq. 1 .and. Mshchng .eq. 0) Iatt = 0 If (Nmest .gt. Nmax) Go To 360 * It Appears That We Can Perform The Desired Selective Mesh * Refinement. * Now Begin Running Through The Mesh, Adding And Possibly Deleting * Points As Indicated By The Irefin Array. * The Integer New Is A Count Of The Number Of Intervals In * The Tentative Mesh Being Generated By The Refinement Strategy. New = 1 * The First Interval Is Treated As A Special Case, Since Xx(1) * Always Remains In The Mesh, And Cannot Be A Fixed Point. Rlen = Xxold(2) - Xx(1) Slen = Rlen If (Irefin(1) .gt. 1) Then Dx = Rlen/Irefin(1) Do 230 J = 2, Irefin(1) New = New + 1 Xx(New) = Xx(1) + (J-1)*Dx If (Xx(New) .eq. Xx(New-1)) Then Iprec = 2 Return Endif 230 Continue Endif * The Fixed Points Specified By The Fixpnt Array Cannot Be * Removed From The Mesh. The Value Fxnext Indicates The 'Next' * Fixed Point. When No Further Fixed Points Remain To Be Processed * (Or If Nfxpnt = 0), Fxnext Is Set To A Value Strictly Larger Than * The Last Mesh Point, So That No Mesh Point Can Equal Fxnext. * This Way We Need To Compare Only The Values Of Xxold(I) * And Fxnext. Ifxcnt = 1 If (Nfxpnt .eq. 0) Then Fxnext = Onep1*Abs(Xxold(Nmsh)) Else Fxnext = Fixpnt(Ifxcnt) Endif * Jtkout Is A Counter Of The Number Of Consecutive Points That * Have Been Removed From The Mesh. Jtkout = 0 Do 330 Im = 2, Ninter Rlold = Rlen Rlen = Xxold(Im+1) - Xxold(Im) * If Xxold(Im) Is The Next Fixed Point, It Cannot Be Removed * And So We Don'T Test Its Error Estimates. If(Xxold(Im) .eq. Fxnext) Then Ifxcnt = Ifxcnt + 1 If(Ifxcnt .gt. Nfxpnt) Then Fxnext = Onep1*Abs(Xxold(Nmsh)) Else Fxnext = Fixpnt(Ifxcnt) Endif Elseif (Irefin(Im) .eq. 1) Then * If Xxold(Im) Is Not A Fixed Point And Irefin(Im) = 1, We May Wish * To Remove Point Im From The Mesh. * If We Are Considering Removing Points And Jtkout = 0, This * Is The First Point In A Possible Consecutive Set To Be Removed, * And We Initialize Phihat, Which Represents A Maximum Of * Certain Estimates. * If Jtkout Is Not Zero, Previous Points Contiguous To This * Point Have Been Removed, And Phihat Retains Its Previous Value. Slen = Slen + Rlen If (Jtkout .eq. 0) Then Ind1 = Ihcomp(Im-1) Phihat = Ermeas(Ind1,Im-1)/(Rlold**Ipow) Endif Phihat = Max(Phihat, * Ermeas(Ihcomp(Im),Im)/(Rlen**Ipow)) Val1 = Phihat*(Slen**Ipow) If (Val1 .le. Phitst * .and. Jtkout .lt. 4) Then * Increment The Counter Of Removed Points. * 'Remove' The Mesh Point Xxold(Im) By Not Including It. Jtkout = Jtkout+1 Go To 330 Endif * End Of Logic For Irefin(Im) = 1. Endif Jtkout = 0 New = New + 1 Xx(New) = Xxold(Im) If (Xx(New) .eq. Xx(New-1)) Then Iprec = 2 Return Endif If (Irefin(Im) .gt. 1) Then Dx = Rlen/Irefin(Im) Do 300 J = 2, Irefin(Im) New = New + 1 Xx(New) = Xxold(Im) + (J-1)*Dx If (Xx(New) .eq. Xx(New-1)) Then Iprec = 2 Return Endif 300 Continue Endif Slen = Rlen * If The New Mesh Contains Too Many Points, Branch Out Of The * Loop To Try Alternative Strategies. If (New .gt. Nmax-1) Goto 360 330 Continue * To End Up Here, We Have Processed The Entire Interval, * And Have Neither Exceeded The Specified Maximum Nor * Exceeded Three Times The Number Of Intervals In The Old * Mesh. The Last Mesh Point Remains Unchanged. New = New + 1 Xx(New) = Xxold(Nmsh) If (Xx(New) .eq. Xx(New-1)) Then Iprec = 2 Return Endif Nmsh = New Maxmsh = .false. Return 360 Continue * To Reach Here, The Number Of Mesh Points Created At Some Stage * Of The Refinement Process Was Larger Than The Maximum Permitted * Value Nmax. * Check Whether The Mesh Can Safely Be Doubled. If ((2*Nmsh-1) .lt. Nmax) Then * Double The Mesh. Call Dblmsh (Nmsh, Nmax, Xx, Nmold, Xxold, Maxmsh) Double = .true. * If The Number Of Intervals Is Too Large And The Mesh Cannot Be * Doubled, Increase The Threshold Thres By A Factor Of Erdcid And * Try The Selective Refinement Again. * If This Happens Three Times Without Success Or If Thres Exceeds * Or Is Equal To Errmax, Stop. (In This Case, We Know Already * That Doubling The Mesh Produces Too Many Points.) Elseif (Thres .lt. Errmax .and. Ithres .lt. 3) Then Ithres = Ithres + 1 Thres = Erdcid*Thres If(Thres .gt. Errmax) Thres = Errmax Call Dcopy(Nmsh, Xxold, 1, Xx, 1) Go To 200 Else Nmsh = 2*Nmsh - 1 Maxmsh = .true. Endif Return 904 Format(1h ,'Nmest, Irefin',(10i5)) 910 Format(1h ,'Ihcomp',(10i5)) End Subroutine Unimsh(Nmsh, Aleft, Aright, Nfxpnt, Fixpnt, Xx) Implicit Double Precision (A-H,O-Z) Integer Nmsh, Nfxpnt Dimension Fixpnt(*), Xx(Nmsh) Intrinsic Max * Given A Left Endpoint Aleft, A Right Endpoint Aright, * A Set Of Nfxpnt Fixed Points Fixpnt(I), I = 1,...,Nfxpnt, * (Where Fixpnt(I) Is Different From Aleft And Aright For All I), * And An Initial Target Number Nmsh Of Mesh Points, * The Subroutine Unimsh Generates A Piecewise Uniform Mesh * Beginning At Aleft, Ending At Aright, And With Equally * Spaced Points Between Aleft And Fixpnt(1), Then Between * Fixpnt(1) And Fixpnt(2), ..., And Finally Between * Fixpnt(Nfxpnt) And Aright. The Final Number Of Intervals * Is The Maximum Of Nfxpnt+2 And The Initial Value Of Nmsh. * In The Simplest Case When Nfxpnt = 0, Unimsh Generates A * Uniform Mesh With Nmsh Intervals In The Closed Interval * (Aleft, Aright). * On Exit, The Integer Nmsh Contains The Number Of Mesh Points * (Which Is The Maximum Of The Initial Nmsh And Nfxpnt). * The Array Xx (Of Dimension Nmsh) Contains The Mesh Points. If (Nfxpnt .eq. 0) Then * If There Are No Interior Fixed Points, The Spacing Is Uniform * Throughout The Interval. Calculate The Spacing Dx * And Set Up The Xx Array. Ninter = Nmsh - 1 Dx = (Aright - Aleft)/Ninter Do 10 I = 1, Ninter Xx(I) = Aleft + (I-1)*Dx 10 Continue Xx(Nmsh) = Aright Return Endif * We Know That There Is At Least One Fixed Point Strictly Between * The Endpoints. If (Nmsh .lt. Nfxpnt+2) Nmsh = Nfxpnt + 2 Ninter = Nmsh - 1 Xx(1) = Aleft Ileft = 1 Xleft = Aleft Totint = Aright - Aleft Ndif = Ninter - Nfxpnt Do 50 J = 1, Nfxpnt + 1 * Deal In Turn With The Subintervals Defined By The Interval * Boundaries And The Fixed Points. If (J .lt. Nfxpnt+1) Then * The J-Th Fixed Point Is Xright. Calculate Where It Should * Fall In The Mesh. Xright = Fixpnt(J) Nmin = Int(Ninter*(Xright-Aleft)/Totint + 1.5d+0) If (Nmin .gt. Ndif+J) Nmin = Ndif + J Iright = Max(Ileft+1, Nmin) Else Xright = Aright Iright = Nmsh Endif * Npt Is The Number Of Equally Spaced Points That Should * Lie Strictly Between The (J-1)-Th And J-Th Fixed Points. Xx(Iright) = Xright Npt = Iright - Ileft - 1 Dx = (Xright - Xleft)/(Npt + 1) Do 30 I = 1, Npt Xx(Ileft+I) = Xleft + I*Dx 30 Continue Ileft = Iright Xleft = Xright 50 Continue Return C End * File Rest.f Subroutine Errest (Ncomp, Nmsh, Ntol, Ltol, Tol, * Nudim, U, Uold, Etest, Errok) Implicit Double Precision (A-H,O-Z) Dimension Ltol(Ntol), Tol(Ntol), U(Nudim,Nmsh), * Uold(Ncomp,Nmsh), Etest(Ntol) Logical Errok Intrinsic Abs, Max Parameter ( One = 1.0d+0 ) * Given Current And Previous Solutions U And Uold On The Same * Mesh, Errest Calculates An Error Measure For Each * Component For Which A Tolerance Is Specified. * The Error Measure Is The Usual Relative Error Normalized * By Dividing By The Tolerance. * The Array Etest Specifies The Error Test To Be Applied To Each * Error Measure. * On Exit, The Logical Flag Errok * -- Is .false. If Any Of The Error Measures Exceeds The * Corresponding Value In The Array Etest * -- Is .true. If All Error Measures Are Less Than The * Corresponding Values Of Etest. Errok = .true. Do 10 Im = 1, Nmsh Do 10 It = 1, Ntol Icmp = Ltol(It) Er = U(Icmp,Im) - Uold(Icmp,Im) Denom = Max(One, Abs(Uold(Icmp,Im))) Errel = Abs(Er/(Tol(It)*Denom)) If (Errel .gt. Etest(It)) Errok = .false. 10 Continue Return End Subroutine Getptq( Debug, Mfsrch, Nout, Alfmax, Alfsml, Alfuzz, * Epsaf, Epsag, Ftry, Oldf, Oldg, * Rmu, Tolabs, Tolrel, Toltny, Imprvd, * Inform, Nfsrch, Alfa, Alfbst, Fbest, * Braktd, Crampd, Extrap,Vset,Wset,Nsamea,Nsameb, * A, B, Fa, Factor, Fv, Fw, Xtry, Xv, Xw ) Implicit Double Precision (A-H,O-Z) Logical Debug, Imprvd Logical Braktd, Crampd, Extrap, Vset, Wset Integer Mfsrch, Nout, Inform, Nfsrch, Nsamea, Nsameb C C ********************************************************************* C Getptq Is A Step-Length Algorithm For Minimizing A Function Of One C Variable. It Will Be Called Repeatedly By A Search Routine Whose C Purpose Is To Estimate A Point Alfa = Alfbst That Minimizes Some C Function F(Alfa) Over The Closed Interval (0, Alfmax). C C Getptq Requires The Function F(Alfa) (But Not Its Gradient) C To Be Evaluated At Various Points Within The Interval. New C Step-Length Estimates Are Computed Using Quadratic Interpolation With C Safeguards. C C Reverse Communication Is Used To Allow The Calling Program To C Evaluate F. Some Of The Parameters Must Be Set Or Tested C By The Calling Program. The Remainder Would Ordinarily Be Local C Variables. C C C Input Parameters (Relevant To The Calling Program) C -------------------------------------------------- C C Debug Specifies Whether Detailed Output Is Wanted. C C Inform Must Be Nonzero On The First Entry (E.g., -1). C It Will Be Altered By Getptq For Later Entries. C C Mfsrch Is An Upper Limit On The Number Of Times Getptq Is C To Be Entered Consecutively With Inform = 0 C (Following An Initial Entry With Inform Lt 0). C C Nout Is The File Number To Be Used For Printed Output C If Debug Is True. C C Alfa Is The First Estimate Of The Step Length. Alfa Is C Subsequently Altered By Getptq (See Below). C C Alfmax Is The Upper Limit Of The Interval To Be Searched. C C Alfsml Is Intended To Prevent Inefficiency When The Optimum C Step Is Very Small, For Cases Where The Calling C Program Would Prefer To Re-Define F(Alfa). Alfsml Is C Allowed To Be Zero. Early Termination Will Occur If C Getptq Determines That The Optimum Step Lies C Somewhere In The Interval (0, Alfsml) (But Not If C Alfmax .le. Alfsml). C C Epsaf Is An Estimate Of The Absolute Precision In The C Computed Values Of F. C C Oldf Is The Value Of F(0). C C Oldg Is An Estimate Of The Gradient Of F At Alfa = 0. C It Should Be Non-Positive. C C Rmu Controls What Is Meant By A Significant Decrease In F. C The Final F(Alfbst) Should Lie On Or Below The Line C L(Alfa) = Oldf + Alfa*Rmu*Oldg. C Rmu Should Be In The Open Interval (0, 0.5). C The Value Rmu = 1.0d-4 Is Good For Most Purposes. C C Tolabs,Tolrel Define A Function Tol(Alfa) = Tolrel*Alfa + Tolabs C Such That If F Has Already Been Evaluated At Step C Alfa, Then It Will Not Be Evaluated At Any Point C Closer Than Tol(Alfa). C These Values May Be Reduced By Getptq If They Seem C To Be Too Large. C C Toltny Is The Smallest Value That Tolabs Is Allowed To Be C Reduced To. C C C Output Parameters (Relevant To The Calling Program) C --------------------------------------------------- C C Imprvd Is True If The Previous Step Alfa Was The Best C Point So Far. Any Related Quantities (E.g., Arrays) C Should Be Saved By The Calling Program Before Paying C Attention To Inform. C C Inform = 0 Means The Calling Program Should Evaluate C Ftry = F(Alfa) C For The New Trial Step Alfa, And Then Re-Enter C Getptq. C C Inform = 1 Means The Search Has Terminated Successfully C With A Step Alfbst That Is Less Than The C Upper Bound Alfmax. C C Inform = 2 Means The Search Has Terminated Successfully C With A Step Alfbst That Is Equal To The C Upper Bound Alfmax. C C Inform = 3 Means That The Search Failed To Find A Point Of C Sufficient Decrease In Mfsrch Functions, But An C Improved Point Was Found. C C Inform = 4 Means Alfmax Is So Small That A Search Should C Not Have Been Done. C C Inform = 5 Means That The Search Was Terminated Prematurely C Because Of The Value Of Alfsml (See Above). C C Inform = 6 Means The Search Has Failed To Find A Useful Step. If C The Subroutine For The Function And Gradient Has Been C Programmed Correctly, This Will Usually Occur If The C Minimum Lies Very Close To Alfa = 0 Or The Gradient C Is Not Sufficiently Accurate. C C Inform = 7 Means That The Value Of G(0) Was Positive On Entry. C C Alfa Is The Step At Which The Next Function Value Must Be C Computed. C C Alfbst Should Be Accepted By The Calling Program As The C Required Step-Length Estimate, Whenever Getptq C Returns Inform = 1, 2 Or 3. C C Fbest Will Be The Corresponding Value Of F. C C C The Following Parameters Retain Information Between Entries C ----------------------------------------------------------- C C Alfuzz Is Such That, If The Final Alfa Lies In The Interval C (0,Alfuzz) And Abs( F(Alfa)-Oldf ) Le Epsaf, Alfa C Cannot Be Guaranteed To Be A Point Of Sufficient C Decrease. C C Braktd Is False If F Has Not Been Evaluated At The Far End C Of The Interval Of Uncertainty. In This Case, The C Point B Will Be At Alfmax + Tol(Alfmax). C C Crampd Is True If Alfmax Is Very Small (Le Tolabs). C If The Search Fails, This Indicates That A Zero C Step Should Be Taken. C C Extrap Is True If Alfbst Has Moved At Least Once And Xv C Lies Outside The Interval Of Uncertainty. In This C Case, Extra Safeguards Are Applied To Allow For C Instability In The Polynomial Fit. C C Vset Records Whether A Third-Best Point Has Been C Determined. C C Wset Records Whether A Second-Best Point Has Been C Determined. It Will Always Be True By The C Time The Convergence Test Is Applied (Label 300). C C Nsamea Is The Number Of Consecutive Times That The Left-Hand C End Of The Interval Of Uncertainty Has Remained The C Same. C C Nsameb Similarly For The Right-Hand End. C C A, B, Alfbst Define The Current Interval Of Uncertainty. C The Required Minimum Lies Somewhere Within The C Closed Interval (Alfbst + A, Alfbst + B). C C Alfbst Is The Best Point So Far. It Is Strictly Within The C The Interval Of Uncertainty Except When It Lies At The C Left-Hand End When Alfbst Has Not Been Moved. C Hence We Have A Le 0, B Gt 0. C C Fbest Is The Value Of F At The Point Alfbst. C C Fa Is The Value Of F At The Point Alfbst + A. C C Factor Controls The Rate At Which Extrapolated Estimates Of C Alfa May Expand Into The Interval Of Uncertainty. C Factor Is Not Used If The Minimum Has Been Bracketed C (I.e., When The Variable Braktd Is True). C C Fv, Fw Are The Values Of F At The Points Alfbst + Xv, C Alfbst + Xw. They Are Not Defined Until Vset C Or Wset (Respectively) Is True. C C Ftry Is The Value Of F At The New Point Alfbst + Xtry. C C Xtry Is The Trial Point Within The Shifted Interval (A, B). C The New Trial Function Value Must Be Computed At The C Point Alfa = Alfbst + Xtry. C C Xv Is Such That Alfbst + Xv Is The Third-Best Point. C It Is Not Defined Until Vset Is True. C C Xw Is Such That Alfbst + Xw Is The Second-Best Point. C It Is Not Defined Until Wset Is True. C In Some Cases, Xw Will Replace A Previous Xw That C Has A Lower Function But Has Just Been Excluded From C The Interval Of Uncertainty. C C C Systems Optimization Laboratory, Stanford University, California. C Original Version February 1982. Rev. May 1983. C ********************************************************************* C Logical Closef, Conv1, Conv2, Conv3, Convrg Logical Moved, Sigdec, Xinxw Data Zero, Point1, Half/ 0.0d+0, 0.1d+0, 0.5d+0/ Data One, Two, Five / 1.0d+0, 2.0d+0, 5.0d+0/ Data Ten, Eleven /10.0d+0, 11.0d+0 / C C C Local Variables C --------------- C C Closef Is True If The Worst Function Fv Is Within Epsaf C Of Fbest (Up Or Down). C C Convrg Will Be Set To True If At Least One Of The Convergence C Conditions Holds At Alfbst. C C Moved Is True If A Better Point Has Been Found (Alfbst Gt 0). C C Sigdec Says Whether Fbest Represents A Significant Decrease C In The Function, Compared To The Initial Value Oldf. C C Xinxw Is True If Xtry Is In (Xw,0) Or (0,Xw). C --------------------------------------------------------------------- C Imprvd = .false. If (Inform .ne. -1) Go To 100 C C --------------------------------------------------------------------- C First Entry. Initialize Various Quantities, Check Input Data And C Prepare To Evaluate The Function At The Initial Step Alfa. C --------------------------------------------------------------------- Nfsrch = 0 Alfbst = Zero Fbest = Oldf If (Oldg .gt. Zero) Go To 970 If (Oldg .ge. (- Epsag)) Go To 960 If (Alfmax .le. Toltny) Go To 940 C Braktd = .false. Crampd = Alfmax .le. Tolabs Extrap = .false. Vset = .false. Wset = .false. Nsamea = 0 Nsameb = 0 Alfuzz = Two*Epsaf/(Rmu*Abs( Oldg )) A = Zero B = Alfmax + (Tolrel*Alfmax + Tolabs) Fa = Oldf Factor = Five Tol = Tolabs Xtry = Alfa If (Debug) Write (Nout, 1000) Alfmax, Oldf, Oldg, Tolabs, * Alfuzz, Epsaf, Epsag, Tolrel, Crampd Go To 800 C C --------------------------------------------------------------------- C Subsequent Entries. C The Function Has Just Been Evaluated At Alfa = Alfbst + Xtry, C Giving Ftry. C --------------------------------------------------------------------- 100 Nsamea = Nsamea + 1 Nsameb = Nsameb + 1 Xtry = Alfa - Alfbst Moved = Alfbst .gt. Zero C C Check If Xtry Is In The Interval (Xw,0) Or (0,Xw). C Xinxw = .false. If (Wset) Xinxw = Zero .lt. Xtry .and. Xtry .le. Xw * .or. Xw .le. Xtry .and. Xtry .lt. Zero C C See If The New Step Is Better. C Deltaf = Ftry - Oldf Ctry = Deltaf - Alfa*Rmu*Oldg If (Alfa .le. Alfuzz) Sigdec = Deltaf .le. (- Epsaf) If (Alfa .gt. Alfuzz) Sigdec = Ctry .le. Epsaf Imprvd = Sigdec .and. ( Ftry - Fbest ) .le. (- Epsaf) C If (Debug) Write (Nout, 1100) Alfa, Ftry, Ctry If (.not. Imprvd) Go To 130 C C We Seem To Have An Improvement. The New Point Becomes The C Origin And Other Points Are Shifted Accordingly. C If (.not. Wset) Go To 110 Xv = Xw - Xtry Fv = Fw Vset = .true. 110 Xw = Zero - Xtry Fw = Fbest Wset = .true. Fbest = Ftry Alfbst = Alfa A = A - Xtry B = B - Xtry Moved = .true. Extrap = .not. Xinxw C C Decrease The Length Of The Interval Of Uncertainty. C If (Xtry .lt. Zero) Go To 120 A = Xw Fa = Fw Nsamea = 0 Go To 300 120 B = Xw Nsameb = 0 Braktd = .true. Go To 300 C C The New Function Value Is No Better Than The Best Point Found So Far. C The Point Xtry Must Be A New End Point Of The Interval Of C Uncertainty. C 130 If (Xtry .ge. Zero) Go To 140 A = Xtry Fa = Ftry Nsamea = 0 Go To 150 140 B = Xtry Nsameb = 0 Braktd = .true. C C The Origin Remains Unchanged But Xtry May Qualify As Xw. C 150 If (.not. Wset) Go To 160 If ((Ftry - Fw) .gt. Epsaf) Go To 170 Xv = Xw Fv = Fw Vset = .true. 160 Xw = Xtry Fw = Ftry Wset = .true. If (Moved) Extrap = Xinxw Go To 300 C C Ftry Is No Better Than Fbest Or Fw. If The Best Point Has Not C Been Moved, There Must Be More Than One Minimum. C 170 If (Moved) Go To 175 Xw = Xtry Fw = Ftry Go To 300 C C Ftry Is No Better Than Fbest Or Fw, But Xtry May Become Xv. C Extrap Has The Value Set In The Previous Entry. C 175 If (.not. Vset) Go To 180 If ((Ftry - Fv) .gt. Epsaf .and. Extrap) Go To 300 180 If (Xinxw) Go To 190 Xv = Xtry Fv = Ftry Vset = .true. Go To 300 190 If (Vset) Xw = Xv If (Vset) Fw = Fv Xv = Xtry Fv = Ftry Vset = .true. C C --------------------------------------------------------------------- C Check The Termination Criteria. C --------------------------------------------------------------------- 300 Tol = Tolrel*Alfbst + Tolabs Deltaf = Fbest - Oldf Cbest = Deltaf - Alfbst*Rmu*Oldg If (Alfbst .le. Alfuzz) Sigdec = Deltaf .le. (- Epsaf) If (Alfbst .gt. Alfuzz) Sigdec = Cbest .le. Epsaf Closef = .false. If (Vset) Closef = Abs( Fbest - Fv ) .le. Epsaf C Conv1 = Max( Abs( A ), B ) .le. (Tol + Tol) Conv2 = Moved .and. Sigdec Conv3 = Closef .and. (Sigdec .or. * (.not. Moved) .and. (B .le. Alfuzz)) Convrg = Conv1 .or. Conv2 .or. Conv3 C Atrue = Alfbst + A Btrue = Alfbst + B Alfaw = Alfbst + Xw Gap = B - A If (Debug) Write (Nout, 1200) Atrue, Btrue, Gap, Tol, * Nsamea, Nsameb, Braktd, Closef, Imprvd, Conv1, Conv2, Conv3, * Extrap, Alfbst, Fbest, Cbest, Alfaw, Fw If (Vset) Alfav = Alfbst + Xv If (Debug .and. Vset) Write (Nout, 1300) Alfav, Fv If (Convrg .and. Moved) Go To 910 C C Exit If The Step Is Too Small. C If (Btrue .lt. Alfsml) Go To 950 If (Nfsrch .ge. Mfsrch) Go To 930 If (.not. Convrg) Go To 400 C C A Better Point Has Not Yet Been Found (The Step Xw Is No Better C Than Step Zero). Check That The Change In F Is Consistent With A C Perturbation In X Of Tol, The Estimate Of The Minimum Spacing C Constant. If The Change In F Is Larger Than Epsaf, The Value C Of Tol Is Reduced. C Tol = Xw/Ten Tolabs = Tol If (Abs(Fw - Oldf) .gt. Epsaf .and. Tol .gt. Toltny) Go To 400 If (Crampd) Go To 940 Go To 960 C C --------------------------------------------------------------------- C Proceed With The Computation Of A Trial Step Length. C The Choices Are... C 1. Parabolic Fit Using Function Values Only. C 2. Damped Parabolic Fit If The Regular Fit Appears To Be C Consistently Over-Estimating The Distance To The Minimum. C 3. Bisection, Geometric Bisection, Or A Step Of Tol If The C Parabolic Fit Is Unsatisfactory. C --------------------------------------------------------------------- 400 Xmidpt = Half*(A + B) Q = Zero S = Zero C C --------------------------------------------------------------------- C Fit A Parabola. C --------------------------------------------------------------------- C C Check If There Are Two Or Three Points For The Parabolic Fit. C Gw = (Fw - Fbest)/Xw If (Vset .and. Moved) Go To 450 C C Only Two Points Available. Use Fbest, Fw And The Derivative C Oldg. C If (.not. Moved) S = Oldg If ( Moved) S = Oldg - Two*Gw Q = Two*(Oldg - Gw) If (Debug) Write (Nout, 2100) Go To 600 C C Three Points Available. Use Fbest, Fw And Fv. C 450 Gv = (Fv - Fbest)/Xv S = Gv - (Xv/Xw)*Gw Q = Two*(Gv - Gw) If (Debug) Write (Nout, 2200) C C --------------------------------------------------------------------- C Construct An Artificial Interval (Artifa, Artifb) In Which The C New Estimate Of The Step Length Must Lie. Set A Default Value Of C Xtry That Will Be Used If The Polynomial Fit Is Rejected. In The C Following, The Interval (A,B) Is Considered The Sum Of Two C Intervals Of Lengths Dtry And Daux, With Common End Point At The C Best Point (Zero). Dtry Is The Length Of The Interval Into Which C The Default Xtry Will Be Placed And Endpnt Denotes Its Non-Zero C End Point. The Magnitude Of Xtry Is Computed So That The Exponents C Of Dtry And Daux Are Approximately Bisected. C --------------------------------------------------------------------- 600 Artifa = A Artifb = B If (Braktd) Go To 610 C C The Minimum Has Not Been Bracketed. Set An Artificial Upper Bound C By Expanding The Interval Xw By A Suitable Factor. C Xtry = - Factor*Xw Artifb = Xtry If (Alfbst + Xtry .lt. Alfmax) Factor = Five*Factor Go To 700 C C The Minimum Has Been Bracketed. C If The Gradient At The Origin Is Being Used For The C Polynomial Fit, The Default Xtry Is One Tenth Of Xw. C 610 If (Vset .and. Moved) Go To 620 Xtry = Xw/Ten If (Debug) Write (Nout, 2400) Xtry Go To 700 C C Three Points Exist In The Interval Of Uncertainty. Check Whether C The Points Are Configured For An Extrapolation Or Interpolation. C 620 If (Extrap) Go To 660 C C If The Interpolation Appears To Be Consistently Over-Estimating The C Distance To The Minimum, Damp The Interpolation Step. C If (Nsamea .lt. 3 .and. Nsameb .lt. 3) Go To 630 Factor = Factor / Five S = Factor * S Go To 640 630 Factor = One C C The Points Are Configured For An Interpolation. The Artificial C Interval Will Be Just (A,B). Set Endpnt So That Xtry C Lies In The Larger Of The Intervals (A,0) And (0,B). C 640 If (Xmidpt .lt. Zero) Endpnt = A If (Xmidpt .gt. Zero) Endpnt = B C C If A Bound Has Remained The Same For Three Iterations, Set Endpnt C So That Xtry Is Likely To Replace The Offending Bound. C If (Nsamea .ge. 3) Endpnt = A If (Nsameb .ge. 3) Endpnt = B Go To 680 C C The Points Are Configured For An Extrapolation. C 660 If (Xw .lt. Zero) Endpnt = B If (Xw .gt. Zero) Endpnt = A C C Compute The Default Value Of Xtry. C 680 Dtry = Abs( Endpnt ) Daux = Gap - Dtry If (Daux .ge. Dtry) Xtry = Five*Dtry*(Point1 + Dtry/Daux)/Eleven If (Daux .lt. Dtry) Xtry = Half*Sqrt( Daux )*Sqrt( Dtry ) If (Endpnt .lt. Zero) Xtry = - Xtry If (Debug) Write (Nout, 2500) Xtry, Daux, Dtry C C If The Points Are Configured For An Extrapolation Set The Artificial C Bounds So That The Artificial Interval Lies Strictly Within (A,B). C If The Polynomial Fit Is Rejected, Xtry Will Remain At The Relevant C Artificial Bound. C If (Extrap .and. Xtry .le. Zero) Artifa = Xtry If (Extrap .and. Xtry .gt. Zero) Artifb = Xtry C C --------------------------------------------------------------------- C The Polynomial Fits Give (S/Q)*Xw As The New Step. C Reject This Step If It Lies Outside (Artifa, Artifb). C --------------------------------------------------------------------- 700 If (Q .eq. Zero) Go To 800 If (Q .lt. Zero) S = - S If (Q .lt. Zero) Q = - Q If (S*Xw .lt. Q*Artifa .or. S*Xw .gt. Q*Artifb) Go To 800 C C Accept The Polynomial Fit. C Xtry = Zero If (Abs( S*Xw ) .ge. Q*Tol) Xtry = (S/Q)*Xw If (Debug) Write (Nout, 2600) Xtry C C --------------------------------------------------------------------- C Test For Xtry Being Larger Than Alfmax Or Too Close To A Or B. C --------------------------------------------------------------------- 800 If (Braktd) Go To 810 C C If The Step Is Close To Or Larger Than Alfmax, Replace It By C Alfmax (To Force Evaluation Of The Function At The Boundary). C Alfa = Alfbst + Xtry If (Alfmax - Alfa .gt. (Tolrel*Alfmax + Tolabs)) Go To 810 Braktd = .true. Xtry = Alfmax - Alfbst Alfa = Alfmax Go To 900 C C Otherwise, The Function Must Not Be Evaluated At A Point Too Close C To A Or B. (It Has Already Been Evaluated At Both Those Points.) C 810 Xmidpt = Half*(A + B) If (Xtry .gt. A + Tol .and. Xtry .lt. B - Tol) Go To 820 If (Xmidpt .gt. Zero) Xtry = Tol If (Xmidpt .le. Zero) Xtry = - Tol C C C F Must Not Be Calculated Too Close To Alfbst. C 820 If (Abs( Xtry ) .lt. Tol .and. Xmidpt .lt. Zero) Xtry = - Tol If (Abs( Xtry ) .lt. Tol .and. Xmidpt .ge. Zero) Xtry = Tol Alfa = Alfbst + Xtry C C --------------------------------------------------------------------- C Exit. C --------------------------------------------------------------------- C C New Function Value Required. C 900 Inform = 0 Go To 990 C C Convergence Test Satisfied. C 910 Inform = 1 If (Alfa .eq. Alfmax) Inform = 2 Go To 990 C C Mfsrch Function Evaluations Without Sufficient Decrease, But An C Improved Point Was Found. C 930 If (.not. Moved) Go To 960 Inform = 3 Go To 990 C C Zero Step (Alfmax Too Small). C 940 Inform = 4 Go To 990 C C Premature Termination. The Step Is Smaller Than Alfsml. C 950 Inform = 5 Go To 990 C C Zero Step (A Sufficiently Better Point Could Not Be Found). C 960 Inform = 6 Go To 990 C C Zero Step (Positive Gradient At The Starting Point). C 970 Inform = 7 C C Exit. C 990 If (Debug) Write (Nout, 3000) Return C 1000 Format(/ 31h Alfmax Oldf Oldg Tolabs, 1p2e22.14, 1p2e16.8 * / 31h Alfuzz Epsaf Epsag Tolrel, 1p2e22.14, 1p2e16.8 * / 31h Crampd , L6) 1100 Format(/ 31h Alfa Ftry Ctry , 1p2e22.14, 1pe16.8) 1200 Format(/ 31h A B B - A Tol , 1p2e22.14, 1p2e16.8 * / 31h Nsamea Nsameb Braktd Closef, 2i3, 2l6 * / 31h Imprvd Convrg Extrap , L6, 3x, 3l1, L6 * / 31h Alfbst Fbest Cbest , 1p2e22.14, 1pe16.8 * / 31h Alfaw Fw , 1p2e22.14) 1300 Format( 31h Alfav Fv , 1p2e22.14 /) 2100 Format(30h Parabolic Fit, Two Points.) 2200 Format(30h Parabolic Fit, Three Points.) 2400 Format(31h Exponent Reduced. Trial Point, 1p1e22.14) 2500 Format(31h Geo. Bisection. Xtry,Daux,Dtry, 1p3e22.14) 2600 Format(31h Polynomial Fit Accepted. Xtry, 1p1e22.14) 3000 Format(53h ---------------------------------------------------- /) C C End Of Getptq End Subroutine Interp(Ncomp, Nmsh, Xx, Nudim, U, * Nmold, Xxold, Uold) Implicit Double Precision (A-H, O-Z) Dimension Xx(*), U(Nudim,*), Xxold(*), Uold(Ncomp,*) * Blas: Dcopy Parameter (Zero = 0.0d+0) * Interp Performs Piecewise Linear Interpolation Of The Old * Solution Uold At The Nmold Old Mesh Points Xxold Onto The Nmsh * New Mesh Points Xx, Producing An Interpolated Solution U. * Note That No Assumption Is Made That The New Mesh Has * More Points Than The Old, Nor That The New And Old Mesh * Points Are Related In A Specific Way (Except That Their First * And Last Points Are Identical). * By Construction, Xx(1) = Xxold(1). Copy The First Ncomp * Components Of Uold Into Those Of U. Call Dcopy(Ncomp, Uold(1,1), 1, U(1,1), 1) I = 2 Do 100 Im = 2, Nmsh-1 50 Continue If (I .gt. Nmold) Return * Check Whether The Im-Th Point In The New Mesh Lies Strictly * To The Right Of, Or To The Left Of (Or Exactly On) The * I-Th Point In The Old Mesh. If (Xx(Im) .gt. Xxold(I)) Then I = I + 1 Go To 50 Else Xdif = Xxold(I) - Xx(Im) If (Xdif .eq. Zero) Then * Xx(Im) And Xxold(I) Are Identical. Call Dcopy(Ncomp, Uold(1,I), 1, U(1,Im), 1) I = I + 1 Else Xint = Xxold(I) - Xxold(I-1) Xrat = Xdif/Xint Do 70 K = 1, Ncomp U(K,Im) = Uold(K,I) + Xrat*(Uold(K,I-1)-Uold(K,I)) 70 Continue Endif Endif 100 Continue Call Dcopy(Ncomp, Uold(1,Nmold), 1, U(1,Nmsh), 1) Return End Double Precision Function D1mach(I) C C Double-Precision Machine Constants C C D1mach( 1) = B**(Emin-1), The Smallest Positive Magnitude. C C D1mach( 2) = B**Emax*(1 - B**(-T)), The Largest Magnitude. C C D1mach( 3) = B**(-T), The Smallest Relative Spacing. C C D1mach( 4) = B**(1-T), The Largest Relative Spacing. C C D1mach( 5) = Log10(B) C C To Alter This Function For A Particular Environment, C The Desired Set Of Data Statements Should Be Activated By C Removing The C From Column 1. C On Rare Machines A Static Statement May Need To Be Added. C (But Probably More Systems Prohibit It Than Require It.) C C For Ieee-Arithmetic Machines (Binary Standard), One Of The First C Two Sets Of Constants Below Should Be Appropriate. If You Do Not C Know Which Set To Use, Try Both And See Which Gives Plausible C Values. C C Where Possible, Decimal, Octal Or Hexadecimal Constants Are Used C To Specify The Constants Exactly. Sometimes This Requires Using C Equivalent Integer Arrays. If Your Compiler Uses Half-Word C Integers By Default (Sometimes Called Integer*2), You May Need To C Change Integer To Integer*4 Or Otherwise Instruct Your Compiler C To Use Full-Word Integers In The Next 5 Declarations. C C Comments Just Before The End Statement (Lines Starting With *) C Give C Source For D1mach. C Integer Small(2) Integer Large(2) Integer Right(2) Integer Diver(2) Integer Log10(2) Integer Sc C Double Precision Dmach(5) C Equivalence (Dmach(1),Small(1)) Equivalence (Dmach(2),Large(1)) Equivalence (Dmach(3),Right(1)) Equivalence (Dmach(4),Diver(1)) Equivalence (Dmach(5),Log10(1)) C C Machine Constants For Big-Endian Ieee Arithmetic (Binary Format) C Machines In Which The Most Significant Byte Is Stored First, C Such As The At&T 3b Series, Motorola 68000 Based Machines (E.g. C Sun 3), And Machines That Use Sparc, Hp, Or Ibm Risc Chips. C C Data Small(1),Small(2) / 1048576, 0 / C Data Large(1),Large(2) / 2146435071, -1 / C Data Right(1),Right(2) / 1017118720, 0 / C Data Diver(1),Diver(2) / 1018167296, 0 / C Data Log10(1),Log10(2) / 1070810131, 1352628735 /, Sc/987/ C C Machine Constants For Little-Endian (Binary) Ieee Arithmetic C Machines In Which The Least Significant Byte Is Stored First, C E.g. Ibm Pcs And Other Machines That Use Intel 80x87 Or Dec C Alpha Chips. C Data Small(1),Small(2) / 0, 1048576 / Data Large(1),Large(2) / -1, 2146435071 / Data Right(1),Right(2) / 0, 1017118720 / Data Diver(1),Diver(2) / 0, 1018167296 / Data Log10(1),Log10(2) / 1352628735, 1070810131 /, Sc/987/ C C Machine Constants For Amdahl Machines. C C Data Small(1),Small(2) / 1048576, 0 / C Data Large(1),Large(2) / 2147483647, -1 / C Data Right(1),Right(2) / 856686592, 0 / C Data Diver(1),Diver(2) / 873463808, 0 / C Data Log10(1),Log10(2) / 1091781651, 1352628735 /, Sc/987/ C C Machine Constants For The Burroughs 1700 System. C C Data Small(1) / Zc00800000 / C Data Small(2) / Z000000000 / C C Data Large(1) / Zdffffffff / C Data Large(2) / Zfffffffff / C C Data Right(1) / Zcc5800000 / C Data Right(2) / Z000000000 / C C Data Diver(1) / Zcc6800000 / C Data Diver(2) / Z000000000 / C C Data Log10(1) / Zd00e730e7 / C Data Log10(2) / Zc77800dc0 /, Sc/987/ C C Machine Constants For The Burroughs 5700 System. C C Data Small(1) / O1771000000000000 / C Data Small(2) / O0000000000000000 / C C Data Large(1) / O0777777777777777 / C Data Large(2) / O0007777777777777 / C C Data Right(1) / O1461000000000000 / C Data Right(2) / O0000000000000000 / C C Data Diver(1) / O1451000000000000 / C Data Diver(2) / O0000000000000000 / C C Data Log10(1) / O1157163034761674 / C Data Log10(2) / O0006677466732724 /, Sc/987/ C C Machine Constants For The Burroughs 6700/7700 Systems. C C Data Small(1) / O1771000000000000 / C Data Small(2) / O7770000000000000 / C C Data Large(1) / O0777777777777777 / C Data Large(2) / O7777777777777777 / C C Data Right(1) / O1461000000000000 / C Data Right(2) / O0000000000000000 / C C Data Diver(1) / O1451000000000000 / C Data Diver(2) / O0000000000000000 / C C Data Log10(1) / O1157163034761674 / C Data Log10(2) / O0006677466732724 /, Sc/987/ C C Machine Constants For Ftn4 On The Cdc 6000/7000 Series. C C Data Small(1) / 00564000000000000000b / C Data Small(2) / 00000000000000000000b / C C Data Large(1) / 37757777777777777777b / C Data Large(2) / 37157777777777777774b / C C Data Right(1) / 15624000000000000000b / C Data Right(2) / 00000000000000000000b / C C Data Diver(1) / 15634000000000000000b / C Data Diver(2) / 00000000000000000000b / C C Data Log10(1) / 17164642023241175717b / C Data Log10(2) / 16367571421742254654b /, Sc/987/ C C Machine Constants For Ftn5 On The Cdc 6000/7000 Series. C C Data Small(1) / O"00564000000000000000" / C Data Small(2) / O"00000000000000000000" / C C Data Large(1) / O"37757777777777777777" / C Data Large(2) / O"37157777777777777774" / C C Data Right(1) / O"15624000000000000000" / C Data Right(2) / O"00000000000000000000" / C C Data Diver(1) / O"15634000000000000000" / C Data Diver(2) / O"00000000000000000000" / C C Data Log10(1) / O"17164642023241175717" / C Data Log10(2) / O"16367571421742254654" /, Sc/987/ C C Machine Constants For Convex C-1 C C Data Small(1),Small(2) / '00100000'X, '00000000'X / C Data Large(1),Large(2) / '7fffffff'X, 'Ffffffff'X / C Data Right(1),Right(2) / '3cc00000'X, '00000000'X / C Data Diver(1),Diver(2) / '3cd00000'X, '00000000'X / C Data Log10(1),Log10(2) / '3ff34413'X, '509f79ff'X /, Sc/987/ C C Machine Constants For The Cray 1, Xmp, 2, And 3. C C Data Small(1) / 201354000000000000000b / C Data Small(2) / 000000000000000000000b / C C Data Large(1) / 577767777777777777777b / C Data Large(2) / 000007777777777777776b / C C Data Right(1) / 376434000000000000000b / C Data Right(2) / 000000000000000000000b / C C Data Diver(1) / 376444000000000000000b / C Data Diver(2) / 000000000000000000000b / C C Data Log10(1) / 377774642023241175717b / C Data Log10(2) / 000007571421742254654b /, Sc/987/ C C Machine Constants For The Data General Eclipse S/200 C C Small, Large, Right, Diver, Log10 Should Be Declared C Integer Small(4), Large(4), Right(4), Diver(4), Log10(4) C C Note - It May Be Appropriate To Include The Following Line - C Static Dmach(5) C C Data Small/20k,3*0/,Large/77777k,3*177777k/ C Data Right/31420k,3*0/,Diver/32020k,3*0/ C Data Log10/40423k,42023k,50237k,74776k/, Sc/987/ C C Machine Constants For The Harris Slash 6 And Slash 7 C C Data Small(1),Small(2) / '20000000, '00000201 / C Data Large(1),Large(2) / '37777777, '37777577 / C Data Right(1),Right(2) / '20000000, '00000333 / C Data Diver(1),Diver(2) / '20000000, '00000334 / C Data Log10(1),Log10(2) / '23210115, '10237777 /, Sc/987/ C C Machine Constants For The Honeywell Dps 8/70 Series. C C Data Small(1),Small(2) / O402400000000, O000000000000 / C Data Large(1),Large(2) / O376777777777, O777777777777 / C Data Right(1),Right(2) / O604400000000, O000000000000 / C Data Diver(1),Diver(2) / O606400000000, O000000000000 / C Data Log10(1),Log10(2) / O776464202324, O117571775714 /, Sc/987/ C C Machine Constants For The Ibm 360/370 Series, C The Xerox Sigma 5/7/9 And The Sel Systems 85/86. C C Data Small(1),Small(2) / Z00100000, Z00000000 / C Data Large(1),Large(2) / Z7fffffff, Zffffffff / C Data Right(1),Right(2) / Z33100000, Z00000000 / C Data Diver(1),Diver(2) / Z34100000, Z00000000 / C Data Log10(1),Log10(2) / Z41134413, Z509f79ff /, Sc/987/ C C Machine Constants For The Interdata 8/32 C With The Unix System Fortran 77 Compiler. C C For The Interdata Fortran Vii Compiler Replace C The Z'S Specifying Hex Constants With Y'S. C C Data Small(1),Small(2) / Z'00100000', Z'00000000' / C Data Large(1),Large(2) / Z'7effffff', Z'Ffffffff' / C Data Right(1),Right(2) / Z'33100000', Z'00000000' / C Data Diver(1),Diver(2) / Z'34100000', Z'00000000' / C Data Log10(1),Log10(2) / Z'41134413', Z'509f79ff' /, Sc/987/ C C Machine Constants For The Pdp-10 (Ka Processor). C C Data Small(1),Small(2) / "033400000000, "000000000000 / C Data Large(1),Large(2) / "377777777777, "344777777777 / C Data Right(1),Right(2) / "113400000000, "000000000000 / C Data Diver(1),Diver(2) / "114400000000, "000000000000 / C Data Log10(1),Log10(2) / "177464202324, "144117571776 /, Sc/987/ C C Machine Constants For The Pdp-10 (Ki Processor). C C Data Small(1),Small(2) / "000400000000, "000000000000 / C Data Large(1),Large(2) / "377777777777, "377777777777 / C Data Right(1),Right(2) / "103400000000, "000000000000 / C Data Diver(1),Diver(2) / "104400000000, "000000000000 / C Data Log10(1),Log10(2) / "177464202324, "047674776746 /, Sc/987/ C C Machine Constants For Pdp-11 Fortrans Supporting C 32-Bit Integers (Expressed In Integer And Octal). C C Data Small(1),Small(2) / 8388608, 0 / C Data Large(1),Large(2) / 2147483647, -1 / C Data Right(1),Right(2) / 612368384, 0 / C Data Diver(1),Diver(2) / 620756992, 0 / C Data Log10(1),Log10(2) / 1067065498, -2063872008 /, Sc/987/ C C Data Small(1),Small(2) / O00040000000, O00000000000 / C Data Large(1),Large(2) / O17777777777, O37777777777 / C Data Right(1),Right(2) / O04440000000, O00000000000 / C Data Diver(1),Diver(2) / O04500000000, O00000000000 / C Data Log10(1),Log10(2) / O07746420232, O20476747770 /, Sc/987/ C C Machine Constants For Pdp-11 Fortrans Supporting C 16-Bit Integers (Expressed In Integer And Octal). C C Small, Large, Right, Diver, Log10 Should Be Declared C Integer Small(4), Large(4), Right(4), Diver(4), Log10(4) C C Data Small(1),Small(2) / 128, 0 / C Data Small(3),Small(4) / 0, 0 / C C Data Large(1),Large(2) / 32767, -1 / C Data Large(3),Large(4) / -1, -1 / C C Data Right(1),Right(2) / 9344, 0 / C Data Right(3),Right(4) / 0, 0 / C C Data Diver(1),Diver(2) / 9472, 0 / C Data Diver(3),Diver(4) / 0, 0 / C C Data Log10(1),Log10(2) / 16282, 8346 / C Data Log10(3),Log10(4) / -31493, -12296 /, Sc/987/ C C Data Small(1),Small(2) / O000200, O000000 / C Data Small(3),Small(4) / O000000, O000000 / C C Data Large(1),Large(2) / O077777, O177777 / C Data Large(3),Large(4) / O177777, O177777 / C C Data Right(1),Right(2) / O022200, O000000 / C Data Right(3),Right(4) / O000000, O000000 / C C Data Diver(1),Diver(2) / O022400, O000000 / C Data Diver(3),Diver(4) / O000000, O000000 / C C Data Log10(1),Log10(2) / O037632, O020232 / C Data Log10(3),Log10(4) / O102373, O147770 /, Sc/987/ C C Machine Constants For The Prime 50 Series Systems C With 32-Bit Integers And 64v Mode Instructions, C Supplied By Igor Bray. C C Data Small(1),Small(2) / :10000000000, :00000100001 / C Data Large(1),Large(2) / :17777777777, :37777677775 / C Data Right(1),Right(2) / :10000000000, :00000000122 / C Data Diver(1),Diver(2) / :10000000000, :00000000123 / C Data Log10(1),Log10(2) / :11504046501, :07674600177 /, Sc/987/ C C Machine Constants For The Sequent Balance 8000 C C Data Small(1),Small(2) / $00000000, $00100000 / C Data Large(1),Large(2) / $ffffffff, $7fefffff / C Data Right(1),Right(2) / $00000000, $3ca00000 / C Data Diver(1),Diver(2) / $00000000, $3cb00000 / C Data Log10(1),Log10(2) / $509f79ff, $3fd34413 /, Sc/987/ C C Machine Constants For The Univac 1100 Series. C C Data Small(1),Small(2) / O000040000000, O000000000000 / C Data Large(1),Large(2) / O377777777777, O777777777777 / C Data Right(1),Right(2) / O170540000000, O000000000000 / C Data Diver(1),Diver(2) / O170640000000, O000000000000 / C Data Log10(1),Log10(2) / O177746420232, O411757177572 /, Sc/987/ C C Machine Constants For The Vax Unix F77 Compiler C C Data Small(1),Small(2) / 128, 0 / C Data Large(1),Large(2) / -32769, -1 / C Data Right(1),Right(2) / 9344, 0 / C Data Diver(1),Diver(2) / 9472, 0 / C Data Log10(1),Log10(2) / 546979738, -805796613 /, Sc/987/ C C Machine Constants For The Vax-11 With C Fortran Iv-Plus Compiler C C Data Small(1),Small(2) / Z00000080, Z00000000 / C Data Large(1),Large(2) / Zffff7fff, Zffffffff / C Data Right(1),Right(2) / Z00002480, Z00000000 / C Data Diver(1),Diver(2) / Z00002500, Z00000000 / C Data Log10(1),Log10(2) / Z209a3f9a, Zcff884fb /, Sc/987/ C C Machine Constants For Vax/Vms Version 2.2 C C Data Small(1),Small(2) / '80'X, '0'X / C Data Large(1),Large(2) / 'Ffff7fff'X, 'Ffffffff'X / C Data Right(1),Right(2) / '2480'X, '0'X / C Data Diver(1),Diver(2) / '2500'X, '0'X / C Data Log10(1),Log10(2) / '209a3f9a'X, 'Cff884fb'X /, Sc/987/ C C *** Issue Stop 779 If All Data Statements Are Commented... If (Sc .ne. 987) Stop 779 C *** Issue Stop 778 If All Data Statements Are Obviously Wrong... If (Dmach(4) .ge. 1.0d0) Stop 778 If (I .lt. 1 .or. I .gt. 5) Goto 999 D1mach = Dmach(I) Return 999 Write(*,1999) I 1999 Format(' D1mach - I Out Of Bounds',I10) Stop End