C.... ***************************************************************** C.... Beginning Of Sample Problem Description. C.... ***************************************************************** C.... This Is A Sample Main Program For Colmod, An Automatic C.... Continuation Code For Solving `Stiff' Boundary Value Problems. C.... The Code Is Fully Documented Within The Subroutine Colmod, Which C.... Follows Directly After This Main Program And The Sample Problem C.... Description. 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 Runcol Implicit Double Precision(A-H,O-Z) Parameter(Ifdim = 500000) Parameter(Iidim = 50000) Dimension Fspace(Ifdim),Ispace(Iidim) Dimension Zeta(2),Tol(2),Ltol(2),Ipar(13),M(1),Fixpnt(1) External Fsub,Dfsub,Gsub,Dgsub,Guess Common /Prob/ Linnon 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.... Note That Both Test Problems Are Solved In Their Second Order C.... Form And Have Not Been Reduced To A First Order System. C.... ***************************************************************** C.... Specify The Desired Final Eps Value. Epsmin = 1.d-9 C.... Notol Is The Number Of Tolerances Notol = 2 Tol(1) = 1.d-8 Tol(2) = 1.d-4 Ltol(1) = 1 Ltol(2) = 2 C.... Specify The Number Of Components Ncomp = 1 M(1) = 2 C.... The User Chooses A Linear Or Nonlinear Test Problem (0 = Linear, C.... 1 = Nonlinear). 10 Write(*,1000) Read*, Linnon C.... Specify The Boundary Points If (Linnon .eq. 1) Then Aleft = 0.d0 Aright = 1.0d0 Zeta(1) = 0.d0 Zeta(2) = 1.0d0 Else If (Linnon .eq. 0) Then Aleft = -1.0d0 Aright = 1.0d0 Zeta(1) = -1.0d0 Zeta(2) = 1.0d0 Else Write(*,2000) Goto 10 Endif C.... Initialise The Variables In The Ipar Array Ipar(1) = Linnon Ipar(2) = 0 Ipar(3) = 5 Ipar(4) = Notol Ipar(5) = Ifdim Ipar(6) = Iidim Ipar(7) = 0 Ipar(8) = 0 Ipar(9) = 0 Ipar(10) = 0 Ipar(11) = 0 Ipar(12) = 0 Ipar(13) = 0 C.... Solve The Problem Call Colmod( Ncomp, M, Aleft, Aright, Zeta, Ipar, Ltol, + Tol, Fixpnt, Ispace, Fspace, Iflag, Eps, Epsmin, + Fsub, Dfsub, Gsub, Dgsub, Guess) C---------------------------------------------------------------------- 1000 Format(//,'Choose A Test Problem (0 = Linear, 1 = Nonlinear).') 2000 Format(/,'** Only 0 Or 1 Is Allowed.') C---------------------------------------------------------------------- End Subroutine Fsub(X,Z,F,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),F(*) Common /Prob/ Linnon Common /Valpi/ Pi If (Linnon .eq. 0) Then Pix = Pi*X F(1) = (Z(1)-Eps*Pi*Pi*Cos(Pix)-Cos(Pix))/Eps Else F(1) = (Z(1)*(1.d0-Z(2)))/Eps Endif Return End Subroutine Dfsub(X,Z,Df,Ncomp,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),Df(Ncomp,*) Common /Prob/ Linnon If (Linnon .eq. 0) Then Df(1,1) = 1.0d0/Eps Df(1,2) = 0.0d0 Else Df(1,1) = (1.d0-Z(2))/Eps Df(1,2) = -Z(1)/Eps Endif Return End Subroutine Gsub(I,Z,G,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*) Common /Prob/ Linnon If (Linnon .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,Z,Dg,Eps) Implicit Double Precision(A-H,O-Z) Dimension Z(*),Dg(*) Dg(1) = 1.d0 Dg(2) = 0.d0 Return End Subroutine Guess(X,Z,Dmval,Eps) Return End C.... ***************************************************************** C.... End Of Sample Problem Description. C.... ***************************************************************** Subroutine Colmod(Ncomp, M, Aleft, Aright, Zeta, Ipar, Ltol, + Tol, Fixpnt, Ispace, Fspace, Iflag, Eps, Epsmin, + Fsub, Dfsub, Gsub, Dgsub, Guess) C.... ****************************************************************** C.... Colmod - Automatic Continuation With Collocation. C.... This Package Solves `Stiff' Boundary Value Problems For Ordinary C.... Differential Equations By Using Continuation, As Described Below. C.... Colmod Is A Revised Version Of The Package Colnew By Ascher C.... And Bader [2], Which In Turn Is A Modification Of The Package C.... Colsys By Ascher, Christiansen And Russell [1]. Colmod Has Been C.... Adapted To Allow An Automatic Continuation Strategy (See [7]) To C.... Be Used. The Mesh Selection Algorithm Used In Colmod Differs From C.... That Used In Colnew. The Details Of The New Mesh Selection C.... Algorithm Are Explained In [6] C.... ****************************************************************** C... A LARGE PART OF THE ORIGINAL COLSYS CODE REMAINS UNCHANGED IN C.... COLMOD, AND WE GRATEFULLY ACKNOWLEDGE THE INPUT OF THE AUTHORS OF C.... COLSYS AND COLNEW. C.... ****************************************************************** C---------------------------------------------------------------------- C P A R T 1 C Main Storage Allocation And Program Control Subroutines -- See [7] C---------------------------------------------------------------------- C.... ****************************************************************** C.... Adapted From Colsys And Colnew By: C.... R. W. Wright C.... Department Of Mathematics, C.... Imperial College Of Science, Technology And Medicine, C.... South Kensington, C.... London Sw7 2bz, C.... England. C.... J. R. Cash C.... Department Of Mathematics, C.... Imperial College Of Science, Technology And Medicine, C.... South Kensington, C.... London Sw7 2bz, C.... England. C.... E-Mail To r.wright@ic.ac.uk Or j.cash@ic.ac.uk C.... ****************************************************************** C.... Purpose C.... This Package Solves A Multi-Point Boundary Value Problem For C.... A Mixed Order System Of Odes Involving A Small Positive C... Parameter (Which We Denote By Eps). Such A System Is Given By C.... (M(I)) C.... U = F ( X; Z(U(X)), Eps ) I = 1, ... ,Ncomp C.... I I C.... Aleft .lt. X .lt. Aright, C.... G ( Zeta(J); Z(U(Zeta(J))), Eps ) = 0, J = 1, ... ,Mstar, C.... J C.... Mstar = M(1)+M(2)+...+M(Ncomp), C.... Where T C.... U = (U , U , ... ,U ) Is The Exact Solution Vector C.... 1 2 Ncomp C.... (Mi) C.... U Is The Mi (= M(I)) Th Derivative Of U W.r.t. X C.... I I C.... (1) (M1-1) C.... Z(U(X)) = ( U (X),U (X), ... ,U (X), ... , U (X), ... C.... 1 1 1 2 C.... (1) (Mncomp-1) C.... ... , U (X), U (X), ... ,U (X) ) C.... Ncomp Ncomp Ncomp C.... F (X,Z(U),Eps) Is A (Generally) Nonlinear Function Of C.... I C.... Z(U) = Z(U(X)). C.... G (Zeta(J);z(U), Eps) Is A (Generally) Nonlinear Function C.... J C.... Used To Represent A Boundary Condition. C.... The Boundary Points Satisfy C.... Aleft .le. Zeta(1) .le. .. .le. Zeta(Mstar) .le. Aright C.... The Orders, M(I), Of The Differential Equations Satisfy C.... 1 .le. M(I) .le. 4, For All 1 <= I <= Ncomp. C.... ****************************************************************** C.... Method C.... The Method Used To Approximate The Solution U Is C.... Collocation At Gaussian Points, With The Requirement That There C.... Are M(I)-1 Continuous Derivatives In The I-Th Component, C.... I = 1, ..., Ncomp. Here, K Is The Number Of Collocation Points C.... (Stages) Per Subinterval And Is Chosen Such That K .ge. Max M(I). C.... A Runge-Kutta-Monomial Basis Representation Is Utilized. C.... The Continuation Algorithm Used Is Based Upon Predicting The C.... Behaviour Of The Monitor Function (Used For Mesh Selection). The C.... Algorithm Is Fully Described In [7]. The Mesh Selection Algorithm C.... Used In Our Continuation Framework Is Described In [6]. C.... ****************************************************************** C.... Restriction On The Choice Of Eps C.... The Type Of Problems Which Colmod Is Designed To Solve Typically C.... Involve A Small Positive Parameter 0 < Eps << 1. As Eps Becomes C.... Progressively Smaller, The Problem Normally Becomes Increasingly C.... Difficult To Approximate Numerically (For Example, Due To The C.... Appearance Of Narrow Transition Layers In The Profile Of The C.... Analytic Solution). The Idea Of Continuation Is To Solve A Chain C.... Of Problems In Which The Parameter Eps Decreases Monotonically C.... Towards Some Desired Value. That Is, We Attempt To Solve A C.... Sequence Of Problems C.... Eps_s > Eps_1 > Eps_2 > Eps_3 > ..... > Eps_min > 0 C.... Where Eps_s Is A User Provided Starting Value And Eps_min Is A C.... User Desired Final Value For The Parameter. Given Eps_s And C.... Eps_min, The Code Automatically Selects The Intermediate Eps C.... Values And Passes On Meshes And Solutions (If Solving A Nonlinear C.... Problem) At Each Step. We Note That The Intermediate Problems Are C.... Not Necessarily Solved To The User Requested Accuracy. However, C.... It Is Expected That The Intermediate Problems Will, In The Worst C.... Cases, Fail The Requested Accuracy By Only A Relatively Small C.... Amount. For A Detailed Discussion Of This, See [8]. C.... The Dependence On A Small Positive Parameter Is Not Unduly C.... Restrictive, Since Problems Involving A Large Parameter C.... Can Simply Be Re-Expressed As Problems Involving The Reciprocal C.... Of A Small Number. Our Experience Is That It Is Possible To C.... Identify Such A Parameter In Most Stiff Problems Of Practical C.... Interest. C.... ****************************************************************** C.... References C.... [1] U. Ascher, J. Christiansen And R. D. Russell, C.... Collocation Software For Boundary-Value Odes, C.... Acm Trans. Math Software 7 (1981), 209-222. C.... This Paper Contains Examples Where Use Of The Code C.... Is Demonstrated. C.... [2] G. Bader And U. Ascher, C.... A New Basis Implementation For A Mixed Order C.... Boundary Value Ode Solver, C.... Siam J. Scient. Stat. Comput. (1987). C.... [3] U. Ascher, J. Christiansen And R. D. Russell, C.... A Collocation Solver For Mixed Order C.... Systems Of Boundary Value Problems, C.... Math. Comp. 33 (1979), 659-679. C.... [4] U. Ascher, J. Christiansen And R. D. Russell, C.... Colsys - A Collocation Code For Boundary C.... Value Problems, C.... Lecture Notes Comp.sc. 76, Springer Verlag, C.... B. Childs Et. Al. (Eds.) (1979), 164-185. C.... [5] C. Deboor And R. Weiss, C.... Solveblok: A Package For Solving Almost Block Diagonal C.... Linear Systems, C.... Acm Trans. Math. Software 6 (1980), 80-87. C.... [6] R. Wright, J. Cash And G. Moore, C.... Mesh Selection For Stiff Two-Point Boundary Value C.... Problems, C.... Numerical Algorithms 7 (1994), 205-224. C.... [7] J. R. Cash, G. Moore And R. W. Wright, C.... An Automatic Continuation Strategy For The Solution Of C.... Singularly Perturbed Linear Two-Point Boundary Value C.... Problems, C.... J. Comp. Phys. 122 (1995), 266-279. C.... [8] J. R. Cash, G. Moore And R. W. Wright, C.... An Automatic Continuation Strategy For The Solution Of C.... Singularly Perturbed Nonlinear Two-Point Boundary Value C.... Problems, C.... To Appear. C.... ****************************************************************** C.... Input To Colmod C.... Variables C.... Ncomp - No. Of Differential Equations (Ncomp .le. 20) C.... M(J) - Order Of The J-Th Differential Equation (M(J) .le. 4) C.... ( Mstar = M(1) + ... + M(Ncomp) .le. 40 ) C.... Aleft - Left End Of Interval C.... Aright - Right End Of Interval C.... Zeta(J) - J-Th Side Condition Point (Boundary Point). Must C.... Have Zeta(J) .le. Zeta(J+1). All Side Condition C.... Points Must Be Mesh Points In All Meshes Used, C.... See Description Of Ipar(10) And Fixpnt Below. C.... Ipar - An Integer Array Dimensioned At Least 13. C.... A List Of The Parameters In Ipar And Their Meaning C.... Follows. Some Parameters Are Renamed In Colmod; These C.... New Names Are Given In Parentheses. C.... Ipar(1) ( = Nonlin ) C.... = 0 If The Problem Is Linear C.... = 1 If The Problem Is Nonlinear C.... Ipar(2) = No. Of Collocation Points Per Subinterval ( = K ) C.... Where Max M(I) .le. K .le. 7 . If Ipar(2) = 0 Then C.... Colmod Sets K = Max ( Max M(I)+1, 5-Max M(I) ) C.... Ipar(3) = No. Of Subintervals In The Initial Mesh. ( = N ) C.... If Ipar(3) = 0 Then Colmod Arbitrarily Sets N = 5. C.... Ipar(4) = No. Of Solution And Derivative Tolerances. ( = Ntol ) C.... We Require 0 .lt. Ntol .le. Mstar. C.... Ipar(5) = Dimension Of Fspace. ( = Ndimf ) C.... Ipar(6) = Dimension Of Ispace. ( = Ndimi ) C.... Ipar(7) - Output Control ( = Iprint ) C.... = -1 For Full Diagnostic Printout C.... = 0 For Selected Printout C.... = 1 For No Printout C.... Ipar(8) ( = Iread ) C.... = 0 Causes Colmod To Generate A Uniform Initial Mesh. C.... = 1 If The Initial Mesh Is Provided By The User. It C.... Is Defined In Fspace As Follows: The Mesh C.... Aleft = X(1) .lt. X(2) .lt. ... .lt. X(N) C.... .lt. X(N+1) = Aright C.... Will Occupy Fspace(1), ..., Fspace(N+1). The C.... User Needs To Supply Only The Interior Mesh C.... Points Fspace(J) = X(J), J = 2, ..., N. C.... Ipar(9) ( = Iguess ) C.... = 0 If No Initial Guess For The Solution Is C.... Provided. C.... = 1 If An Initial Guess Is Provided By The User C.... In Subroutine Guess. C.... = 2 If An Initial Mesh And Approximate Solution C.... Coefficients Are Provided By The User In Fspace. C.... (The Former And New Mesh Are The Same). C.... = 3 If A Former Mesh And Approximate Solution C.... Coefficients Are Provided By The User In Fspace, C.... And The New Mesh Is To Be Taken Twice As Coarse; C.... I.e.,Every Second Point From The Former Mesh. C.... = 4 If In Addition To A Former Initial Mesh And C.... Approximate Solution Coefficients, A New Mesh C.... Is Provided In Fspace As Well. C.... The Discussion Directly Below Describes How The C.... Variable Ipar(9) Might Be Used. However, As We C.... Are Already Working In A Continuation Framework, It C.... Is Unlikely That The User Will Want To Use Anything C.... Other Than Ipar(9) = 0. An Exception To This Might C.... Be The Case Where The User Wants To Solve The Same C.... Problem With A Different Value Of Eps_min. In This C.... Case, The Mesh And Solution For One Value Of Eps_min C.... Might Be Provided As An Initial Mesh And Initial C.... Guess For The Next (Smaller) Value Of Eps_min (By C.... Setting Ipar(9) = 2 Or 3). C.... [ A Formerly Obtained Solution Can Be Used As The C.... First Approximation For The Nonlinear Iteration For A C.... New Problem By Setting Ipar(9) = 2, 3 Or 4. C.... If The Former Solution Has Just Been Obtained Then The C.... Values Needed To Define The First Approximation Are C.... Already In Ispace And Fspace. C.... Alternatively, If The Former Solution Was Obtained In A C.... Previous Run, And Its Coefficients Were Saved, Then Those C.... Coefficients Must Be Put Back Into C.... Ispace(1),..., Ispace(7+Ncomp) And C.... Fspace(1),..., Fspace(Ispace(7)). C.... For Ipar(9) = 2 Or 3 Set Ipar(3) = Ispace(1) ( = The C.... Size Of The Previous Mesh ). C.... For Ipar(9) = 4 The User Specifies A New Mesh Of N C.... Subintervals As Follows. C.... The Values In Fspace(1),...,Fspace(Ispace(7)) Have To Be C.... Moved By N+1 Locations To Fspace(N+2),..,Fspace(Ispace(7)+N+1) C.... And The New Mesh Is Specified In Fspace(1),..., Fspace(N+1). C.... Also Set Ipar(3) = N. ] C.... Ipar(10) = No. Of Fixed Points In The Mesh Other Than Aleft C.... And Aright. ( = Nfxpnt , The Dimension Of Fixpnt) C.... The Code Requires That All Side Condition Points C.... Other Than Aleft And Aright (See Description Of C.... Zeta ) Be Included As Fixed Points In Fixpnt. C.... Ipar(11) - ( = Maxcon ) C.... The Maximum Number Of Continuation Steps Taken When C.... Attempting To Solve A Particular Problem. The C.... Default Value Is Maxcon = 50 (When Ipar(11) = 0). C.... Ipar(12) - ( = Itsaim ) C.... Used Only For Nonlinear Problems. The Continuation C.... Steps Are Chosen With The Aim That The Number Of C.... Newton Iterations Required For Convergence On The C.... First Mesh, For Each Continuation Problem, Is Less C.... Than Or Equal To Itsaim (See [8] For More Details). C.... The Default Value Is Itsaim = 7 (If The User C.... Specifies Ipar(12) = 0). The User Should Note That C.... Small Values Of Itsaim (E.g. Itsaim = 1,2) May C.... Restrict The Size Of The Continuation Steps C.... Significantly. C.... Ipar(13) - Indicates Whether The Initial Value Of Eps Is C.... Provided By The User. If Ipar(13) .eq. 1 The User C.... Must Specify The Initial Eps Value Before Calling C.... Subroutine Colmod. If Ipar(13) .eq. 0, Eps Takes C.... The Default Starting Value Of Eps = 0.5. C.... Ltol - An Array Of Dimension Ipar(4). Ltol(J) = L Specifies C.... That The J-Th Tolerance In Tol Controls The Error C.... In The L-Th Component Of Z(U). We Also Require That C.... 1 .le. Ltol(1) .lt. Ltol(2) .lt. ... C.... .lt. Ltol(Ntol) .le. Mstar C.... Tol - An Array Of Dimension Ipar(4). Tol(J) Is The Mixed C.... Relative/Absolute Error Tolerance On The Ltol(J)-Th C.... Component Of Z(U). Thus, The Code Attempts To Satisfy C.... For J = 1,...,Ntol On Each Subinterval C.... Abs(Z(V)-Z(U)) .le. Tol(J)*Max(1,Abs(Z(U)) ) C.... Ltol(J) Ltol(J) C.... C.... If V(X) Is The Approximate Solution Vector. C.... Fixpnt - An Array Of Dimension Ipar(10). It Contains C.... The Points, Other Than Aleft And Aright, Which C.... Are To Be Included In Every Mesh. C.... Ispace - An Integer Work Array Of Dimension Ipar(6). C.... Its Size Provides A Constraint On Nmax, C.... The Maximum Number Of Subintervals. Choose C.... Ipar(6) According To The Formula C.... Ipar(6) .ge. Nmax*Nsizei C.... Where C.... Nsizei = 3 + Kdm C.... With C.... Kdm = Kd + Mstar ; Kd = K * Ncomp ; C.... Nrec = No. Of Right End Boundary Conditions. C.... Fspace - A Real Work Array Of Dimension Ipar(5). C.... Its Size Provides A Constraint On Nmax. C.... Choose Ipar(5) According To The Formula C.... Ipar(5) .ge. Nmax*Nsizef C.... Where, If We Are Solving A Linear Problem, C.... Nsizef = 7 + 3 * Mstar + (5+Kd) * Kdm + C.... (2*Mstar-Nrec) * 2*Mstar, C.... And If We Are Solving A Nonlinear Problem C.... Nsizef = 8 + 4 * Mstar + (5+Kd) * Kdm + C.... (2*Mstar-Nrec) * 2*Mstar + Kd. C.... Iflag - The Mode Of Return From Colmod. C.... = 1 For Normal Return C.... = 0 If The Collocation Matrix Is Singular For The Final C.... Continuation Problem. C.... = -1 If The Expected No. Of Subintervals Exceeds Storage C.... Specifications. C.... = -2 If The Nonlinear Iteration Has Not Converged For The C.... Final Continuation Problem. C.... = -3 If There Is An Input Data Error. C.... Eps - The Initial Value Of The Continuation Parameter. Eps Must C.... Be Set Before Calling Colmod If Ipar(13) .eq. 1. (If C.... Ipar(13) .eq. 0 Eps Takes The Default Starting Value Of C.... Eps = 0.5.) For Many Singular Perturbation Type Problems, C.... The Choice Of 0.1 < Eps < 1 Represents A (Fairly) Easy C.... Problem. The User Should Attempt To Specify An Initial C.... Problem That Is Not `Too' Challenging. The Code Assumes C.... That Eps = 1 Represents An `Easy' Problem. On Return From C.... Colmod The Variable Eps Contains The Value Used When C.... Solving The Final Continuation Problem (Usually Epsmin). C.... If Ipar(13) .eq. 1, Eps Must Be Initialised Strictly Less C.... Than 1 And Greater Than 0. C.... Epsmin - The Desired Value Of Eps For Which The User Would C.... Like To Solve The Problem. Epsmin Must Be Less Than C.... Or Equal To Eps. C.... ****************************************************************** C.... User Supplied Subroutines C.... The Following Subroutines Must Be Declared External In The C.... Main Program Which Calls Colmod. C.... Fsub - Name Of Subroutine For Evaluating F(X,Z(U(X)),Eps) = C.... T C.... (F ,...,F ) At A Point X In (Aleft,Aright). It C.... 1 Ncomp C.... Should Have The Heading C.... Subroutine Fsub (X , Z , F, Eps) C.... Where F Is The Vector Containing The Value Of Fi(X,Z(U)) C.... In The I-Th Component And T C.... Z(U(X)) = (Z(1),...,Z(Mstar)) C.... Is Defined As Above Under Purpose . C.... Dfsub - Name Of Subroutine For Evaluating The Jacobian Of C.... F(X,Z(U),Eps) At A Point X. It Should Have The Heading C.... Subroutine Dfsub (X , Z , Df, Ncomp, Eps) C.... Where Z(U(X)) Is Defined As For Fsub And The (Ncomp) By C.... (Mstar) Array Df Should Be Filled By The Partial Deriv- C.... Atives Of F, Viz, For A Particular Call One Calculates C.... Df(I,J) = Dfi / Dzj, I = 1,...,Ncomp C.... J = 1,...,Mstar. C.... Gsub - Name Of Subroutine For Evaluating The I-Th Component Of C.... G(X,Z(U(X)),Eps) = G (Zeta(I),Z(U(Zeta(I))),Eps) At Point C.... I C.... X = Zeta(I) Where 1 .le. I .le. Mstar. It Should Have The C.... Heading C.... Subroutine Gsub (I , Z , G, Eps) C.... Where Z(U) Is As For Fsub, And I And G = G Are As Above. C.... I C.... Note That In Contrast To F In Fsub , Here Only One C.... Value Per Call Is Returned In G. C.... Dgsub - Name Of Subroutine For Evaluating The I-Th Row Of C.... The Jacobian Of G(X,U(X),Eps). It Has The Heading C.... Subroutine Dgsub (I , Z , Dg, Eps) C.... Where Z(U) Is As For Fsub, I As For Gsub And The Mstar- C.... Vector Dg Should Be Filled With The Partial Derivatives C.... Of G, Viz, For A Particular Call One Calculates C.... Dg(I,J) = Dgi / Dzj J = 1,...,Mstar. C.... Guess - Name Of Subroutine To Evaluate The Initial C.... Approximation For Z(U(X)) And For Dmval(U(X)) = Vector C.... Of The Mj-Th Derivatives Of U(X). It Should Have The C.... Heading C.... Subroutine Guess (X , Z , Dmval, Eps) C.... Note That This Subroutine Is Needed Only If Using C.... Ipar(9) = 1, And Then All Mstar Components Of Z C.... And Ncomp Components Of Dmval Should Be Specified C.... For Any X, Aleft .le. X .le. Aright . C.... ****************************************************************** C.... Use Of Output From Colmod - Solution Evaluation C.... On Return From Colmod, The Arrays Fspace And Ispace C.... Contain Information Specifying The Approximate Solution. C.... The User Can Produce The Solution Vector Z( U(X) ) At C.... Any Point X, Aleft .le. X .le. Aright, By The Statement, C.... Call Appsln (X, Z, Fspace, Ispace) C.... When Saving The Coefficients For Later Reference, Only C.... Ispace(1),...,Ispace(7+Ncomp) And C.... Fspace(1),...,Fspace(Ispace(7)) Need To Be Saved As C.... These Are The Quantities Used By Appsln. C.... ****************************************************************** C.... Package Subroutines C.... The Following Description Gives A Brief Overview Of How The C.... Procedure Is Broken Down Into The Subroutines Which Make Up C.... The Package Called Colmod . For Further Details The C.... User Should Refer To Documentation In The Various Subroutines C.... And To The References Cited Above. C.... The Subroutines Fall Into Four Groups: C Part 1 - The Main Storage Allocation And Program Control Subroutines C.... Colmod - Tests Input Values, Does Initialization And Breaks Up C.... The Work Areas, Fspace And Ispace, Into The Arrays C.... Used By The Program. After Initialisation, This C.... Subroutine Contains The Actual Continuation Algorithm. C.... The Continuation Step Sizes Are Selected In Colmod. C.... Contrl - Is The Actual Driver Of The Package. This Routine C.... Contains The Strategy For Nonlinear Equation Solving. C.... Skale - Provides Scaling For The Control C.... Of Convergence In The Nonlinear Iteration. C Part 2 - Mesh Selection And Error Estimation Subroutines C.... Consts - Is Called Once By Colmod To Initialize Constants C.... Used For Error Estimation And Mesh Selection. C.... Newmsh - Generates Meshes. C.... Errchk - Produces Error Estimates And Checks Against The C.... Tolerances At Each Subinterval C Part 3 - Collocation System Set-Up Subroutines C.... Lsyslv - Controls The Set-Up And Solution Of The Linear C.... Algebraic Systems Of Collocation Equations Which C.... Arise At Each Newton Iteration. C.... Gderiv - Is Used By Lsyslv To Set Up The Equation Associated C.... With A Side Condition Point. C.... Vwblok - Is Used By Lsyslv To Set Up The Equation(S) Associated C.... With A Collocation Point. C.... Gblock - Is Used By Lsyslv To Construct A Block Of The Global C.... Collocation Matrix Or The Corresponding Right Hand C.... Side. C Part 4 - Service Subroutines C.... Appsln - Sets Up A Standard Call To Approx . C.... Approx - Evaluates A Piecewise Polynomial Solution. C.... Rkbas - Evaluates The Mesh Independent Runge-Kutta Basis C.... Vmonde - Solves A Vandermonde System For Given Right Hand C.... Side C.... Horder - Evaluates The Highest Order Derivatives Of The C.... Current Collocation Solution Used For Mesh Refinement. C Part 5 - Linear Algebra Subroutines C.... To Solve The Global Linear Systems Of Collocation Equations C.... Constructed In Part 3, Colmod Uses A Column Oriented Version C.... Of The Package Solveblok Originally Due To De Boor And Weiss. C C.... To Solve The Linear Systems For Static Parameter Condensation C.... In Each Block Of The Collocation Equations, The Linpack C.... Routines Dgefa And Dgesl Are Included. But These C.... May Be Replaced When Solving Problems On Vector Processors C.... Or When Solving Large Scale Sparse Jacobian Problems. C---------------------------------------------------------------------- Implicit Double Precision (A-H,O-Z) Dimension M(*), Zeta(*), Ipar(*), Ltol(*), Tol(*), Dummy(1), + Fixpnt(*), Ispace(*), Fspace(*) Dimension Phi(3), E(3), Pmax(2), Hord(2) 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 ) Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Convg/ Nits Common /Mshvar/ Pmax,Hord,Hsml Common /Colout/ Precis, Iout, Iprint Common /Colloc/ Rho(7), Coef(49) Common /Colord/ K, Nc, Mstar, Kd, Mmax, Mt(20) Common /Colapr/ N, Nold, Nmax, Nz, Ndmz, Mshflg Common /Colsid/ Tzeta(40), Tleft, Tright, Izeta, Idum Common /Colnln/ Nonlin, Iter, Limit, Iguess Common /Colest/ Wgtmsh(40), Wgterr(40), Tolin(40), + Root(40), Jtol(40), Lttol(40), Ntol External Fsub, Dfsub, Gsub, Dgsub, Guess C.... ***************************************************************** C.... The Subroutine Colmod Serves As An Interface With C.... The Package Of Subroutines Referred To Collectively As C.... Colmod. The Subroutine Serves To Test Some Of The Input C.... Parameters, Rename Some Of The Parameters (To Make Under- C.... Standing Of The Coding Easier), To Do Some Initialization, C.... And To Break The Work Areas Fspace And Ispace Up Into The C.... Arrays Needed By The Program. C.... The Second Part Of This Subroutine Contains The Actual Algorithm C.... For Selecting The Continuation Parameter. C.... ****************************************************************** C.... Specify Machine Dependent Output Unit Iout And Compute Machine C.... Dependent Constant Precis = 100 * Machine Unit Roundoff Iout = 6 Precis = One 10 Precis = Precis / Two Precp1 = Precis + One If ( Precp1 .gt. One ) Go To 10 Precis = Precis * 100.d0 C C.... In Case Incorrect Input Data Is Detected, The Program Returns C.... Immediately With Iflag = -3. C Iflag = -3 If ( Ncomp .lt. 1 .or. Ncomp .gt. 20 ) Return Do 20 I = 1,Ncomp If ( M(I) .lt. 1 .or. M(I) .gt. 4 ) Return 20 Continue C C.... Rename Some Of The Parameters And Set Default Values. C Nonlin = Ipar(1) K = Ipar(2) N = Ipar(3) If ( N .eq. 0 ) N = 5 Iread = Ipar(8) Iguess = Ipar(9) If ( Nonlin .eq. 0 .and. Iguess .eq. 1 ) Iguess = 0 If ( Iguess .ge. 2 .and. Iread .eq. 0 ) Iread = 1 Ntol = Ipar(4) Ndimf = Ipar(5) Ndimi = Ipar(6) Nfxpnt = Ipar(10) Iprint = Ipar(7) Maxcon = Ipar(11) If ( Maxcon .eq. 0 ) Maxcon = 50 Itsaim = Ipar(12) If ( Itsaim .eq. 0 ) Itsaim = 7 If ( Ipar(13) .eq. 0 ) Eps = 0.5d0 Mstar = 0 Mmax = 0 Do 30 I = 1, Ncomp Mmax = Max ( Mmax, M(I) ) Mstar = Mstar + M(I) Mt(I) = M(I) 30 Continue If ( K .eq. 0 ) K = Max( Mmax + 1 , 5 - Mmax ) Do 40 I = 1, Mstar 40 Tzeta(I) = Zeta(I) Do 50 I = 1, Ntol Lttol(I) = Ltol(I) 50 Tolin(I) = Tol(I) Tleft = Aleft Tright = Aright Nc = Ncomp Kd = K * Ncomp C.... Print The Input Data For Checking. If ( Iprint .gt. 0 ) Go To 80 If ( Nonlin .gt. 0 ) Go To 60 Write (Iout,1000) Ncomp, (M(Ip), Ip = 1,Ncomp) Go To 70 60 Write (Iout,1001) Ncomp, (M(Ip), Ip = 1,Ncomp) 70 Write (Iout,1002) (Zeta(Ip), Ip = 1,Mstar) If ( Nfxpnt .gt. 0 ) + Write (Iout,1003) Nfxpnt, (Fixpnt(Ip), Ip = 1,Nfxpnt) Write (Iout,1004) K Write (Iout,1005) (Ltol(Ip), Ip = 1,Ntol) Write (Iout,1006) (Tol(Ip), Ip = 1,Ntol) If (Iguess .ge. 2) Write (Iout,1007) Write (Iout,1008) Eps, Epsmin 80 Continue C.... Check For Correctness Of Data If ( K .lt. 0 .or. K .gt. 7 ) Return If ( N .lt. 0 ) Return If ( Maxcon .lt. 0 ) Return If ( Iread .lt. 0 .or. Iread .gt. 2 ) Return If ( Iguess .lt. 0 .or. Iguess .gt. 4 ) Return If ( Ntol .lt. 0 .or. Ntol .gt. Mstar ) Return If ( Nfxpnt .lt. 0 ) Return If ( Iprint .lt. (-1) .or. Iprint .gt. 1 ) Return If ( Mstar .lt. 0 .or. Mstar .gt. 40 ) Return If ( Eps .le. Zero .or. Eps .ge. One ) Return If ( Eps .lt. Epsmin .or. Epsmin .le. Zero) Return If ( Nonlin .eq. 1 .and. Itsaim .lt. 0 ) Return Ip = 1 Do 100 I = 1, Mstar If ( Abs(Zeta(I) - Aleft) .lt. Precis .or. + Abs(Zeta(I) - Aright) .lt. Precis ) Go To 100 90 If ( Ip .gt. Nfxpnt ) Return If ( Zeta(I) - Precis .lt. Fixpnt(Ip) ) Go To 95 Ip = Ip + 1 Go To 90 95 If ( Zeta(I) + Precis .lt. Fixpnt(Ip) ) Return 100 Continue C.... Set Limits On Iterations And Initialize Counters. C.... Limit = Maximum Number Of Newton Iterations Per Mesh. C.... See Subroutine Newmsh For The Role Of Mshflg. Mshflg = 0 Limit = 40 C.... Compute The Maxium Possible N For The Given Sizes Of C.... Ispace And Fspace. Nrec = 0 Do 110 I = 1, Mstar Ib = Mstar + 1 - I If ( Zeta(Ib) .ge. Aright ) Nrec = I 110 Continue Nfixi = Mstar Nsizei = 3 + Kd + Mstar If (Nonlin .eq. 1) Then Nfixf = Nrec * (2*Mstar) + 6 * Mstar + 6 Nsizef = 8 + 4 * Mstar + (Kd+5) * (Kd+Mstar) + + (2*Mstar-Nrec) * 2*Mstar + Kd Else Nfixf = Nrec * (2*Mstar) + 5 * Mstar + 5 Nsizef = 7 + 3 * Mstar + (Kd+5) * (Kd+Mstar) + + (2*Mstar-Nrec) * 2*Mstar Endif Nmaxf = (Ndimf - Nfixf) / Nsizef Nmaxi = (Ndimi - Nfixi) / Nsizei If ( Iprint .lt. 0 ) Write(Iout,1009) Nmaxf, Nmaxi Nmax = Min( Nmaxf, Nmaxi ) If ( Nmax .lt. N ) Return If ( Nmax .lt. Nfxpnt+1 ) Return If (Nmax .lt. 2*Nfxpnt+2 .and. Iprint .lt. 1) Write(Iout,1010) C.... Generate Pointers To Break Up Fspace And Ispace. Lxi = 1 Lg = Lxi + Nmax + 1 Lxiold = Lg + 2*Mstar * (Nmax * (2*Mstar-Nrec) + Nrec) Lw = Lxiold + Nmax + 1 Lv = Lw + Kd**2 * Nmax Lz = Lv + Mstar * Kd * Nmax Ldmz = Lz + Mstar * (Nmax + 1) Ldelz = Ldmz + Kd * Nmax Ldeldz = Ldelz + Mstar * (Nmax + 1) Ldqz = Ldeldz + Kd * Nmax Ldqdmz = Ldqz + Mstar * (Nmax + 1) Lrhs = Ldqdmz + Kd * Nmax Lvalst = Lrhs + Kd * Nmax + Mstar Lslope = Lvalst + 4 * Mstar * Nmax Laccum = Lslope + Nmax Lscl = Laccum + Nmax + 1 Ldscl = Lscl + Mstar * (Nmax + 1) Lext1 = Ldscl + Kd * Nmax Lext2 = Lext1 + Nmax Lext3 = Lext2 + Nmax + 1 Lpvtg = 1 Lpvtw = Lpvtg + Mstar * (Nmax + 1) Linteg = Lpvtw + Kd * Nmax C.... If Iguess .ge. 2, Move Xiold, Z, And Dmz To Their Proper C.... Locations In Fspace. If ( Iguess .lt. 2 ) Go To 160 Nold = N If (Iguess .eq. 4) Nold = Ispace(1) Nz = Mstar * (Nold + 1) Ndmz = Kd * Nold Np1 = N + 1 If ( Iguess .eq. 4 ) Np1 = Np1 + Nold + 1 Do 120 I = 1,Nz 120 Fspace( Lz+I-1 ) = Fspace( Np1+I ) Idmz = Np1 + Nz Do 125 I = 1,Ndmz 125 Fspace( Ldmz+I-1 ) = Fspace( Idmz+I ) Np1 = Nold + 1 If ( Iguess .eq. 4 ) Go To 140 Do 130 I = 1,Np1 130 Fspace( Lxiold+I-1 ) = Fspace( Lxi+I-1 ) Go To 160 140 Do 150 I = 1,Np1 150 Fspace( Lxiold+I-1 ) = Fspace( N+1+I ) 160 Continue C.... Initialize Collocation Points And Constants. Call Consts ( K, Rho, Coef ) C.... Initialize Mesh. Call Newmsh (3+Iread, Fspace(Lxi), Fspace(Lxiold), Dummy, + Dummy, Dummy, Dummy, Dummy, Nfxpnt, Fixpnt, Dummy, Dummy) C.... Determine First Approximation, If The Problem Is Nonlinear. If (Iguess .ge. 2) Go To 200 Np1 = N + 1 Do 170 I = 1, Np1 170 Fspace( I + Lxiold - 1 ) = Fspace( I + Lxi - 1 ) Nold = N If ( Nonlin .eq. 0 .or. Iguess .eq. 1 ) Go To 200 C.... System Provides First Approximation Of The Solution. C.... Choose Z(J) = 0 For J = 1,...,Mstar. Do 180 I = 1, Nz 180 Fspace( Lz-1+I ) = 0.5d0 Do 190 I = 1, Ndmz 190 Fspace( Ldmz-1+I ) = 0.5d0 200 Continue If (Iguess .ge. 2) Iguess = 0 C.... ***************************************************************** C.... Initialisation And Explanation Of Variables Used In The C.... Continuation Strategy Begin Here. C.... ***************************************************************** C.... Save Details Of Initial Mesh And Initial Guess In Case We Need C.... To Backtrack Nbk1 = N Nbk2 = Nold Do 202, J = 1,Nbk1+1 Fspace(Lext3 + J - 1) = Fspace(J) 202 Continue If (Nonlin .eq. 1) Then Indx = Lext3 + Nbk1 Do 204, J = 1,Nbk2+1 Fspace(Indx + J) = Fspace(Lxiold+J-1) 204 Continue Indx = Indx + Nbk2 + 1 Nz = Mstar * (Nbk2 + 1) Do 206, J = 1,Nz Fspace(Indx + J) = Fspace(Lz+J-1) 206 Continue Indx = Indx + Nz Ndmz = Kd * Nbk2 Do 208, J = 1,Ndmz Fspace(Indx + J) = Fspace(Ldmz+J-1) 208 Continue Endif C.... Initialise Extrapolation Variables Do 210 J = 1,3 Phi(J) = Zero E(J) = Zero 210 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. Ifinal = 0 Iback = 0 C.... Istep Is A Flag Which Limits The Size Of The Continuation Steps C.... After Experiencing A Failure For Some Step. Istep = 0 C.... Ncs Counts The Total Number Of Continuation Steps Taken. C.... Nss Counts The Number Of Successful Continuation Steps Taken. Ncs = 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. C.... Iprec Is A Flag Used To Indicate Whether The Bounds Imposed By C.... Machine Precision Are Being Approached Or Passed. C.... Epsp Is The Value Of Eps Beyond Which The Machine Precision Is C.... (Possibly) Not Sufficient To Solve The Given Problem. Idc = 0 Iextrap = 0 Istuk = 0 Iprec = 0 Epsp = Zero 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 = 1.0d+2/Precis Do 220 J = 1,Ipar(4) Tolmax = Min(Tol(J),Tolmax) 220 Continue Phimax = Phimax*Tolmax Phiaim = Zero Phialt = Zero Epold = Zero Hsmlpv = 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 230 For Every Continuation Step. C.... ***************************************************************** 230 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 Ncs. Ncs = Ncs+1 If (Iprint .lt. 1) Write(Iout,1011) Ncs, Eps Iflag = 1 C.... If We Have Reached The Maximum Number Of Continuation Steps C.... Then This Will Be The Last Problem We Attempt. If (Ncs .eq. Maxcon) Then If (Iprint .lt. 1) Write(Iout,1012) 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 C.... Attempt To Solve The Latest Continuation Problem. Call Contrl (Fspace(Lxi),Fspace(Lxiold),Fspace(Lz),Fspace(Ldmz), + Fspace(Lrhs),Fspace(Ldelz),Fspace(Ldeldz),Fspace(Ldqz), + Fspace(Ldqdmz),Fspace(Lg),Fspace(Lw),Fspace(Lv), + Fspace(Lvalst),Fspace(Lslope),Fspace(Lscl),Fspace(Ldscl), + Fspace(Laccum),Ispace(Lpvtg),Ispace(Linteg),Ispace(Lpvtw), + Nfxpnt,Fixpnt,Iflag,Fsub,Dfsub,Gsub,Dgsub,Guess, + Fspace(Lext1),Fspace(Lext2),Eps) C.... ***************************************************************** C.... The Logic For A Successful Continuation Step Starts Here C.... ***************************************************************** If (Iflag .eq. 1) 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 .lt. 1) Write(Iout,1013) Eps Goto 340 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 .lt. 1) Write(Iout,1014) Eps Epsp = Eps Iprec = 1 Endif C.... Save Details Of Last Problem Solved Successfully In Case We C.... Need To Backtrack Nbk1 = N Nbk2 = Nold Do 240, J = 1,Nbk1+1 Fspace(Lext3 + J - 1) = Fspace(J) 240 Continue If (Nonlin .eq. 1) Then Indx = Lext3 + Nbk1 Do 250, J = 1,Nbk2+1 Fspace(Indx + J) = Fspace(Lxiold+J-1) 250 Continue Indx = Indx + Nbk2 + 1 Nz = Mstar * (Nbk2 + 1) Do 260, J = 1,Nz Fspace(Indx + J) = Fspace(Lz+J-1) 260 Continue Indx = Indx + Nz Ndmz = Kd * Nbk2 Do 270, J = 1,Ndmz Fspace(Indx + J) = Fspace(Ldmz+J-1) 270 Continue Endif 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 290 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 290 Else Idc = 1 E(3) = Ep Phi(3) = Phit Endif C.... Otherwise Update Extrapolation Data Else Idc = 0 Do 280 J = 1,2 E(J) = E(J+1) Phi(J) = Phi(J+1) 280 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. (N .gt. (Nmax/2))) Then Ep = E(3) Epsmin = One/Ep If (Iprint .lt. 1) Then If (Dele .lt. 0.01d0*E(3)) Then Write(Iout,1015) Epsmin Else Write(Iout,1016) Epsmin Endif Endif Emin = Ep Iback = 1 Goto 290 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 (Nold .gt. 3*N/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/Hsmlpv,One) Hsmlpv = Hsml Bbm = Max(1.5d0*Bmax,Bmax+Three) If (Nonlin .eq. 1) 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 (Nonlin .eq. 0) 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 290 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 .lt. 1) Then If (Epsp .ne. Zero) Then Write(Iout,1017) Eps,Iflag,Eps,Epsp Else Write(Iout,1018) Eps,Iflag,Eps Endif Endif Goto 340 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 .lt. 1) Write(Iout,1019) Ifinal = 0 N = Nbk1 Ep = (Ep+E(3))/Two If ((Ep-E(3)) .lt. 0.01d0*E(3)) Then Ep = E(3) Epsmin = One/Ep If (Iprint .lt. 1) Write(Iout,1015) Epsmin Emin = Ep Iback = 1 Iflag = 1 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. N = Nbk1 Nold = Nbk2 Do 300, J = 1,N+1 Fspace(J) = Fspace(Lext3+J-1) 300 Continue If (Nonlin .eq. 1) Then Indx = Lext3 + N Do 310, J = 1,Nbk2+1 Fspace(Lxiold+J-1) = Fspace(Indx+J) 310 Continue Indx = Indx + Nbk2 + 1 Nz = Mstar * (Nbk2 + 1) Do 320, J = 1,Nz Fspace(Lz+J-1) = Fspace(Indx + J) 320 Continue Indx = Indx + Nz Ndmz = Kd * Nbk2 Do 330, J = 1,Ndmz Fspace(Ldmz+J-1) = Fspace(Indx + J) 330 Continue Endif Endif C C.... ***************************************************************** C.... End Of Logic For An Unsuccessful Continuation Step. C.... ***************************************************************** C C.... Set Continuation Variables For Colmod C Iguess = 0 Iread = 1 Nz = Mstar * (N + 1) Ndmz = Kd * N C.... ***************************************************************** C.... The Program Loops Back To Line 230 For Every Continuation Step. C.... ***************************************************************** Goto 230 C.... ***************************************************************** C.... Before Exiting, Prepare Output. C.... ***************************************************************** 340 Ispace(1) = N Ispace(2) = K Ispace(3) = Ncomp Ispace(4) = Mstar Ispace(5) = Mmax Ispace(6) = Nz + Ndmz + N + 2 K2 = K * K Ispace(7) = Ispace(6) + K2 - 1 Do 350 I = 1, Ncomp 350 Ispace(7+I) = M(I) Do 360 I = 1, Nz 360 Fspace( N+1+I ) = Fspace( Lz-1+I ) Idmz = N + 1 + Nz Do 370 I = 1, Ndmz 370 Fspace( Idmz+I ) = Fspace( Ldmz-1+I ) Ic = Idmz + Ndmz Do 380 I = 1, K2 380 Fspace( Ic+I ) = Coef(I) If (Iflag .eq. 1 .and. Iprint .eq. 0) Then Write(Iout,1020) ('Z',Ltol(J), J=1,Ntol) Write(Iout,*) Jstep = Max(N/30,1) Iadd = Jstep*Mstar Ind = N + 1 Do 390 I = 1, N, Jstep Write(Iout,1021) I,Fspace(I),(Fspace(Ind+Ltol(J)),J=1,Ntol) Ind = Ind + Iadd 390 Continue Ind = N+1 + Nz - Mstar Write(Iout,1021) N+1,Fspace(N+1),(Fspace(Ind+Ltol(J)),J=1,Ntol) Write(Iout,1022) Endif Return C----------------------------------------------------------------------- 1000 Format(/ 37h The Number Of (Linear) Diff Eqns Is , I3/, + 17h Their Orders Are, 20i3) 1001 Format(/ 40h The Number Of (Nonlinear) Diff Eqns Is , I3/, + 17h Their Orders Are, 20i3) 1002 Format(27h Side Condition Points Zeta, 8f10.6, 4( / 27x, 8f10.6)) 1003 Format(10h There Are ,I5,27h Fixed Points In The Mesh - , + 10(6f10.6/)) 1004 Format(37h Number Of Colloc Pts Per Interval Is, I3) 1005 Format(39h Components Of Z Requiring Tolerances -,8(7x,I2,1x), + 4(/38x,8i10)) 1006 Format(33h Corresponding Error Tolerances -,6x,8d10.2, + 4(/39x,8d10.2)) 1007 Format(44h Initial Mesh(Es) And Z,Dmz Provided By User) 1008 Format (33h The Initial Value Of Epsilon Is ,D11.4,/, + 39h The Desired Final Value Of Epsilon Is ,D11.4) 1009 Format(44h The Maximum Number Of Subintervals Is Min (, I4, + 23h (Allowed From Fspace),,I4, 24h (Allowed From Ispace) )) 1010 Format(/53h Insufficient Space To Double Mesh For Error Estimate) 1011 Format (/,1x,82('*'),/,' Continuation Step ',I2, + ', Epsilon = ',D9.4,/,1x,82('*')) 1012 Format (/,1x,'This Is The Final Continuation Step Since ', + 'Maxcon =',I4) 1013 Format (/,1x,82('$'),//,' Tolerances Satisfied For Final Problem', + ' Epsilon = ',D9.4,//,1x,82('$'),/) 1014 Format(/,' ** Machine Precision (Possibly) Not Sufficient For', + ' Epsilon Less Than ',D9.4) 1015 Format (/,' Continuation Steps Too Small, Change Epsmin To ',D9.4) 1016 Format (/,' Storage Limit Being Approached, Change Epsmin To ', + D9.4) 1017 Format (/,1x,82('$'),//,' Final Problem Epsilon = ',D10.4,/, + ' Not Solved, Iflag =',I3,/, ' Try Running Problem Again', + ' With Epsmin Greater Than ',D9.4,/, ' Machine Precision', + ' (Possibly) Not Sufficient For Eps Less Than ',D9.4,//, + 1x,82('$'),/) 1018 Format (/,1x,82('$'),//,' Final Problem Epsilon = ',D10.4,/, + ' Not Solved, Iflag =',I3,/, ' Try Running Problem Again', + ' With Epsmin Greater Than ',D9.4,//,1x,82('$'),/) 1019 Format (/,' ** Failed Step - Bactracking For Larger Value ', + 'Of Epsilon') 1020 Format(' The Final Mesh And Solution Components Are:',//, + 5x,'I',10x,'X(I) ',40(14x,A,'(',I2,')')) 1021 Format(I6,41d19.8) 1022 Format(/,1x,82('$'),/) C----------------------------------------------------------------------- End Subroutine Contrl (Xi, Xiold, Z, Dmz, Rhs, Delz, Deldmz, + Dqz, Dqdmz, G, W, V, Valstr, Slope, Scale, Dscale, + Accum, Ipvtg, Integs, Ipvtw, Nfxpnt, Fixpnt, Iflag, + Fsub, Dfsub, Gsub, Dgsub, Guess, Slpold, Voldmsh, Eps ) C C********************************************************************** C C Purpose C This Subroutine Is The Actual Driver. The Nonlinear Iteration C Strategy Is Controlled Here ( See [4] ). Upon Convergence, Errchk C Is Called To Test For Satisfaction Of The Requested Tolerances. C C Variables C C Check - Maximum Tolerance Value, Used As Part Of Criteria For C Checking For Nonlinear Iteration Convergence C Relax - The Relaxation Factor For Damped Newton Iteration C Relmin - Minimum Allowable Value For Relax (Otherwise The C Jacobian Is Considered Singular). C Rlxold - Previous Relax C Rstart - Initial Value For Relax When Problem Is Sensitive C Ifrz - Number Of Fixed Jacobian Iterations C Lmtfrz - Maximum Value For Ifrz Before Performing A Reinversion C Iter - Number Of Iterations (Counted Only When Jacobian C Reinversions Are Performed). C Xi - Current Mesh C Xiold - Previous Mesh C Ipred = 0 If Relax Is Determined By A Correction C = 1 If Relax Is Determined By A Prediction C Ifreez = 0 If The Jacobian Is To Be Updated C = 1 If The Jacobian Is Currently Fixed (Frozen) C Iconv = 0 If No Previous Convergence Has Been Obtained C = 1 If Convergence On A Previous Mesh Has Been Obtained C Rnorm - Norm Of Rhs (Right Hand Side) For Current Iteration C Rnold - Norm Of Rhs For Previous Iteration C Anscl - Scaled Norm Of Newton Correction C Anfix - Scaled Norm Of Newton Correction At Next Step C Anorm - Scaled Norm Of A Correction Obtained With Jacobian Fixed C Nz - Number Of Components Of Z (See Subroutine Approx) C Ndmz - Number Of Components Of Dmz (See Subroutine Approx) C Imesh - A Control Variable For Subroutines Newmsh And Errchk C = 1 The Current Mesh Resulted From Mesh Selection C Or Is The Initial Mesh. C = 2 The Current Mesh Resulted From Doubling The C Previous Mesh C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Xi(*), Xiold(*), Z(*), Dmz(*), Rhs(*) Dimension G(*), W(*), V(*), Valstr(*), Slope(*), Accum(*) Dimension Delz(*), Deldmz(*), Dqz(*), Dqdmz(*) , Fixpnt(*) Dimension Dummy(1), Scale(*), Dscale(*) Dimension Integs(*), Ipvtg(*), Ipvtw(*) Dimension Slpold(*), Voldmsh(*) C Common /Colout/ Precis, Iout, Iprint Common /Colord/ K, Ncomp, Mstar, Kd, Mmax, M(20) Common /Colapr/ N, Nold, Nmax, Nz, Ndmz, Mshflg Common /Colnln/ Nonlin, Iter, Limit, Iguess Common /Colest/ Wgtmsh(40), Wgterr(40), Tolin(40), + Root(40), Jtol(40), Ltol(40), Ntol C Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Convg/ Nits C External Fsub, Dfsub, Gsub, Dgsub, Guess * The Parameter Inumb Is A Counter Used To Limit To Three The Number * Of Mesh Selections Performed For The Final Continuation Problem. Inumb = 0 Relmin = 1.d-3 Rstart = 1.d-2 Lmtfrz = 4 C C.... Compute The Maximum Tolerance C Check = 0.d0 Do 10 I = 1, Ntol 10 Check = Max ( Tolin(I), Check ) Imesh = 1 Iconv = 0 If ( Nonlin .eq. 0 ) Iconv = 1 Icor = 0 Msing = 0 C C.... The Main Iteration Begins Here . C.... Loop 20 Is Executed Until Error Tolerances Are Satisfied Or C.... The Code Fails (Due To A Singular Matrix Or Storage Limitations) C 20 Continue C C.... Initialization For A New Mesh C Iter = 0 If ( Nonlin .gt. 0 ) Go To 50 C C.... The Linear Case. C.... Set Up And Solve Equations C Call Lsyslv (Msing, Xi, Xiold, Dummy, Dummy, Z, Dmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, 0, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Check For A Singular Matrix C If ( Msing .eq. 0 ) Go To 400 30 If ( Msing .lt. 0 ) Go To 40 If ( Iprint .lt. 1 ) Write (Iout,1000) Go To 460 40 If ( Iprint .lt. 1 ) Write (Iout,1001) Iflag = 0 Return C C.... Iteration Loop For Nonlinear Case C.... Define The Initial Relaxation Parameter ( = Relax) C 50 Relax = 1.d0 C C.... Check For Previous Convergence C If ( Iconv .eq. 0 ) Go To 160 C C.... Convergence On A Previous Mesh Has Been Obtained. Thus C.... We Have A Very Good Initial Approximation For The Newton C.... Process. Proceed With One Full Newton And Then Iterate C.... With A Fixed Jacobian. C Ifreez = 0 C C.... Evaluate Right Hand Side And Its Norm And C.... Find The First Newton Correction C Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, G, + W, V, Rhs, Dqdmz, Integs, Ipvtg, Ipvtw, Rnold, 1, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C If ( Iprint .lt. 0 ) Write(Iout,1002) If ( Iprint .lt. 0 ) Write (Iout,1003) Iter, Rnold Go To 70 C C.... Solve For The Next Iterate . C.... The Value Of Ifreez Determines Whether This Is A Full C.... Newton Step ( = 0) Or A Fixed Jacobian Iteration ( = 1). C 60 If ( Iprint .lt. 0 ) Write (Iout,1003) Iter, Rnorm Rnold = Rnorm Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, + 3+Ifreez, Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Check For A Singular Matrix C 70 If ( Msing .ne. 0 ) Go To 30 If ( Ifreez .eq. 1 ) Go To 80 C C.... A Full Newton Step C Iter = Iter + 1 Ifrz = 0 80 Continue C C.... Update Z And Dmz , Compute New Rhs And Its Norm C Do 90 I = 1, Nz Z(I) = Z(I) + Delz(I) 90 Continue Do 100 I = 1, Ndmz Dmz(I) = Dmz(I) + Deldmz(I) 100 Continue Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, 2, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Check Monotonicity. If The Norm Of Rhs Gets Smaller, C.... Proceed With A Fixed Jacobian; Else Proceed Cautiously, C.... As If Convergence Has Not Been Obtained Before (Iconv = 0). C If ( Rnorm .lt. Precis ) Go To 390 If ( Rnorm .gt. Rnold ) Go To 130 If ( Ifreez .eq. 1 ) Go To 110 Ifreez = 1 Go To 60 C C.... Verify That The Linear Convergence With Fixed Jacobian C.... Is Fast Enough. C 110 Ifrz = Ifrz + 1 If ( Ifrz .ge. Lmtfrz ) Ifreez = 0 If ( Rnold .lt. 4.d0*Rnorm ) Ifreez = 0 C C.... Check Convergence (Iconv = 1). C Do 120 It = 1, Ntol Inz = Ltol(It) Do 120 Iz = Inz, Nz, Mstar If ( Abs(Delz(Iz)) .gt. + Tolin(It) * (Abs(Z(Iz)) + 1.d0)) Go To 60 120 Continue C C.... Convergence Obtained C If ( Iprint .eq. -1 ) Write (Iout,1004) Iter If ( Iprint .eq. 0 ) Write (Iout,1016) Iter If (Iatt .eq. -1) Nits = Iter Go To 400 C C.... Convergence Of Fixed Jacobian Iteration Failed. C 130 If ( Iprint .lt. 0 ) Write (Iout,1003) Iter, Rnorm If ( Iprint .lt. 0 ) Write (Iout,1005) Iconv = 0 Relax = Rstart Do 140 I = 1, Nz Z(I) = Z(I) - Delz(I) 140 Continue Do 150 I = 1, Ndmz Dmz(I) = Dmz(I) - Deldmz(I) 150 Continue C C.... Update Old Mesh C Np1 = N + 1 Do 155 I = 1, Np1 Xiold(I) = Xi(I) 155 Continue Nold = N C Iter = 0 C C.... No Previous Convergence Has Been Obtained. Proceed C.... With The Damped Newton Method. C.... Evaluate Rhs And Find The First Newton Correction. C 160 If(Iprint .lt. 0) Write (Iout,1006) Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, G, + W, V, Rhs, Dqdmz, Integs, Ipvtg, Ipvtw, Rnold, 1, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Check For A Singular Matrix C If ( Msing .ne. 0 ) Go To 30 C C.... Bookkeeping For First Mesh C If ( Iguess .eq. 1 ) Iguess = 0 C C.... Find Initial Scaling C Call Skale (N, Mstar, Kd, Z, Xi, Scale, Dscale) Go To 220 C C.... Main Iteration Loop C 170 Rnold = Rnorm If ( Iter .ge. Limit ) Go To 430 C C.... Update Scaling C Call Skale (N, Mstar, Kd, Z, Xi, Scale, Dscale) C C.... Compute Norm Of Newton Correction With New Scaling C Anscl = 0.d0 Do 180 I = 1, Nz Anscl = Anscl + (Delz(I) * Scale(I))**2 180 Continue Do 190 I = 1, Ndmz Anscl = Anscl + (Deldmz(I) * Dscale(I))**2 190 Continue Anscl = Sqrt(Anscl / Dfloat(Nz+Ndmz)) C C.... Find A Newton Direction C Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, 3, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Check For A Singular Matrix C If ( Msing .ne. 0 ) Go To 30 C C.... Predict Relaxation Factor For Newton Step. C Andif = 0.d0 Do 200 I = 1, Nz Andif = Andif + ((Dqz(I) - Delz(I)) * Scale(I))**2 200 Continue Do 210 I = 1, Ndmz Andif = Andif + ((Dqdmz(I) - Deldmz(I)) * Dscale(I))**2 210 Continue Andif = Sqrt(Andif/Dfloat(Nz+Ndmz) + Precis) Relax = Relax * Anscl / Andif If ( Relax .gt. 1.d0 ) Relax = 1.d0 220 Rlxold = Relax Ipred = 1 Iter = Iter + 1 C C.... Determine A New Z And Dmz And Find New Rhs And Its Norm C Do 230 I = 1, Nz Z(I) = Z(I) + Relax * Delz(I) 230 Continue Do 240 I = 1, Ndmz Dmz(I) = Dmz(I) + Relax * Deldmz(I) 240 Continue 250 Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Dqz, Dqdmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, 2, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Compute A Fixed Jacobian Iterate (Used To Control Relax) C Call Lsyslv (Msing, Xi, Xiold, Z, Dmz, Dqz, Dqdmz, G, + W, V, Rhs, Dummy, Integs, Ipvtg, Ipvtw, Rnorm, 4, + Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C C.... Find Scaled Norms Of Various Terms Used To Correct Relax C Anorm = 0.d0 Anfix = 0.d0 Do 260 I = 1, Nz Anorm = Anorm + (Delz(I) * Scale(I))**2 Anfix = Anfix + (Dqz(I) * Scale(I))**2 260 Continue Do 270 I = 1, Ndmz Anorm = Anorm + (Deldmz(I) * Dscale(I))**2 Anfix = Anfix + (Dqdmz(I) * Dscale(I))**2 270 Continue Anorm = Sqrt(Anorm / Dfloat(Nz+Ndmz)) Anfix = Sqrt(Anfix / Dfloat(Nz+Ndmz)) If ( Icor .eq. 1 ) Go To 280 If (Iprint .lt. 0) Write (Iout,1007) Iter, Relax, Anorm, + Anfix, Rnold, Rnorm Go To 290 280 If (Iprint .lt. 0) Write (Iout,1008) Relax, Anorm, Anfix, + Rnold, Rnorm 290 Icor = 0 C C.... Check For Monotonic Decrease In Delz And Deldmz. C If (Anfix .lt. Precis .or. Rnorm .lt. Precis) Go To 390 If ( Anfix .gt. Anorm ) Go To 300 C C.... We Have A Decrease. C.... If Dqz And Dqdmz Small, Check For Convergence C If ( Anfix .le. Check ) Go To 350 C C.... Correct The Predicted Relax Unless The Corrected C.... Value Is Within 10 Percent Of The Predicted One. C If ( Ipred .ne. 1 ) Go To 170 300 If ( Iter .ge. Limit ) Go To 430 C C.... Correct The Relaxation Factor. C Ipred = 0 Arg = (Anfix/Anorm - 1.d0) / Relax + 1.d0 If ( Arg .lt. 0.d0 ) Go To 170 If (Arg .le. .25d0*Relax+.125d0*Relax**2) Go To 310 Factor = -1.d0 + Sqrt (1.d0+8.d0 * Arg) If ( Abs(Factor-1.d0) .lt. .1d0*Factor ) Go To 170 If ( Factor .lt. 0.5d0 ) Factor = 0.5d0 Relax = Relax / Factor Go To 320 310 If ( Relax .ge. .9d0 ) Go To 170 Relax = 1.d0 320 Icor = 1 If ( Relax .lt. Relmin ) Go To 440 Fact = Relax - Rlxold Do 330 I = 1, Nz Z(I) = Z(I) + Fact * Delz(I) 330 Continue Do 340 I = 1, Ndmz Dmz(I) = Dmz(I) + Fact * Deldmz(I) 340 Continue Rlxold = Relax Go To 250 C C.... Check Convergence (Iconv = 0). C 350 Continue Do 360 It = 1, Ntol Inz = Ltol(It) Do 360 Iz = Inz, Nz, Mstar If ( Abs(Dqz(Iz)) .gt. + Tolin(It) * (Abs(Z(Iz)) + 1.d0) ) Go To 170 360 Continue C C.... Convergence Obtained C If ( Iprint .eq. -1 ) Write (Iout,1004) Iter If ( Iprint .eq. 0 ) Write (Iout,1016) Iter If (Iatt .eq. -1) Nits = Iter C C.... Since Convergence Obtained, Update Z And Dmz With Term C.... From The Fixed Jacobian Iteration. C Do 370 I = 1, Nz Z(I) = Z(I) + Dqz(I) 370 Continue Do 380 I = 1, Ndmz Dmz(I) = Dmz(I) + Dqdmz(I) 380 Continue 390 If (Anfix .lt. Precis .or. Rnorm .lt. Precis) Then If (Iatt .eq. -1) Nits = Iter If ( Iprint .eq. -1 ) Write (Iout,1004) Iter If ( Iprint .eq. 0 ) Write (Iout,1016) Iter Endif Iconv = 1 C C.... If Full Output Has Been Requested, Print Values Of The C.... Solution Components Z At The Meshpoints. C 400 If ( Iprint .ge. 0 ) Go To 420 Do 410 J = 1, Mstar Write(Iout,1009) J Write(Iout,1010) (Z(Lj), Lj = J, Nz, Mstar) 410 Continue C.... Check For Error Tolerance Satisfaction 420 Ifin = 1 If (Imesh .eq. 2) Call Errchk (Xi, Z, Dmz, Valstr, Ifin) If ( Imesh .eq. 1 .or. Ifin .eq. 0 ) Go To 460 C Iflag = 1 Return C C.... Diagnostics For Failure Of Nonlinear Iteration. C 430 If ( Iprint .lt. 1 ) Write (Iout,1011) Iter Go To 450 440 If( Iprint .lt. 1 ) Write(Iout,1012) Relax, Relmin 450 Iflag = -2 Return C C.... Update Old Mesh C 460 Np1 = N + 1 Do 470 I = 1, Np1 Xiold(I) = Xi(I) 470 Continue Nold = N C.... Pick A New Mesh Imesh = 1 If (Ifinal .eq. 1) Then If (Iatt .ge. 1) Then If (N .gt. Nmax/2) Then Iflag = -1 Return Else Imesh = 2 Inumb = Inumb+1 Endif If (Inumb .eq. 4 .and. Iback .eq. 0) Then Iflag = -1 Return Endif Endif Else If (Iatt .eq. 0) Then Call Newmsh (Imesh, Xi, Xiold, Z, Dmz, Valstr, + Slope, Accum, Nfxpnt, Fixpnt, Slpold, Voldmsh) If (Iprec .eq. 2) Then If (Iprint .lt. 1) Write(Iout,1013) Iflag = -1 Else Iflag = 1 Endif Return Endif Endif Call Newmsh (Imesh, Xi, Xiold, Z, Dmz, Valstr, + Slope, Accum, Nfxpnt, Fixpnt, Slpold, Voldmsh) If (Iprec .eq. 2) Then If (Iprint .lt. 1) Write(Iout,1013) Iflag = -1 Return Endif Iatt = Iatt+1 C.... Exit If Expected N Is Too Large (But May Try N = Nmax Once) If ( N .le. Nmax ) Go To 480 N = N / 2 Iflag = -1 If ( Iconv .eq. 0 .and. Iprint .lt. 1 ) Write (Iout,1014) If ( Iconv .eq. 1 .and. Iprint .lt. 1 ) Write (Iout,1015) Return 480 If ( Iconv .eq. 0 ) Imesh = 1 Go To 20 C --------------------------------------------------------------- 1000 Format(/40h A Local Elimination Matrix Is Singular ) 1001 Format(/35h The Global Bvp-Matrix Is Singular ) 1002 Format(/27h Fixed Jacobian Iterations,) 1003 Format(13h Iteration = , I3, 15h Norm (Rhs) = , D10.2) 1004 Format(/18h Convergence After , I3,11h Iterations ) 1005 Format(/35h Switch To Damped Newton Iteration,) 1006 Format(/30h Full Damped Newton Iteration,) 1007 Format(13h Iteration = ,I3,22h Relaxation Factor = ,D10.2 + /33h Norm Of Scaled Rhs Changes From ,D10.2,3h To,D10.2 + /33h Norm Of Rhs Changes From ,D10.2,3h To,D10.2, + D10.2) 1008 Format(40h Relaxation Factor Corrected To Relax = , D10.2 + /33h Norm Of Scaled Rhs Changes From ,D10.2,3h To,D10.2 + /33h Norm Of Rhs Changes From ,D10.2,3h To,D10.2 + ,D10.2) 1009 Format(/19h Mesh Values For Z(, I2, 2h), ) 1010 Format(1h , 8d15.7) 1011 Format(/22h No Convergence After , I3, 11h Iterations) 1012 Format(/37h No Convergence. Relaxation Factor =,D10.3 + ,24h Is Too Small (Less Than, D10.3, 1h)) 1013 Format(/,1x,'** Mesh Cannot Be Defined Within The Bounds Imposed', + ' By The Machine Precision') 1014 Format(18h (No Convergence) ) 1015 Format(50h (Probably Tolerances Too Stringent, Or Nmax Too + ,6hsmall) ) 1016 Format(18h Convergence After , I3,11h Iterations ) End Subroutine Skale (N, Mstar, Kd, Z, Xi, Scale, Dscale) C C********************************************************************** C C Purpose C Provide A Proper Scaling Of The State Variables, Used C To Control The Damping Factor For A Newton Iteration [2]. C C Variables C C N = Number Of Mesh Subintervals C Mstar = Number Of Unknomns In Z(U(X)) C Kd = Number Of Unknowns In Dmz C Z = The Global Unknown Vector C Xi = The Current Mesh C Scale = Scaling Vector For Z C Dscale = Scaling Vector For Dmz C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Z(Mstar,*), Scale(Mstar,*), Dscale(Kd,*) Dimension Xi(*), Basm(5) C Common /Colord/ K, Ncomp, Id1, Id2, Mmax, M(20) C Basm(1) = 1.d0 Do 50 J = 1,N Iz = 1 H = Xi(J+1) - Xi(J) Do 10 L = 1, Mmax Basm(L+1) = Basm(L) * H / Dfloat(L) 10 Continue Do 40 Icomp = 1, Ncomp Scal = (Abs(Z(Iz,J)) + Abs(Z(Iz,J+1))) * .5d0 + 1.d0 Mj = M(Icomp) Do 20 L = 1, Mj Scale(Iz,J) = Basm(L) / Scal Iz = Iz + 1 20 Continue Scal = Basm(Mj+1) / Scal Do 30 Idmz = Icomp, Kd, Ncomp Dscale(Idmz,J) = Scal 30 Continue 40 Continue 50 Continue Np1 = N + 1 Do 60 Iz = 1, Mstar Scale(Iz,Np1) = Scale(Iz,N) 60 Continue Return End C---------------------------------------------------------------------- C P A R T 2 C Mesh Selection, Error Estimation, (And Related C Constant Assignment) Routines -- See [3], [4], [6] C---------------------------------------------------------------------- C Subroutine Newmsh (Mode, Xi, Xiold, Z, Dmz, Valstr, + Slope, Accum, Nfxpnt, Fixpnt, Slpold, Voldmsh) C C********************************************************************** C C Purpose C Select A Mesh On Which A Collocation Solution Is To Be C Determined C C There Are 4 Possible Modes Of Action: C Mode = 4,3 - Deal Mainly With Definition Of An Initial C Mesh For The Current Boundary Value Problem C = 2,1 - Deal With Definition Of A New Mesh, Either C By Simple Mesh Halving Or By Mesh Selection C More Specifically, For C Mode = 4 An Initial (Generally Nonuniform) Mesh Is C Defined By The User C = 3 A Simple Uniform Mesh (Except Possibly For Some C Fixed Points) Is Defined; N = No. Of Subintervals C = 1 The Automatic Mesh Selection Procedure Is Used C (See [6] For Details) C = 2 A Simple Mesh Halving Is Performed C C********************************************************************** C C Variables C C N = Number Of Mesh Subintervals C Nold = Number Of Subintervals For Former Mesh C Xi - Mesh Point Array C Xiold - Former Mesh Point Array C Mshflg = 1 The Mesh Is A Halving Of Its Former Mesh C (So An Error Estimate Has Been Calculated) C = 0 Otherwise C Iguess - Ipar(9) In Subroutine Colmod. It Is Used C Here Only For Mode = 5 And 4, Where C = 2 The Subroutine Sets Xi = Xiold. This Is C Used E.g. If Continuation Is Being Per- C Formed, And A Mesh For The Old Differen- C Tial Equation Is Being Used C = 3 Same As For = 2, Except Xi Uses Every Other C Point Of Xiold (So Mesh Xiold Is Mesh Xi C Halved) C = 4 Xi Has Been Defined By The User, And An Old C Mesh Xiold Is Also Available C Otherwise, Xi Has Been Defined By The User C And We Set Xiold = Xi In This Subroutine C Slope - An Approximate Quantity To Be Equidistributed For C Mesh Selection (See [3]), Viz, C . (K+Mj) C Slope(I) = Max (Weight(L) *U (Xi(I))) C 1 .le. L .le. Ntol J C C Where J = Jtol(L) C Slphmx - Maximum Of Slope(I)*(Xiold(I+1)-Xiold(I)) For C I = 1 ,..., Nold. C Accum - Accum(I) Is The Integral Of Slope From Aleft C To Xiold(I). C Valstr - Is Assigned Values Needed In Errchk For The C Error Estimate. C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension D1(40), D2(40), Slope(*), Accum(*), Valstr(*), D3(40) Dimension Xi(*), Xiold(*), Z(*), Dmz(*), Fixpnt(*), Dummy(1) Dimension Slpold(*), Voldmsh(*) Dimension Pmax(2), Hord(2) C Common /Colout/ Precis, Iout, Iprint Common /Colord/ K, Ncomp, Mstar, Kd, Mmax, M(20) Common /Colapr/ N, Nold, Nmax, Nz, Ndmz, Mshflg Common /Colnln/ Nonlin, Iter, Limit, Iguess Common /Colsid/ Zeta(40), Aleft, Aright, Izeta, Idum Common /Colbas/ B(28), Acol(28,7), Asave(28,4) Common /Colest/ Wgtmsh(40), Wgterr(40), Tolin(40), + Root(40), Jtol(40), Ltol(40), Ntol Common /Flags/ Ifinal,Iatt,Iback,Iprec Common /Mshvar/ Pmax,Hord,Hsml C Nfxp1 = Nfxpnt +1 Iprec = Min(Iprec,1) Go To (180, 100, 50, 20), Mode C C.... Mode = 4 The User-Specified Initial Mesh Is Already In Place. C 20 If ( Iguess .lt. 2 ) Go To 40 C C.... Iguess = 2, 3 Or 4. C Noldp1 = Nold + 1 If (Iprint .lt. 0) Write(Iout,1000) Nold,(Xiold(I), I = 1,Noldp1) If ( Iguess .ne. 3 ) Go To 40 C C.... If Iread ( Ipar(8) ) .eq. 1 And Iguess ( Ipar(9) ) .eq. 3 C.... Then The First Mesh Is Every Second Point Of The C.... Mesh In Xiold . C N = Nold /2 I = 0 Do 30 J = 1, Nold, 2 I = I + 1 30 Xi(I) = Xiold(J) 40 Continue Np1 = N + 1 Xi(1) = Aleft Xi(Np1) = Aright Go To 320 C C.... Mode = 3 Generate A (Piecewise) Uniform Mesh. If There Are C.... Fixed Points Then Ensure That The N Being Used Is Large Enough. C 50 If ( N .lt. Nfxp1 ) N = Nfxp1 Np1 = N + 1 Xi(1) = Aleft Ileft = 1 Xleft = Aleft C C.... Loop Over The Subregions Between Fixed Points. C Do 90 J = 1, Nfxp1 Xright = Aright Iright = Np1 If ( J .eq. Nfxp1 ) Go To 60 Xright = Fixpnt(J) C C.... Determine Where The J-Th Fixed Point Should Fall In The C.... New Mesh - This Is Xi(Iright) And The (J-1)St Fixed C.... Point Is In Xi(Ileft) C Nmin = Int((Xright-Aleft) / (Aright-Aleft) * Dble(N) + 1.5d0) If (Nmin .gt. N-Nfxpnt+J) Nmin = N - Nfxpnt + J Iright = Max (Ileft+1, Nmin) 60 Xi(Iright) = Xright C C.... Generate Equally Spaced Points Between The J-1st And The C.... J-Th Fixed Points. C Nregn = Iright - Ileft - 1 If ( Nregn .eq. 0 ) Go To 80 Dx = (Xright - Xleft) / Dfloat(Nregn+1) Do 70 I = 1, Nregn 70 Xi(Ileft+I) = Xleft + Dfloat(I) * Dx 80 Ileft = Iright Xleft = Xright 90 Continue Go To 320 C C.... Mode = 2 Halve The Current Mesh (I.e. Double Its Size) C 100 N2 = 2 * N C C.... Check That N Does Not Exceed Storage Limitations C 112 If ( N2 .le. Nmax ) Go To 120 C C.... If Possible, Try With N = Nmax. Redistribute First. C If ( Mode .eq. 2 ) Go To 110 N = Nmax / 2 Go To 220 110 If ( Iprint .lt. 1 ) Write(Iout,1001) N = N2 Return C C.... Calculate The Old Approximate Solution Values At C.... Points To Be Used In Errchk For Error Estimates. C.... If Mshflg = 1 An Error Estimate Was Obtained For C.... For The Old Approximation So Half The Needed Values C.... Will Already Be In Valstr . C 120 If ( Mshflg .eq. 0 ) Go To 140 C C.... Save In Valstr The Values Of The Old Solution C.... At The Relative Positions 1/6 And 5/6 In Each Subinterval. C Kstore = 1 Do 130 I = 1, Nold Hd6 = (Xiold(I+1) - Xiold(I)) / 6.d0 X = Xiold(I) + Hd6 Call Approx (I, X, Valstr(Kstore), Asave(1,1), Dummy, Xiold, + Nold, Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) X = X + 4.d0 * Hd6 Kstore = Kstore + 3 * Mstar Call Approx (I, X, Valstr(Kstore), Asave(1,4), Dummy, Xiold, + Nold, Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) Kstore = Kstore + Mstar 130 Continue Go To 160 C C.... Save In Valstr The Values Of The Old Solution C.... At The Relative Positions 1/6, 2/6, 4/6 And 5/6 In C.... Each Subinterval. C 140 Kstore = 1 Do 150 I = 1, N X = Xi(I) Hd6 = (Xi(I+1) - Xi(I)) / 6.d0 Do 150 J = 1, 4 X = X + Hd6 If ( J .eq. 3 ) X = X + Hd6 Call Approx (I, X, Valstr(Kstore), Asave(1,J), Dummy, Xiold, + Nold, Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) Kstore = Kstore + Mstar 150 Continue 160 Mshflg = 0 C C.... Generate The Halved Mesh. C J = 2 Do 170 I = 1, N Xi(J) = (Xiold(I) + Xiold(I+1)) / 2.d0 Xi(J+1) = Xiold(I+1) 170 J = J + 2 N = N2 Go To 320 C C.... Mode = 1 We Do Mesh Selection If It Is Deemed Worthwhile C 180 If ( Nold .eq. 1 ) Go To 100 If ( Nold .le. 2*Nfxpnt ) Go To 100 C C.... The First Interval Has To Be Treated Separately From The C.... Other Intervals (Generally The Solution On The (I-1)St And Ith C.... Intervals Will Be Used To Approximate The Needed Derivative, But C.... Here The 1st And Second Intervals Are Used.) C I = 1 Hiold = Xiold(2) - Xiold(1) Call Horder (1, D1, Hiold, Dmz, Ncomp, K) Hiold = Xiold(3) - Xiold(2) Call Horder (2, D2, Hiold, Dmz, Ncomp, K) Accum(1) = 0.d0 Slope(1) = 0.d0 Oneovh = 2.d0 / ( Xiold(3) - Xiold(1) ) Do 190 J = 1, Ntol Jj = Jtol(J) Jz = Ltol(J) Jz1 = Jz + Mstar Zap = Min(Abs(Z(Jz)),Abs(Z(Jz1))) D3(Jj) = D2(Jj) Slope(1) = Max(Slope(1),(Abs(D2(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Max(Zap,1.d0)) **Root(J)) 190 Continue Iflip = 1 C C.... Go Through The Remaining Intervals Generating Slope C.... And Accum. C X2 = Xiold(3) Hiold = Xiold(2) - Xiold(1) Hiold1 = X2 - Xiold(2) Do 210 I = 2, Nold-1 X1 = X2 X2 = Xiold(I+2) Hioldm1 = Hiold Hiold = Hiold1 Hiold1 = X2 - X1 Do 195 J = 1,Ntol Jj = Jtol(J) If ( Iflip .eq. -1 ) D1(Jj) = D3(Jj) If ( Iflip .eq. 1 ) D2(Jj) = D3(Jj) 195 Continue Call Horder ( I+1, D3, Hiold1, Dmz, Ncomp, K) Rat1 = Max(Hiold/Hioldm1,Hioldm1/Hiold) Rat2 = Max(Hiold/Hiold1,Hiold1/Hiold) If (Rat1 .le. 2.d0 .and. Rat2 .le. 2.d0) Then Lsrs = -1 Oneovh = 2.d0 / ( Hiold + Hioldm1 ) Oneovh1 = 2.d0 / ( Hiold1 + Hiold ) Else If (Rat1 .lt. Rat2) Then Lsrs = 0 Oneovh = 2.d0 / ( Hiold + Hioldm1 ) Else Lsrs = 1 Oneovh = 2.d0 / ( Hiold1 + Hiold ) Endif Slope(I) = 0.d0 C.... Evaluate Function To Be Equidistributed Do 200 J = 1, Ntol Jj = Jtol(J) Jz = Ltol(J) + (I-1)*Mstar Jz1 = Ltol(J) + I*Mstar Zap = Min(Abs(Z(Jz)),Abs(Z(Jz1))) Zap = Max(Zap,1.d0) If (Lsrs .eq. 0) Then Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Zap) **Root(J)) Else If ((Lsrs .eq. 1) .and. (Iflip .eq. 1)) Then Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D3(Jj))*Wgtmsh(J)* + Oneovh/Zap) **Root(J)) Else If ((Lsrs .eq. 1) .and. (Iflip .eq. -1)) Then Slope(I) = Max(Slope(I),(Abs(D3(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Zap) **Root(J)) Else If ((Lsrs .eq. -1) .and. (Iflip .eq. 1)) Then Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Zap) **Root(J)) Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D3(Jj))*Wgtmsh(J)* + Oneovh1/Zap) **Root(J)) Else If ((Lsrs .eq. -1) .and. (Iflip .eq. -1)) Then Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Zap) **Root(J)) Slope(I) = Max(Slope(I),(Abs(D3(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh1/Zap) **Root(J)) Endif 200 Continue Iflip = -Iflip 210 Continue C.... Similarly To First Subinterval, Treat Last Subinterval Separately, I = Nold Slope(I) = 0.d0 Oneovh = 2.d0/(X2-Xiold(I-1)) Do 215 J = 1, Ntol Jj = Jtol(J) Jz = Ltol(J) + (I-1)*Mstar Jz1 = Ltol(J) + I * Mstar Zap = Min(Abs(Z(Jz)),Abs(Z(Jz1))) Zap = Max(Zap,1.d0) If (Iflip .eq. -1) Then Slope(I) = Max(Slope(I),(Abs(D2(Jj)-D3(Jj))*Wgtmsh(J)* + Oneovh/Max(Zap,1.d0)) **Root(J)) Else Slope(I) = Max(Slope(I),(Abs(D3(Jj)-D1(Jj))*Wgtmsh(J)* + Oneovh/Max(Zap,1.d0)) **Root(J)) Endif 215 Continue C C.... Accumulate Approximate Integral Of Function To Be Equidistributed C Accum(1) = 0.0d0 Slphmx = 0.0d0 Philrg = 0.d0 Hordlrg = 0.d0 Do 225 J = 1,Nold Temp = Slope(J)*(Xiold(J+1)-Xiold(J)) Slphmx = Max(Slphmx,Temp) Accum(J+1) = Accum(J) + Temp If (Iatt .eq. -1) Then Slpold(J) = Slope(J) Voldmsh(J) = Xiold(J) Else If (Iatt .eq. 0 .and. Slope(J) .ge. Philrg) Then Philrg = Slope(J) Hordlrg = Max(Hordlrg,Temp) Imreg = J Endif 225 Continue If (Iatt .eq. -1) Then Voldmsh(Nold+1) = Xiold(Nold+1) Nvold = Nold Endif If (Iatt .eq. 0) Then Pmax(2) = Philrg Hord(2) = Hordlrg Xloc1 = Xiold(Imreg) Xloc2 = Xiold(Imreg+1) Ichkpt = Nvold/2 If (Xloc1 .lt. Voldmsh(Ichkpt)) Ichkpt = 1 Xb = Voldmsh(Ichkpt) Do 227 J = Ichkpt, Nvold 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) = Slpold(J) Hord(1) = (Xb-Xa)*Pmax(1) Else Pmax(1) = Slpold(J+1) Hord(1) = (Voldmsh(J+2)-Xb)*Pmax(1) Endif Goto 228 Endif 227 Continue 228 Endif Avrg = Accum(Nold+1)/Dfloat(Nold) Degequ = Avrg/Max(Slphmx,Precis) C C.... Naccum = Expected N To Achieve .1x User Requested Tolerances C Naccum = Int(Accum(Nold+1) + 1.d0) If ( Iprint .lt. 0 ) Write(Iout,1002) Degequ, Naccum C C.... This Assures That Halving Will Be Possible Later (For Error Est) C Nmax2 = Nmax / 2 C C.... We Do Not Know In Advance Exactly How Many Points Will Be In C.... The New Mesh. The Parameter Nsafety Has Been Introduced So That C.... We Do Not Violate The Maximum Mesh Size. C If (Ifinal .eq. 1 .and. Iatt .eq. 0) Then If ( Iprint .lt. 1 ) Write(Iout,1007) N = Min(Nmax2-10,Naccum/2) Else N = Min(Nmax-20,Naccum) Endif 220 Noldp1 = Nold + 1 Nsafety = N If ( N .lt. Nfxp1 ) N = Nfxp1 Mshflg = 0 C C.... Having Decided To Generate A New Mesh With N Subintervals We Now C.... Do So, Taking Into Account That The Nfxpnt Points In The Array C.... Fixpnt Must Be Included In The New Mesh. C In = 1 Accl = 0.d0 Lold = 2 Xi(1) = Aleft Xi(N+1) = Aright Do 310 I = 1, Nfxp1 If ( I .eq. Nfxp1 ) Go To 250 Do 230 J = Lold, Noldp1 Lnew = J If ( Fixpnt(I) .le. Xiold(J) ) Go To 240 230 Continue 240 Continue Accr = Accum(Lnew) + (Fixpnt(I)-Xiold(Lnew))*Slope(Lnew-1) Nregn = Int((Accr-Accl) / Accum(Noldp1) * Dble(N) - 0.5d0) Nregn = Min(Nregn, N - In - Nfxp1 + I) Xi(In + Nregn + 1) = Fixpnt(I) Go To 260 250 Accr = Accum(Noldp1) Lnew = Noldp1 Nregn = N - In 260 If ( Nregn .eq. 0 ) Go To 300 Temp = Accl Tsum = (Accr - Accl) / Dfloat(Nregn+1) Do 290 J = 1, Nregn In = In + 1 Temp = Temp + Tsum Do 270 L = Lold, Lnew Lcarry = L If ( Temp .le. Accum(L) ) Go To 280 270 Continue 280 Continue Lold = Lcarry 290 Xi(In) = Xiold(Lold-1) + (Temp - Accum(Lold-1)) / + Slope(Lold-1) 300 In = In + 1 Accl = Accr Lold = Lnew 310 Continue If (N .eq. Nmax2) Goto 320 If (N .gt. Nmax2-30) Goto 321 C C.... Check That This New Mesh Will Not Create Any (Innacurately) Large C.... Subinterval Errors. If It Does Then Extra Points Must Be Added To C.... The Mesh. C I = 1 Jpt1 = 1 X2 = Xi(1) Slprgt = Slope(1) Equcon = Accum(Noldp1)/Dble(N) Equmax = 1.5d0*Equcon If (Iatt .eq. 0) Hsml = Equcon/Slope(Imreg) C 331 X1 = X2 Jpt = Jpt1 Slplft = Slprgt 335 X2 = Xi(I+1) Xrpt = Xiold(Jpt) Do 332 J = Jpt,Nold Xlpt = Xrpt Xrpt = Xiold(J+1) If (X2 .ge. Xlpt .and. X2 .le. Xrpt) Then Slprgt = Slope(J) Jpt1 = J Goto 333 Endif 332 Continue 333 If (Slplft .gt. Slprgt) Then Iend = 1 Appint = Slplft*(X2-X1) Else Iend = 2 Appint = Slprgt*(X2-X1) Endif If (Appint .gt. Equmax) Then N = N+1 Do 334 Jj = N,I+1,-1 Xi(Jj+1) = Xi(Jj) 334 Continue Requ = Max(2.d0,Appint/Equcon) Equaim = Appint/Requ If (Iend .eq. 1) Xi(I+1) = X1+Equaim/Slplft If (Iend .eq. 2) Xi(I+1) = X2-Equaim/Slprgt If (X2 .eq. Xi(I+1) .or. X1 .eq. Xi(I+1)) Then Iprec = 2 Goto 320 Endif Goto 335 Else I = I+1 If (I .lt. N+1) Goto 331 Endif 321 Continue C C.... Check Two Subintervals Approximately Same Size C If (N .eq. 1) Then Xpt = (Xi(1)+Xi(2))/2.d0 Xi(3) = Xi(2) Xi(2) = Xpt N = N+1 Endif I = 1 X1 = Xi(I) X2 = Xi(I+1) X3 = Xi(I+2) Hiold = X2 - X1 Hiold1 = X3 - X2 If (X1 .eq. X2 .or. X2 .eq. X3) Then Iprec = 2 Goto 320 Endif Rat2 = Max(Hiold1/Hiold,Hiold/Hiold1) If (Rat2 .gt. 2.d0) Then Xpt = (X1+X2)/2.d0 N = N+1 Do 400 Jj = N,2,-1 Xi(Jj+1) = Xi(Jj) 400 Continue Xi(2) = Xpt Hiold = Hiold/2.d0 I = I+1 Endif If (Rat2 .le. 2.d0 .and. N .eq. 2) Goto 320 410 I = I+1 X1 = X2 X2 = X3 X3 = Xi(I+2) If (X2 .eq. X3) Then Iprec = 2 Goto 320 Endif Hioldm1 = Hiold Hiold = Hiold1 Hiold1 = X3 - X2 Rat1 = Max(Hiold/Hioldm1,Hioldm1/Hiold) Rat2 = Max(Hiold/Hiold1,Hiold1/Hiold) If ((Rat1 .gt. 2.d0) .and. (Rat2 .gt. 2.d0)) Then Xpt = (X1+X2)/2.d0 N = N+1 Do 420 Jj = N,I+1,-1 Xi(Jj+1) = Xi(Jj) 420 Continue Xi(I+1) = Xpt Hiold = Hiold/2.d0 I = I+1 Endif If (N .eq. Nmax2) Goto 320 If (I .lt. N-1) Goto 410 Hioldm1 = Hiold Hiold = Hiold1 Rat1 = Max(Hioldm1/Hiold,Hiold/Hioldm1) If (Rat1 .gt. 2.d0) Then Xpt = (X2+X3)/2.d0 N = N+1 Xi(N+1) = X3 Xi(N) = Xpt Endif C If (N .gt. Nmax) Then N = Nsafety - 20 Goto 220 Endif 320 Continue Np1 = N + 1 If (Mode .le. 2) Then If ( Iprint .eq. -1 ) Write(Iout,1003) N, (Xi(I),I = 1,Np1) If ( Iprint .eq. 0 ) Write(Iout,1004) N Else If ( Iprint .eq. -1 ) Write(Iout,1005) N, (Xi(I),I = 1,Np1) If ( Iprint .eq. 0 ) Write(Iout,1006) N Endif Nz = Mstar * (N + 1) Ndmz = Kd * N Return C---------------------------------------------------------------- 1000 Format(/20h The Former Mesh (Of,I5,15h Subintervals),, + 600(/8f12.6)) 1001 Format (/23h Expected N Too Large ) 1002 Format(/21h Mesh Selection Info,/30h Degree Of Equidistribution = + , F8.5, 28h Prediction For Required N = , I8) 1003 Format(/17h The New Mesh (Of,I5,16h Subintervals), ,600(/8f12.6)) 1004 Format(17h The New Mesh Has,I5,13h Subintervals) 1005 Format(/21h The Initial Mesh (Of,I5,16h Subintervals), , + 600(/8f12.6)) 1006 Format(/21h The Initial Mesh Has,I5,13h Subintervals) 1007 Format(' Halving Mesh (And Then Doubling) In Order To', + ' Calculate An Error Estimate') End Subroutine Consts (K, Rho, Coef) C C********************************************************************** C C Purpose C Assign (Once) Values To Various Array Constants. C C Arrays Assigned During Compilation: C Cnsts1 - Weights For Extrapolation Error Estimate C Cnsts2 - Weights For Mesh Selection C (The Above Weights Come From The Theoretical Form For C The Collocation Error -- See [3]) C C Arrays Assigned During Execution: C Wgterr - The Particular Values Of Cnsts1 Used For Current Run C (Depending On K, M) C Wgtmsh - Gotten From The Values Of Cnsts2 Which In Turn Are C The Constants In The Theoretical Expression For The C Errors. The Quantities In Wgtmsh Are 10x The Values C In Cnsts2 So That The Mesh Selection Algorithm C Is Aiming For Errors .1x As Large As The User C Requested Tolerances. C Jtol - Components Of Differential System To Which Tolerances C Refer (Viz, If Ltol(I) Refers To A Derivative Of U(J), C Then Jtol(I) = J) C Root - Reciprocals Of Expected Rates Of Convergence Of Compo- C Nents Of Z(J) For Which Tolerances Are Specified C Rho - The K Collocation Points On (0,1) C Coef - C Acol - The Runge-Kutta Coefficients Values At Collocation C Points C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Rho(7), Coef(K,*), Cnsts1(28), Cnsts2(28), Dummy(1) C Common /Colord/ Kdum, Ncomp, Mstar, Kd, Mmax, M(20) Common /Colbas/ B(28), Acol(28,7), Asave(28,4) Common /Colest/ Wgtmsh(40), Wgterr(40), Tolin(40), + Root(40), Jtol(40), Ltol(40), Ntol C Data Cnsts1 / .25d0, .625d-1, 7.2169d-2, 1.8342d-2, + 1.9065d-2, 5.8190d-2, 5.4658d-3, 5.3370d-3, 1.8890d-2, + 2.7792d-2, 1.6095d-3, 1.4964d-3, 7.5938d-3, 5.7573d-3, + 1.8342d-2, 4.673d-3, 4.150d-4, 1.919d-3, 1.468d-3, + 6.371d-3, 4.610d-3, 1.342d-4, 1.138d-4, 4.889d-4, + 4.177d-4, 1.374d-3, 1.654d-3, 2.863d-3 / Data Cnsts2 / 1.25d-1, 2.604d-3, 8.019d-3, 2.170d-5, + 7.453d-5, 5.208d-4, 9.689d-8, 3.689d-7, 3.100d-6, + 2.451d-5, 2.691d-10, 1.120d-9, 1.076d-8, 9.405d-8, + 1.033d-6, 5.097d-13, 2.290d-12, 2.446d-11, 2.331d-10, + 2.936d-9, 3.593d-8, 7.001d-16, 3.363d-15, 3.921d-14, + 4.028d-13, 5.646d-12, 7.531d-11, 1.129d-9 / C C.... Assign Weights For Error Estimate C Koff = K * ( K + 1 ) / 2 Iz = 1 Do 10 J = 1, Ncomp Mj = M(J) Do 10 L = 1, Mj Wgterr(Iz) = Cnsts1(Koff - Mj + L) Iz = Iz + 1 10 Continue C C.... Assign Array Values For Mesh Selection: Wgtmsh, Jtol, And Root C Jcomp = 1 Mtot = M(1) Do 40 I = 1, Ntol Ltoli = Ltol(I) 20 Continue If ( Ltoli .le. Mtot ) Go To 30 Jcomp = Jcomp + 1 Mtot = Mtot + M(Jcomp) Go To 20 30 Continue Jtol(I) = Jcomp Wgtmsh(I) = 1.d1 * Cnsts2(Koff+Ltoli-Mtot) / Tolin(I) Root(I) = 1.d0 / Dfloat(K+Mtot-Ltoli+1) 40 Continue C C.... Specify Collocation Points C Go To (50,60,70,80,90,100,110), K 50 Rho(1) = 0.d0 Go To 120 60 Rho(2) = .57735026918962576451d0 Rho(1) = - Rho(2) Go To 120 70 Rho(3) = .77459666924148337704d0 Rho(2) = .0d0 Rho(1) = - Rho(3) Go To 120 80 Rho(4) = .86113631159405257523d0 Rho(3) = .33998104358485626480d0 Rho(2) = - Rho(3) Rho(1) = - Rho(4) Go To 120 90 Rho(5) = .90617984593866399280d0 Rho(4) = .53846931010568309104d0 Rho(3) = .0d0 Rho(2) = - Rho(4) Rho(1) = - Rho(5) Go To 120 100 Rho(6) = .93246951420315202781d0 Rho(5) = .66120938646626451366d0 Rho(4) = .23861918608319690863d0 Rho(3) = -Rho(4) Rho(2) = -Rho(5) Rho(1) = -Rho(6) Go To 120 110 Rho(7) = .949107991234275852452d0 Rho(6) = .74153118559939443986d0 Rho(5) = .40584515137739716690d0 Rho(4) = 0.d0 Rho(3) = -Rho(5) Rho(2) = -Rho(6) Rho(1) = -Rho(7) 120 Continue C C.... Map (-1,1) To (0,1) By T = .5 * (1. + X) C Do 130 J = 1, K Rho(J) = .5d0 * (1.d0 + Rho(J)) 130 Continue C C.... Now Find Runge-Kutta Coeffitients B, Acol And Asave C.... The Values Of Asave Are To Be Used In Newmsh And Errchk . C Do 140 J = 1, K Do 135 I = 1, K 135 Coef(I,J) = 0.d0 Coef(J,J) = 1.d0 Call Vmonde (Rho, Coef(1,J), K) 140 Continue Call Rkbas ( 1.d0, Coef, K, Mmax, B, Dummy, 0) Do 150 I = 1, K Call Rkbas ( Rho(I), Coef, K, Mmax, Acol(1,I), Dummy, 0) 150 Continue Call Rkbas ( 1.d0/6.d0, Coef, K, Mmax, Asave(1,1), Dummy, 0) Call Rkbas ( 1.d0/3.d0, Coef, K, Mmax, Asave(1,2), Dummy, 0) Call Rkbas ( 2.d0/3.d0, Coef, K, Mmax, Asave(1,3), Dummy, 0) Call Rkbas ( 5.d0/6.d0, Coef, K, Mmax, Asave(1,4), Dummy, 0) Return End Subroutine Errchk (Xi, Z, Dmz, Valstr, Ifin) C C********************************************************************** C C Purpose C Determine The Error Estimates And Test To See If The C Error Tolerances Are Satisfied. C C Variables C Xi - Current Mesh Points C Valstr - Values Of The Previous Solution Which Are Needed C For The Extrapolation- Like Error Estimate. C Wgterr - Weights Used In The Extrapolation-Like Error C Estimate. The Array Values Are Assigned In C Subroutine Consts. C Errest - Storage For Error Estimates C Err - Temporary Storage Used For Error Estimates C Z - Approximate Solution On Mesh Xi C Ifin - A 0-1 Variable. On Return It Indicates Whether C The Error Tolerances Were Satisfied C Mshflg - Is Set By Errchk To Indicate To Newmsh Whether C Any Values Of The Current Solution Are Stored In C The Array Valstr. (0 For No, 1 For Yes) C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Err(40), Errest(40), Dummy(1) Dimension Xi(*), Z(*), Dmz(*), Valstr(*) C Common /Colout/ Precis, Iout, Iprint Common /Colord/ K, Ncomp, Mstar, Kd, Mmax, M(20) Common /Colapr/ N, Nold, Nmax, Nz, Ndmz, Mshflg Common /Colbas/ B(28), Acol(28,7), Asave(28,4) Common /Colest/ Wgtmsh(40), Wgterr(40), Tolin(40), + Root(40), Jtol(40), Ltol(40), Ntol C C.... Error Estimates Are To Be Generated And Tested C.... To See If The Tolerance Requirements Are Satisfied. C Ifin = 1 Mshflg = 1 Do 10 J = 1, Mstar 10 Errest(J) = 0.d0 Do 60 Iback = 1, N I = N + 1 - Iback C C.... The Error Estimates Are Obtained By Combining Values Of C.... The Numerical Solutions For Two Meshes. C.... For Each Value Of Iback We Will Consider The Two C.... Approximations At 2 Points In Each Of C.... The New Subintervals. We Work Backwards Through C.... The Subinterval So That New Values Can Be Stored C.... In Valstr In Case They Prove To Be Needed Later C.... For An Error Estimate. The Routine Newmsh C.... Filled In The Needed Values Of The Old Solution C.... In Valstr. C Knew = ( 4 * (I-1) + 2 ) * Mstar + 1 Kstore = ( 2 * (I-1) + 1 ) * Mstar + 1 X = Xi(I) + (Xi(I+1)-Xi(I)) * 2.d0 / 3.d0 Call Approx (I, X, Valstr(Knew), Asave(1,3), Dummy, Xi, + N, Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) Do 20 L = 1,Mstar Err(L) = Wgterr(L) * Abs(Valstr(Knew) - + Valstr(Kstore)) Knew = Knew + 1 Kstore = Kstore + 1 20 Continue Knew = ( 4 * (I-1) + 1 ) * Mstar + 1 Kstore = 2 * (I-1) * Mstar + 1 X = Xi(I) + (Xi(I+1)-Xi(I)) / 3.d0 Call Approx (I, X, Valstr(Knew), Asave(1,2), Dummy, Xi, + N, Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) Do 30 L = 1,Mstar Err(L) = Err(L) + Wgterr(L) * Abs(Valstr(Knew) - + Valstr(Kstore)) Knew = Knew + 1 Kstore = Kstore + 1 30 Continue C C.... Find Component-Wise Maximum Error Estimate C Do 40 L = 1,Mstar Lzpt = L + (I-1) * Mstar Errest(L) = Max(Errest(L),Err(L)/Max(Abs(Z(Lzpt)),1.d0)) 40 Continue C C.... Test Whether The Tolerance Requirements Are Satisfied C.... In The I-Th Interval. C If ( Ifin .eq. 0 ) Go To 60 Do 50 J = 1, Ntol Ltolj = Ltol(J) Ltjz = Ltolj + (I-1) * Mstar If ( Err(Ltolj) .gt. + Tolin(J) * Max(Abs(Z(Ltjz)),1.d0) ) Ifin = 0 50 Continue 60 Continue If ( Iprint .gt. 0 ) Return If ( Ifin .eq. 0 .and. Iprint .eq. 0) Return Write(Iout,1000) Lj = 1 Do 70 J = 1,Ncomp Mj = Lj - 1 + M(J) Write(Iout,1001) J, (Errest(L), L = Lj, Mj) Lj = Mj + 1 70 Continue Return C--------------------------------------------------------------------- 1000 Format (/52h The Estimated (Mixed Relative/Absolute) Errors Are,) 1001 Format (3h U(, I2, 3h) -,4d12.4) End C--------------------------------------------------------------------- C P A R T 3 C Collocation System Setup Routines C--------------------------------------------------------------------- C Subroutine Lsyslv (Msing, Xi, Xiold, Z, Dmz, Delz, Deldmz, + G, W, V, Rhs, Dmzo, Integs, Ipvtg, Ipvtw, Rnorm, + Mode, Fsub, Dfsub, Gsub, Dgsub, Guess, Eps ) C********************************************************************* C C Purpose C This Routine Controls The Set Up And Solution Of A Linear C System Of Collocation Equations. C The Matrix G Is Cast Into An Almost Block Diagonal C Form By An Appropriate Ordering Of The Columns And Solved C Using The Package Of De Boor-Weiss [5]. The Matrix Is Composed C Of N Blocks. The I-Th Block Has The Size C Integs(1,I) * Integs(2,I). C It Contains In Its Last Rows The Linearized Collocation C Equations, Condensed As Described In [2], C And The Linearized Side Conditions Corresponding To C The I-Th Subinterval. Integs(3,I) Steps Of Gaussian C Elimination Are Applied To It To Achieve A Partial Plu C Decomposition. The Right Hand Side Vector Is Put Into Rhs C And The Solution Vector Is Returned In Delz And Deldmz. C C Lsyslv Operates According To One Of 5 Modes: C Mode = 0 - Set Up The Collocation Matrices V , W , G C And The Right Hand Side Rhs , And Solve. C (For Linear Problems Only.) C Mode = 1 - Set Up The Collocation Matrices V , W , G C And The Right Hand Sides Rhs And Dmzo , C And Solve. Also Set Up Integs . C (First Iteration Of Nonlinear Problems Only). C Mode = 2 - Set Up Rhs Only And Compute Its Norm. C Mode = 3 - Set Up V, W, G Only And Solve System. C Mode = 4 - Perform Forward And Backward Substitution Only C (Do Not Set Up The Matrices Nor Form The Rhs). C C Variables C C Ig,Izeta - Pointers To G,Zeta Respectively C (Necessary To Keep Track Of Blocks Of G C During Matrix Manipulations) C Idmz,Irhs,Iv,Iw - Pointers To Rhs,V,W Rspectively C Df - Partial Derivatives Of F From Dfsub C Rnorm - Euclidean Norm Of Rhs C Lside - Number Of Side Conditions In Current And Previous Blocks C Iguess = 1 When Current Soln Is User Specified Via Guess C = 0 Otherwise C C********************************************************************* Implicit Double Precision (A-H,O-Z) Dimension Z(*), Dmz(*), Delz(*), Deldmz(*), Xi(*), Xiold(*) Dimension G(*), W(*), V(*), Rhs(*) , Dmzo(*), Dummy(1) Dimension Integs(3,*), Ipvtg(*), Ipvtw(*) Dimension Zval(40), F(40), Dgz(40), Dmval(20), Df(800), At(28) C Common /Colout/ Precis, Iout, Iprint Common /Colloc/ Rho(7), Coef(49) Common /Colord/ K, Ncomp, Mstar, Kd, Mmax, M(20) Common /Colsid/ Zeta(40), Aleft, Aright, Izeta, Izsave Common /Colapr/ N, Nold, Nmax, Nz, Ndmz, Mshflg Common /Colnln/ Nonlin, Iter, Limit, Iguess Common /Colbas/ B(28), Acol(28,7), Asave(28,4) C External Dfsub, Dgsub C M1 = Mode + 1 Go To (10, 30, 30, 30, 310), M1 C C.... Linear Problem Initialization C 10 Do 20 I = 1,Mstar 20 Zval(I) = 0.d0 C C.... Initialization C 30 Idmz = 1 Idmzo = 1 Irhs = 1 Ig = 1 Iw = 1 Iv = 1 Izeta = 1 Lside = 0 Iold = 1 Ncol = 2 * Mstar Rnorm = 0.d0 If ( Mode .gt. 1 ) Go To 80 C C.... Build Integs (Describing Block Structure Of Matrix) C Do 70 I = 1,N Integs(2,I) = Ncol If (I .lt. N) Go To 40 Integs(3,N) = Ncol Lside = Mstar Go To 60 40 Integs(3,I) = Mstar 50 If( Lside .eq. Mstar ) Go To 60 If ( Zeta(Lside+1) .ge. Xi(I)+Precis ) Go To 60 Lside = Lside + 1 Go To 50 60 Nrow = Mstar + Lside 70 Integs(1,I) = Nrow 80 Continue If ( Mode .eq. 2 ) Go To 90 C C.... Zero The Matrices To Be Computed C Lw = Kd * Kd * N Do 84 L = 1, Lw 84 W(L) = 0.d0 C C.... The Do Loop 290 Sets Up The Linear System Of Equations. C 90 Continue Do 290 I = 1, N C C.... Construct A Block Of A And A Corresponding Piece Of Rhs. C Xii = Xi(I) H = Xi(I+1) - Xi(I) Nrow = Integs(1,I) C C.... Go Thru The Ncomp Collocation Equations And Side Conditions C.... In The I-Th Subinterval C 100 If ( Izeta .gt. Mstar ) Go To 140 If ( Zeta(Izeta) .gt. Xii + Precis ) Go To 140 C C.... Build Equation For A Side Condition. C If ( Mode .eq. 0 ) Go To 110 If ( Iguess .ne. 1 ) Go To 102 C C.... Case Where User Provided Current Approximation C Call Guess (Xii, Zval, Dmval, Eps) Go To 110 C C.... Other Nonlinear Case C 102 If ( Mode .ne. 1 ) Go To 106 Call Approx (Iold, Xii, Zval, At, Coef, Xiold, Nold, + Z, Dmz, K, Ncomp, Mmax, M, Mstar, 2, Dummy, 0) Go To 110 106 Call Approx (I, Xii, Zval, At, Dummy, Xi, N, Z, Dmz, + K, Ncomp, Mmax, M, Mstar, 1, Dummy, 0) 108 If ( Mode .eq. 3 ) Go To 120 C C.... Find Rhs Boundary Value. C 110 Call Gsub (Izeta, Zval, Gval, Eps) Rhs(Ndmz+Izeta) = -Gval Rnorm = Rnorm + Gval**2 If ( Mode .eq. 2 ) Go To 130 C C.... Build A Row Of A Corresponding To A Boundary Point C 120 Call Gderiv (G(Ig), Nrow, Izeta, Zval, Dgz, 1, Dgsub, Eps) 130 Izeta = Izeta + 1 Go To 100 C C.... Assemble Collocation Equations C 140 Do 220 J = 1, K Hrho = H * Rho(J) Xcol = Xii + Hrho C C.... This Value Corresponds To A Collocation (Interior) C.... Point. Build The Corresponding Ncomp Equations. C If ( Mode .eq. 0 ) Go To 200 If ( Iguess .ne. 1 ) Go To 160 C C.... Use Initial Approximation Provided By The User. C Call Guess (Xcol, Zval, Dmzo(Irhs), Eps ) Go To 170 C C.... Find Rhs Values C 160 If ( Mode .ne. 1 ) Go To 190 Call Approx (Iold, Xcol, Zval, At, Coef, Xiold, Nold, + Z, Dmz, K, Ncomp, Mmax, M, Mstar, 2, Dmzo(Irhs), 1) C 170 Call Fsub (Xcol, Zval, F, Eps) Do 180 Jj = 1, Ncomp Value = Dmzo(Irhs) - F(Jj) Rhs(Irhs) = - Value Rnorm = Rnorm + Value**2 Irhs = Irhs + 1 180 Continue Go To 210 C C.... Evaluate Former Collocation Solution C 190 Call Approx (I, Xcol, Zval, Acol(1,J), Coef, Xi, N, + Z, Dmz, K, Ncomp, Mmax, M, Mstar, 4, Dummy, 0) If ( Mode .eq. 3 ) Go To 210 C C.... Fill In Rhs Values (And Accumulate Its Norm). C Call Fsub (Xcol, Zval, F, Eps) Do 195 Jj = 1, Ncomp Value = Dmz(Irhs) - F(Jj) Rhs(Irhs) = - Value Rnorm = Rnorm + Value**2 Irhs = Irhs + 1 195 Continue Go To 220 C C.... The Linear Case C 200 Call Fsub (Xcol, Zval, Rhs(Irhs), Eps) Irhs = Irhs + Ncomp C C.... Fill In Ncomp Rows Of W And V C 210 Call Vwblok (Xcol, Hrho, J, W(Iw), V(Iv), Ipvtw(Idmz), Kd, + Zval, Df, Acol(1,J), Dmzo(Idmzo), Ncomp, Dfsub, Msing, Eps) If ( Msing .ne. 0 ) Return 220 Continue C C.... Build Global Bvp Matrix G C If ( Mode .ne. 2 ) + Call Gblock (H, G(Ig), Nrow, Izeta, W(Iw), V(Iv), Kd, + Dummy, Deldmz(Idmz), Ipvtw(Idmz), 1 ) If ( I .lt. N ) Go To 280 Izsave = Izeta 240 If ( Izeta .gt. Mstar ) Go To 290 C C.... Build Equation For A Side Condition. C If ( Mode .eq. 0 ) Go To 250 If ( Iguess .ne. 1 ) Go To 245 C C.... Case Where User Provided Current Approximation C Call Guess (Aright, Zval, Dmval, Eps) Go To 250 C C.... Other Nonlinear Case C 245 If ( Mode .ne. 1 ) Go To 246 Call Approx (Nold+1, Aright, Zval, At, Coef, Xiold, Nold, + Z, Dmz, K, Ncomp, Mmax, M, Mstar, 1, Dummy, 0) Go To 250 246 Call Approx (N+1, Aright, Zval, At, Coef, Xi, N, + Z, Dmz, K, Ncomp, Mmax, M, Mstar, 1, Dummy, 0) 248 If ( Mode .eq. 3 ) Go To 260 C C.... Find Rhs Boundary Value. C 250 Call Gsub (Izeta, Zval, Gval, Eps) Rhs(Ndmz+Izeta) = - Gval Rnorm = Rnorm + Gval**2 If ( Mode .eq. 2 ) Go To 270 C C.... Build A Row Of A Corresponding To A Boundary Point C 260 Izm = Izeta+Mstar Call Gderiv (G(Ig), Nrow, Izm, Zval, Dgz, 2, Dgsub, Eps) 270 Izeta = Izeta + 1 Go To 240 C C.... Update Counters -- I-Th Block Completed C 280 Ig = Ig + Nrow * Ncol Iv = Iv + Kd * Mstar Iw = Iw + Kd * Kd Idmz = Idmz + Kd If ( Mode .eq. 1 ) Idmzo = Idmzo + Kd 290 Continue C C.... Assembly Process Completed C If ( Mode .eq. 0 .or. Mode .eq. 3 ) Go To 300 Rnorm = Sqrt(Rnorm / Dfloat(Nz+Ndmz)) If ( Mode .ne. 2 ) Go To 300 Return C C.... Solve The Linear System. C C.... Matrix Decomposition C 300 Call Fcblok (G, Integs, N, Ipvtg, Df, Msing) C C.... Check For Singular Matrix C Msing = - Msing If( Msing .ne. 0 ) Return C C.... Perform Forward And Backward Substitution For Mode = 4 Only. C 310 Continue Do 311 L = 1, Ndmz Deldmz(L) = Rhs(L) 311 Continue Iz = 1 Idmz = 1 Iw = 1 Izet = 1 Do 320 I = 1, N Nrow = Integs(1,I) Izeta = Nrow + 1 - Mstar If ( I .eq. N ) Izeta = Izsave 322 If ( Izet .eq. Izeta ) Go To 324 Delz(Iz-1+Izet) = Rhs(Ndmz+Izet) Izet = Izet + 1 Go To 322 324 H = Xi(I+1) - Xi(I) Call Gblock (H, G(1), Nrow, Izeta, W(Iw), V(1), Kd, + Delz(Iz), Deldmz(Idmz), Ipvtw(Idmz), 2 ) Iz = Iz + Mstar Idmz = Idmz + Kd Iw = Iw + Kd * Kd If ( I .lt. N ) Go To 320 326 If ( Izet .gt. Mstar ) Go To 320 Delz(Iz-1+Izet) = Rhs(Ndmz+Izet) Izet = Izet + 1 Go To 326 320 Continue C C.... Perform Forward And Backward Substitution For Mode = 0,2, Or 3. C Call Sbblok (G, Integs, N, Ipvtg, Delz) C C.... Finaly Find Deldmz C Call Dmzsol (Kd, Mstar, N, V, Delz, Deldmz) C If ( Mode .ne. 1 ) Return Do 321 L = 1, Ndmz Dmz(L) = Dmzo(L) 321 Continue Iz = 1 Idmz = 1 Iw = 1 Izet = 1 Do 350 I = 1, N Nrow = Integs(1,I) Izeta = Nrow + 1 - Mstar If ( I .eq. N ) Izeta = Izsave 330 If ( Izet .eq. Izeta ) Go To 340 Z(Iz-1+Izet) = Dgz(Izet) Izet = Izet + 1 Go To 330 340 H = Xi(I+1) - Xi(I) Call Gblock (H, G(1), Nrow, Izeta, W(Iw), Df, Kd, + Z(Iz), Dmz(Idmz), Ipvtw(Idmz), 2 ) Iz = Iz + Mstar Idmz = Idmz + Kd Iw = Iw + Kd * Kd If ( I .lt. N ) Go To 350 342 If ( Izet .gt. Mstar ) Go To 350 Z(Iz-1+Izet) = Dgz(Izet) Izet = Izet + 1 Go To 342 350 Continue Call Sbblok (G, Integs, N, Ipvtg, Z) C C.... Finaly Find Dmz C Call Dmzsol (Kd, Mstar, N, V, Z, Dmz) C Return End Subroutine Gderiv ( Gi, Nrow, Irow, Zval, Dgz, Mode, Dgsub, Eps) C C********************************************************************** C C Purpose: C C Construct A Collocation Matrix Row According To Mode: C Mode = 1 - A Row Corresponding To A Initial Condition C (I.e. At The Left End Of The Subinterval). C Mode = 2 - A Row Corresponding To A Final Condition. C C Variables: C C Gi - The Sub-Block Of The Global Bvp Matrix In C Which The Equations Are To Be Formed. C Nrow - No. Of Rows In Gi. C Irow - The Row In Gi To Be Used For Equations. C Zval - Z(Xi) C Dg - The Derivatives Of The Side Condition. C C********************************************************************** Implicit Double Precision (A-H,O-Z) Dimension Gi(Nrow,*), Zval(*), Dgz(*), Dg(40) C Common /Colord/ Kdum, Ndum, Mstar, Kd, Mmax, M(20) Common /Colsid/ Zeta(40), Aleft, Aright, Izeta, Idum Common /Colnln/ Nonlin, Iter, Limit, Iguess C C.... Zero Jacobian Dg C Do 10 J = 1,Mstar 10 Dg(J) = 0.d0 C C.... Evaluate Jacobian Dg C Call Dgsub (Izeta, Zval, Dg, Eps) C C.... Evaluate Dgz = Dg * Zval Once For A New Mesh C If (Nonlin .eq. 0 .or. Iter .gt. 0) Go To 30 Dot = 0.d0 Do 20 J = 1, Mstar 20 Dot = Dot + Dg(J) * Zval(J) Dgz(Izeta) = Dot C C.... Branch According To M O D E C 30 If ( Mode .eq. 2 ) Go To 50 C C.... Provide Coefficients Of The J-Th Linearized Side Condition. C.... Specifically, At X = Zeta(J) The J-Th Side Condition Reads C.... Dg(1)*Z(1) + ... +Dg(Mstar)*Z(Mstar) + G = 0 C C C.... Handle An Initial Condition C Do 40 J = 1, Mstar Gi(Irow,J) = Dg(J) 40 Gi(Irow,Mstar+J) = 0.d0 Return C C.... Handle A Final Condition C 50 Do 60 J = 1, Mstar Gi(Irow,J) = 0.d0 60 Gi(Irow,Mstar+J) = Dg(J) Return End Subroutine Vwblok (Xcol, Hrho, Jj, Wi, Vi, Ipvtw, Kd, Zval, + Df, Acol, Dmzo, Ncomp, Dfsub, Msing, Eps) C C********************************************************************** C C Purpose: C C Construct A Group Of Ncomp Rows Of The Matrices Wi And Vi. C Corresponding To An Interior Collocation Point. C C C Variables: C C Xcol - The Location Of The Collocation Point. C Jj - Xcol Is The Jj-Th Of K Collocation Points C In The I-Th Subinterval. C Wi,Vi - The I-Th Block Of The Collocation Matrix C Before Parameter Condensation. C Kd - No. Of Rows In Vi And Wi . C Zval - Z(Xcol) C Df - The Jacobian At Xcol . C Jcomp - Counter For The Component Being Dealt With. C C********************************************************************** Implicit Double Precision (A-H,O-Z) Dimension Wi(Kd,*), Vi(Kd,*), Zval(*), Dmzo(*), Df(Ncomp,*) Dimension Ipvtw(*), Ha(7,4), Acol(7,4), Basm(5) C Common /Colord/ K, Ncdum, Mstar, Kdum, Mmax, M(20) Common /Colnln/ Nonlin, Iter, Limit, Iguess C C.... If Jj = 1 Initialize Wi . C If ( Jj .gt. 1 ) Go To 30 Do 10 Id = 1, Kd Wi(Id,Id) = 1.d0 10 Continue C C.... Calculate Local Basis C 30 Fact = 1.d0 Do 150 L = 1,Mmax Fact = Fact * Hrho / Dfloat(L) Basm(L) = Fact Do 150 J = 1,K Ha(J,L) = Fact * Acol(J,L) 150 Continue C C....zero Jacobian C Do 40 Jcol = 1, Mstar Do 40 Ir = 1, Ncomp 40 Df(Ir,Jcol) = 0.d0 C C.... Build Ncomp Rows For Interior Collocation Point X. C.... The Linear Expressions To Be Constructed Are: C.... (M(Id)) C.... U - Df(Id,1)*Z(1) - ... - Df(Id,Mstar)*Z(Mstar) C.... Id C.... For Id = 1 To Ncomp. C Call Dfsub (Xcol, Zval, Df, Ncomp, Eps) I0 = (Jj-1) * Ncomp I1 = I0 + 1 I2 = I0 + Ncomp C C.... Evaluate Dmzo = Dmz - Df * Zval Once For A New Mesh C If (Nonlin .eq. 0 .or. Iter .gt. 0) Go To 60 Do 50 J = 1, Mstar Fact = - Zval(J) Do 50 Id = 1, Ncomp Dmzo(I0+Id) = Dmzo(I0+Id) + Fact * Df(Id,J) 50 Continue C C.... Loop Over The Ncomp Expressions To Be Set Up For The C.... Current Collocation Point. C 60 Do 70 J = 1, Mstar Do 70 Id = 1, Ncomp Vi(I0+Id,J) = Df(Id,J) 70 Continue Jn = 1 Do 140 Jcomp = 1, Ncomp Mj = M(Jcomp) Jn = Jn + Mj Do 130 L = 1, Mj Jv = Jn - L Jw = Jcomp Do 90 J = 1, K Ajl = - Ha(J,L) Do 80 Iw = I1, I2 Wi(Iw,Jw) = Wi(Iw,Jw) + Ajl * Vi(Iw,Jv) 80 Continue 90 Jw = Jw + Ncomp Lp1 = L + 1 If ( L .eq. Mj ) Go To 130 Do 110 Ll = Lp1, Mj Jdf = Jn - Ll Bl = Basm(Ll-L) Do 100 Iw = I1, I2 Vi(Iw,Jv) = Vi(Iw,Jv) + Bl * Vi(Iw,Jdf) 100 Continue 110 Continue 130 Continue 140 Continue If ( Jj .lt. K ) Return C C ...decompose The Wi Block And Solve For The Mstar Columns Of Vi C C C.... Do Parameter Condensation C Msing = 0 Call Dgefa (Wi, Kd, Kd, Ipvtw, Msing) C C.... Check For Singularity C If ( Msing .ne. 0 ) Return Do 250 J = 1,Mstar Call Dgesl (Wi, Kd, Kd, Ipvtw, Vi(1,J), 0) 250 Continue Return End Subroutine Gblock (H, Gi, Nrow, Irow, Wi, Vi, Kd, + Rhsz, Rhsdmz, Ipvtw, Mode) C C********************************************************************** C C Purpose: C C Construct Collocation Matrix Rows According To Mode: C Mode = 1 - A Group Of Mstar Rows Corresponding C An Interior Mesh Interval. C = 2 - Corresponding Right Hand Side C C Variables: C C H - The Local Stepsize. C Gi - The Sub-Block Of The Collocation Matrix In C Which The Equations Are To Be Formed. C Wi - The Sub-Block Of Noncondensed Collocation Equations, C Left-Hand Side Part. C Vi - The Sub-Block Of Noncondensed Collocation Equations, C Right-Hand Side Part. C Rhsdmz - The Inhomogenous Term Of The Uncondensed Collocation C Equations. C Rhsz - The Inhomogenous Term Of The Condensed Collocation C Equations. C Nrow - No. Of Rows In Gi. C Irow - The First Row In Gi To Be Used For Equations. C C********************************************************************** Implicit Double Precision (A-H,O-Z) Dimension Hb(7,4), Basm(5) Dimension Gi(Nrow,*), Wi(*), Vi(Kd,*) Dimension Rhsz(*), Rhsdmz(*), Ipvtw(*) C Common /Colord/ K, Ncomp, Mstar, Kdum, Mmax, M(20) Common /Colbas/ B(7,4), Acol(28,7), Asave(28,4) C C.... Compute Local Basis C Fact = 1.d0 Basm(1) = 1.d0 Do 30 L = 1,Mmax Fact = Fact * H / Dfloat(L) Basm(L+1) = Fact Do 20 J = 1,K 20 Hb(J,L) = Fact * B(J,L) 30 Continue C C.... Branch According To M O D E C Go To (40, 110), Mode C C.... Set Right Gi-Block To Identity C 40 Continue Do 60 J = 1, Mstar Do 50 Ir = 1, Mstar Gi(Irow-1+Ir,J) = 0.d0 50 Gi(Irow-1+Ir,Mstar+J) = 0.d0 60 Gi(Irow-1+J,Mstar+J) = 1.d0 C C.... Compute The Block Gi C Ir = Irow Do 100 Icomp = 1, Ncomp Mj = M(Icomp) Ir = Ir + Mj Do 90 L = 1, Mj Id = Ir - L Do 80 Jcol = 1, Mstar Ind = Icomp Rsum = 0.d0 Do 70 J = 1, K Rsum = Rsum - Hb(J,L) * Vi(Ind,Jcol) 70 Ind = Ind + Ncomp Gi(Id,Jcol) = Rsum 80 Continue Jd = Id - Irow Do 85 Ll = 1, L Gi(Id,Jd+Ll) = Gi(Id,Jd+Ll) - Basm(Ll) 85 Continue 90 Continue 100 Continue Return C C.... Compute The Appropriate Piece Of Rhsz C 110 Continue Call Dgesl (Wi, Kd, Kd, Ipvtw, Rhsdmz, 0) Ir = Irow Do 140 Jcomp = 1, Ncomp Mj = M(Jcomp) Ir = Ir + Mj Do 130 L = 1, Mj Ind = Jcomp Rsum = 0.d0 Do 120 J = 1, K Rsum = Rsum + Hb(J,L) * Rhsdmz(Ind) 120 Ind = Ind + Ncomp Rhsz(Ir-L) = Rsum 130 Continue 140 Continue Return End C C---------------------------------------------------------------------- C P A R T 4 C Polynomial And Service Routines C---------------------------------------------------------------------- C Subroutine Appsln (X, Z, Fspace, Ispace) C C***************************************************************** C C Purpose C C Set Up A Standard Call To Approx To Evaluate The C Approximate Solution Z = Z( U(X) ) At A Point X C (It Has Been Computed By A Call To Colmod ). C The Parameters Needed For Approx Are Retrieved C From The Work Arrays Ispace And Fspace . C C***************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Z(*), Fspace(*), Ispace(*), A(28), Dummy(1) Is6 = Ispace(6) Is5 = Ispace(1) + 2 Is4 = Is5 + Ispace(4) * (Ispace(1) + 1) I = 1 Call Approx (I, X, Z, A, Fspace(Is6), Fspace(1), Ispace(1), + Fspace(Is5), Fspace(Is4), Ispace(2), Ispace(3), + Ispace(5), Ispace(8), Ispace(4), 2, Dummy, 0) Return End Subroutine Approx (I, X, Zval, A, Coef, Xi, N, Z, Dmz, K, + Ncomp, Mmax, M, Mstar, Mode, Dmval, Modm ) C C********************************************************************** C C Purpose C (1) (M1-1) (Mncomp-1) C Evaluate Z(U(X))=(U (X),U (X),...,U (X),...,U (X) ) C 1 1 1 Mncomp C At One Point X. C C Variables C A - Array Of Mesh Independent Rk-Basis Coefficients C Basm - Array Of Mesh Dependent Monomial Coefficients C Xi - The Current Mesh (Having N Subintervals) C Z - The Current Solution Vector C Dmz - The Array Of Mj-Th Derivatives Of The Current Solution C Mode - Determines The Amount Of Initialization Needed C = 4 Forms Z(U(X)) Using Z, Dmz And Ha C = 3 As In = 4, But Computes Local Rk-Basis C = 2 As In = 3, But Determines I Such That C Xi(I) .le. X .lt. Xi(I+1) (Unless X = Xi(N+1)) C = 1 Retrieve Z = Z(U(X(I))) Directly C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Zval(*), Dmval(*), Xi(*), M(*), A(7,*), Dm(7) Dimension Z(*), Dmz(*), Bm(4), Coef(*) C Common /Colout/ Precis, Iout, Iprint C Go To (10, 30, 80, 90), Mode C C.... Mode = 1 , Retrieve Z( U(X) ) Directly For X = Xi(I). C 10 X = Xi(I) Iz = (I-1) * Mstar Do 20 J = 1, Mstar Iz = Iz + 1 Zval(J) = Z(Iz) 20 Continue Return C C.... Mode = 2 , Locate I So Xi(I) .le. X .lt. Xi(I+1) C 30 Continue If ( X .ge. Xi(1)-Precis .and. X .le. Xi(N+1)+Precis ) + Go To 40 If (Iprint .lt. 1) Write(Iout,1000) X, Xi(1), Xi(N+1) If ( X .lt. Xi(1) ) X = Xi(1) If ( X .gt. Xi(N+1) ) X = Xi(N+1) 40 If ( I .gt. N .or. I .lt. 1 ) I = (N+1) / 2 Ileft = I If ( X .lt. Xi(Ileft) ) Go To 60 Do 50 L = Ileft, N I = L If ( X .lt. Xi(L+1) ) Go To 80 50 Continue Go To 80 60 Iright = Ileft - 1 Do 70 L = 1, Iright I = Iright + 1 - L If ( X .ge. Xi(I) ) Go To 80 70 Continue C C.... Mode = 2 Or 3 , Compute Mesh Independent Rk-Basis. C 80 Continue S = (X - Xi(I)) / (Xi(I+1) - Xi(I)) Call Rkbas ( S, Coef, K, Mmax, A, Dm, Modm ) C C.... Mode = 2, 3, Or 4 , Compute Mesh Dependent Rk-Basis. C 90 Continue Bm(1) = X - Xi(I) Do 95 L = 2, Mmax Bm(L) = Bm(1) / Dfloat(L) 95 Continue C C.... Evaluate Z( U(X) ). C 100 Ir = 1 Iz = (I-1) * Mstar + 1 Idmz = (I-1) * K * Ncomp Do 140 Jcomp = 1, Ncomp Mj = M(Jcomp) Ir = Ir + Mj Iz = Iz + Mj Do 130 L = 1, Mj Ind = Idmz + Jcomp Zsum = 0.d0 Do 110 J = 1, K Zsum = Zsum + A(J,L) * Dmz(Ind) 110 Ind = Ind + Ncomp Do 120 Ll = 1, L Lb = L + 1 - Ll 120 Zsum = Zsum * Bm(Lb) + Z(Iz-Ll) 130 Zval(Ir-L) = Zsum 140 Continue If ( Modm .eq. 0 ) Return C C.... For Modm = 1 Evaluate Dmval(J) = Mj-Th Derivative Of Uj. C Do 150 Jcomp = 1, Ncomp 150 Dmval(Jcomp) = 0.d0 Idmz = Idmz + 1 Do 170 J = 1, K Fact = Dm(J) Do 160 Jcomp = 1, Ncomp Dmval(Jcomp) = Dmval(Jcomp) + Fact * Dmz(Idmz) Idmz = Idmz + 1 160 Continue 170 Continue Return C-------------------------------------------------------------------- 1000 Format(37h ****** Domain Error In Approx ****** + /4h X = ,D20.10, 10h Aleft = ,D20.10, + 11h Aright = ,D20.10) End Subroutine Rkbas (S, Coef, K, M, Rkb, Dm, Mode) C C********************************************************************** C C Purpose C Evaluate Mesh Independent Runge-Kutta Basis For Given S C C Variables C S - Argument, I.e. The Relative Position For Which C The Basis Is To Be Evaluated ( 0. .le. S .le. 1. ). C Coef - Precomputed Derivatives Of The Basis C K - Number Of Collocatin Points Per Subinterval C M - Maximal Order Of The Differential Equation C Rkb - The Runge-Kutta Basis (0-Th To (M-1)-Th Derivatives ) C Dm - Basis Elements For M-Th Derivative C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Coef(K,*), Rkb(7,*), Dm(*), T(10) C If ( K .eq. 1 ) Go To 70 Kpm1 = K + M - 1 Do 10 I = 1, Kpm1 10 T(I) = S / Dfloat(I) Do 40 L = 1, M Lb = K + L + 1 Do 30 I = 1, K P = Coef(1,I) Do 20 J = 2, K P = P * T(Lb-J) + Coef(J,I) 20 Continue Rkb(I,L) = P 30 Continue 40 Continue If ( Mode .eq. 0 ) Return Do 60 I = 1, K P = Coef(1,I) Do 50 J = 2, K 50 P = P * T(K+1-J) + Coef(J,I) Dm(I) = P 60 Continue Return 70 Rkb(1,1) = 1.0d0 Dm(1) = 1.0d0 Return End Subroutine Vmonde ( Rho, Coef, K ) C C********************************************************************** C C Purpose C Solve Vandermonde System V * X = E C With V(I,J) = Rho(J)**(I-1)/(I-1)! . C C********************************************************************** C Integer K, I,Ifac,J,Km1,Kmi Double Precision Rho(K), Coef(K) C If ( K .eq. 1 ) Return Km1 = K - 1 Do 10 I = 1, Km1 Kmi = K - I Do 10 J = 1, Kmi Coef(J) = (Coef(J+1) - Coef(J)) / (Rho(J+I) - Rho(J)) 10 Continue C Ifac = 1 Do 40 I = 1, Km1 Kmi = K + 1 - I Do 30 J = 2, Kmi 30 Coef(J) = Coef(J) - Rho(J+I-1) * Coef(J-1) Coef(Kmi) = Dfloat(Ifac) * Coef(Kmi) Ifac = Ifac * I 40 Continue Coef(1) = Dfloat(Ifac) * Coef(1) Return End Subroutine Horder (I, Uhigh, Hi, Dmz, Ncomp, K) C C********************************************************************** C C Purpose C Determine Highest Order (Piecewise Constant) Derivatives C Of The Current Collocation Solution C C Variables C Hi - The Stepsize, Hi = Xi(I+1) - Xi(I) C Dmz - Vector Of Mj-Th Derivative Of The Solution C Uhigh - The Array Of Highest Order (Piecewise Constant) C Derivatives Of The Approximate Solution On C (Xi(I),Xi(I+1)), Viz, C (K+Mj-1) C Uhigh(J) = U (X) On (Xi(I),Xi(I+1)) C J C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension Uhigh(*), Dmz(*) C Common /Colloc/ Rho(7), Coef(49) C Dn = 1.d0 / Hi**(K-1) C C.... Loop Over The Ncomp Solution Components C Do 10 Id = 1, Ncomp Uhigh(Id) = 0.d0 10 Continue Kin = 1 Idmz = (I-1) * K * Ncomp + 1 Do 30 J = 1, K Fact = Dn * Coef(Kin) Do 20 Id = 1, Ncomp Uhigh(Id) = Uhigh(Id) + Fact * Dmz(Idmz) Idmz = Idmz + 1 20 Continue Kin = Kin + K 30 Continue Return End Subroutine Dmzsol (Kd, Mstar, N, V, Z, Dmz) C C********************************************************************** C C Purpose C Compute Dmz In A Blockwise Manner C Dmz(I) = Dmz(I) + V(I) * Z(I), I = 1,...,N C C********************************************************************** C Implicit Double Precision (A-H,O-Z) Dimension V(Kd,*), Dmz(Kd,*), Z(*) C Jz = 1 Do 30 I = 1, N Do 20 J = 1, Mstar Fact = Z(Jz) Do 10 L = 1, Kd Dmz(L,I) = Dmz(L,I) + Fact * V(L,Jz) 10 Continue Jz = Jz + 1 20 Continue 30 Continue Return End C---------------------------------------------------------------------- C P A R T 5 C We List Here A Modified (Column Oriented, Faster) C Version Of The Package Solveblok Of De Boor - Weiss [5]. C We Also Give A Listing Of The Linpack C Routines Dgefa Und Dgesl Used By Colmod. C---------------------------------------------------------------------- C Subroutine Fcblok (Bloks, Integs, Nbloks, Ipivot, Scrtch, Info) C C C Calls Subroutines Factrb And Shiftb . C C Fcblok Supervises The Plu Factorization With Pivoting Of C Scaled Rows Of The Almost Block Diagonal Matrix Stored In The C Arrays Bloks And Integs . C C Factrb = Subprogram Which Carries Out Steps 1,...,Last Of Gauss C Elimination (With Pivoting) For An Individual Block. C Shiftb = Subprogram Which Shifts The Remaining Rows To The Top Of C The Next Block C C Parameters C Bloks An Array That Initially Contains The Almost Block Diago- C Nal Matrix A To Be Factored, And On Return Contains The C Computed Factorization Of A . C Integs An Integer Array Describing The Block Structure Of A . C Nbloks The Number Of Blocks In A . C Ipivot An Integer Array Of Dimension Sum (Integs(3,N) ; N = 1, C ...,Nbloks) Which, On Return, Contains The Pivoting Stra- C Tegy Used. C Scrtch Work Area Required, Of Length Max (Integs(1,N) ; N = 1, C ...,Nbloks). C Info Output Parameter; C = 0 In Case Matrix Was Found To Be Nonsingular. C Otherwise, C = N If The Pivot Element In The Nth Gauss Step Is Zero. C C********************************************************************** C Integer Integs(3,Nbloks),Ipivot(*),Info, I,Index,Indexn,Last, + Ncol,Nrow Double Precision Bloks(*),Scrtch(*) Info = 0 Indexx = 1 Indexn = 1 I = 1 C C.... Loop Over The Blocks. I Is Loop Index C 10 Index = Indexn Nrow = Integs(1,I) Ncol = Integs(2,I) Last = Integs(3,I) C C.... Carry Out Elimination On The I-Th Block Until Next Block C.... Enters, I.e., For Columns 1,...,Last Of I-Th Block. C Call Factrb ( Bloks(Index), Ipivot(Indexx), Scrtch, Nrow, + Ncol, Last, Info) C C.... Check For Having Reached A Singular Block Or The Last Block C If ( Info .ne. 0 ) Go To 20 If ( I .eq. Nbloks ) Return I = I+1 Indexn = Nrow * Ncol + Index Indexx = Indexx + Last C C.... Put The Rest Of The I-Th Block Onto The Next Block C Call Shiftb ( Bloks(Index), Nrow, Ncol, Last, + Bloks(Indexn), Integs(1,I), Integs(2,I) ) Go To 10 20 Info = Info + Indexx - 1 Return End Subroutine Factrb ( W, Ipivot, D, Nrow, Ncol, Last, Info) C C******************************************************************** C C Adapted From P.132 Of Element.numer.analysis By Conte-De Boor C C Constructs A Partial Plu Factorization, Corresponding To Steps C 1,..., Last In Gauss Elimination, For The Matrix W Of C Order ( Nrow , Ncol ), Using Pivoting Of Scaled Rows. C C Parameters C W Contains The (Nrow,Ncol) Matrix To Be Partially Factored C On Input, And The Partial Factorization On Output. C Ipivot An Integer Array Of Length Last Containing A Record Of C The Pivoting Strategy Used; Explicit Interchanges C Are Used For Pivoting. C D A Work Array Of Length Nrow Used To Store Row Sizes C Temporarily. C Nrow Number Of Rows Of W. C Ncol Number Of Columns Of W. C Last Number Of Elimination Steps To Be Carried Out. C Info On Output, Zero If The Matrix Is Found To Be Non- C Singular, In Case A Zero Pivot Was Encountered In Row C N, Info = N On Output. C C********************************************************************** C Integer Ipivot(Nrow),Ncol,Last,Info, I,J,K,L,Kp1 Double Precision W(Nrow,Ncol),D(Nrow), Colmax,T,S Double Precision Abs,Max C C.... Initialize D C Do 10 I = 1, Nrow D(I) = 0.d0 10 Continue Do 20 J = 1, Ncol Do 20 I = 1, Nrow D(I) = Max( D(I) , Abs(W(I,J))) 20 Continue C C.... Gauss Elimination With Pivoting Of Scaled Rows, Loop Over C.... K = 1,.,Last C K = 1 C C.... As Pivot Row For K-Th Step, Pick Among The Rows Not Yet Used, C.... I.e., From Rows K ,..., Nrow , The One Whose K-Th Entry C.... (Compared To The Row Size) Is Largest. Then, If This Row C.... Does Not Turn Out To Be Row K, Interchange Row K With This C.... Particular Row And Redefine Ipivot(K). C 30 Continue If ( D(K) .eq. 0.d0 ) Go To 90 If (K .eq. Nrow) Go To 80 L = K Kp1 = K+1 Colmax = Abs(W(K,K)) / D(K) C C.... Find The (Relatively) Largest Pivot C Do 40 I = Kp1, Nrow If ( Abs(W(I,K)) .le. Colmax * D(I) ) Go To 40 Colmax = Abs(W(I,K)) / D(I) L = I 40 Continue Ipivot(K) = L T = W(L,K) S = D(L) If ( L .eq. K ) Go To 50 W(L,K) = W(K,K) W(K,K) = T D(L) = D(K) D(K) = S 50 Continue C C.... If Pivot Element Is Too Small In Absolute Value, Declare C.... Matrix To Be Noninvertible And Quit. C If ( Abs(T)+D(K) .le. D(K) ) Go To 90 C C.... Otherwise, Subtract The Appropriate Multiple Of The Pivot C.... Row From Remaining Rows, I.e., The Rows (K+1),..., (Nrow) C.... To Make K-Th Entry Zero. Save The Multiplier In Its Place. C.... For High Performance Do This Operations Column Oriented. C T = -1.0d0/T Do 60 I = Kp1, Nrow 60 W(I,K) = W(I,K) * T Do 70 J = Kp1,Ncol T = W(L,J) If ( L .eq. K ) Go To 62 W(L,J) = W(K,J) W(K,J) = T 62 If ( T .eq. 0.d0 ) Go To 70 Do 64 I = Kp1, Nrow 64 W(I,J) = W(I,J) + W(I,K) * T 70 Continue K = Kp1 C C.... Check For Having Reached The Next Block. C If ( K .le. Last ) Go To 30 Return C C.... If Last .eq. Nrow , Check Now That Pivot Element In Last Row C.... Is Nonzero. C 80 If( Abs(W(Nrow,Nrow))+D(Nrow) .gt. D(Nrow) ) Return C C.... Singularity Flag Set C 90 Info = K Return End Subroutine Shiftb (Ai, Nrowi, Ncoli, Last, Ai1, Nrowi1, Ncoli1) C C********************************************************************* C C Shifts The Rows In Current Block, Ai, Not Used As Pivot Rows, If C Any, I.e., Rows (Last+1),..., (Nrowi), Onto The First Mmax = C = Nrow-Last Rows Of The Next Block, Ai1, With Column Last+J Of C Ai Going To Column J , J = 1,...,Jmax = Ncoli-Last. The Remaining C Columns Of These Rows Of Ai1 Are Zeroed Out. C C Picture C C Original Situation After Results In A New Block I+1 C Last = 2 Columns Have Been Created And Ready To Be C Done In Factrb (Assuming No Factored By Next Factrb C Interchanges Of Rows) Call. C 1 C X X 1x X X X X X X X C 1 C 0 X 1x X X 0 X X X X C Block I 1 --------------- C Nrowi = 4 0 0 1x X X 0 0 1x X X 0 01 C Ncoli = 5 1 1 1 C Last = 2 0 0 1x X X 0 0 1x X X 0 01 C ------------------------------- 1 1 New C 1x X X X X 1x X X X X1 Block C 1 1 1 I+1 C Block I+1 1x X X X X 1x X X X X1 C Nrowi1 = 5 1 1 1 C Ncoli1 = 5 1x X X X X 1x X X X X1 C ------------------------------- 1-------------1 C 1 C C********************************************************************* C Integer Last, J,Jmax,Jmaxp1,M,Mmax Double Precision Ai(Nrowi,Ncoli),Ai1(Nrowi1,Ncoli1) Mmax = Nrowi - Last Jmax = Ncoli - Last If (Mmax .lt. 1 .or. Jmax .lt. 1) Return C C.... Put The Remainder Of Block I Into Ai1 C Do 10 J = 1,Jmax Do 10 M = 1,Mmax 10 Ai1(M,J) = Ai(Last+M,Last+J) If (Jmax .eq. Ncoli1) Return C C.... Zero Out The Upper Right Corner Of Ai1 C Jmaxp1 = Jmax + 1 Do 20 J = Jmaxp1,Ncoli1 Do 20 M = 1,Mmax 20 Ai1(M,J) = 0.d0 Return End Subroutine Sbblok ( Bloks, Integs, Nbloks, Ipivot, X ) C C********************************************************************** C C Calls Subroutines Subfor And Subbak . C C Supervises The Solution (By Forward And Backward Substitution) Of C The Linear System A*X = B For X, With The Plu Factorization Of C A Already Generated In Fcblok . Individual Blocks Of C Equations Are Solved Via Subfor And Subbak . C C Parameters C Bloks, Integs, Nbloks, Ipivot Are As On Return From Fcblok. C X On Input: The Right Hand Side, In Dense Storage C On Output: The Solution Vector C C********************************************************************* C Integer Integs(3,Nbloks),Ipivot(*), I,Index,Indexx,J,Last, + Nbp1,Ncol,Nrow Double Precision Bloks(*), X(*) C C.... Forward Substitution Pass C Index = 1 Indexx = 1 Do 10 I = 1, Nbloks Nrow = Integs(1,I) Last = Integs(3,I) Call Subfor ( Bloks(Index), Ipivot(Indexx), Nrow, Last, + X(Indexx) ) Index = Nrow * Integs(2,I) + Index 10 Indexx = Indexx + Last C C.... Back Substitution Pass C Nbp1 = Nbloks + 1 Do 20 J = 1, Nbloks I = Nbp1 - J Nrow = Integs(1,I) Ncol = Integs(2,I) Last = Integs(3,I) Index = Index - Nrow * Ncol Indexx = Indexx - Last 20 Call Subbak ( Bloks(Index), Nrow, Ncol, Last, X(Indexx) ) Return End Subroutine Subfor ( W, Ipivot, Nrow, Last, X ) C C********************************************************************** C C Carries Out The Forward Pass Of Substitution For The Current C Block, I.e., The Action On The Right Side Corresponding To The C Elimination Carried Out In Factrb For This Block. C C Parameters C W, Ipivot, Nrow, Last Are As On Return From Factrb. C X(J) Is Expected To Contain, On Input, The Right Side Of J-Th C Equation For This Block, J = 1,...,Nrow. C X(J) Contains, On Output, The Appropriately Modified Right C Side Of Equation (J) In This Block, J = 1,...,Last And C For J = Last+1,...,Nrow. C C********************************************************************* C Integer Ipivot(Last), Ip,K,Kp1,Lstep Double Precision W(Nrow,Last), X(Nrow), T C If ( Nrow .eq. 1 ) Return Lstep = Min( Nrow-1 , Last ) Do 20 K = 1, Lstep Kp1 = K + 1 Ip = Ipivot(K) T = X(Ip) X(Ip) = X(K) X(K) = T If ( T .eq. 0.d0 ) Go To 20 Do 10 I = Kp1, Nrow 10 X(I) = X(I) + W(I,K) * T 20 Continue 30 Return End Subroutine Subbak ( W, Nrow, Ncol, Last, X ) C C********************************************************************* C C Carries Out Backsubstitution For Current Block. C C Parameters C W, Ipivot, Nrow, Ncol, Last Are As On Return From Factrb. C X(1),...,X(Ncol) Contains, On Input, The Right Side For The C Equations In This Block After Backsubstitution Has Been C Carried Up To But Not Including Equation (Last). C Means That X(J) Contains The Right Side Of Equation (J) C As Modified During Elimination, J = 1,...,Last, While C For J .gt. Last, X(J) Is Already A Component Of The C Solution Vector. C X(1),...,X(Ncol) Contains, On Output, The Components Of The C Solution Corresponding To The Present Block. C C********************************************************************* C Integer Last, I,J,K,Km1,Lm1,Lp1 Double Precision W(Nrow,Ncol),X(Ncol), T C Lp1 = Last + 1 If ( Lp1 .gt. Ncol ) Go To 30 Do 20 J = Lp1, Ncol T = - X(J) If ( T .eq. 0.d0 ) Go To 20 Do 10 I = 1, Last 10 X(I) = X(I) + W(I,J) * T 20 Continue 30 If ( Last .eq. 1 ) Go To 60 Lm1 = Last - 1 Do 50 Kb = 1, Lm1 Km1 = Last - Kb K = Km1 + 1 X(K) = X(K)/W(K,K) T = - X(K) If ( T .eq. 0.d0 ) Go To 50 Do 40 I = 1, Km1 40 X(I) = X(I) + W(I,K) * T 50 Continue 60 X(1) = X(1)/W(1,1) Return End Subroutine Dgefa(A,Lda,N,Ipvt,Info) Integer Lda,N,Ipvt(*),Info Double Precision A(Lda,*) C C Dgefa Factors A Double Precision Matrix By Gaussian Elimination. C C Dgefa Is Usually Called By Dgeco, But It Can Be Called C Directly With A Saving In Time If Rcond Is Not Needed. C (Time For Dgeco) = (1 + 9/N)*(Time For Dgefa) . C C On Entry C C A Double Precision(Lda, N) C The Matrix To Be Factored. C C Lda Integer C The Leading Dimension Of The Array A . C C N Integer C The Order Of The Matrix A . C C On Return C C A An Upper Triangular Matrix And The Multipliers C Which Were Used To Obtain It. C The Factorization Can Be Written A = L*U Where C L Is A Product Of Permutation And Unit Lower C Triangular Matrices And U Is Upper Triangular. C C Ipvt Integer(N) C An Integer Vector Of Pivot Indices. C C Info Integer C = 0 Normal Value. C = K If U(K,K) .eq. 0.0 . This Is Not An Error C Condition For This Subroutine, But It Does C Indicate That Dgesl Or Dgedi Will Divide By Zero C If Called. Use Rcond In Dgeco For A Reliable C Indication Of Singularity. C C Linpack. This Version Dated 08/14/78 . C Cleve Moler, University Of New Mexico, Argonne National Lab. C C Subroutines And Functions C C Blas Daxpy,Dscal,Idamax C C Internal Variables C Double Precision T Integer Idamax,J,K,Kp1,L,Nm1 C C C Gaussian Elimination With Partial Pivoting C Info = 0 Nm1 = N - 1 If (Nm1 .lt. 1) Go To 70 Do 60 K = 1, Nm1 Kp1 = K + 1 C C Find L = Pivot Index C L = Idamax(N-K+1,A(K,K),1) + K - 1 Ipvt(K) = L C C Zero Pivot Implies This Column Already Triangularized C If (A(L,K) .eq. 0.0d0) Go To 40 C C Interchange If Necessary C If (L .eq. K) Go To 10 T = A(L,K) A(L,K) = A(K,K) A(K,K) = T 10 Continue C C Compute Multipliers C T = -1.0d0/A(K,K) Call Dscal(N-K,T,A(K+1,K),1) C C Row Elimination With Column Indexing C Do 30 J = Kp1, N T = A(L,J) If (L .eq. K) Go To 20 A(L,J) = A(K,J) A(K,J) = T 20 Continue Call Daxpy(N-K,T,A(K+1,K),1,A(K+1,J),1) 30 Continue Go To 50 40 Continue Info = K 50 Continue 60 Continue 70 Continue Ipvt(N) = N If (A(N,N) .eq. 0.0d0) Info = N Return End Subroutine Dgesl(A,Lda,N,Ipvt,B,Job) Integer Lda,N,Ipvt(*),Job Double Precision A(Lda,*),B(*) C C Dgesl Solves The Double Precision System C A * X = B Or Trans(A) * X = B C Using The Factors Computed By Dgeco Or Dgefa. C C On Entry C C A Double Precision(Lda, N) C The Output From Dgeco Or Dgefa. C C Lda Integer C The Leading Dimension Of The Array A . C C N Integer C The Order Of The Matrix A . C C Ipvt Integer(N) C The Pivot Vector From Dgeco Or Dgefa. C C B Double Precision(N) C The Right Hand Side Vector. C C Job Integer C = 0 To Solve A*X = B , C = Nonzero To Solve Trans(A)*X = B Where C Trans(A) Is The Transpose. C C On Return C C B The Solution Vector X . C C Error Condition C C A Division By Zero Will Occur If The Input Factor Contains A C Zero On The Diagonal. Technically This Indicates Singularity C But It Is Often Caused By Improper Arguments Or Improper C Setting Of Lda . It Will Not Occur If The Subroutines Are C Called Correctly And If Dgeco Has Set Rcond .gt. 0.0 C Or Dgefa Has Set Info .eq. 0 . C C To Compute Inverse(A) * C Where C Is A Matrix C With P Columns C Call Dgeco(A,Lda,N,Ipvt,Rcond,Z) C If (Rcond Is Too Small) Go To ... C Do 10 J = 1, P C Call Dgesl(A,Lda,N,Ipvt,C(1,J),0) C 10 Continue C C Linpack. This Version Dated 08/14/78 . C Cleve Moler, University Of New Mexico, Argonne National Lab. C C Subroutines And Functions C C Blas Daxpy,Ddot C C Internal Variables C Double Precision Ddot,T Integer K,Kb,L,Nm1 C Nm1 = N - 1 If (Job .ne. 0) Go To 50 C C Job = 0 , Solve A * X = B C First Solve L*Y = B C If (Nm1 .lt. 1) Go To 30 Do 20 K = 1, Nm1 L = Ipvt(K) T = B(L) If (L .eq. K) Go To 10 B(L) = B(K) B(K) = T 10 Continue Call Daxpy(N-K,T,A(K+1,K),1,B(K+1),1) 20 Continue 30 Continue C C Now Solve U*X = Y C Do 40 Kb = 1, N K = N + 1 - Kb B(K) = B(K)/A(K,K) T = -B(K) Call Daxpy(K-1,T,A(1,K),1,B(1),1) 40 Continue Go To 100 50 Continue C C Job = Nonzero, Solve Trans(A) * X = B C First Solve Trans(U)*Y = B C Do 60 K = 1, N T = Ddot(K-1,A(1,K),1,B(1),1) B(K) = (B(K) - T)/A(K,K) 60 Continue C C Now Solve Trans(L)*X = Y C If (Nm1 .lt. 1) Go To 90 Do 80 Kb = 1, Nm1 K = N - Kb B(K) = B(K) + Ddot(N-K,A(K+1,K),1,B(K+1),1) L = Ipvt(K) If (L .eq. K) Go To 70 T = B(L) B(L) = B(K) B(K) = T 70 Continue 80 Continue 90 Continue 100 Continue Return 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 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