*DECK DERKF SUBROUTINE DERKF (F, NEQ, T, Y, TOUT, INFO, RTOL, ATOL, IDID, + RWORK, LRW, IWORK, LIW, RPAR, IPAR) C***BEGIN PROLOGUE DERKF C***PURPOSE Solve an initial value problem in ordinary differential C equations using a Runge-Kutta-Fehlberg scheme. C***LIBRARY SLATEC (DEPAC) C***CATEGORY I1A1A C***TYPE SINGLE PRECISION (DERKF-S, DDERKF-D) C***KEYWORDS DEPAC, INITIAL VALUE PROBLEMS, ODE, C ORDINARY DIFFERENTIAL EQUATIONS, RKF, C RUNGE-KUTTA-FEHLBERG METHODS C***AUTHOR Watts, H. A., (SNLA) C Shampine, L. F., (SNLA) C***DESCRIPTION C C This is the Runge-Kutta code in the package of differential equation C solvers DEPAC, consisting of the codes DERKF, DEABM, and DEBDF. C Design of the package was by L. F. Shampine and H. A. Watts. C It is documented in C SAND-79-2374 , DEPAC - Design of a User Oriented Package of ODE C Solvers. C DERKF is a driver for a modification of the code RKF45 written by C H. A. Watts and L. F. Shampine C Sandia Laboratories C Albuquerque, New Mexico 87185 C C ********************************************************************** C ** DEPAC PACKAGE OVERVIEW ** C ********************************************************************** C C You have a choice of three differential equation solvers from C DEPAC. The following brief descriptions are meant to aid you C in choosing the most appropriate code for your problem. C C DERKF is a fifth order Runge-Kutta code. It is the simplest of C the three choices, both algorithmically and in the use of the C code. DERKF is primarily designed to solve non-stiff and mild- C ly stiff differential equations when derivative evaluations are C not expensive. It should generally not be used to get high C accuracy results nor answers at a great many specific points. C Because DERKF has very low overhead costs, it will usually C result in the least expensive integration when solving C problems requiring a modest amount of accuracy and having C equations that are not costly to evaluate. DERKF attempts to C discover when it is not suitable for the task posed. C C DEABM is a variable order (one through twelve) Adams code. Its C complexity lies somewhere between that of DERKF and DEBDF. C DEABM is primarily designed to solve non-stiff and mildly C stiff differential equations when derivative evaluations are C expensive, high accuracy results are needed or answers at C many specific points are required. DEABM attempts to discover C when it is not suitable for the task posed. C C DEBDF is a variable order (one through five) backward C differentiation formula code. It is the most complicated of C the three choices. DEBDF is primarily designed to solve stiff C differential equations at crude to moderate tolerances. C If the problem is very stiff at all, DERKF and DEABM will be C quite inefficient compared to DEBDF. However, DEBDF will be C inefficient compared to DERKF and DEABM on non-stiff problems C because it uses much more storage, has a much larger overhead, C and the low order formulas will not give high accuracies C efficiently. C C The concept of stiffness cannot be described in a few words. C If you do not know the problem to be stiff, try either DERKF C or DEABM. Both of these codes will inform you of stiffness C when the cost of solving such problems becomes important. C C ********************************************************************** C ** ABSTRACT ** C ********************************************************************** C C Subroutine DERKF uses a Runge-Kutta-Fehlberg (4,5) method to C integrate a system of NEQ first order ordinary differential C equations of the form C DU/DX = F(X,U) C when the vector Y(*) of initial values for U(*) at X=T is given. C The subroutine integrates from T to TOUT. It is easy to continue the C integration to get results at additional TOUT. This is the interval C mode of operation. It is also easy for the routine to return with C the solution at each intermediate step on the way to TOUT. This is C the intermediate-output mode of operation. C C DERKF uses subprograms DERKFS, DEFEHL, HSTART, HVNRM, R1MACH, and C the error handling routine XERMSG. The only machine dependent C parameters to be assigned appear in R1MACH. C C ********************************************************************** C ** DESCRIPTION OF THE ARGUMENTS TO DERKF (AN OVERVIEW) ** C ********************************************************************** C C The Parameters are: C C F -- This is the name of a subroutine which you provide to C define the differential equations. C C NEQ -- This is the number of (first order) differential C equations to be integrated. C C T -- This is a value of the independent variable. C C Y(*) -- This array contains the solution components at T. C C TOUT -- This is a point at which a solution is desired. C C INFO(*) -- The basic task of the code is to integrate the C differential equations from T to TOUT and return an C answer at TOUT. INFO(*) is an INTEGER array which is used C to communicate exactly how you want this task to be C carried out. C C RTOL, ATOL -- These quantities represent relative and absolute C error tolerances which you provide to indicate how C accurately you wish the solution to be computed. You may C choose them to be both scalars or else both vectors. C C IDID -- This scalar quantity is an indicator reporting what C the code did. You must monitor this INTEGER variable to C decide what action to take next. C C RWORK(*), LRW -- RWORK(*) is a REAL work array of length LRW C which provides the code with needed storage space. C C IWORK(*), LIW -- IWORK(*) is an INTEGER work array of length LIW C which provides the code with needed storage space and an C across call flag. C C RPAR, IPAR -- These are REAL and INTEGER parameter arrays which C you can use for communication between your calling C program and the F subroutine. C C Quantities which are used as input items are C NEQ, T, Y(*), TOUT, INFO(*), C RTOL, ATOL, LRW and LIW. C C Quantities which may be altered by the code are C T, Y(*), INFO(1), RTOL, ATOL, C IDID, RWORK(*) and IWORK(*). C C ********************************************************************** C ** INPUT -- What to do On The First Call To DERKF ** C ********************************************************************** C C The first call of the code is defined to be the start of each new C problem. Read through the descriptions of all the following items, C provide sufficient storage space for designated arrays, set C appropriate variables for the initialization of the problem, and C give information about how you want the problem to be solved. C C C F -- Provide a subroutine of the form C F(X,U,UPRIME,RPAR,IPAR) C to define the system of first order differential equations C which is to be solved. For the given values of X and the C vector U(*)=(U(1),U(2),...,U(NEQ)) , the subroutine must C evaluate the NEQ components of the system of differential C equations DU/DX=F(X,U) and store the derivatives in the C array UPRIME(*), that is, UPRIME(I) = * DU(I)/DX * for C equations I=1,...,NEQ. C C Subroutine F must not alter X or U(*). You must declare C the name F in an external statement in your program that C calls DERKF. You must dimension U and UPRIME in F. C C RPAR and IPAR are REAL and INTEGER parameter arrays which C you can use for communication between your calling program C and subroutine F. They are not used or altered by DERKF. C If you do not need RPAR or IPAR, ignore these parameters C by treating them as dummy arguments. If you do choose to C use them, dimension them in your calling program and in F C as arrays of appropriate length. C C NEQ -- Set it to the number of differential equations. C (NEQ .GE. 1) C C T -- Set it to the initial point of the integration. C You must use a program variable for T because the code C changes its value. C C Y(*) -- Set this vector to the initial values of the NEQ solution C components at the initial point. You must dimension Y at C least NEQ in your calling program. C C TOUT -- Set it to the first point at which a solution C is desired. You can take TOUT = T, in which case the code C will evaluate the derivative of the solution at T and C return. Integration either forward in T (TOUT .GT. T) or C backward in T (TOUT .LT. T) is permitted. C C The code advances the solution from T to TOUT using C step sizes which are automatically selected so as to C achieve the desired accuracy. If you wish, the code will C return with the solution and its derivative following C each intermediate step (intermediate-output mode) so that C you can monitor them, but you still must provide TOUT in C accord with the basic aim of the code. C C The first step taken by the code is a critical one C because it must reflect how fast the solution changes near C the initial point. The code automatically selects an C initial step size which is practically always suitable for C the problem. By using the fact that the code will not C step past TOUT in the first step, you could, if necessary, C restrict the length of the initial step size. C C For some problems it may not be permissible to integrate C past a point TSTOP because a discontinuity occurs there C or the solution or its derivative is not defined beyond C TSTOP. Since DERKF will never step past a TOUT point, C you need only make sure that no TOUT lies beyond TSTOP. C C INFO(*) -- Use the INFO array to give the code more details about C how you want your problem solved. This array should be C dimensioned of length 15 to accommodate other members of C DEPAC or possible future extensions, though DERKF uses C only the first three entries. You must respond to all of C the following items which are arranged as questions. The C simplest use of the code corresponds to answering all C questions as YES ,i.e. setting all entries of INFO to 0. C C INFO(1) -- This parameter enables the code to initialize C itself. You must set it to indicate the start of every C new problem. C C **** Is this the first call for this problem ... C YES -- Set INFO(1) = 0 C NO -- Not applicable here. C See below for continuation calls. **** C C INFO(2) -- How much accuracy you want of your solution C is specified by the error tolerances RTOL and ATOL. C The simplest use is to take them both to be scalars. C To obtain more flexibility, they can both be vectors. C The code must be told your choice. C C **** Are both error tolerances RTOL, ATOL scalars ... C YES -- Set INFO(2) = 0 C and input scalars for both RTOL and ATOL C NO -- Set INFO(2) = 1 C and input arrays for both RTOL and ATOL **** C C INFO(3) -- The code integrates from T in the direction C of TOUT by steps. If you wish, it will return the C computed solution and derivative at the next C intermediate step (the intermediate-output mode). C This is a good way to proceed if you want to see the C behavior of the solution. If you must have solutions at C a great many specific TOUT points, this code is C INEFFICIENT. The code DEABM in DEPAC handles this task C more efficiently. C C **** Do you want the solution only at C TOUT (and not at the next intermediate step) ... C YES -- Set INFO(3) = 0 C NO -- Set INFO(3) = 1 **** C C RTOL, ATOL -- You must assign relative (RTOL) and absolute (ATOL) C error tolerances to tell the code how accurately you want C the solution to be computed. They must be defined as C program variables because the code may change them. You C have two choices -- C Both RTOL and ATOL are scalars. (INFO(2)=0) C Both RTOL and ATOL are vectors. (INFO(2)=1) C In either case all components must be non-negative. C C The tolerances are used by the code in a local error test C at each step which requires roughly that C ABS(LOCAL ERROR) .LE. RTOL*ABS(Y)+ATOL C for each vector component. C (More specifically, a maximum norm is used to measure C the size of vectors, and the error test uses the average C of the magnitude of the solution at the beginning and end C of the step.) C C The true (global) error is the difference between the true C solution of the initial value problem and the computed C approximation. Practically all present day codes, C including this one, control the local error at each step C and do not even attempt to control the global error C directly. Roughly speaking, they produce a solution Y(T) C which satisfies the differential equations with a C residual R(T), DY(T)/DT = F(T,Y(T)) + R(T) , C and, almost always, R(T) is bounded by the error C tolerances. Usually, but not always, the true accuracy of C the computed Y is comparable to the error tolerances. This C code will usually, but not always, deliver a more accurate C solution if you reduce the tolerances and integrate again. C By comparing two such solutions you can get a fairly C reliable idea of the true error in the solution at the C bigger tolerances. C C Setting ATOL=0. results in a pure relative error test on C that component. Setting RTOL=0. yields a pure absolute C error test on that component. A mixed test with non-zero C RTOL and ATOL corresponds roughly to a relative error C test when the solution component is much bigger than ATOL C and to an absolute error test when the solution component C is smaller than the threshold ATOL. C C Proper selection of the absolute error control parameters C ATOL requires you to have some idea of the scale of the C solution components. To acquire this information may mean C that you will have to solve the problem more than once. In C the absence of scale information, you should ask for some C relative accuracy in all the components (by setting RTOL C values non-zero) and perhaps impose extremely small C absolute error tolerances to protect against the danger of C a solution component becoming zero. C C The code will not attempt to compute a solution at an C accuracy unreasonable for the machine being used. It will C advise you if you ask for too much accuracy and inform C you as to the maximum accuracy it believes possible. C If you want relative accuracies smaller than about C 10**(-8), you should not ordinarily use DERKF. The code C DEABM in DEPAC obtains stringent accuracies more C efficiently. C C RWORK(*) -- Dimension this REAL work array of length LRW in your C calling program. C C LRW -- Set it to the declared length of the RWORK array. C You must have LRW .GE. 33+7*NEQ C C IWORK(*) -- Dimension this INTEGER work array of length LIW in C your calling program. C C LIW -- Set it to the declared length of the IWORK array. C You must have LIW .GE. 34 C C RPAR, IPAR -- These are parameter arrays, of REAL and INTEGER C type, respectively. You can use them for communication C between your program that calls DERKF and the F C subroutine. They are not used or altered by DERKF. If C you do not need RPAR or IPAR, ignore these parameters by C treating them as dummy arguments. If you do choose to use C them, dimension them in your calling program and in F as C arrays of appropriate length. C C ********************************************************************** C ** OUTPUT -- After any return from DERKF ** C ********************************************************************** C C The principal aim of the code is to return a computed solution at C TOUT, although it is also possible to obtain intermediate results C along the way. To find out whether the code achieved its goal C or if the integration process was interrupted before the task was C completed, you must check the IDID parameter. C C C T -- The solution was successfully advanced to the C output value of T. C C Y(*) -- Contains the computed solution approximation at T. C You may also be interested in the approximate derivative C of the solution at T. It is contained in C RWORK(21),...,RWORK(20+NEQ). C C IDID -- Reports what the code did C C *** Task Completed *** C Reported by positive values of IDID C C IDID = 1 -- A step was successfully taken in the C intermediate-output mode. The code has not C yet reached TOUT. C C IDID = 2 -- The integration to TOUT was successfully C completed (T=TOUT) by stepping exactly to TOUT. C C *** Task Interrupted *** C Reported by negative values of IDID C C IDID = -1 -- A large amount of work has been expended. C (500 steps attempted) C C IDID = -2 -- The error tolerances are too stringent. C C IDID = -3 -- The local error test cannot be satisfied C because you specified a zero component in ATOL C and the corresponding computed solution C component is zero. Thus, a pure relative error C test is impossible for this component. C C IDID = -4 -- The problem appears to be stiff. C C IDID = -5 -- DERKF is being used very inefficiently C because the natural step size is being C restricted by too frequent output. C C IDID = -6,-7,..,-32 -- Not applicable for this code but C used by other members of DEPAC or possible C future extensions. C C *** Task Terminated *** C Reported by the value of IDID=-33 C C IDID = -33 -- The code has encountered trouble from which C it cannot recover. A message is printed C explaining the trouble and control is returned C to the calling program. For example, this C occurs when invalid input is detected. C C RTOL, ATOL -- These quantities remain unchanged except when C IDID = -2. In this case, the error tolerances have been C increased by the code to values which are estimated to be C appropriate for continuing the integration. However, the C reported solution at T was obtained using the input values C of RTOL and ATOL. C C RWORK, IWORK -- Contain information which is usually of no C interest to the user but necessary for subsequent calls. C However, you may find use for C C RWORK(11)--which contains the step size H to be C attempted on the next step. C C RWORK(12)--If the tolerances have been increased by the C code (IDID = -2) , they were multiplied by the C value in RWORK(12). C C RWORK(20+I)--which contains the approximate derivative C of the solution component Y(I). In DERKF, it C is always obtained by calling subroutine F to C evaluate the differential equation using T and C Y(*). C C ********************************************************************** C ** INPUT -- What To Do To Continue The Integration ** C ** (calls after the first) ** C ********************************************************************** C C This code is organized so that subsequent calls to continue the C integration involve little (if any) additional effort on your C part. You must monitor the IDID parameter to determine C what to do next. C C Recalling that the principal task of the code is to integrate C from T to TOUT (the interval mode), usually all you will need C to do is specify a new TOUT upon reaching the current TOUT. C C Do not alter any quantity not specifically permitted below, C in particular do not alter NEQ, T, Y(*), RWORK(*), IWORK(*) or C the differential equation in subroutine F. Any such alteration C constitutes a new problem and must be treated as such, i.e. C you must start afresh. C C You cannot change from vector to scalar error control or vice C versa (INFO(2)) but you can change the size of the entries of C RTOL, ATOL. Increasing a tolerance makes the equation easier C to integrate. Decreasing a tolerance will make the equation C harder to integrate and should generally be avoided. C C You can switch from the intermediate-output mode to the C interval mode (INFO(3)) or vice versa at any time. C C The parameter INFO(1) is used by the code to indicate the C beginning of a new problem and to indicate whether integration C is to be continued. You must input the value INFO(1) = 0 C when starting a new problem. You must input the value C INFO(1) = 1 if you wish to continue after an interrupted task. C Do not set INFO(1) = 0 on a continuation call unless you C want the code to restart at the current T. C C *** Following a Completed Task *** C If C IDID = 1, call the code again to continue the integration C another step in the direction of TOUT. C C IDID = 2, define a new TOUT and call the code again. C TOUT must be different from T. You cannot change C the direction of integration without restarting. C C *** Following an Interrupted Task *** C To show the code that you realize the task was C interrupted and that you want to continue, you C must take appropriate action and reset INFO(1) = 1 C If C IDID = -1, the code has attempted 500 steps. C If you want to continue, set INFO(1) = 1 and C call the code again. An additional 500 steps C will be allowed. C C IDID = -2, the error tolerances RTOL, ATOL have been C increased to values the code estimates appropriate C for continuing. You may want to change them C yourself. If you are sure you want to continue C with relaxed error tolerances, set INFO(1)=1 and C call the code again. C C IDID = -3, a solution component is zero and you set the C corresponding component of ATOL to zero. If you C are sure you want to continue, you must first C alter the error criterion to use positive values C for those components of ATOL corresponding to zero C solution components, then set INFO(1)=1 and call C the code again. C C IDID = -4, the problem appears to be stiff. It is very C inefficient to solve such problems with DERKF. C Code DEBDF in DEPAC handles this task efficiently. C If you are absolutely sure you want to continue C with DERKF, set INFO(1)=1 and call the code again. C C IDID = -5, you are using DERKF very inefficiently by C choosing output points TOUT so close together that C the step size is repeatedly forced to be rather C smaller than necessary. If you are willing to C accept solutions at the steps chosen by the code, C a good way to proceed is to use the intermediate C output mode (setting INFO(3)=1). If you must have C solutions at so many specific TOUT points, the C code DEABM in DEPAC handles this task C efficiently. If you want to continue with DERKF, C set INFO(1)=1 and call the code again. C C IDID = -6,-7,..,-32 --- cannot occur with this code but C used by other members of DEPAC or possible future C extensions. C C *** Following a Terminated Task *** C If C IDID = -33, you cannot continue the solution of this C problem. An attempt to do so will result in your C run being terminated. C C ********************************************************************** C *Long Description: C C ********************************************************************** C ** DEPAC Package Overview ** C ********************************************************************** C C .... You have a choice of three differential equation solvers from C .... DEPAC. The following brief descriptions are meant to aid you in C .... choosing the most appropriate code for your problem. C C .... DERKF is a fifth order Runge-Kutta code. It is the simplest of C .... the three choices, both algorithmically and in the use of the C .... code. DERKF is primarily designed to solve non-stiff and C .... mildly stiff differential equations when derivative evaluations C .... are not expensive. It should generally not be used to get high C .... accuracy results nor answers at a great many specific points. C .... Because DERKF has very low overhead costs, it will usually C .... result in the least expensive integration when solving C .... problems requiring a modest amount of accuracy and having C .... equations that are not costly to evaluate. DERKF attempts to C .... discover when it is not suitable for the task posed. C C .... DEABM is a variable order (one through twelve) Adams code. C .... Its complexity lies somewhere between that of DERKF and C .... DEBDF. DEABM is primarily designed to solve non-stiff and C .... mildly stiff differential equations when derivative evaluations C .... are expensive, high accuracy results are needed or answers at C .... many specific points are required. DEABM attempts to discover C .... when it is not suitable for the task posed. C C .... DEBDF is a variable order (one through five) backward C .... differentiation formula code. it is the most complicated of C .... the three choices. DEBDF is primarily designed to solve stiff C .... differential equations at crude to moderate tolerances. C .... If the problem is very stiff at all, DERKF and DEABM will be C .... quite inefficient compared to DEBDF. However, DEBDF will be C .... inefficient compared to DERKF and DEABM on non-stiff problems C .... because it uses much more storage, has a much larger overhead, C .... and the low order formulas will not give high accuracies C .... efficiently. C C .... The concept of stiffness cannot be described in a few words. C .... If you do not know the problem to be stiff, try either DERKF C .... or DEABM. Both of these codes will inform you of stiffness C .... when the cost of solving such problems becomes important. C C ********************************************************************* C C***REFERENCES L. F. Shampine and H. A. Watts, DEPAC - design of a user C oriented package of ODE solvers, Report SAND79-2374, C Sandia Laboratories, 1979. C L. F. Shampine and H. A. Watts, Practical solution of C ordinary differential equations by Runge-Kutta C methods, Report SAND76-0585, Sandia Laboratories, C 1976. C***ROUTINES CALLED DERKFS, XERMSG C***REVISION HISTORY (YYMMDD) C 800501 DATE WRITTEN C 890831 Modified array declarations. (WRB) C 891024 Changed references from VNORM to HVNRM. (WRB) C 891024 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900510 Convert XERRWV calls to XERMSG calls, change Prologue C comments to agree with DDERKF. (RWC) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE DERKF