SLATEC Routines --- DDERKF ---
*DECK DDERKF
SUBROUTINE DDERKF (DF, NEQ, T, Y, TOUT, INFO, RTOL, ATOL, IDID,
+ RWORK, LRW, IWORK, LIW, RPAR, IPAR)
C***BEGIN PROLOGUE DDERKF
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 DOUBLE 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 DDERKF, DDEABM, and DDEBDF.
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 DDERKF 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 ** DDEPAC PACKAGE OVERVIEW **
C **********************************************************************
C
C You have a choice of three differential equation solvers from
C DDEPAC. The following brief descriptions are meant to aid you
C in choosing the most appropriate code for your problem.
C
C DDERKF 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. DDERKF 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 DDERKF 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. DDERKF attempts to
C discover when it is not suitable for the task posed.
C
C DDEABM is a variable order (one through twelve) Adams code. Its
C complexity lies somewhere between that of DDERKF and DDEBDF.
C DDEABM 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. DDEABM attempts to discover
C when it is not suitable for the task posed.
C
C DDEBDF is a variable order (one through five) backward
C differentiation formula code. It is the most complicated of
C the three choices. DDEBDF is primarily designed to solve stiff
C differential equations at crude to moderate tolerances.
C If the problem is very stiff at all, DDERKF and DDEABM will be
C quite inefficient compared to DDEBDF. However, DDEBDF will be
C inefficient compared to DDERKF and DDEABM 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 DDERKF
C or DDEABM. 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 DDERKF 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 = DF(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 DDERKF uses subprograms DRKFS, DFEHL, DHSTRT, DHVNRM, D1MACH, and
C the error handling routine XERMSG. The only machine dependent
C parameters to be assigned appear in D1MACH.
C
C **********************************************************************
C ** DESCRIPTION OF THE ARGUMENTS TO DDERKF (AN OVERVIEW) **
C **********************************************************************
C
C The Parameters are:
C
C DF -- 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 DOUBLE PRECISION value of the independent
C variable.
C
C Y(*) -- This DOUBLE PRECISION array contains the solution
C components at T.
C
C TOUT -- This is a DOUBLE PRECISION point at which a solution is
C 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 DOUBLE PRECISION quantities represent
C relative and absolute error tolerances which you provide
C to indicate how accurately you wish the solution to be
C computed. You may choose them to be both scalars or else
C 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 DOUBLE PRECISION work array of
C length LRW which provides the code with needed storage
C 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 DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C calling program and the DF 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 DDERKF **
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 DF -- Provide a subroutine of the form
C DF(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=DF(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 DF must not alter X or U(*). You must declare
C the name DF in an external statement in your program that
C calls DDERKF. You must dimension U and UPRIME in DF.
C
C RPAR and IPAR are DOUBLE PRECISION and INTEGER parameter
C arrays which you can use for communication between your
C calling program and subroutine DF. They are not used or
C altered by DDERKF. If you do not need RPAR or IPAR,
C ignore these parameters by treating them as dummy
C arguments. If you do choose to use them, dimension them in
C your calling program and in DF as arrays of appropriate
C 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 DDERKF 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 DDERKF 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 DDEABM 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 = DF(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 DDERKF. The code
C DDEABM in DEPAC obtains stringent accuracies more
C efficiently.
C
C RWORK(*) -- Dimension this DOUBLE PRECISION work array of length
C LRW in your 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 DOUBLE PRECISION and
C INTEGER type, respectively. You can use them for
C communication between your program that calls DDERKF and
C the DF subroutine. They are not used or altered by
C DDERKF. If you do not need RPAR or IPAR, ignore these
C parameters by treating them as dummy arguments. If you do
C choose to use them, dimension them in your calling program
C and in DF as arrays of appropriate length.
C
C **********************************************************************
C ** OUTPUT -- After any return from DDERKF **
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 -- DDERKF 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 DDERKF, it
C is always obtained by calling subroutine DF 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 DF. 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 DDERKF.
C The code DDEBDF in DEPAC handles this task
C efficiently. If you are absolutely sure you want
C to continue with DDERKF, set INFO(1)=1 and call
C the code again.
C
C IDID = -5, you are using DDERKF 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 DDEABM in DEPAC handles this task
C efficiently. If you want to continue with DDERKF,
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 .... DDERKF 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. DDERKF 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 DDERKF 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. DDERKF attempts to
C .... discover when it is not suitable for the task posed.
C
C .... DDEABM is a variable order (one through twelve) Adams code.
C .... Its complexity lies somewhere between that of DDERKF and
C .... DDEBDF. DDEABM 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. DDEABM attempts to discover
C .... when it is not suitable for the task posed.
C
C .... DDEBDF is a variable order (one through five) backward
C .... differentiation formula code. it is the most complicated of
C .... the three choices. DDEBDF is primarily designed to solve stiff
C .... differential equations at crude to moderate tolerances.
C .... If the problem is very stiff at all, DDERKF and DDEABM will be
C .... quite inefficient compared to DDEBDF. However, DDEBDF will be
C .... inefficient compared to DDERKF and DDEABM 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 DDERKF
C .... or DDEABM. 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 DRKFS, XERMSG
C***REVISION HISTORY (YYMMDD)
C 820301 DATE WRITTEN
C 890831 Modified array declarations. (WRB)
C 891024 Changed references from DVNORM to DHVNRM. (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, make Prologue comments
C consistent with DERKF. (RWC)
C 920501 Reformatted the REFERENCES section. (WRB)
C***END PROLOGUE DDERKF