IVPRK

Solves an initial-value problem for ordinary differential equations using the Runge-Kutta-Verner fifth-order and sixth-order method.

Required Arguments

IDO — Flag indicating the state of the computation. (Input/Output)

IDO | State |
---|---|

1 | Initial entry |

2 | Normal re-entry |

3 | Final call to release workspace |

4 | Return because of interrupt 1 |

5 | Return because of interrupt 2 with step accepted |

6 | Return because of interrupt 2 with step rejected |

Normally, the initial call is made with IDO = 1. The routine then sets IDO = 2, and this value is used for all but the last call that is made with IDO = 3. This final call is used to release workspace, which was automatically allocated by the initial call with IDO = 1. No integration is performed on this final call. See Comment 3 for a description of the other interrupts.

FCN — User-supplied subroutine to evaluate functions. The usage is CALL FCN (N, T, Y, YPRIME), where

N – Number of equations. (Input)

T – Independent variable, t. (Input)

Y – Array of size N containing the dependent variable values, y. (Input)

YPRIME – Array of size N containing the values of the vector yʹ evaluated at (t, y). (Output)

FCN must be declared EXTERNAL in the calling program.

T — Independent variable. (Input/Output)

On input, T contains the initial value. On output, T is replaced by TEND unless error conditions have occurred. See IDO for details.

On input, T contains the initial value. On output, T is replaced by TEND unless error conditions have occurred. See IDO for details.

TEND — Value of t where the solution is required. (Input)

The value TEND may be less than the initial value of t.

The value TEND may be less than the initial value of t.

Y — Array of size NEQ of dependent variables. (Input/Output)

On input, Y contains the initial values. On output, Y contains the approximate solution.

On input, Y contains the initial values. On output, Y contains the approximate solution.

Optional Arguments

NEQ — Number of differential equations. (Input)

Default: NEQ = size (Y,1).

Default: NEQ = size (Y,1).

TOL — Tolerance for error control. (Input)

An attempt is made to control the norm of the local error such that the global error is proportional to TOL.

Default: TOL = machine precision.

An attempt is made to control the norm of the local error such that the global error is proportional to TOL.

Default: TOL = machine precision.

PARAM — A floating-point array of size 50 containing optional parameters. (Input/ Output)

If a parameter is zero, then a default value is used. These default values are given below. Parameters that concern values of step size are applied in the direction of integration. The following parameters may be set by the user:

If a parameter is zero, then a default value is used. These default values are given below. Parameters that concern values of step size are applied in the direction of integration. The following parameters may be set by the user:

PARAM | Meaning | |
---|---|---|

1 | HINIT | Initial value of the step size. Default: 10.0 * MAX (AMACH (1), AMACH(4) * MAX(ABS(TEND), ABS(T))) |

2 | HMIN | Minimum value of the step size. Default: 0.0 |

3 | HMAX | Maximum value of the step size. Default: 2.0 |

4 | MXSTEP | Maximum number of steps allowed. Default: 500 |

5 | MXFCN | Maximum number of function evaluations allowed. Default: No enforced limit. |

6 | Not used. | |

7 | INTRP1 | If nonzero, then return with IDO = 4 before each step. See Comment 3. Default: 0. |

8 | INTRP2 | If nonzero, then return with IDO = 5 after every successful step and with IDO = 6 after every unsuccessful step. See Comment 3. Default: 0. |

9 | SCALE | A measure of the scale of the problem, such as an approximation to the average value of a norm of the Jacobian matrix along the solution. Default: 1.0 |

10 | INORM | Switch determining error norm. In the following, ei is the absolute value of an estimate of the error in yi(t). Default: 0.0 − min(absolute error, relative error) = max(ei/wi); i = 1, …, NEQ, where wi = max(|yi(t)|, 1.0). 1 − absolute error = max(ei), i = 1, …, NEQ. 2− max(ei/wi), i = 1, …, NEQ where wi = max(∣yi (t)∣, FLOOR), and FLOOR is PARAM(11). 3 − Scaled Euclidean norm defined as where wi = max(|yi (t)|, 1.0). Other definitions of YMAX can be specified by the user, as explained in Comment 1. |

11 | FLOOR | Used in the norm computation associated with parameter INORM. Default: 1.0. |

12-30 | Not used. |

The following entries in PARAM are set by the program.

PARAM | Meaning | |
---|---|---|

31 | HTRIAL | Current trial step size. |

32 | HMINC | Computed minimum step size allowed. |

33 | HMAXC | Computed maximum step size allowed. |

34 | NSTEP | Number of steps taken. |

35 | NFCN | Number of function evaluations used. |

36-50 | Not used. |

FORTRAN 90 Interface

Generic: CALL IVPRK (IDO, FCN, T, TEND, Y [, …])

Specific: The specific interface names are S_IVPRK and D_IVPRK.

FORTRAN 77 Interface

Single: CALL IVPRK (IDO, NEQ, FCN, T, TEND, TOL, PARAM, Y)

Double: The double precision name is DIVPRK.

Description

Routine IVPRK finds an approximation to the solution of a system of first-order differential equations of the form y0 = f (t, y) with given initial data. The routine attempts to keep the global error proportional to a user-specified tolerance. This routine is efficient for nonstiff systems where the derivative evaluations are not expensive.

The routine IVPRK is based on a code designed by Hull, Enright and Jackson (1976, 1977). It uses Runge-Kutta formulas of order five and six developed by J. H. Verner.

Comments

1. Workspace may be explicitly provided, if desired, by use of I2PRK/DI2PRK. The reference is:

CALL I2PRK (IDO, NEQ, FCN, T, TEND, TOL, PARAM, Y, VNORM, WK)

The additional arguments are as follows:

VNORM — A Fortran subroutine to compute the norm of the error. (Input)

The routine may be provided by the user, or the IMSL routine I3PRK/DI3PRK may be used. In either case, the name must be declared in a Fortran EXTERNAL statement. If usage of the IMSL routine is intended, then the name I3PRK/DI3PRK should be used. The usage of the error norm routine is CALL VNORM (N, V, Y, YMAX, ENORM), where

The routine may be provided by the user, or the IMSL routine I3PRK/DI3PRK may be used. In either case, the name must be declared in a Fortran EXTERNAL statement. If usage of the IMSL routine is intended, then the name I3PRK/DI3PRK should be used. The usage of the error norm routine is CALL VNORM (N, V, Y, YMAX, ENORM), where

Arg | Definition |
---|---|

N | Number of equations. (Input). |

V | Array of size N containing the vector whose norm is to be computed. (Input) |

Y | Array of size N containing the values of the dependent variable. (Input) |

YMAX | Array of size N containing the maximum values of |y(t)|. (Input). |

ENORM | Norm of the vector V. (Output). |

VNORM must be declared EXTERNAL in the calling program.

WK — Work array of size 10N using the working precision. The contents of WK must not be changed from the first call with IDO = 1 until after the final call with IDO = 3.

2. Informational errors

Type | Code | Description |
---|---|---|

4 | 1 | Cannot satisfy error condition. The value of TOL may be too small. |

4 | 2 | Too many function evaluations needed. |

4 | 3 | Too many steps needed. The problem may be stiff. |

3. If PARAM(7) is nonzero, the subroutine returns with IDO = 4 and will resume calculation at the point of interruption if re-entered with IDO = 4. If PARAM(8) is nonzero, the subroutine will interrupt the calculations immediately after it decides whether or not to accept the result of the most recent trial step. The values used are IDO = 5 if the routine plans to accept, or IDO = 6 if it plans to reject the step. The values of IDO may be changed by the user (by changing IDO from 6 to 5) in order to force acceptance of a step that would otherwise be rejected. Some parameters the user might want to examine after return from an interrupt are IDO, HTRIAL, NSTEP, NFCN, T, and Y. The array Y contains the newly computed trial value for y(t), accepted or not.

Examples

Example 1

Consider a predator-prey problem with rabbits and foxes. Let r be the density of rabbits and let f be the density of foxes. In the absence of any predator-prey interaction, the rabbits would increase at a rate proportional to their number, and the foxes would die of starvation at a rate proportional to their number. Mathematically,

r ʹ = 2r

f ʹ = − f

The rate at which the rabbits are eaten by the foxes is 2r f, and the rate at which the foxes increase, because they are eating the rabbits, is r f. So, the model to be solved is

r ʹ = 2r − 2r f

f ʹ = − f + r f

The initial conditions are r(0) = 1 and f(0) = 3 over the interval 0 ≤ t ≤ 10.

In the program Y(1) = r and Y(2) = f. Note that the parameter vector PARAM is first set to zero with IMSL routine SSET (Chapter 9, “Basic Matrix/Vector Operations”). Then, absolute error control is selected by setting PARAM(10) = 1.0.

The last call to IVPRK with IDO = 3 deallocates IMSL workspace allocated on the first call to IVPRK. It is not necessary to release the workspace in this example because the program ends after solving a single problem. The call to release workspace is made as a model of what would be needed if the program included further calls to IMSL routines.

USE IVPRK_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER MXPARM, N

PARAMETER (MXPARM=50, N=2)

! SPECIFICATIONS FOR LOCAL VARIABLES

INTEGER IDO, ISTEP, NOUT

REAL PARAM(MXPARM), T, TEND, TOL, Y(N)

! SPECIFICATIONS FOR SUBROUTINES

EXTERNAL FCN

!

CALL UMACH (2, NOUT)

! Set initial conditions

T = 0.0

Y(1) = 1.0

Y(2) = 3.0

! Set error tolerance

TOL = 0.0005

! Set PARAM to default

PARAM = 0.E0

! Select absolute error control

PARAM(10) = 1.0

! Print header

WRITE (NOUT,99999)

IDO = 1

ISTEP = 0

10 CONTINUE

ISTEP = ISTEP + 1

TEND = ISTEP

CALL IVPRK (IDO, FCN, T, TEND, Y, TOL=TOL, PARAM=PARAM)

IF (ISTEP .LE. 10) THEN

WRITE (NOUT,'(I6,3F12.3)') ISTEP, T, Y

! Final call to release workspace

IF (ISTEP .EQ. 10) IDO = 3

GO TO 10

END IF

99999 FORMAT (4X, 'ISTEP', 5X, 'Time', 9X, 'Y1', 11X, 'Y2')

END

SUBROUTINE FCN (N, T, Y, YPRIME)

! SPECIFICATIONS FOR ARGUMENTS

INTEGER N

REAL T, Y(N), YPRIME(N)

!

YPRIME(1) = 2.0*Y(1) - 2.0*Y(1)*Y(2)

YPRIME(2) = -Y(2) + Y(1)*Y(2)

RETURN

END

Output

ISTEP Time Y1 Y2

1 1.000 0.078 1.465

2 2.000 0.085 0.578

3 3.000 0.292 0.250

4 4.000 1.449 0.187

5 5.000 4.046 1.444

6 6.000 0.176 2.256

7 7.000 0.066 0.908

8 8.000 0.148 0.367

9 9.000 0.655 0.188

10 10.000 3.157 0.352

Example 2

This is a mildly stiff problem (F2) from the test set of Enright and Pryce (1987). It is included here because it illustrates the inefficiency of requiring more function evaluations with a nonstiff solver, for a requested accuracy, than would be required using a stiff solver. Also, see IVPAG Example 2, where the problem is solved using a BDF method. The number of function evaluations may vary, depending on the accuracy and other arithmetic characteristics of the computer. The test problem has n = 2 equations:

USE IVPRK_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER MXPARM, N

PARAMETER (MXPARM=50, N=2)

! SPECIFICATIONS FOR LOCAL VARIABLES

INTEGER IDO, ISTEP, NOUT

REAL PARAM(MXPARM), T, TEND, TOL, Y(N)

! SPECIFICATIONS FOR SUBROUTINES

! SPECIFICATIONS FOR FUNCTIONS

EXTERNAL FCN

!

CALL UMACH (2, NOUT)

! Set initial conditions

T = 0.0

Y(1) = 1.0

Y(2) = 0.0

! Set error tolerance

TOL = 0.001

! Set PARAM to default

PARAM = 0.0E0

! Select absolute error control

PARAM(10) = 1.0

! Print header

WRITE (NOUT,99998)

IDO = 1

ISTEP = 0

10 CONTINUE

ISTEP = ISTEP + 24

TEND = ISTEP

CALL IVPRK (IDO, FCN, T, TEND, Y, TOL=TOL, PARAM=PARAM)

IF (ISTEP .LE. 240) THEN

WRITE (NOUT,'(I6,3F12.3)') ISTEP/24, T, Y

! Final call to release workspace

IF (ISTEP .EQ. 240) IDO = 3

GO TO 10

END IF

! Show number of function calls.

WRITE (NOUT,99999) PARAM(35)

99998 FORMAT (4X, 'ISTEP', 5X, 'Time', 9X, 'Y1', 11X, 'Y2')

99999 FORMAT (4X, 'Number of fcn calls with IVPRK =', F6.0)

END

SUBROUTINE FCN (N, T, Y, YPRIME)

! SPECIFICATIONS FOR ARGUMENTS

INTEGER N

REAL T, Y(N), YPRIME(N)

! SPECIFICATIONS FOR DATA VARIABLES

REAL AK1, AK2, AK3

!

DATA AK1, AK2, AK3/294.0E0, 3.0E0, 0.01020408E0/

!

YPRIME(1) = -Y(1) - Y(1)*Y(2) + AK1*Y(2)

YPRIME(2) = -AK2*Y(2) + AK3*(1.0E0-Y(2))*Y(1)

RETURN

END

Output

ISTEP Time Y1 Y2

1 24.000 0.688 0.002

2 48.000 0.634 0.002

3 72.000 0.589 0.002

4 96.000 0.549 0.002

5 120.000 0.514 0.002

6 144.000 0.484 0.002

7 168.000 0.457 0.002

8 192.000 0.433 0.001

9 216.000 0.411 0.001

10 240.000 0.391 0.001

Number of fcn calls with IVPRK = 2153.