Integrates a function over an infinite or semi-infinite interval.

Required Arguments

F — User-supplied FUNCTION to be integrated. The form is
F(X), where
                                X − Independent variable.   (Input)
                                F − The function value.   (Output)
F must be declared EXTERNAL in the calling program.

BOUND — Finite bound of the integration range.   (Input)
Ignored if INTERV = 2.

INTERV — Flag indicating integration interval.   (Input)

INTERV         Interval

−1                               (−∞, BOUND)

1                                  (BOUND, + ∞)

2                                  (−∞, + ∞)

RESULT — Estimate of the integral from A to B of F.   (Output)

Optional Arguments

ERRABS — Absolute accuracy desired.   (Input)
Default: ERRABS = 1.e-3 for single precision and 1.d-8 for double precision.

ERRREL — Relative accuracy desired.   (Input)
Default: ERRREL = 1.e-3 for single precision and 1.d-8 for double precision.

ERREST — Estimate of the absolute value of the error.   (Output)

FORTRAN 90 Interface

Generic:          CALL QDAGI (F, BOUND, INTERV, RESULT [,…])

Specific:         The specific interface names are S_QDAGI and D_QDAGI.

FORTRAN 77 Interface

Single:            CALL QDAGI (F, BOUND, INTERV, ERRABS, ERRREL, RESULT, ERREST)

Double:          The double precision name is DQDAGI.

Description

The routine QDAGI uses a globally adaptive scheme in an attempt to reduce the absolute error. It initially transforms an infinite or semi-infinite interval into the finite interval [0, 1]. Then, QDAGI uses a 21-point Gauss-Kronrod rule to estimate the integral and the error. It bisects any interval with an unacceptable error estimate and continues this process until termination. This routine is designed to handle endpoint singularities. In addition to the general strategy described in QDAG, this subroutine employs an extrapolation procedure known as the ɛ-algorithm. The routine QDAGI is an implementation of the subroutine QAGI, which is fully documented by Piessens et al. (1983).

Comments

1.         Workspace may be explicitly provided, if desired, by use of Q2AGI/DQ2AGI. The reference is

CALL Q2AGI (F, BOUND, INTERV, ERRABS, ERRREL, RESULT, ERREST, MAXSUB, NEVAL, NSUBIN, ALIST, BLIST, RLIST, ELIST, IORD)

The additional arguments are as follows:

MAXSUB — Number of subintervals allowed.   (Input)
A value of 500 is used by QDAGI.

NEVAL — Number of evaluations of F.   (Output)

NSUBIN — Number of subintervals generated.   (Output)

ALIST — Array of length MAXSUB containing a list of the NSUBIN left endpoints.   (Output)

BLIST — Array of length MAXSUB containing a list of the NSUBIN right endpoints.   (Output)

RLIST — Array of length MAXSUB containing approximations to the NSUBIN integrals over the intervals defined by ALIST, BLIST.   (Output)

ELIST — Array of length MAXSUB containing the error estimates of the NSUBIN values in RLIST.   (Output)

IORD — Array of length MAXSUB.   (Output)
Let K be NSUBIN if NSUBIN .LE.(MAXSUB/2 + 2), MAXSUB +
1 − NSUBIN otherwise. The first K locations contain pointers to the error estimates over the subintervals, such that ELIST(IORD(1)), …, ELIST(IORD(K)) form a decreasing sequence.

2.         Informational errors
Type                                   Code

   4                  1       The maximum number of subintervals allowed has been reached.

   3                  2       Roundoff error, preventing the requested tolerance from being achieved, has been detected.

   3                  3       A degradation in precision has been detected.

   3                  4       Roundoff error in the extrapolation table, preventing the requested tolerance from being achieved, has been detected.

   4                  5       Integral is divergent or slowly convergent.

3.         If EXACT is the exact value, QDAGI attempts to find RESULT such that
ABS(EXACT − RESULT).LE.MAX(ERRABS, ERRREL * ABS(EXACT)). To specify only a relative error, set ERRABS to zero. Similarly, to specify only an absolute error, set ERRREL to zero.

Example

The value of

is estimated. The values of the actual and estimated error are machine dependent. Note that we have requested an absolute error of 0 and a relative error of .001. The effect of these requests, as documented in Comment 3 above, is to ignore the absolute error requirement.

 

      USE QDAGI_INT

      USE UMACH_INT

      USE CONST_INT

 

      IMPLICIT   NONE

      INTEGER    INTERV, NOUT

      REAL       ABS, ALOG, BOUND, ERRABS, ERREST, ERROR, &

                ERRREL, EXACT, F, PI, RESULT

      INTRINSIC  ABS, ALOG

      EXTERNAL   F

!                                 Get output unit number

      CALL UMACH (2, NOUT)

!                                 Set limits of integration

      BOUND  = 0.0

      INTERV = 1

!                                 Set error tolerances

      ERRABS = 0.0

      CALL QDAGI (F, BOUND, INTERV, RESULT, ERRABS=ERRABS,  &

                 ERREST=ERREST)

!                                 Print results

      PI    = CONST('PI')

      EXACT = -PI*ALOG(10.)/20.

      ERROR = ABS(RESULT-EXACT)

      WRITE (NOUT,99999) RESULT, EXACT, ERREST, ERROR

99999 FORMAT (' Computed =', F8.3, 13X, ' Exact =', F8.3//' Error ', &

            'estimate =', 1PE10.3, 6X, 'Error =', 1PE10.3)

      END

!

      REAL FUNCTION F (X)

      REAL       X

      REAL       ALOG

      INTRINSIC  ALOG

      F = ALOG(X)/(1.+(10.*X)**2)

      RETURN

      END

Output

 

Computed =  -0.362              Exact =  -0.362

Error estimate = 2.652E-06      Error = 5.960E-08

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