QDAGS

The routine QDAGS is a general-purpose integrator that uses a globally adaptive scheme to reduce the absolute error. It subdivides the interval [A, B] and uses a 21-point Gauss-Kronrod rule to estimate the integral over each subinterval. The error for each subinterval is estimated by comparison with the 10-point Gauss quadrature rule. This routine is designed to handle functions with endpoint singularities. However, the performance on functions, which are well-behaved at the endpoints, is quite good also. In addition to the general strategy described in QDAG, this routine uses an extrapolation procedure known as the ɛ-algorithm. The routine QDAGS is an implementation of the routine QAGS, which is fully documented by Piessens et al. (1983). Should QDAGS fail to produce acceptable results, then either IMSL routines QDAG or QDAG* may be appropriate. These routines are documented in this chapter.

Comments

1. Workspace may be explicitly provided, if desired, by use of Q2AGS/DQ2AGS. The reference is

CALL Q2AGS (F, A, B, 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 QDAGS.

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 ≤ (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	Description
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 probably divergent or slowly convergent.

3. If EXACT is the exact value, QDAGS attempts to find RESULT such that ∣EXACT − RESULT∣ ≤ max(ERRABS, ERRREL * ∣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.

USE QDAGS_INT

USE UMACH_INT

IMPLICIT NONE

INTEGER NOUT

REAL A, ABS, B, ERRABS, ERREST, ERROR, ERRREL, EXACT, F, &

RESULT

INTRINSIC ABS

EXTERNAL F

! Get output unit number

CALL UMACH (2, NOUT)

! Set limits of integration

A = 0.0

B = 1.0

! Set error tolerances

ERRABS = 0.0

CALL QDAGS (F, A, B, RESULT, ERRABS=ERRABS, ERREST=ERREST)

! Print results

EXACT = -4.0

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, SQRT

INTRINSIC ALOG, SQRT

F = ALOG(X)/SQRT(X)

RETURN

END

Output

Computed = -4.000 Exact = -4.000

Error estimate = 1.519E-04 Error = 2.098E-05