Integrates a function on a hyper-rectangle.
Required Arguments
F — User-supplied
FUNCTION to be
integrated. The form is F(N, X),
where
N – The
dimension of the hyper-rectangle.
(Input)
X – The
independent variable of dimension N.
(Input)
F – The value of
the integrand at X.
(Output)
F
must be declared EXTERNAL in the
calling program.
N — The dimension
of the hyper-rectangle. (Input)
N must be less than or
equal to 20.
A — Vector of
length N. (Input)
Lower limits of integration.
B — Vector of
length N. (Input)
Upper limits of integration.
RESULT — Estimate
of the integral from A to B of F.
(Output)
The integral of F is approximated over
the N-dimensional
hyper-rectangle A.LE.X.LE.B.
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.
MAXFCN —
Approximate maximum number of function evaluations to be permitted.
(Input)
MAXFCN cannot be
greater than 256N or IMACH(5) if N is greater than
3.
Default: MAXFCN = 32**N.
ERREST — Estimate of the absolute value of the error. (Output)
FORTRAN 90 Interface
Generic: CALL QAND (F, N, A, B, RESULT [,…])
Specific: The specific interface names are S_QAND and D_QAND.
FORTRAN 77 Interface
Single: CALL QAND (F, N, A, B, ERRABS, ERRREL, MAXFCN, RESULT, ERREST)
Double: The double precision name is DQAND.
Description
The routine QAND approximates the n-dimensional iterated integral
with the approximation returned in RESULT. An estimate of the error is returned in ERREST. The approximation is achieved by iterated applications of product Gauss formulas. The integral is first estimated by a two-point tensor product formula in each direction. Then for i = 1, …, n the routine calculates a new estimate by doubling the number of points in the i-th direction, but halving the number immediately afterwards if the new estimate does not change appreciably. This process is repeated until either one complete sweep results in no increase in the number of sample points in any dimension, or the number of Gauss points in one direction exceeds 256, or the number of function evaluations needed to complete a sweep would exceed MAXFCN.
Comments
1.
Informational
errors
Type
Code
3 1 MAXFCN was set greater than 256N.
4 2 The maximum number of function evaluations has been reached, and convergence has not been attained.
2.
If EXACT is the
exact value, QAND 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
In this example, we approximate the integral of
on an expanding cube. The values of the error estimates are machine dependent. The exact integral over
USE QAND_INT
USE UMACH_INT
IMPLICIT NONE
INTEGER I, J, MAXFCN, N, NOUT
REAL A(3), B(3), CNST, ERRABS, ERREST, ERRREL, F, RESULT
EXTERNAL F
! Get output unit number
CALL UMACH (2, NOUT)
!
N = 3
MAXFCN = 100000
! Set error tolerances
ERRABS = 0.0001
ERRREL = 0.001
!
DO 20 I=1, 6
CNST = I/2.0
! Set limits of integration
! As CNST approaches infinity, the
! answer approaches PI**1.5
DO 10 J=1, 3
A(J) = -CNST
B(J) = CNST
10 CONTINUE
CALL QAND (F, N, A, B, RESULT, ERRABS, ERRREL, MAXFCN, ERREST)
WRITE (NOUT,99999) CNST, RESULT, ERREST
20 CONTINUE
99999 FORMAT (1X, 'For CNST = ', F4.1, ', result = ', F7.3, ' with ', &
'error estimate ', 1PE10.3)
END
!
REAL FUNCTION F (N, X)
INTEGER N
REAL X(N)
REAL EXP
INTRINSIC EXP
F = EXP(-(X(1)*X(1)+X(2)*X(2)+X(3)*X(3)))
RETURN
END
Output
For CNST = 0.5, result = 0.785 with error
estimate 3.934E-06
For CNST = 1.0, result = 3.332
with error estimate 2.100E-03
For CNST = 1.5, result
= 5.021 with error estimate 1.192E-05
For CNST = 2.0,
result = 5.491 with error estimate 2.413E-04
For CNST
= 2.5, result = 5.561 with error estimate
4.232E-03
For CNST = 3.0, result = 5.568 with error
estimate 2.580E-04
|
Visual Numerics, Inc. PHONE: 713.784.3131 FAX:713.781.9260 |