QAWO(3)

SYNOPSIS
SUBROUTINE QAWO (F, A, B, OMEGA, INTEGR, EPSABS, EPSREL, RESULT, + ABSERR, NEVAL, IER, LENIW, MAXP1, LENW, LAST, IWORK, WORK)

NAME
qawo - Calculate an approximation to a given definite integral I = Integral of F(X)*W(X) over (A,B), where W(X) = COS(OMEGA*X) or W(X) = SIN(OMEGA*X), hopefully satisfying the following claim for accuracy ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).

LIBRARY
SLATEC (QUADPACK)

CATEGORY
H2A2A1

TYPE
SINGLE PRECISION (QAWO-S, DQAWO-D)

KEYWORDS
AUTOMATIC INTEGRATOR, CLENSHAW-CURTIS METHOD, EXTRAPOLATION, GLOBALLY ADAPTIVE, INTEGRAND WITH OSCILLATORY COS OR SIN FACTOR, QUADPACK, QUADRATURE, SPECIAL-PURPOSE

AUTHOR
Piessens, Robert Applied Mathematics and Programming Division K. U. Leuven de Doncker, Elise Applied Mathematics and Programming Division K. U. Leuven

DESCRIPTION
Computation of oscillatory integrals Standard fortran subroutine Real version PARAMETERS ON ENTRY F - Real Function subprogram defining the function F(X). The actual name for F needs to be declared E X T E R N A L in the driver program. A - Real Lower limit of integration B - Real Upper limit of integration OMEGA - Real Parameter in the integrand weight function INTEGR - Integer Indicates which of the weight functions is used INTEGR = 1 W(X) = COS(OMEGA*X) INTEGR = 2 W(X) = SIN(OMEGA*X) If INTEGR.NE.1.AND.INTEGR.NE.2, the routine will end with IER = 6. EPSABS - Real Absolute accuracy requested EPSREL - Real Relative accuracy requested If EPSABS.LE.0 and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28), the routine will end with IER = 6. ON RETURN RESULT - Real Approximation to the integral ABSERR - Real Estimate of the modulus of the absolute error, which should equal or exceed ABS(I-RESULT) NEVAL - Integer Number of integrand evaluations IER - Integer IER = 0 Normal and reliable termination of the routine. It is assumed that the requested accuracy has been achieved. - IER.GT.0 Abnormal termination of the routine. The estimates for integral and error are less reliable. It is assumed that the requested accuracy has not been achieved. ERROR MESSAGES IER = 1 Maximum number of subdivisions allowed has been achieved(= LENIW/2). One can allow more subdivisions by increasing the value of LENIW (and taking the according dimension adjustments into account). However, if this yields no improvement it is advised to analyze the integrand in order to determine the integration difficulties. If the position of a local difficulty can be determined (e.g. SINGULARITY, DISCONTINUITY within the interval) one will probably gain from splitting up the interval at this point and calling the integrator on the subranges. If possible, an appropriate special-purpose integrator should be used which is designed for handling the type of difficulty involved. = 2 The occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved. The error may be under-estimated. = 3 Extremely bad integrand behaviour occurs at some interior points of the integration interval. = 4 The algorithm does not converge. Roundoff error is detected in the extrapolation table. It is presumed that the requested tolerance cannot be achieved due to roundoff in the extrapolation table, and that the returned result is the best which can be obtained. = 5 The integral is probably divergent, or slowly convergent. It must be noted that divergence can occur with any other value of IER. = 6 The input is invalid, because (EPSABS.LE.0 and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28)) or (INTEGR.NE.1 AND INTEGR.NE.2), or LENIW.LT.2 OR MAXP1.LT.1 or LENW.LT.LENIW*2+MAXP1*25. RESULT, ABSERR, NEVAL, LAST are set to zero. Except when LENIW, MAXP1 or LENW are invalid, WORK(LIMIT*2+1), WORK(LIMIT*3+1), IWORK(1), IWORK(LIMIT+1) are set to zero, WORK(1) is set to A and WORK(LIMIT+1) to B. DIMENSIONING PARAMETERS LENIW - Integer Dimensioning parameter for IWORK. LENIW/2 equals the maximum number of subintervals allowed in the partition of the given integration interval (A,B), LENIW.GE.2. If LENIW.LT.2, the routine will end with IER = 6. MAXP1 - Integer Gives an upper bound on the number of Chebyshev moments which can be stored, i.e. for the intervals of lengths ABS(B-A)*2**(-L), L=0,1, ..., MAXP1-2, MAXP1.GE.1 If MAXP1.LT.1, the routine will end with IER = 6. LENW - Integer Dimensioning parameter for WORK LENW must be at least LENIW*2+MAXP1*25. If LENW.LT.(LENIW*2+MAXP1*25), the routine will end with IER = 6. LAST - Integer On return, LAST equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the WORK ARRAYS. WORK ARRAYS IWORK - Integer Vector of dimension at least LENIW on return, the first K elements of which contain pointers to the error estimates over the subintervals, such that WORK(LIMIT*3+IWORK(1)), .. WORK(LIMIT*3+IWORK(K)) form a decreasing sequence, with LIMIT = LENW/2 , and K = LAST if LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST otherwise. Furthermore, IWORK(LIMIT+1), ..., IWORK(LIMIT+ LAST) indicate the subdivision levels of the subintervals, such that IWORK(LIMIT+I) = L means that the subinterval numbered I is of length ABS(B-A)*2**(1-L). WORK - Real Vector of dimension at least LENW On return WORK(1), ..., WORK(LAST) contain the left end points of the subintervals in the partition of (A,B), WORK(LIMIT+1), ..., WORK(LIMIT+LAST) contain the right end points, WORK(LIMIT*2+1), ..., WORK(LIMIT*2+LAST) contain the integral approximations over the subintervals, WORK(LIMIT*3+1), ..., WORK(LIMIT*3+LAST) contain the error estimates. WORK(LIMIT*4+1), ..., WORK(LIMIT*4+MAXP1*25) Provide space for storing the Chebyshev moments. Note that LIMIT = LENW/2.

REFERENCES
(NONE)

ROUTINES CALLED
QAWOE, XERMSG

REVISION HISTORY (YYMMDD)
800101 DATE WRITTEN
890831 Modified array declarations. (WRB)
890831 REVISION DATE from Version 3.2
891214 Prologue converted to Version 4.0 format. (BAB)
900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)