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OpenLibm/slatec/qawce.f
Viral B. Shah c977aa998f Add Makefile.extras to build libopenlibm-extras. 
Replace amos with slatec
2012-12-31 16:37:05 -05:00

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*DECK QAWCE
SUBROUTINE QAWCE (F, A, B, C, EPSABS, EPSREL, LIMIT, RESULT,
+ ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, IORD, LAST)
C***BEGIN PROLOGUE QAWCE
C***PURPOSE The routine calculates an approximation result to a
C CAUCHY PRINCIPAL VALUE I = Integral of F*W over (A,B)
C (W(X) = 1/(X-C), (C.NE.A, C.NE.B), hopefully satisfying
C following claim for accuracy
C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I))
C***LIBRARY SLATEC (QUADPACK)
C***CATEGORY H2A2A1, J4
C***TYPE SINGLE PRECISION (QAWCE-S, DQAWCE-D)
C***KEYWORDS AUTOMATIC INTEGRATOR, CAUCHY PRINCIPAL VALUE,
C CLENSHAW-CURTIS METHOD, QUADPACK, QUADRATURE,
C SPECIAL-PURPOSE
C***AUTHOR Piessens, Robert
C Applied Mathematics and Programming Division
C K. U. Leuven
C de Doncker, Elise
C Applied Mathematics and Programming Division
C K. U. Leuven
C***DESCRIPTION
C
C Computation of a CAUCHY PRINCIPAL VALUE
C Standard fortran subroutine
C Real version
C
C PARAMETERS
C ON ENTRY
C F - Real
C Function subprogram defining the integrand
C function F(X). The actual name for F needs to be
C declared E X T E R N A L in the driver program.
C
C A - Real
C Lower limit of integration
C
C B - Real
C Upper limit of integration
C
C C - Real
C Parameter in the WEIGHT function, C.NE.A, C.NE.B
C If C = A OR C = B, the routine will end with
C IER = 6.
C
C EPSABS - Real
C Absolute accuracy requested
C EPSREL - Real
C Relative accuracy requested
C If EPSABS.LE.0
C and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
C the routine will end with IER = 6.
C
C LIMIT - Integer
C Gives an upper bound on the number of subintervals
C in the partition of (A,B), LIMIT.GE.1
C
C ON RETURN
C RESULT - Real
C Approximation to the integral
C
C ABSERR - Real
C Estimate of the modulus of the absolute error,
C which should equal or exceed ABS(I-RESULT)
C
C NEVAL - Integer
C Number of integrand evaluations
C
C IER - Integer
C IER = 0 Normal and reliable termination of the
C routine. It is assumed that the requested
C accuracy has been achieved.
C IER.GT.0 Abnormal termination of the routine
C the estimates for integral and error are
C less reliable. It is assumed that the
C requested accuracy has not been achieved.
C ERROR MESSAGES
C IER = 1 Maximum number of subdivisions allowed
C has been achieved. One can allow more sub-
C divisions by increasing the value of
C LIMIT. However, if this yields no
C improvement it is advised to analyze the
C the integrand, in order to determine the
C the integration difficulties. If the
C position of a local difficulty can be
C determined (e.g. SINGULARITY,
C DISCONTINUITY within the interval) one
C will probably gain from splitting up the
C interval at this point and calling
C appropriate integrators on the subranges.
C = 2 The occurrence of roundoff error is detec-
C ted, which prevents the requested
C tolerance from being achieved.
C = 3 Extremely bad integrand behaviour
C occurs at some interior points of
C the integration interval.
C = 6 The input is invalid, because
C C = A or C = B or
C (EPSABS.LE.0 and
C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28))
C or LIMIT.LT.1.
C RESULT, ABSERR, NEVAL, RLIST(1), ELIST(1),
C IORD(1) and LAST are set to zero. ALIST(1)
C and BLIST(1) are set to A and B
C respectively.
C
C ALIST - Real
C Vector of dimension at least LIMIT, the first
C LAST elements of which are the left
C end points of the subintervals in the partition
C of the given integration range (A,B)
C
C BLIST - Real
C Vector of dimension at least LIMIT, the first
C LAST elements of which are the right
C end points of the subintervals in the partition
C of the given integration range (A,B)
C
C RLIST - Real
C Vector of dimension at least LIMIT, the first
C LAST elements of which are the integral
C approximations on the subintervals
C
C ELIST - Real
C Vector of dimension LIMIT, the first LAST
C elements of which are the moduli of the absolute
C error estimates on the subintervals
C
C IORD - Integer
C Vector of dimension at least LIMIT, the first K
C elements of which are pointers to the error
C estimates over the subintervals, so that
C ELIST(IORD(1)), ..., ELIST(IORD(K)) with K = LAST
C If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST
C otherwise, form a decreasing sequence
C
C LAST - Integer
C Number of subintervals actually produced in
C the subdivision process
C
C***REFERENCES (NONE)
C***ROUTINES CALLED QC25C, QPSRT, R1MACH
C***REVISION HISTORY (YYMMDD)
C 800101 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 890831 Modified array declarations. (WRB)
C 890831 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C***END PROLOGUE QAWCE
C
REAL A,AA,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1,A2,B,BB,BLIST,
1 B1,B2,C,R1MACH,ELIST,EPMACH,EPSABS,EPSREL,ERRBND,ERRMAX,ERROR1,
2 ERROR2,ERRSUM,F,RESULT,RLIST,UFLOW
INTEGER IER,IORD,IROFF1,IROFF2,K,KRULE,LAST,LIMIT,MAXERR,NEV,
1 NEVAL,NRMAX
C
DIMENSION ALIST(*),BLIST(*),RLIST(*),ELIST(*),
1 IORD(*)
C
EXTERNAL F
C
C LIST OF MAJOR VARIABLES
C -----------------------
C
C ALIST - LIST OF LEFT END POINTS OF ALL SUBINTERVALS
C CONSIDERED UP TO NOW
C BLIST - LIST OF RIGHT END POINTS OF ALL SUBINTERVALS
C CONSIDERED UP TO NOW
C RLIST(I) - APPROXIMATION TO THE INTEGRAL OVER
C (ALIST(I),BLIST(I))
C ELIST(I) - ERROR ESTIMATE APPLYING TO RLIST(I)
C MAXERR - POINTER TO THE INTERVAL WITH LARGEST
C ERROR ESTIMATE
C ERRMAX - ELIST(MAXERR)
C AREA - SUM OF THE INTEGRALS OVER THE SUBINTERVALS
C ERRSUM - SUM OF THE ERRORS OVER THE SUBINTERVALS
C ERRBND - REQUESTED ACCURACY MAX(EPSABS,EPSREL*
C ABS(RESULT))
C *****1 - VARIABLE FOR THE LEFT SUBINTERVAL
C *****2 - VARIABLE FOR THE RIGHT SUBINTERVAL
C LAST - INDEX FOR SUBDIVISION
C
C
C MACHINE DEPENDENT CONSTANTS
C ---------------------------
C
C EPMACH IS THE LARGEST RELATIVE SPACING.
C UFLOW IS THE SMALLEST POSITIVE MAGNITUDE.
C
C***FIRST EXECUTABLE STATEMENT QAWCE
EPMACH = R1MACH(4)
UFLOW = R1MACH(1)
C
C
C TEST ON VALIDITY OF PARAMETERS
C ------------------------------
C
IER = 6
NEVAL = 0
LAST = 0
ALIST(1) = A
BLIST(1) = B
RLIST(1) = 0.0E+00
ELIST(1) = 0.0E+00
IORD(1) = 0
RESULT = 0.0E+00
ABSERR = 0.0E+00
IF (C.EQ.A.OR.C.EQ.B.OR.(EPSABS.LE.0.0E+00.AND.
1 EPSREL.LT.MAX(0.5E+02*EPMACH,0.5E-14))) GO TO 999
C
C FIRST APPROXIMATION TO THE INTEGRAL
C -----------------------------------
C
AA=A
BB=B
IF (A.LE.B) GO TO 10
AA=B
BB=A
10 IER=0
KRULE = 1
CALL QC25C(F,AA,BB,C,RESULT,ABSERR,KRULE,NEVAL)
LAST = 1
RLIST(1) = RESULT
ELIST(1) = ABSERR
IORD(1) = 1
ALIST(1) = A
BLIST(1) = B
C
C TEST ON ACCURACY
C
ERRBND = MAX(EPSABS,EPSREL*ABS(RESULT))
IF(LIMIT.EQ.1) IER = 1
IF(ABSERR.LT.MIN(0.1E-01*ABS(RESULT),ERRBND)
1 .OR.IER.EQ.1) GO TO 70
C
C INITIALIZATION
C --------------
C
ALIST(1) = AA
BLIST(1) = BB
RLIST(1) = RESULT
ERRMAX = ABSERR
MAXERR = 1
AREA = RESULT
ERRSUM = ABSERR
NRMAX = 1
IROFF1 = 0
IROFF2 = 0
C
C MAIN DO-LOOP
C ------------
C
DO 40 LAST = 2,LIMIT
C
C BISECT THE SUBINTERVAL WITH NRMAX-TH LARGEST
C ERROR ESTIMATE.
C
A1 = ALIST(MAXERR)
B1 = 0.5E+00*(ALIST(MAXERR)+BLIST(MAXERR))
B2 = BLIST(MAXERR)
IF(C.LE.B1.AND.C.GT.A1) B1 = 0.5E+00*(C+B2)
IF(C.GT.B1.AND.C.LT.B2) B1 = 0.5E+00*(A1+C)
A2 = B1
KRULE = 2
CALL QC25C(F,A1,B1,C,AREA1,ERROR1,KRULE,NEV)
NEVAL = NEVAL+NEV
CALL QC25C(F,A2,B2,C,AREA2,ERROR2,KRULE,NEV)
NEVAL = NEVAL+NEV
C
C IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL
C AND ERROR AND TEST FOR ACCURACY.
C
AREA12 = AREA1+AREA2
ERRO12 = ERROR1+ERROR2
ERRSUM = ERRSUM+ERRO12-ERRMAX
AREA = AREA+AREA12-RLIST(MAXERR)
IF(ABS(RLIST(MAXERR)-AREA12).LT.0.1E-04*ABS(AREA12)
1 .AND.ERRO12.GE.0.99E+00*ERRMAX.AND.KRULE.EQ.0)
2 IROFF1 = IROFF1+1
IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX.AND.KRULE.EQ.0)
1 IROFF2 = IROFF2+1
RLIST(MAXERR) = AREA1
RLIST(LAST) = AREA2
ERRBND = MAX(EPSABS,EPSREL*ABS(AREA))
IF(ERRSUM.LE.ERRBND) GO TO 15
C
C TEST FOR ROUNDOFF ERROR AND EVENTUALLY
C SET ERROR FLAG.
C
IF(IROFF1.GE.6.AND.IROFF2.GT.20) IER = 2
C
C SET ERROR FLAG IN THE CASE THAT NUMBER OF INTERVAL
C BISECTIONS EXCEEDS LIMIT.
C
IF(LAST.EQ.LIMIT) IER = 1
C
C SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR
C AT A POINT OF THE INTEGRATION RANGE.
C
IF(MAX(ABS(A1),ABS(B2)).LE.(0.1E+01+0.1E+03*EPMACH)
1 *(ABS(A2)+0.1E+04*UFLOW)) IER = 3
C
C APPEND THE NEWLY-CREATED INTERVALS TO THE LIST.
C
15 IF(ERROR2.GT.ERROR1) GO TO 20
ALIST(LAST) = A2
BLIST(MAXERR) = B1
BLIST(LAST) = B2
ELIST(MAXERR) = ERROR1
ELIST(LAST) = ERROR2
GO TO 30
20 ALIST(MAXERR) = A2
ALIST(LAST) = A1
BLIST(LAST) = B1
RLIST(MAXERR) = AREA2
RLIST(LAST) = AREA1
ELIST(MAXERR) = ERROR2
ELIST(LAST) = ERROR1
C
C CALL SUBROUTINE QPSRT TO MAINTAIN THE DESCENDING ORDERING
C IN THE LIST OF ERROR ESTIMATES AND SELECT THE
C SUBINTERVAL WITH NRMAX-TH LARGEST ERROR ESTIMATE (TO BE
C BISECTED NEXT).
C
30 CALL QPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX)
C ***JUMP OUT OF DO-LOOP
IF(IER.NE.0.OR.ERRSUM.LE.ERRBND) GO TO 50
40 CONTINUE
C
C COMPUTE FINAL RESULT.
C ---------------------
C
50 RESULT = 0.0E+00
DO 60 K=1,LAST
RESULT = RESULT+RLIST(K)
60 CONTINUE
ABSERR = ERRSUM
70 IF (AA.EQ.B) RESULT=-RESULT
999 RETURN
END
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