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OpenLibm/slatec/qawse.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 QAWSE
SUBROUTINE QAWSE (F, A, B, ALFA, BETA, INTEGR, EPSABS, EPSREL,
+ LIMIT, RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST,
+ IORD, LAST)
C***BEGIN PROLOGUE QAWSE
C***PURPOSE The routine calculates an approximation result to a given
C definite integral I = Integral of F*W over (A,B),
C (where W shows a singular behaviour at the end points,
C see parameter INTEGR).
C Hopefully satisfying following claim for accuracy
C ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
C***LIBRARY SLATEC (QUADPACK)
C***CATEGORY H2A2A1
C***TYPE SINGLE PRECISION (QAWSE-S, DQAWSE-D)
C***KEYWORDS ALGEBRAIC-LOGARITHMIC END POINT SINGULARITIES,
C AUTOMATIC INTEGRATOR, CLENSHAW-CURTIS METHOD, QUADPACK,
C QUADRATURE, 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 Integration of functions having algebraico-logarithmic
C end point singularities
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, B.GT.A
C If B.LE.A, the routine will end with IER = 6.
C
C ALFA - Real
C Parameter in the WEIGHT function, ALFA.GT.(-1)
C If ALFA.LE.(-1), the routine will end with
C IER = 6.
C
C BETA - Real
C Parameter in the WEIGHT function, BETA.GT.(-1)
C If BETA.LE.(-1), the routine will end with
C IER = 6.
C
C INTEGR - Integer
C Indicates which WEIGHT function is to be used
C = 1 (X-A)**ALFA*(B-X)**BETA
C = 2 (X-A)**ALFA*(B-X)**BETA*LOG(X-A)
C = 3 (X-A)**ALFA*(B-X)**BETA*LOG(B-X)
C = 4 (X-A)**ALFA*(B-X)**BETA*LOG(X-A)*LOG(B-X)
C If INTEGR.LT.1 or INTEGR.GT.4, the routine
C will end with 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.2
C If LIMIT.LT.2, the routine will end with IER = 6.
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 the integral and error
C are less reliable. It is assumed that the
C requested accuracy has not been achieved.
C ERROR MESSAGES
C = 1 Maximum number of subdivisions allowed
C has been achieved. One can allow more
C subdivisions by increasing the value of
C LIMIT. However, if this yields no
C improvement, it is advised to analyze the
C integrand in order to determine the
C integration difficulties which prevent the
C requested tolerance from being achieved.
C In case of a jump DISCONTINUITY or a local
C SINGULARITY of algebraico-logarithmic type
C at one or more interior points of the
C integration range, one should proceed by
C splitting up the interval at these
C points and calling the integrator on the
C subranges.
C = 2 The occurrence of roundoff error is
C detected, which prevents the requested
C tolerance from being achieved.
C = 3 Extremely bad integrand behaviour occurs
C at some points of the integration
C interval.
C = 6 The input is invalid, because
C B.LE.A or ALFA.LE.(-1) or BETA.LE.(-1), or
C INTEGR.LT.1 or INTEGR.GT.4, or
C (EPSABS.LE.0 and
C EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
C or LIMIT.LT.2.
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 at least LIMIT, the first
C LAST elements of which are the moduli of the
C absolute error estimates on the subintervals
C
C IORD - Integer
C Vector of dimension at least LIMIT, the first K
C 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 QC25S, QMOMO, 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 QAWSE
C
REAL A,ABSERR,ALFA,ALIST,AREA,AREA1,AREA12,
1 AREA2,A1,A2,B,BETA,BLIST,B1,B2,CENTRE,
2 R1MACH,ELIST,EPMACH,EPSABS,EPSREL,ERRBND,ERRMAX,
3 ERROR1,ERRO12,ERROR2,ERRSUM,F,RESAS1,RESAS2,RESULT,RG,RH,RI,RJ,
4 RLIST,UFLOW
INTEGER IER,INTEGR,IORD,IROFF1,IROFF2,K,LAST,
1 LIMIT,MAXERR,NEV,NEVAL,NRMAX
C
EXTERNAL F
C
DIMENSION ALIST(*),BLIST(*),RLIST(*),ELIST(*),
1 IORD(*),RI(25),RJ(25),RH(25),RG(25)
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 QAWSE
EPMACH = R1MACH(4)
UFLOW = R1MACH(1)
C
C TEST ON VALIDITY OF PARAMETERS
C ------------------------------
C
IER = 6
NEVAL = 0
LAST = 0
RLIST(1) = 0.0E+00
ELIST(1) = 0.0E+00
IORD(1) = 0
RESULT = 0.0E+00
ABSERR = 0.0E+00
IF (B.LE.A.OR.(EPSABS.EQ.0.0E+00.AND.
1 EPSREL.LT.MAX(0.5E+02*EPMACH,0.5E-14)).OR.ALFA.LE.(-0.1E+01)
2 .OR.BETA.LE.(-0.1E+01).OR.INTEGR.LT.1.OR.INTEGR.GT.4.OR.
3 LIMIT.LT.2) GO TO 999
IER = 0
C
C COMPUTE THE MODIFIED CHEBYSHEV MOMENTS.
C
CALL QMOMO(ALFA,BETA,RI,RJ,RG,RH,INTEGR)
C
C INTEGRATE OVER THE INTERVALS (A,(A+B)/2)
C AND ((A+B)/2,B).
C
CENTRE = 0.5E+00*(B+A)
CALL QC25S(F,A,B,A,CENTRE,ALFA,BETA,RI,RJ,RG,RH,AREA1,
1 ERROR1,RESAS1,INTEGR,NEV)
NEVAL = NEV
CALL QC25S(F,A,B,CENTRE,B,ALFA,BETA,RI,RJ,RG,RH,AREA2,
1 ERROR2,RESAS2,INTEGR,NEV)
LAST = 2
NEVAL = NEVAL+NEV
RESULT = AREA1+AREA2
ABSERR = ERROR1+ERROR2
C
C TEST ON ACCURACY.
C
ERRBND = MAX(EPSABS,EPSREL*ABS(RESULT))
C
C INITIALIZATION
C --------------
C
IF(ERROR2.GT.ERROR1) GO TO 10
ALIST(1) = A
ALIST(2) = CENTRE
BLIST(1) = CENTRE
BLIST(2) = B
RLIST(1) = AREA1
RLIST(2) = AREA2
ELIST(1) = ERROR1
ELIST(2) = ERROR2
GO TO 20
10 ALIST(1) = CENTRE
ALIST(2) = A
BLIST(1) = B
BLIST(2) = CENTRE
RLIST(1) = AREA2
RLIST(2) = AREA1
ELIST(1) = ERROR2
ELIST(2) = ERROR1
20 IORD(1) = 1
IORD(2) = 2
IF(LIMIT.EQ.2) IER = 1
IF(ABSERR.LE.ERRBND.OR.IER.EQ.1) GO TO 999
ERRMAX = ELIST(1)
MAXERR = 1
NRMAX = 1
AREA = RESULT
ERRSUM = ABSERR
IROFF1 = 0
IROFF2 = 0
C
C MAIN DO-LOOP
C ------------
C
DO 60 LAST = 3,LIMIT
C
C BISECT THE SUBINTERVAL WITH LARGEST ERROR ESTIMATE.
C
A1 = ALIST(MAXERR)
B1 = 0.5E+00*(ALIST(MAXERR)+BLIST(MAXERR))
A2 = B1
B2 = BLIST(MAXERR)
C
CALL QC25S(F,A,B,A1,B1,ALFA,BETA,RI,RJ,RG,RH,AREA1,
1 ERROR1,RESAS1,INTEGR,NEV)
NEVAL = NEVAL+NEV
CALL QC25S(F,A,B,A2,B2,ALFA,BETA,RI,RJ,RG,RH,AREA2,
1 ERROR2,RESAS2,INTEGR,NEV)
NEVAL = NEVAL+NEV
C
C IMPROVE PREVIOUS APPROXIMATIONS INTEGRAL AND ERROR
C AND TEST FOR ACCURACY.
C
AREA12 = AREA1+AREA2
ERRO12 = ERROR1+ERROR2
ERRSUM = ERRSUM+ERRO12-ERRMAX
AREA = AREA+AREA12-RLIST(MAXERR)
IF(A.EQ.A1.OR.B.EQ.B2) GO TO 30
IF(RESAS1.EQ.ERROR1.OR.RESAS2.EQ.ERROR2) GO TO 30
C
C TEST FOR ROUNDOFF ERROR.
C
IF(ABS(RLIST(MAXERR)-AREA12).LT.0.1E-04*ABS(AREA12)
1 .AND.ERRO12.GE.0.99E+00*ERRMAX) IROFF1 = IROFF1+1
IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF2 = IROFF2+1
30 RLIST(MAXERR) = AREA1
RLIST(LAST) = AREA2
C
C TEST ON ACCURACY.
C
ERRBND = MAX(EPSABS,EPSREL*ABS(AREA))
IF(ERRSUM.LE.ERRBND) GO TO 35
C
C SET ERROR FLAG IN THE CASE THAT THE NUMBER OF INTERVAL
C BISECTIONS EXCEEDS LIMIT.
C
IF(LAST.EQ.LIMIT) IER = 1
C
C
C SET ERROR FLAG IN THE CASE OF ROUNDOFF ERROR.
C
IF(IROFF1.GE.6.OR.IROFF2.GE.20) IER = 2
C
C SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR
C AT INTERIOR POINTS OF 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
35 IF(ERROR2.GT.ERROR1) GO TO 40
ALIST(LAST) = A2
BLIST(MAXERR) = B1
BLIST(LAST) = B2
ELIST(MAXERR) = ERROR1
ELIST(LAST) = ERROR2
GO TO 50
40 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 LARGEST ERROR ESTIMATE (TO BE
C BISECTED NEXT).
C
50 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 70
60 CONTINUE
C
C COMPUTE FINAL RESULT.
C ---------------------
C
70 RESULT = 0.0E+00
DO 80 K=1,LAST
RESULT = RESULT+RLIST(K)
80 CONTINUE
ABSERR = ERRSUM
999 RETURN
END
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