src/numerical/slatec/fortran/dqagpe.f

   1 *DECK DQAGPE
   2       SUBROUTINE DQAGPE (F, A, B, NPTS2, POINTS, EPSABS, EPSREL, LIMIT,
   3      +   RESULT, ABSERR, NEVAL, IER, ALIST, BLIST, RLIST, ELIST, PTS,
   4      +   IORD, LEVEL, NDIN, LAST)
   5 C***BEGIN PROLOGUE  DQAGPE
   6 C***PURPOSE  Approximate a given definite integral I = Integral of F
   7 C            over (A,B), hopefully satisfying the accuracy claim:
   8 C                 ABS(I-RESULT).LE.MAX(EPSABS,EPSREL*ABS(I)).
   9 C            Break points of the integration interval, where local
  10 C            difficulties of the integrand may occur (e.g. singularities
  11 C            or discontinuities) are provided by the user.
  12 C***LIBRARY   SLATEC (QUADPACK)
  13 C***CATEGORY  H2A2A1
  14 C***TYPE      DOUBLE PRECISION (QAGPE-S, DQAGPE-D)
  15 C***KEYWORDS  AUTOMATIC INTEGRATOR, EXTRAPOLATION, GENERAL-PURPOSE,
  16 C             GLOBALLY ADAPTIVE, QUADPACK, QUADRATURE,
  17 C             SINGULARITIES AT USER SPECIFIED POINTS
  18 C***AUTHOR  Piessens, Robert
  19 C             Applied Mathematics and Programming Division
  20 C             K. U. Leuven
  21 C           de Doncker, Elise
  22 C             Applied Mathematics and Programming Division
  23 C             K. U. Leuven
  24 C***DESCRIPTION
  25 C
  26 C        Computation of a definite integral
  27 C        Standard fortran subroutine
  28 C        Double precision version
  29 C
  30 C        PARAMETERS
  31 C         ON ENTRY
  32 C            F      - Double precision
  33 C                     Function subprogram defining the integrand
  34 C                     function F(X). The actual name for F needs to be
  35 C                     declared E X T E R N A L in the driver program.
  36 C
  37 C            A      - Double precision
  38 C                     Lower limit of integration
  39 C
  40 C            B      - Double precision
  41 C                     Upper limit of integration
  42 C
  43 C            NPTS2  - Integer
  44 C                     Number equal to two more than the number of
  45 C                     user-supplied break points within the integration
  46 C                     range, NPTS2.GE.2.
  47 C                     If NPTS2.LT.2, the routine will end with IER = 6.
  48 C
  49 C            POINTS - Double precision
  50 C                     Vector of dimension NPTS2, the first (NPTS2-2)
  51 C                     elements of which are the user provided break
  52 C                     POINTS. If these POINTS do not constitute an
  53 C                     ascending sequence there will be an automatic
  54 C                     sorting.
  55 C
  56 C            EPSABS - Double precision
  57 C                     Absolute accuracy requested
  58 C            EPSREL - Double precision
  59 C                     Relative accuracy requested
  60 C                     If  EPSABS.LE.0
  61 C                     and EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28),
  62 C                     the routine will end with IER = 6.
  63 C
  64 C            LIMIT  - Integer
  65 C                     Gives an upper bound on the number of subintervals
  66 C                     in the partition of (A,B), LIMIT.GE.NPTS2
  67 C                     If LIMIT.LT.NPTS2, the routine will end with
  68 C                     IER = 6.
  69 C
  70 C         ON RETURN
  71 C            RESULT - Double precision
  72 C                     Approximation to the integral
  73 C
  74 C            ABSERR - Double precision
  75 C                     Estimate of the modulus of the absolute error,
  76 C                     which should equal or exceed ABS(I-RESULT)
  77 C
  78 C            NEVAL  - Integer
  79 C                     Number of integrand evaluations
  80 C
  81 C            IER    - Integer
  82 C                     IER = 0 Normal and reliable termination of the
  83 C                             routine. It is assumed that the requested
  84 C                             accuracy has been achieved.
  85 C                     IER.GT.0 Abnormal termination of the routine.
  86 C                             The estimates for integral and error are
  87 C                             less reliable. It is assumed that the
  88 C                             requested accuracy has not been achieved.
  89 C            ERROR MESSAGES
  90 C                     IER = 1 Maximum number of subdivisions allowed
  91 C                             has been achieved. One can allow more
  92 C                             subdivisions by increasing the value of
  93 C                             LIMIT (and taking the according dimension
  94 C                             adjustments into account). However, if
  95 C                             this yields no improvement it is advised
  96 C                             to analyze the integrand in order to
  97 C                             determine the integration difficulties. If
  98 C                             the position of a local difficulty can be
  99 C                             determined (i.e. SINGULARITY,
 100 C                             DISCONTINUITY within the interval), it
 101 C                             should be supplied to the routine as an
 102 C                             element of the vector points. If necessary
 103 C                             an appropriate special-purpose integrator
 104 C                             must be used, which is designed for
 105 C                             handling the type of difficulty involved.
 106 C                         = 2 The occurrence of roundoff error is
 107 C                             detected, which prevents the requested
 108 C                             tolerance from being achieved.
 109 C                             The error may be under-estimated.
 110 C                         = 3 Extremely bad integrand behaviour occurs
 111 C                             At some points of the integration
 112 C                             interval.
 113 C                         = 4 The algorithm does not converge.
 114 C                             Roundoff error is detected in the
 115 C                             extrapolation table. It is presumed that
 116 C                             the requested tolerance cannot be
 117 C                             achieved, and that the returned result is
 118 C                             the best which can be obtained.
 119 C                         = 5 The integral is probably divergent, or
 120 C                             slowly convergent. It must be noted that
 121 C                             divergence can occur with any other value
 122 C                             of IER.GT.0.
 123 C                         = 6 The input is invalid because
 124 C                             NPTS2.LT.2 or
 125 C                             Break points are specified outside
 126 C                             the integration range or
 127 C                             (EPSABS.LE.0 and
 128 C                              EPSREL.LT.MAX(50*REL.MACH.ACC.,0.5D-28))
 129 C                             or LIMIT.LT.NPTS2.
 130 C                             RESULT, ABSERR, NEVAL, LAST, RLIST(1),
 131 C                             and ELIST(1) are set to zero. ALIST(1) and
 132 C                             BLIST(1) are set to A and B respectively.
 133 C
 134 C            ALIST  - Double precision
 135 C                     Vector of dimension at least LIMIT, the first
 136 C                      LAST  elements of which are the left end points
 137 C                     of the subintervals in the partition of the given
 138 C                     integration range (A,B)
 139 C
 140 C            BLIST  - Double precision
 141 C                     Vector of dimension at least LIMIT, the first
 142 C                      LAST  elements of which are the right end points
 143 C                     of the subintervals in the partition of the given
 144 C                     integration range (A,B)
 145 C
 146 C            RLIST  - Double precision
 147 C                     Vector of dimension at least LIMIT, the first
 148 C                      LAST  elements of which are the integral
 149 C                     approximations on the subintervals
 150 C
 151 C            ELIST  - Double precision
 152 C                     Vector of dimension at least LIMIT, the first
 153 C                      LAST  elements of which are the moduli of the
 154 C                     absolute error estimates on the subintervals
 155 C
 156 C            PTS    - Double precision
 157 C                     Vector of dimension at least NPTS2, containing the
 158 C                     integration limits and the break points of the
 159 C                     interval in ascending sequence.
 160 C
 161 C            LEVEL  - Integer
 162 C                     Vector of dimension at least LIMIT, containing the
 163 C                     subdivision levels of the subinterval, i.e. if
 164 C                     (AA,BB) is a subinterval of (P1,P2) where P1 as
 165 C                     well as P2 is a user-provided break point or
 166 C                     integration limit, then (AA,BB) has level L if
 167 C                     ABS(BB-AA) = ABS(P2-P1)*2**(-L).
 168 C
 169 C            NDIN   - Integer
 170 C                     Vector of dimension at least NPTS2, after first
 171 C                     integration over the intervals (PTS(I)),PTS(I+1),
 172 C                     I = 0,1, ..., NPTS2-2, the error estimates over
 173 C                     some of the intervals may have been increased
 174 C                     artificially, in order to put their subdivision
 175 C                     forward. If this happens for the subinterval
 176 C                     numbered K, NDIN(K) is put to 1, otherwise
 177 C                     NDIN(K) = 0.
 178 C
 179 C            IORD   - Integer
 180 C                     Vector of dimension at least LIMIT, the first K
 181 C                     elements of which are pointers to the
 182 C                     error estimates over the subintervals,
 183 C                     such that ELIST(IORD(1)), ..., ELIST(IORD(K))
 184 C                     form a decreasing sequence, with K = LAST
 185 C                     If LAST.LE.(LIMIT/2+2), and K = LIMIT+1-LAST
 186 C                     otherwise
 187 C
 188 C            LAST   - Integer
 189 C                     Number of subintervals actually produced in the
 190 C                     subdivisions process
 191 C
 192 C***REFERENCES  (NONE)
 193 C***ROUTINES CALLED  D1MACH, DQELG, DQK21, DQPSRT
 194 C***REVISION HISTORY  (YYMMDD)
 195 C   800101  DATE WRITTEN
 196 C   890531  Changed all specific intrinsics to generic.  (WRB)
 197 C   890831  Modified array declarations.  (WRB)
 198 C   890831  REVISION DATE from Version 3.2
 199 C   891214  Prologue converted to Version 4.0 format.  (BAB)
 200 C***END PROLOGUE  DQAGPE
 201       DOUBLE PRECISION A,ABSEPS,ABSERR,ALIST,AREA,AREA1,AREA12,AREA2,A1,
 202      1  A2,B,BLIST,B1,B2,CORREC,DEFABS,DEFAB1,DEFAB2,
 203      2  DRES,D1MACH,ELIST,EPMACH,EPSABS,EPSREL,ERLARG,ERLAST,ERRBND,
 204      3  ERRMAX,ERROR1,ERRO12,ERROR2,ERRSUM,ERTEST,F,OFLOW,POINTS,PTS,
 205      4  RESA,RESABS,RESEPS,RESULT,RES3LA,RLIST,RLIST2,SIGN,TEMP,UFLOW
 206       INTEGER I,ID,IER,IERRO,IND1,IND2,IORD,IP1,IROFF1,IROFF2,IROFF3,J,
 207      1  JLOW,JUPBND,K,KSGN,KTMIN,LAST,LEVCUR,LEVEL,LEVMAX,LIMIT,MAXERR,
 208      2  NDIN,NEVAL,NINT,NINTP1,NPTS,NPTS2,NRES,NRMAX,NUMRL2
 209       LOGICAL EXTRAP,NOEXT
 210 C
 211 C
 212       DIMENSION ALIST(*),BLIST(*),ELIST(*),IORD(*),
 213      1  LEVEL(*),NDIN(*),POINTS(*),PTS(*),RES3LA(3),
 214      2  RLIST(*),RLIST2(52)
 215 C
 216       EXTERNAL F
 217 C
 218 C            THE DIMENSION OF RLIST2 IS DETERMINED BY THE VALUE OF
 219 C            LIMEXP IN SUBROUTINE EPSALG (RLIST2 SHOULD BE OF DIMENSION
 220 C            (LIMEXP+2) AT LEAST).
 221 C
 222 C
 223 C            LIST OF MAJOR VARIABLES
 224 C            -----------------------
 225 C
 226 C           ALIST     - LIST OF LEFT END POINTS OF ALL SUBINTERVALS
 227 C                       CONSIDERED UP TO NOW
 228 C           BLIST     - LIST OF RIGHT END POINTS OF ALL SUBINTERVALS
 229 C                       CONSIDERED UP TO NOW
 230 C           RLIST(I)  - APPROXIMATION TO THE INTEGRAL OVER
 231 C                       (ALIST(I),BLIST(I))
 232 C           RLIST2    - ARRAY OF DIMENSION AT LEAST LIMEXP+2
 233 C                       CONTAINING THE PART OF THE EPSILON TABLE WHICH
 234 C                       IS STILL NEEDED FOR FURTHER COMPUTATIONS
 235 C           ELIST(I)  - ERROR ESTIMATE APPLYING TO RLIST(I)
 236 C           MAXERR    - POINTER TO THE INTERVAL WITH LARGEST ERROR
 237 C                       ESTIMATE
 238 C           ERRMAX    - ELIST(MAXERR)
 239 C           ERLAST    - ERROR ON THE INTERVAL CURRENTLY SUBDIVIDED
 240 C                       (BEFORE THAT SUBDIVISION HAS TAKEN PLACE)
 241 C           AREA      - SUM OF THE INTEGRALS OVER THE SUBINTERVALS
 242 C           ERRSUM    - SUM OF THE ERRORS OVER THE SUBINTERVALS
 243 C           ERRBND    - REQUESTED ACCURACY MAX(EPSABS,EPSREL*
 244 C                       ABS(RESULT))
 245 C           *****1    - VARIABLE FOR THE LEFT SUBINTERVAL
 246 C           *****2    - VARIABLE FOR THE RIGHT SUBINTERVAL
 247 C           LAST      - INDEX FOR SUBDIVISION
 248 C           NRES      - NUMBER OF CALLS TO THE EXTRAPOLATION ROUTINE
 249 C           NUMRL2    - NUMBER OF ELEMENTS IN RLIST2. IF AN APPROPRIATE
 250 C                       APPROXIMATION TO THE COMPOUNDED INTEGRAL HAS
 251 C                       BEEN OBTAINED, IT IS PUT IN RLIST2(NUMRL2) AFTER
 252 C                       NUMRL2 HAS BEEN INCREASED BY ONE.
 253 C           ERLARG    - SUM OF THE ERRORS OVER THE INTERVALS LARGER
 254 C                       THAN THE SMALLEST INTERVAL CONSIDERED UP TO NOW
 255 C           EXTRAP    - LOGICAL VARIABLE DENOTING THAT THE ROUTINE
 256 C                       IS ATTEMPTING TO PERFORM EXTRAPOLATION. I.E.
 257 C                       BEFORE SUBDIVIDING THE SMALLEST INTERVAL WE
 258 C                       TRY TO DECREASE THE VALUE OF ERLARG.
 259 C           NOEXT     - LOGICAL VARIABLE DENOTING THAT EXTRAPOLATION IS
 260 C                       NO LONGER ALLOWED (TRUE-VALUE)
 261 C
 262 C            MACHINE DEPENDENT CONSTANTS
 263 C            ---------------------------
 264 C
 265 C           EPMACH IS THE LARGEST RELATIVE SPACING.
 266 C           UFLOW IS THE SMALLEST POSITIVE MAGNITUDE.
 267 C           OFLOW IS THE LARGEST POSITIVE MAGNITUDE.
 268 C
 269 C***FIRST EXECUTABLE STATEMENT  DQAGPE
 270       EPMACH = D1MACH(4)
 271 C
 272 C            TEST ON VALIDITY OF PARAMETERS
 273 C            -----------------------------
 274 C
 275       IER = 0
 276       NEVAL = 0
 277       LAST = 0
 278       RESULT = 0.0D+00
 279       ABSERR = 0.0D+00
 280       ALIST(1) = A
 281       BLIST(1) = B
 282       RLIST(1) = 0.0D+00
 283       ELIST(1) = 0.0D+00
 284       IORD(1) = 0
 285       LEVEL(1) = 0
 286       NPTS = NPTS2-2
 287       IF(NPTS2.LT.2.OR.LIMIT.LE.NPTS.OR.(EPSABS.LE.0.0D+00.AND.
 288      1  EPSREL.LT.MAX(0.5D+02*EPMACH,0.5D-28))) IER = 6
 289       IF(IER.EQ.6) GO TO 999
 290 C
 291 C            IF ANY BREAK POINTS ARE PROVIDED, SORT THEM INTO AN
 292 C            ASCENDING SEQUENCE.
 293 C
 294       SIGN = 1.0D+00
 295       IF(A.GT.B) SIGN = -1.0D+00
 296       PTS(1) = MIN(A,B)
 297       IF(NPTS.EQ.0) GO TO 15
 298       DO 10 I = 1,NPTS
 299         PTS(I+1) = POINTS(I)
 300    10 CONTINUE
 301    15 PTS(NPTS+2) = MAX(A,B)
 302       NINT = NPTS+1
 303       A1 = PTS(1)
 304       IF(NPTS.EQ.0) GO TO 40
 305       NINTP1 = NINT+1
 306       DO 20 I = 1,NINT
 307         IP1 = I+1
 308         DO 20 J = IP1,NINTP1
 309           IF(PTS(I).LE.PTS(J)) GO TO 20
 310           TEMP = PTS(I)
 311           PTS(I) = PTS(J)
 312           PTS(J) = TEMP
 313    20 CONTINUE
 314       IF(PTS(1).NE.MIN(A,B).OR.PTS(NINTP1).NE.MAX(A,B)) IER = 6
 315       IF(IER.EQ.6) GO TO 999
 316 C
 317 C            COMPUTE FIRST INTEGRAL AND ERROR APPROXIMATIONS.
 318 C            ------------------------------------------------
 319 C
 320    40 RESABS = 0.0D+00
 321       DO 50 I = 1,NINT
 322         B1 = PTS(I+1)
 323         CALL DQK21(F,A1,B1,AREA1,ERROR1,DEFABS,RESA)
 324         ABSERR = ABSERR+ERROR1
 325         RESULT = RESULT+AREA1
 326         NDIN(I) = 0
 327         IF(ERROR1.EQ.RESA.AND.ERROR1.NE.0.0D+00) NDIN(I) = 1
 328         RESABS = RESABS+DEFABS
 329         LEVEL(I) = 0
 330         ELIST(I) = ERROR1
 331         ALIST(I) = A1
 332         BLIST(I) = B1
 333         RLIST(I) = AREA1
 334         IORD(I) = I
 335         A1 = B1
 336    50 CONTINUE
 337       ERRSUM = 0.0D+00
 338       DO 55 I = 1,NINT
 339         IF(NDIN(I).EQ.1) ELIST(I) = ABSERR
 340         ERRSUM = ERRSUM+ELIST(I)
 341    55 CONTINUE
 342 C
 343 C           TEST ON ACCURACY.
 344 C
 345       LAST = NINT
 346       NEVAL = 21*NINT
 347       DRES = ABS(RESULT)
 348       ERRBND = MAX(EPSABS,EPSREL*DRES)
 349       IF(ABSERR.LE.0.1D+03*EPMACH*RESABS.AND.ABSERR.GT.ERRBND) IER = 2
 350       IF(NINT.EQ.1) GO TO 80
 351       DO 70 I = 1,NPTS
 352         JLOW = I+1
 353         IND1 = IORD(I)
 354         DO 60 J = JLOW,NINT
 355           IND2 = IORD(J)
 356           IF(ELIST(IND1).GT.ELIST(IND2)) GO TO 60
 357           IND1 = IND2
 358           K = J
 359    60   CONTINUE
 360         IF(IND1.EQ.IORD(I)) GO TO 70
 361         IORD(K) = IORD(I)
 362         IORD(I) = IND1
 363    70 CONTINUE
 364       IF(LIMIT.LT.NPTS2) IER = 1
 365    80 IF(IER.NE.0.OR.ABSERR.LE.ERRBND) GO TO 999
 366 C
 367 C           INITIALIZATION
 368 C           --------------
 369 C
 370       RLIST2(1) = RESULT
 371       MAXERR = IORD(1)
 372       ERRMAX = ELIST(MAXERR)
 373       AREA = RESULT
 374       NRMAX = 1
 375       NRES = 0
 376       NUMRL2 = 1
 377       KTMIN = 0
 378       EXTRAP = .FALSE.
 379       NOEXT = .FALSE.
 380       ERLARG = ERRSUM
 381       ERTEST = ERRBND
 382       LEVMAX = 1
 383       IROFF1 = 0
 384       IROFF2 = 0
 385       IROFF3 = 0
 386       IERRO = 0
 387       UFLOW = D1MACH(1)
 388       OFLOW = D1MACH(2)
 389       ABSERR = OFLOW
 390       KSGN = -1
 391       IF(DRES.GE.(0.1D+01-0.5D+02*EPMACH)*RESABS) KSGN = 1
 392 C
 393 C           MAIN DO-LOOP
 394 C           ------------
 395 C
 396       DO 160 LAST = NPTS2,LIMIT
 397 C
 398 C           BISECT THE SUBINTERVAL WITH THE NRMAX-TH LARGEST ERROR
 399 C           ESTIMATE.
 400 C
 401         LEVCUR = LEVEL(MAXERR)+1
 402         A1 = ALIST(MAXERR)
 403         B1 = 0.5D+00*(ALIST(MAXERR)+BLIST(MAXERR))
 404         A2 = B1
 405         B2 = BLIST(MAXERR)
 406         ERLAST = ERRMAX
 407         CALL DQK21(F,A1,B1,AREA1,ERROR1,RESA,DEFAB1)
 408         CALL DQK21(F,A2,B2,AREA2,ERROR2,RESA,DEFAB2)
 409 C
 410 C           IMPROVE PREVIOUS APPROXIMATIONS TO INTEGRAL
 411 C           AND ERROR AND TEST FOR ACCURACY.
 412 C
 413         NEVAL = NEVAL+42
 414         AREA12 = AREA1+AREA2
 415         ERRO12 = ERROR1+ERROR2
 416         ERRSUM = ERRSUM+ERRO12-ERRMAX
 417         AREA = AREA+AREA12-RLIST(MAXERR)
 418         IF(DEFAB1.EQ.ERROR1.OR.DEFAB2.EQ.ERROR2) GO TO 95
 419         IF(ABS(RLIST(MAXERR)-AREA12).GT.0.1D-04*ABS(AREA12)
 420      1  .OR.ERRO12.LT.0.99D+00*ERRMAX) GO TO 90
 421         IF(EXTRAP) IROFF2 = IROFF2+1
 422         IF(.NOT.EXTRAP) IROFF1 = IROFF1+1
 423    90   IF(LAST.GT.10.AND.ERRO12.GT.ERRMAX) IROFF3 = IROFF3+1
 424    95   LEVEL(MAXERR) = LEVCUR
 425         LEVEL(LAST) = LEVCUR
 426         RLIST(MAXERR) = AREA1
 427         RLIST(LAST) = AREA2
 428         ERRBND = MAX(EPSABS,EPSREL*ABS(AREA))
 429 C
 430 C           TEST FOR ROUNDOFF ERROR AND EVENTUALLY SET ERROR FLAG.
 431 C
 432         IF(IROFF1+IROFF2.GE.10.OR.IROFF3.GE.20) IER = 2
 433         IF(IROFF2.GE.5) IERRO = 3
 434 C
 435 C           SET ERROR FLAG IN THE CASE THAT THE NUMBER OF
 436 C           SUBINTERVALS EQUALS LIMIT.
 437 C
 438         IF(LAST.EQ.LIMIT) IER = 1
 439 C
 440 C           SET ERROR FLAG IN THE CASE OF BAD INTEGRAND BEHAVIOUR
 441 C           AT A POINT OF THE INTEGRATION RANGE
 442 C
 443         IF(MAX(ABS(A1),ABS(B2)).LE.(0.1D+01+0.1D+03*EPMACH)*
 444      1  (ABS(A2)+0.1D+04*UFLOW)) IER = 4
 445 C
 446 C           APPEND THE NEWLY-CREATED INTERVALS TO THE LIST.
 447 C
 448         IF(ERROR2.GT.ERROR1) GO TO 100
 449         ALIST(LAST) = A2
 450         BLIST(MAXERR) = B1
 451         BLIST(LAST) = B2
 452         ELIST(MAXERR) = ERROR1
 453         ELIST(LAST) = ERROR2
 454         GO TO 110
 455   100   ALIST(MAXERR) = A2
 456         ALIST(LAST) = A1
 457         BLIST(LAST) = B1
 458         RLIST(MAXERR) = AREA2
 459         RLIST(LAST) = AREA1
 460         ELIST(MAXERR) = ERROR2
 461         ELIST(LAST) = ERROR1
 462 C
 463 C           CALL SUBROUTINE DQPSRT TO MAINTAIN THE DESCENDING ORDERING
 464 C           IN THE LIST OF ERROR ESTIMATES AND SELECT THE SUBINTERVAL
 465 C           WITH NRMAX-TH LARGEST ERROR ESTIMATE (TO BE BISECTED NEXT).
 466 C
 467   110   CALL DQPSRT(LIMIT,LAST,MAXERR,ERRMAX,ELIST,IORD,NRMAX)
 468 C ***JUMP OUT OF DO-LOOP
 469         IF(ERRSUM.LE.ERRBND) GO TO 190
 470 C ***JUMP OUT OF DO-LOOP
 471         IF(IER.NE.0) GO TO 170
 472         IF(NOEXT) GO TO 160
 473         ERLARG = ERLARG-ERLAST
 474         IF(LEVCUR+1.LE.LEVMAX) ERLARG = ERLARG+ERRO12
 475         IF(EXTRAP) GO TO 120
 476 C
 477 C           TEST WHETHER THE INTERVAL TO BE BISECTED NEXT IS THE
 478 C           SMALLEST INTERVAL.
 479 C
 480         IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160
 481         EXTRAP = .TRUE.
 482         NRMAX = 2
 483   120   IF(IERRO.EQ.3.OR.ERLARG.LE.ERTEST) GO TO 140
 484 C
 485 C           THE SMALLEST INTERVAL HAS THE LARGEST ERROR.
 486 C           BEFORE BISECTING DECREASE THE SUM OF THE ERRORS OVER
 487 C           THE LARGER INTERVALS (ERLARG) AND PERFORM EXTRAPOLATION.
 488 C
 489         ID = NRMAX
 490         JUPBND = LAST
 491         IF(LAST.GT.(2+LIMIT/2)) JUPBND = LIMIT+3-LAST
 492         DO 130 K = ID,JUPBND
 493           MAXERR = IORD(NRMAX)
 494           ERRMAX = ELIST(MAXERR)
 495 C ***JUMP OUT OF DO-LOOP
 496           IF(LEVEL(MAXERR)+1.LE.LEVMAX) GO TO 160
 497           NRMAX = NRMAX+1
 498   130   CONTINUE
 499 C
 500 C           PERFORM EXTRAPOLATION.
 501 C
 502   140   NUMRL2 = NUMRL2+1
 503         RLIST2(NUMRL2) = AREA
 504         IF(NUMRL2.LE.2) GO TO 155
 505         CALL DQELG(NUMRL2,RLIST2,RESEPS,ABSEPS,RES3LA,NRES)
 506         KTMIN = KTMIN+1
 507         IF(KTMIN.GT.5.AND.ABSERR.LT.0.1D-02*ERRSUM) IER = 5
 508         IF(ABSEPS.GE.ABSERR) GO TO 150
 509         KTMIN = 0
 510         ABSERR = ABSEPS
 511         RESULT = RESEPS
 512         CORREC = ERLARG
 513         ERTEST = MAX(EPSABS,EPSREL*ABS(RESEPS))
 514 C ***JUMP OUT OF DO-LOOP
 515         IF(ABSERR.LT.ERTEST) GO TO 170
 516 C
 517 C           PREPARE BISECTION OF THE SMALLEST INTERVAL.
 518 C
 519   150   IF(NUMRL2.EQ.1) NOEXT = .TRUE.
 520         IF(IER.GE.5) GO TO 170
 521   155   MAXERR = IORD(1)
 522         ERRMAX = ELIST(MAXERR)
 523         NRMAX = 1
 524         EXTRAP = .FALSE.
 525         LEVMAX = LEVMAX+1
 526         ERLARG = ERRSUM
 527   160 CONTINUE
 528 C
 529 C           SET THE FINAL RESULT.
 530 C           ---------------------
 531 C
 532 C
 533   170 IF(ABSERR.EQ.OFLOW) GO TO 190
 534       IF((IER+IERRO).EQ.0) GO TO 180
 535       IF(IERRO.EQ.3) ABSERR = ABSERR+CORREC
 536       IF(IER.EQ.0) IER = 3
 537       IF(RESULT.NE.0.0D+00.AND.AREA.NE.0.0D+00)GO TO 175
 538       IF(ABSERR.GT.ERRSUM)GO TO 190
 539       IF(AREA.EQ.0.0D+00) GO TO 210
 540       GO TO 180
 541   175 IF(ABSERR/ABS(RESULT).GT.ERRSUM/ABS(AREA))GO TO 190
 542 C
 543 C           TEST ON DIVERGENCE.
 544 C
 545   180 IF(KSGN.EQ.(-1).AND.MAX(ABS(RESULT),ABS(AREA)).LE.
 546      1  RESABS*0.1D-01) GO TO 210
 547       IF(0.1D-01.GT.(RESULT/AREA).OR.(RESULT/AREA).GT.0.1D+03.OR.
 548      1  ERRSUM.GT.ABS(AREA)) IER = 6
 549       GO TO 210
 550 C
 551 C           COMPUTE GLOBAL INTEGRAL SUM.
 552 C
 553   190 RESULT = 0.0D+00
 554       DO 200 K = 1,LAST
 555         RESULT = RESULT+RLIST(K)
 556   200 CONTINUE
 557       ABSERR = ERRSUM
 558   210 IF(IER.GT.2) IER = IER-1
 559       RESULT = RESULT*SIGN
 560   999 RETURN
 561       END