*DECK QNC79 SUBROUTINE QNC79 (FUN, A, B, ERR, ANS, IERR, K) C***BEGIN PROLOGUE QNC79 C***PURPOSE Integrate a function using a 7-point adaptive Newton-Cotes C quadrature rule. C***LIBRARY SLATEC C***CATEGORY H2A1A1 C***TYPE SINGLE PRECISION (QNC79-S, DQNC79-D) C***KEYWORDS ADAPTIVE QUADRATURE, INTEGRATION, NEWTON-COTES C***AUTHOR Kahaner, D. K., (NBS) C Jones, R. E., (SNLA) C***DESCRIPTION C C Abstract C QNC79 is a general purpose program for evaluation of C one dimensional integrals of user defined functions. C QNC79 will pick its own points for evaluation of the C integrand and these will vary from problem to problem. C Thus, QNC79 is not designed to integrate over data sets. C Moderately smooth integrands will be integrated efficiently C and reliably. For problems with strong singularities, C oscillations etc., the user may wish to use more sophis- C ticated routines such as those in QUADPACK. One measure C of the reliability of QNC79 is the output parameter K, C giving the number of integrand evaluations that were needed. C C Description of Arguments C C --Input-- C FUN - name of external function to be integrated. This name C must be in an EXTERNAL statement in your calling C program. You must write a Fortran function to evaluate C FUN. This should be of the form C REAL FUNCTION FUN (X) C C C C X can vary from A to B C C FUN(X) should be finite for all X on interval. C C C FUN = ... C RETURN C END C A - lower limit of integration C B - upper limit of integration (may be less than A) C ERR - is a requested error tolerance. Normally, pick a value C 0 .LT. ERR .LT. 1.0E-3. C C --Output-- C ANS - computed value of the integral. Hopefully, ANS is C accurate to within ERR * integral of ABS(FUN(X)). C IERR - a status code C - Normal codes C 1 ANS most likely meets requested error tolerance. C -1 A equals B, or A and B are too nearly equal to C allow normal integration. ANS is set to zero. C - Abnormal code C 2 ANS probably does not meet requested error tolerance. C K - the number of function evaluations actually used to do C the integration. A value of K .GT. 1000 indicates a C difficult problem; other programs may be more efficient. C QNC79 will gracefully give up if K exceeds 2000. C C***REFERENCES (NONE) C***ROUTINES CALLED I1MACH, R1MACH, XERMSG C***REVISION HISTORY (YYMMDD) C 790601 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ) C 920218 Code and prologue polished. (WRB) C***END PROLOGUE QNC79 C .. Scalar Arguments .. REAL A, ANS, B, ERR INTEGER IERR, K C .. Function Arguments .. REAL FUN EXTERNAL FUN C .. Local Scalars .. REAL AE, AREA, BANK, BLOCAL, C, CE, EE, EF, EPS, Q13, Q7, Q7L, + SQ2, TEST, TOL, VR, W1, W2, W3, W4 INTEGER I, KML, KMX, L, LMN, LMX, NBITS, NIB, NLMN, NLMX LOGICAL FIRST C .. Local Arrays .. REAL AA(40), F(13), F1(40), F2(40), F3(40), F4(40), F5(40), + F6(40), F7(40), HH(40), Q7R(40), VL(40) INTEGER LR(40) C .. External Functions .. REAL R1MACH INTEGER I1MACH EXTERNAL R1MACH, I1MACH C .. External Subroutines .. EXTERNAL XERMSG C .. Intrinsic Functions .. INTRINSIC ABS, LOG, MAX, MIN, SIGN, SQRT C .. Save statement .. SAVE NBITS, NLMX, FIRST, SQ2, W1, W2, W3, W4 C .. Data statements .. DATA KML /7/, KMX /2000/, NLMN /2/ DATA FIRST /.TRUE./ C***FIRST EXECUTABLE STATEMENT QNC79 IF (FIRST) THEN W1 = 41.0E0/140.0E0 W2 = 216.0E0/140.0E0 W3 = 27.0E0/140.0E0 W4 = 272.0E0/140.0E0 NBITS = R1MACH(5)*I1MACH(11)/0.30102000E0 NLMX = MIN(40,(NBITS*4)/5) SQ2 = SQRT(2.0E0) ENDIF FIRST = .FALSE. ANS = 0.0E0 IERR = 1 CE = 0.0E0 IF (A .EQ. B) GO TO 260 LMX = NLMX LMN = NLMN IF (B .EQ. 0.0E0) GO TO 100 IF (SIGN(1.0E0,B)*A .LE. 0.0E0) GO TO 100 C = ABS(1.0E0-A/B) IF (C .GT. 0.1E0) GO TO 100 IF (C .LE. 0.0E0) GO TO 260 NIB = 0.5E0 - LOG(C)/LOG(2.0E0) LMX = MIN(NLMX,NBITS-NIB-4) IF (LMX .LT. 2) GO TO 260 LMN = MIN(LMN,LMX) 100 TOL = MAX(ABS(ERR),2.0E0**(5-NBITS)) IF (ERR .EQ. 0.0E0) TOL = SQRT(R1MACH(4)) EPS = TOL HH(1) = (B-A)/12.0E0 AA(1) = A LR(1) = 1 DO 110 I = 1,11,2 F(I) = FUN(A+(I-1)*HH(1)) 110 CONTINUE BLOCAL = B F(13) = FUN(BLOCAL) K = 7 L = 1 AREA = 0.0E0 Q7 = 0.0E0 EF = 256.0E0/255.0E0 BANK = 0.0E0 C C Compute refined estimates, estimate the error, etc. C 120 DO 130 I = 2,12,2 F(I) = FUN(AA(L)+(I-1)*HH(L)) 130 CONTINUE K = K + 6 C C Compute left and right half estimates C Q7L = HH(L)*((W1*(F(1)+F(7))+W2*(F(2)+F(6)))+ + (W3*(F(3)+F(5))+W4*F(4))) Q7R(L) = HH(L)*((W1*(F(7)+F(13))+W2*(F(8)+F(12)))+ + (W3*(F(9)+F(11))+W4*F(10))) C C Update estimate of integral of absolute value C AREA = AREA + (ABS(Q7L)+ABS(Q7R(L))-ABS(Q7)) C C Do not bother to test convergence before minimum refinement level C IF (L .LT. LMN) GO TO 180 C C Estimate the error in new value for whole interval, Q13 C Q13 = Q7L + Q7R(L) EE = ABS(Q7-Q13)*EF C C Compute nominal allowed error C AE = EPS*AREA C C Borrow from bank account, but not too much C TEST = MIN(AE+0.8E0*BANK,10.0E0*AE) C C Don't ask for excessive accuracy C TEST = MAX(TEST,TOL*ABS(Q13),0.00003E0*TOL*AREA) C C Now, did this interval pass or not? C IF (EE-TEST) 150,150,170 C C Have hit maximum refinement level -- penalize the cumulative error C 140 CE = CE + (Q7-Q13) GO TO 160 C C On good intervals accumulate the theoretical estimate C 150 CE = CE + (Q7-Q13)/255.0 C C Update the bank account. Don't go into debt. C 160 BANK = BANK + (AE-EE) IF (BANK .LT. 0.0E0) BANK = 0.0E0 C C Did we just finish a left half or a right half? C IF (LR(L)) 190,190,210 C C Consider the left half of next deeper level C 170 IF (K .GT. KMX) LMX = MIN(KML,LMX) IF (L .GE. LMX) GO TO 140 180 L = L + 1 EPS = EPS*0.5E0 IF (L .LE. 17) EF = EF/SQ2 HH(L) = HH(L-1)*0.5E0 LR(L) = -1 AA(L) = AA(L-1) Q7 = Q7L F1(L) = F(7) F2(L) = F(8) F3(L) = F(9) F4(L) = F(10) F5(L) = F(11) F6(L) = F(12) F7(L) = F(13) F(13) = F(7) F(11) = F(6) F(9) = F(5) F(7) = F(4) F(5) = F(3) F(3) = F(2) GO TO 120 C C Proceed to right half at this level C 190 VL(L) = Q13 200 Q7 = Q7R(L-1) LR(L) = 1 AA(L) = AA(L) + 12.0E0*HH(L) F(1) = F1(L) F(3) = F2(L) F(5) = F3(L) F(7) = F4(L) F(9) = F5(L) F(11) = F6(L) F(13) = F7(L) GO TO 120 C C Left and right halves are done, so go back up a level C 210 VR = Q13 220 IF (L .LE. 1) GO TO 250 IF (L .LE. 17) EF = EF*SQ2 EPS = EPS*2.0E0 L = L - 1 IF (LR(L)) 230,230,240 230 VL(L) = VL(L+1) + VR GO TO 200 240 VR = VL(L+1) + VR GO TO 220 C C Exit C 250 ANS = VR IF (ABS(CE) .LE. 2.0E0*TOL*AREA) GO TO 270 IERR = 2 CALL XERMSG ('SLATEC', 'QNC79', + 'ANS is probably insufficiently accurate.', 2, 1) GO TO 270 260 IERR = -1 CALL XERMSG ('SLATEC', 'QNC79', + 'A and B are too nearly equal to allow normal integration. \$\$' + // 'ANS is set to zero and IERR to -1.', -1, -1) 270 RETURN END