*DECK GAUS8
SUBROUTINE GAUS8 (FUN, A, B, ERR, ANS, IERR)
C***BEGIN PROLOGUE GAUS8
C***PURPOSE Integrate a real function of one variable over a finite
C interval using an adaptive 8-point Legendre-Gauss
C algorithm. Intended primarily for high accuracy
C integration or integration of smooth functions.
C***LIBRARY SLATEC
C***CATEGORY H2A1A1
C***TYPE SINGLE PRECISION (GAUS8-S, DGAUS8-D)
C***KEYWORDS ADAPTIVE QUADRATURE, AUTOMATIC INTEGRATOR,
C GAUSS QUADRATURE, NUMERICAL INTEGRATION
C***AUTHOR Jones, R. E., (SNLA)
C***DESCRIPTION
C
C Abstract
C GAUS8 integrates real functions of one variable over finite
C intervals using an adaptive 8-point Legendre-Gauss algorithm.
C GAUS8 is intended primarily for high accuracy integration
C or integration of smooth functions.
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 the calling program.
C FUN must be a REAL function of one REAL argument. The
C value of the argument to FUN is the variable of
C integration which ranges from A to B.
C A - lower limit of integration
C B - upper limit of integration (may be less than A)
C ERR - is a requested pseudorelative error tolerance. Normally
C pick a value of ABS(ERR) so that STOL .LT. ABS(ERR) .LE.
C 1.0E-3 where STOL is the single precision unit roundoff
C R1MACH(4). ANS will normally have no more error than
C ABS(ERR) times the integral of the absolute value of
C FUN(X). Usually, smaller values for ERR yield more
C accuracy and require more function evaluations.
C
C A negative value for ERR causes an estimate of the
C absolute error in ANS to be returned in ERR. Note that
C ERR must be a variable (not a constant) in this case.
C Note also that the user must reset the value of ERR
C before making any more calls that use the variable ERR.
C
C Output--
C ERR - will be an estimate of the absolute error in ANS if the
C input value of ERR was negative. (ERR is unchanged if
C the input value of ERR was non-negative.) The estimated
C error is solely for information to the user and should
C not be used as a correction to the computed integral.
C ANS - computed value of integral
C IERR- a status code
C --Normal codes
C 1 ANS most likely meets requested error tolerance,
C or A=B.
C -1 A and B are too nearly equal to allow normal
C integration. ANS is set to zero.
C --Abnormal code
C 2 ANS probably does not meet requested error tolerance.
C
C***REFERENCES (NONE)
C***ROUTINES CALLED I1MACH, R1MACH, XERMSG
C***REVISION HISTORY (YYMMDD)
C 810223 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 900326 Removed duplicate information from DESCRIPTION section.
C (WRB)
C***END PROLOGUE GAUS8
INTEGER IERR, K, KML, KMX, L, LMN, LMX, LR, MXL, NBITS,
1 NIB, NLMN, NLMX
INTEGER I1MACH
REAL A, AA, AE, ANIB, ANS, AREA, B, C, CE, EE, EF, EPS, ERR, EST,
1 GL, GLR, GR, HH, SQ2, TOL, VL, VR, W1, W2, W3, W4, X1, X2, X3,
2 X4, X, H
REAL R1MACH, G8, FUN
DIMENSION AA(30), HH(30), LR(30), VL(30), GR(30)
SAVE X1, X2, X3, X4, W1, W2, W3, W4, SQ2,
1 NLMN, KMX, KML
DATA X1, X2, X3, X4/
1 1.83434642495649805E-01, 5.25532409916328986E-01,
2 7.96666477413626740E-01, 9.60289856497536232E-01/
DATA W1, W2, W3, W4/
1 3.62683783378361983E-01, 3.13706645877887287E-01,
2 2.22381034453374471E-01, 1.01228536290376259E-01/
DATA SQ2/1.41421356E0/
DATA NLMN/1/,KMX/5000/,KML/6/
G8(X,H)=H*((W1*(FUN(X-X1*H) + FUN(X+X1*H))
1 +W2*(FUN(X-X2*H) + FUN(X+X2*H)))
2 +(W3*(FUN(X-X3*H) + FUN(X+X3*H))
3 +W4*(FUN(X-X4*H) + FUN(X+X4*H))))
C***FIRST EXECUTABLE STATEMENT GAUS8
C
C Initialize
C
K = I1MACH(11)
ANIB = R1MACH(5)*K/0.30102000E0
NBITS = ANIB
NLMX = MIN(30,(NBITS*5)/8)
ANS = 0.0E0
IERR = 1
CE = 0.0E0
IF (A .EQ. B) GO TO 140
LMX = NLMX
LMN = NLMN
IF (B .EQ. 0.0E0) GO TO 10
IF (SIGN(1.0E0,B)*A .LE. 0.0E0) GO TO 10
C = ABS(1.0E0-A/B)
IF (C .GT. 0.1E0) GO TO 10
IF (C .LE. 0.0E0) GO TO 140
ANIB = 0.5E0 - LOG(C)/0.69314718E0
NIB = ANIB
LMX = MIN(NLMX,NBITS-NIB-7)
IF (LMX .LT. 1) GO TO 130
LMN = MIN(LMN,LMX)
10 TOL = MAX(ABS(ERR),2.0E0**(5-NBITS))/2.0E0
IF (ERR .EQ. 0.0E0) TOL = SQRT(R1MACH(4))
EPS = TOL
HH(1) = (B-A)/4.0E0
AA(1) = A
LR(1) = 1
L = 1
EST = G8(AA(L)+2.0E0*HH(L),2.0E0*HH(L))
K = 8
AREA = ABS(EST)
EF = 0.5E0
MXL = 0
C
C Compute refined estimates, estimate the error, etc.
C
20 GL = G8(AA(L)+HH(L),HH(L))
GR(L) = G8(AA(L)+3.0E0*HH(L),HH(L))
K = K + 16
AREA = AREA + (ABS(GL)+ABS(GR(L))-ABS(EST))
C IF (L .LT. LMN) GO TO 11
GLR = GL + GR(L)
EE = ABS(EST-GLR)*EF
AE = MAX(EPS*AREA,TOL*ABS(GLR))
IF (EE-AE) 40, 40, 50
30 MXL = 1
40 CE = CE + (EST-GLR)
IF (LR(L)) 60, 60, 80
C
C Consider the left half of this level
C
50 IF (K .GT. KMX) LMX = KML
IF (L .GE. LMX) GO TO 30
L = L + 1
EPS = EPS*0.5E0
EF = EF/SQ2
HH(L) = HH(L-1)*0.5E0
LR(L) = -1
AA(L) = AA(L-1)
EST = GL
GO TO 20
C
C Proceed to right half at this level
C
60 VL(L) = GLR
70 EST = GR(L-1)
LR(L) = 1
AA(L) = AA(L) + 4.0E0*HH(L)
GO TO 20
C
C Return one level
C
80 VR = GLR
90 IF (L .LE. 1) GO TO 120
L = L - 1
EPS = EPS*2.0E0
EF = EF*SQ2
IF (LR(L)) 100, 100, 110
100 VL(L) = VL(L+1) + VR
GO TO 70
110 VR = VL(L+1) + VR
GO TO 90
C
C Exit
C
120 ANS = VR
IF ((MXL.EQ.0) .OR. (ABS(CE).LE.2.0E0*TOL*AREA)) GO TO 140
IERR = 2
CALL XERMSG ('SLATEC', 'GAUS8',
+ 'ANS is probably insufficiently accurate.', 3, 1)
GO TO 140
130 IERR = -1
CALL XERMSG ('SLATEC', 'GAUS8',
+ 'A and B are too nearly equal to allow normal integration. $$'
+ // 'ANS is set to zero and IERR to -1.', 1, -1)
140 IF (ERR .LT. 0.0E0) ERR = CE
RETURN
END