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SampleFMintegrate.f95
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PROGRAM SAMPLE
USE FMZM
IMPLICIT NONE
! Examples and advice for using FM_INTEGRATE.
TYPE (FM), SAVE :: A, B, CHECK, ERR, PI, R1, R2, RESULT, SEVEN, TOL
TYPE (FM), EXTERNAL :: F
INTEGER :: K, KPRT, N, N_ERRORS, NW
CHARACTER(80) :: ST1
N_ERRORS = 0
! 1. Start with an integral without singularities on the interval of integration.
! integrate log(t) * cos(t) from pi/4 to pi/2.
! Set the tolerance to get at least 40 significant digits.
! FM_INTEGRATE does a sequence of iterations using the tanh-sinh quadrature formula.
! Each iteration uses more points until the last two iterates agree within the
! specified tolerance. Since the next-to-last iterate satisfies the tolerance
! and the last iterate (returned as RESULT) is even more accurate, the value
! returned from FM_INTEGRATE is usually slightly more accurate than requested.
! In this case, the error check below shows that RESULT is actually correct
! to about 60 digits.
! It is usually best to set FM's precision level to be slightly higher than the
! number of digits requested with the tolerance. Here we set precision to 20
! more digits.
N = 1
! Call FM_SET to define FM's precision level before any multiple precision variables
! are defined. This sets 60-digit precision and TOL is 1.0e-40.
K = 40
CALL FM_SET(K+20)
TOL = TO_FM(10) ** (-K)
WRITE (*,"(//)")
CALL FM_PI(PI)
! A and B are the limits for the integral.
A = PI / 4
B = PI / 2
! KPRT controls trace printing in FM_INTEGRATE. Setting it to 1 will print a summary
! of the call, giving the result, number of function evaluations, and time.
! NW is the unit number for this trace output.
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,N,A,B,TOL,RESULT,KPRT,NW)
! For these sample problems the integrals have known closed-form results, so
! we can check the accuracy of FM_INTEGRATE.
CHECK = LOG(PI/2) - LOG(PI/4)/SQRT(TO_FM(2)) + SIN_INTEGRAL(PI/4) - SIN_INTEGRAL(PI/2)
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 2. Next do an integral with a singularity at zero
! (from "Integrals of Powers of LogGamma" by T. Amdeberhan, M. Coffey, O. Espinosa,
! C. Koutschan, D. Manna, and V. Moll, in Proc. Amer. Math. Soc., #139, 2011)
! integrate log( gamma(t) ) from 0 to 1.
! Leave precision and tolerance the same as above.
! The tanh-sinh algorithm is good at handling singularities at the endpoints,
! so this case takes about the same number of function evaluations as case 1.
N = 2
WRITE (*,"(//)")
A = 0
B = 1
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,N,A,B,TOL,RESULT,KPRT,NW)
CHECK = LOG( SQRT( 2*PI ) )
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 3. This integral has a pole at pi/2 and a sqrt singularity at zero.
! (from "A Comparison of Three High-Precision Quadrature Schemes" by D. H. Bailey,
! K. Jeyabalan, and X. S. Li, in Experimental Mathematics, Vol 14 (2005), No. 3)
! integrate sqrt( tan(t) ) from 0 to pi/2.
! Set tolerance to give 100 digits.
! Here the fact that the pi/2 endpoint is not exactly representable in floating
! point form causes a problem. FM_INTEGRATE will increase precision above the
! user's level while computing the integral. But the endpoints A and B are input
! values that were defined at the user's precision, and their extra digits will be
! zeros when precision is raised in FM_INTEGRATE.
! That is fine for A = 0, but B = pi/2 will still be accurate only to the user's
! precision, not to the higher intermediate precision. The fact that B is a
! singularity for the function means that FM_INTEGRATE will need to know the
! position of B to higher precision to evaluate the integral accurately.
! The fix is to make a change of variables to get an equivalent integral where
! both endpoints are exact in floating point. Leaving a singular endpoint inexact
! will usually cause FM_INTEGRATE to run much slower, and sometimes fail.
! Let u = t * 2 / pi. Then t = u * pi / 2 and dt = pi/2 du.
! The new form of this integral becomes:
! integrate sqrt( abs( tan( u * pi / 2 ) ) ) * pi / 2 from 0 to 1.
! Now pi will be computed inside function F, so it will be done at whatever higher
! precision FM_INTEGRATE uses.
! When changing variables in this case, we also need to defend against rounding
! errors when computing u * pi / 2. When u is very close to 1, rounding could
! cause u * pi / 2 to round up, giving a value slightly greater than pi/2.
! Then tan would return a negative value and then sqrt would return unknown,
! causing the integration to fail. The fix is to take the absolute value before
! doing the square root.
N = 3
K = 100
CALL FM_SET(K+20)
! Precision has increased, so we must get pi at the new precision.
CALL FM_PI(PI)
WRITE (*,"(//)")
A = 0
B = 1
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,N,A,B,TOL,RESULT,KPRT,NW)
CHECK = PI * SQRT( TO_FM(2) ) / 2
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 4. integrate exp( -t**2 / 2 ) from 0 to infinity.
! (from "A Comparison of Three High-Precision Quadrature Schemes" by D. H. Bailey,
! K. Jeyabalan, and X. S. Li, in Experimental Mathematics, Vol 14 (2005), No. 3)
! Set tolerance to give 100 digits.
! Infinite regions must be converted to finite ones. Let u = 1/(t+1) to get:
! integrate exp( -(1/u - 1)**2 / 2 ) / u**2 from 0 to 1.
! Exponential functions pose another problem for FM_INTEGRATE.
! When u is very close to zero the exponential can underflow. FM does not flush
! underflows to zero like most floating point systems, so when that value is
! then divided by the small u**2 FM detects the possibility that this result
! could be above the underflow threshold. Since FM can't be sure whether the
! true function value is below the underflow threshold, unknown is returned.
! The fix in this case is to see that whenever underflow occurs in this integration
! the final function value is too small to change the integral. That is the usual
! situation whenever F underflows and the final value of the integral is greater
! than 10**(-10**6) in magnitude, because FM's underflow is less than 10**(-10**8).
! So we check for underflow after doing the exponential in function F and replace
! underflowed function values by zero.
N = 4
K = 100
CALL FM_SET(K+20)
CALL FM_PI(PI)
WRITE (*,"(//)")
A = 0
B = 1
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,N,A,B,TOL,RESULT,KPRT,NW)
CHECK = SQRT( PI / 2 )
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 5. integrate log( abs( ( tan(t) + sqrt(7) ) / ( tan(t) - sqrt(7) ) ) )
! from pi/3 to pi/2.
! (from "High-Precision Numerical Integration: Progress and Challenges"
! by D. H. Bailey and J. M. Borwein (2009))
! Set tolerance to give 150 digits.
! There is only one singularity, but it is atan(sqrt(7)), which is not an endpoint.
! FM_INTEGRATE will initially have very slow convergence and then will try to
! isolate the singularity and split into two integrals with the singularity at
! endpoints.
! This strategy works here, but it is slower and doesn't always succeed.
! Case 6 shows a better way to handle interior singularities.
N = 5
K = 150
CALL FM_SET(K+20)
CALL FM_PI(PI)
WRITE (*,"(//)")
A = PI/3
B = PI/2
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,N,A,B,TOL,RESULT,KPRT,NW)
SEVEN = 7
CHECK = ( SQRT(SEVEN) / 168 ) * ( POLYGAMMA(1,1/SEVEN) + POLYGAMMA(1,2/SEVEN) - &
POLYGAMMA(1,3/SEVEN) + POLYGAMMA(1,4/SEVEN) - &
POLYGAMMA(1,5/SEVEN) - POLYGAMMA(1,6/SEVEN) )
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 6. Same integral as case 5.
! integrate log( abs( ( tan(t) + sqrt(7) ) / ( tan(t) - sqrt(7) ) ) )
! from pi/3 to pi/2.
! Set tolerance to give 150 digits.
! Split into two integrals and change variables to make the endpoints exact.
! This will be faster than making FM_INTEGRATE search for the interior singularity
! as in case 5.
! Call the two function numbers 61 and 62.
! 1. from pi/3 to atan(sqrt(7)).
! Let u = ( t - pi/3 ) / ( atan( sqrt(7) ) - pi/3 )
! 2. from atan(sqrt(7)) to pi/2.
! Let v = ( t - atan(sqrt(7)) ) / ( pi/2 - atan( sqrt(7) ) )
! This gives two integrals from 0 to 1, then we add the two results.
N = 6
K = 150
CALL FM_SET(K+20)
CALL FM_PI(PI)
WRITE (*,"(//)")
A = 0
B = 1
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,61,A,B,TOL,R1,KPRT,NW)
CALL FM_INTEGRATE(F,62,A,B,TOL,R2,KPRT,NW)
RESULT = R1 + R2
WRITE (*,*) ' '
WRITE (*,*) ' Adding these last two integrals gives the case 6 result:'
WRITE (*,*) ' '
CALL FM_PRINT(RESULT)
SEVEN = 7
CHECK = ( SQRT(SEVEN) / 168 ) * ( POLYGAMMA(1,1/SEVEN) + POLYGAMMA(1,2/SEVEN) - &
POLYGAMMA(1,3/SEVEN) + POLYGAMMA(1,4/SEVEN) - &
POLYGAMMA(1,5/SEVEN) - POLYGAMMA(1,6/SEVEN) )
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
! 7. Same integral as cases 5 and 6.
! Combine these two integrals into one, so only one call to FM_INTEGRATE is needed.
! This will be faster than doing two calls as in case 6.
! integrate log( abs( ( tan(t) + sqrt(7) ) / ( tan(t) - sqrt(7) ) ) )
! from pi/3 to pi/2.
! Set tolerance to give 150 digits.
! Split into two integrals and change variables to make the endpoints exact.
! Both new integrals are from 0 to 1.
! 1. from pi/3 to atan(sqrt(7)).
! Let u = ( t - pi/3 ) / ( atan( sqrt(7) ) - pi/3 )
! 2. from atan(sqrt(7)) to pi/2.
! Let v = ( t - atan(sqrt(7)) ) / ( pi/2 - atan( sqrt(7) ) )
N = 7
K = 150
CALL FM_SET(K+20)
CALL FM_PI(PI)
WRITE (*,"(//)")
A = 0
B = 1
TOL = TO_FM(10) ** (-K)
KPRT = 1
NW = 6
CALL FM_INTEGRATE(F,7,A,B,TOL,RESULT,KPRT,NW)
SEVEN = 7
CHECK = ( SQRT(SEVEN) / 168 ) * ( POLYGAMMA(1,1/SEVEN) + POLYGAMMA(1,2/SEVEN) - &
POLYGAMMA(1,3/SEVEN) + POLYGAMMA(1,4/SEVEN) - &
POLYGAMMA(1,5/SEVEN) - POLYGAMMA(1,6/SEVEN) )
ERR = ABS( RESULT - CHECK )
CALL FM_FORM('ES12.4',ERR,ST1)
WRITE (*,"(/10X,A,I2,A,A)") ' Error for case ',N,' = ',TRIM(ST1)
IF (ERR > TOL) N_ERRORS = N_ERRORS + 1
WRITE (*,*) ' '
WRITE (*,*) ' '
IF (N_ERRORS == 0) THEN
WRITE (*,*) ' All results were ok -- no errors were found.'
ELSE
WRITE (*,*) N_ERRORS,' error(s) were found.'
ENDIF
WRITE (*,*) ' '
STOP
END PROGRAM SAMPLE
FUNCTION F(X,N)
USE FMZM
IMPLICIT NONE
TYPE (FM) :: F, X
TYPE (FM), SAVE :: C1, C2, PI, SQRT7, TANX
INTEGER :: N
CALL FM_ENTER_USER_FUNCTION(F)
IF (N == 1) THEN
F = LOG(X) * COS(X)
ELSE IF (N == 2) THEN
F = LOG( GAMMA( X ) )
ELSE IF (N == 3) THEN
! The original limits from 0 to pi/2 have been changed to 0 to 1.
CALL FM_PI(PI)
F = PI * SQRT( ABS( TAN( PI * X / 2 ) ) ) / 2
ELSE IF (N == 4) THEN
! Exp could underflow and then make F unknown.
! Check for the underflow and set F = 0 in that case.
F = EXP( -(1 - 1/X)**2 / 2 )
IF ( IS_UNDERFLOW(F) ) THEN
F = 0
ELSE
F = F / X**2
ENDIF
ELSE IF (N == 5) THEN
SQRT7 = SQRT(TO_FM(7))
TANX = TAN(X)
F = LOG( ABS( ( TANX + SQRT7 ) / ( TANX - SQRT7 ) ) )
ELSE IF (N == 61) THEN
CALL FM_PI(PI)
! It is tempting to compute constants like C1, C2, SQRT7 once and then save them for
! use in subsequent calls to F. That can be done, but it is trickier than it seems,
! since FM_INTEGRATE may call F with different precision levels during one integration,
! so it is easy to not have the right precision in a saved variable.
! Here we just compute them each time, making the logic straightforward while the
! function evaluations are somewhat slower.
SQRT7 = SQRT(TO_FM(7))
C1 = ATAN(SQRT7) - PI/3
TANX = TAN( C1*X + PI/3 )
F = C1 * LOG( ABS( ( TANX + SQRT7 ) / ( TANX - SQRT7 ) ) )
ELSE IF (N == 62) THEN
CALL FM_PI(PI)
SQRT7 = SQRT(TO_FM(7))
C2 = ATAN(SQRT7)
C1 = PI/2 - C2
TANX = TAN( C1*X + C2 )
F = C1 * LOG( ABS( ( TANX + SQRT7 ) / ( TANX - SQRT7 ) ) )
ELSE IF (N == 7) THEN
! Combine the two integrals into one.
CALL FM_PI(PI)
SQRT7 = SQRT(TO_FM(7))
C2 = ATAN(SQRT7)
C1 = C2 - PI/3
TANX = TAN( C1*X + PI/3 )
F = C1 * LOG( ABS( ( TANX + SQRT7 ) / ( TANX - SQRT7 ) ) )
C1 = PI/2 - C2
TANX = TAN( C1*X + C2 )
F = F + C1 * LOG( ABS( ( TANX + SQRT7 ) / ( TANX - SQRT7 ) ) )
ENDIF
CALL FM_EXIT_USER_FUNCTION(F)
END FUNCTION F
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