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beta_gamma_psi.F
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beta_gamma_psi.F
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!--------------------------------------------------------------------------------------------------!
! CP2K: A general program to perform molecular dynamics simulations !
! Copyright 2000-2024 CP2K developers group <https://cp2k.org> !
! !
! SPDX-License-Identifier: GPL-2.0-or-later !
!--------------------------------------------------------------------------------------------------!
! **************************************************************************************************
MODULE beta_gamma_psi
! not tested in the case where dp would stand for single precision
! Routines for the beta function are untested
USE kinds, ONLY: dp
#include "./base/base_uses.f90"
IMPLICIT NONE
PRIVATE
PUBLIC :: psi
CONTAINS
! **************************************************************************************************
!> \brief ...
!> \param i ...
!> \return ...
! **************************************************************************************************
FUNCTION ipmpar(i) RESULT(fn_val)
!-----------------------------------------------------------------------
! IPMPAR PROVIDES THE INTEGER MACHINE CONSTANTS FOR THE COMPUTER
! THAT IS USED. IT IS ASSUMED THAT THE ARGUMENT I IS AN INTEGER
! HAVING ONE OF THE VALUES 1-10. IPMPAR(I) HAS THE VALUE ...
! INTEGERS.
! ASSUME INTEGERS ARE REPRESENTED IN THE N-DIGIT, BASE-A FORM
! SIGN ( X(N-1)*A**(N-1) + ... + X(1)*A + X(0) )
! WHERE 0 .LE. X(I) .LT. A FOR I=0,...,N-1.
! IPMPAR(1) = A, THE BASE (radix).
! IPMPAR(2) = N, THE NUMBER OF BASE-A DIGITS (digits).
! IPMPAR(3) = A**N - 1, THE LARGEST MAGNITUDE (huge).
! FLOATING-POINT NUMBERS.
! IT IS ASSUMED THAT THE SINGLE AND DOUBLE PRECISION FLOATING
! POINT ARITHMETICS HAVE THE SAME BASE, SAY B, AND THAT THE
! NONZERO NUMBERS ARE REPRESENTED IN THE FORM
! SIGN (B**E) * (X(1)/B + ... + X(M)/B**M)
! WHERE X(I) = 0,1,...,B-1 FOR I=1,...,M,
! X(1) .GE. 1, AND EMIN .LE. E .LE. EMAX.
! IPMPAR(4) = B, THE BASE.
! SINGLE-PRECISION
! IPMPAR(5) = M, THE NUMBER OF BASE-B DIGITS.
! IPMPAR(6) = EMIN, THE SMALLEST EXPONENT E.
! IPMPAR(7) = EMAX, THE LARGEST EXPONENT E.
! DOUBLE-PRECISION
! IPMPAR(8) = M, THE NUMBER OF BASE-B DIGITS.
! IPMPAR(9) = EMIN, THE SMALLEST EXPONENT E.
! IPMPAR(10) = EMAX, THE LARGEST EXPONENT E.
!-----------------------------------------------------------------------
INTEGER, INTENT(IN) :: i
INTEGER :: fn_val
SELECT CASE (i)
CASE (1)
fn_val = RADIX(i)
CASE (2)
fn_val = DIGITS(i)
CASE (3)
fn_val = HUGE(i)
CASE (4)
fn_val = RADIX(1.0)
CASE (5)
fn_val = DIGITS(1.0)
CASE (6)
fn_val = MINEXPONENT(1.0)
CASE (7)
fn_val = MAXEXPONENT(1.0)
CASE (8)
fn_val = DIGITS(1.0e0_dp)
CASE (9)
fn_val = MINEXPONENT(1.0e0_dp)
CASE (10)
fn_val = MAXEXPONENT(1.0e0_dp)
CASE DEFAULT
CPABORT("unknown case")
END SELECT
END FUNCTION ipmpar
! **************************************************************************************************
!> \brief ...
!> \param i ...
!> \return ...
! **************************************************************************************************
FUNCTION dpmpar(i) RESULT(fn_val)
!-----------------------------------------------------------------------
! DPMPAR PROVIDES THE DOUBLE PRECISION MACHINE CONSTANTS FOR
! THE COMPUTER BEING USED. IT IS ASSUMED THAT THE ARGUMENT
! I IS AN INTEGER HAVING ONE OF THE VALUES 1, 2, OR 3. IF THE
! DOUBLE PRECISION ARITHMETIC BEING USED HAS M BASE B DIGITS AND
! ITS SMALLEST AND LARGEST EXPONENTS ARE EMIN AND EMAX, THEN
! DPMPAR(1) = B**(1 - M), THE MACHINE PRECISION,
! DPMPAR(2) = B**(EMIN - 1), THE SMALLEST MAGNITUDE,
! DPMPAR(3) = B**EMAX*(1 - B**(-M)), THE LARGEST MAGNITUDE.
!-----------------------------------------------------------------------
INTEGER, INTENT(IN) :: i
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: one = 1._dp
! Local variable
SELECT CASE (i)
CASE (1)
fn_val = EPSILON(one)
CASE (2)
fn_val = TINY(one)
CASE (3)
fn_val = HUGE(one)
END SELECT
END FUNCTION dpmpar
! **************************************************************************************************
!> \brief ...
!> \param l ...
!> \return ...
! **************************************************************************************************
FUNCTION dxparg(l) RESULT(fn_val)
!--------------------------------------------------------------------
! IF L = 0 THEN DXPARG(L) = THE LARGEST POSITIVE W FOR WHICH
! DEXP(W) CAN BE COMPUTED.
!
! IF L IS NONZERO THEN DXPARG(L) = THE LARGEST NEGATIVE W FOR
! WHICH THE COMPUTED VALUE OF DEXP(W) IS NONZERO.
!
! NOTE... ONLY AN APPROXIMATE VALUE FOR DXPARG(L) IS NEEDED.
!--------------------------------------------------------------------
INTEGER, INTENT(IN) :: l
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: one = 1._dp
! Local variable
IF (l == 0) THEN
fn_val = LOG(HUGE(one))
ELSE
fn_val = LOG(TINY(one))
END IF
END FUNCTION dxparg
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \return ...
! **************************************************************************************************
FUNCTION alnrel(a) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF THE FUNCTION LN(1 + A)
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: half = 0.5e0_dp, one = 1.e0_dp, p1 = -.129418923021993e+01_dp, &
p2 = .405303492862024e+00_dp, p3 = -.178874546012214e-01_dp, &
q1 = -.162752256355323e+01_dp, q2 = .747811014037616e+00_dp, &
q3 = -.845104217945565e-01_dp, two = 2.e0_dp, zero = 0.e0_dp
REAL(dp) :: t, t2, w, x
!--------------------------
IF (ABS(a) <= 0.375e0_dp) THEN
t = a/(a + two)
t2 = t*t
w = (((p3*t2 + p2)*t2 + p1)*t2 + one)/(((q3*t2 + q2)*t2 + q1)*t2 + one)
fn_val = two*t*w
ELSE
x = one + a
IF (a < zero) x = (a + half) + half
fn_val = LOG(x)
END IF
END FUNCTION alnrel
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \return ...
! **************************************************************************************************
FUNCTION gam1(a) RESULT(fn_val)
!-----------------------------------------------------------------------
! COMPUTATION OF 1/GAMMA(A+1) - 1 FOR -0.5 <= A <= 1.5
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: p(7) = (/.577215664901533e+00_dp, -.409078193005776e+00_dp, &
-.230975380857675e+00_dp, .597275330452234e-01_dp, .766968181649490e-02_dp, &
-.514889771323592e-02_dp, .589597428611429e-03_dp/), q(5) = (/.100000000000000e+01_dp, &
.427569613095214e+00_dp, .158451672430138e+00_dp, .261132021441447e-01_dp, &
.423244297896961e-02_dp/), r(9) = (/-.422784335098468e+00_dp, -.771330383816272e+00_dp, &
-.244757765222226e+00_dp, .118378989872749e+00_dp, .930357293360349e-03_dp, &
-.118290993445146e-01_dp, .223047661158249e-02_dp, .266505979058923e-03_dp, &
-.132674909766242e-03_dp/)
REAL(dp), PARAMETER :: s1 = .273076135303957e+00_dp, s2 = .559398236957378e-01_dp
REAL(dp) :: bot, d, t, top, w
t = a
d = a - 0.5e0_dp
IF (d > 0.0e0_dp) t = d - 0.5e0_dp
IF (t > 0.e0_dp) THEN
top = (((((p(7)*t + p(6))*t + p(5))*t + p(4))*t + p(3))*t + p(2))*t + p(1)
bot = (((q(5)*t + q(4))*t + q(3))*t + q(2))*t + 1.0e0_dp
w = top/bot
IF (d > 0.0e0_dp) THEN
fn_val = (t/a)*((w - 0.5e0_dp) - 0.5e0_dp)
ELSE
fn_val = a*w
END IF
ELSE IF (t < 0.e0_dp) THEN
top = (((((((r(9)*t + r(8))*t + r(7))*t + r(6))*t + r(5))*t &
+ r(4))*t + r(3))*t + r(2))*t + r(1)
bot = (s2*t + s1)*t + 1.0e0_dp
w = top/bot
IF (d > 0.0e0_dp) THEN
fn_val = t*w/a
ELSE
fn_val = a*((w + 0.5e0_dp) + 0.5e0_dp)
END IF
ELSE
fn_val = 0.0e0_dp
END IF
END FUNCTION gam1
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \param b ...
!> \return ...
! **************************************************************************************************
FUNCTION algdiv(a, b) RESULT(fn_val)
!-----------------------------------------------------------------------
! COMPUTATION OF LN(GAMMA(B)/GAMMA(A+B)) WHEN B >= 8
! --------
! IN THIS ALGORITHM, DEL(X) IS THE FUNCTION DEFINED BY
! LN(GAMMA(X)) = (X - 0.5)*LN(X) - X + 0.5*LN(2*PI) + DEL(X).
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a, b
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: c0 = .833333333333333e-01_dp, c1 = -.277777777760991e-02_dp, &
c2 = .793650666825390e-03_dp, c3 = -.595202931351870e-03_dp, &
c4 = .837308034031215e-03_dp, c5 = -.165322962780713e-02_dp
REAL(dp) :: c, d, h, s11, s3, s5, s7, s9, t, u, v, &
w, x, x2
IF (a > b) THEN
h = b/a
c = 1.0e0_dp/(1.0e0_dp + h)
x = h/(1.0e0_dp + h)
d = a + (b - 0.5e0_dp)
ELSE
h = a/b
c = h/(1.0e0_dp + h)
x = 1.0e0_dp/(1.0e0_dp + h)
d = b + (a - 0.5e0_dp)
END IF
! SET SN = (1 - X**N)/(1 - X)
x2 = x*x
s3 = 1.0e0_dp + (x + x2)
s5 = 1.0e0_dp + (x + x2*s3)
s7 = 1.0e0_dp + (x + x2*s5)
s9 = 1.0e0_dp + (x + x2*s7)
s11 = 1.0e0_dp + (x + x2*s9)
! SET W = DEL(B) - DEL(A + B)
t = (1.0e0_dp/b)**2
w = ((((c5*s11*t + c4*s9)*t + c3*s7)*t + c2*s5)*t + c1*s3)*t + c0
w = w*(c/b)
! COMBINE THE RESULTS
u = d*alnrel(a/b)
v = a*(LOG(b) - 1.0e0_dp)
IF (u > v) THEN
fn_val = (w - v) - u
ELSE
fn_val = (w - u) - v
END IF
END FUNCTION algdiv
! **************************************************************************************************
!> \brief ...
!> \param x ...
!> \return ...
! **************************************************************************************************
FUNCTION rexp(x) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF THE FUNCTION EXP(X) - 1
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: x
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: p1 = .914041914819518e-09_dp, p2 = .238082361044469e-01_dp, &
q1 = -.499999999085958e+00_dp, q2 = .107141568980644e+00_dp, &
q3 = -.119041179760821e-01_dp, q4 = .595130811860248e-03_dp
REAL(dp) :: e
IF (ABS(x) < 0.15e0_dp) THEN
fn_val = x*(((p2*x + p1)*x + 1.0e0_dp)/((((q4*x + q3)*x + q2)*x + q1)*x + 1.0e0_dp))
RETURN
END IF
IF (x < 0.0e0_dp) THEN
IF (x > -37.0e0_dp) THEN
fn_val = (EXP(x) - 0.5e0_dp) - 0.5e0_dp
ELSE
fn_val = -1.0e0_dp
END IF
ELSE
e = EXP(x)
fn_val = e*(0.5e0_dp + (0.5e0_dp - 1.0e0_dp/e))
END IF
END FUNCTION rexp
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \param b ...
!> \param x ...
!> \param y ...
!> \param w ...
!> \param eps ...
!> \param ierr ...
! **************************************************************************************************
SUBROUTINE bgrat(a, b, x, y, w, eps, ierr)
!-----------------------------------------------------------------------
! ASYMPTOTIC EXPANSION FOR IX(A,B) WHEN A IS LARGER THAN B.
! THE RESULT OF THE EXPANSION IS ADDED TO W. IT IS ASSUMED
! THAT A <= 15 AND B <= 1. EPS IS THE TOLERANCE USED.
! IERR IS A VARIABLE THAT REPORTS THE STATUS OF THE RESULTS.
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a, b, x, y
REAL(dp), INTENT(INOUT) :: w
REAL(dp), INTENT(IN) :: eps
INTEGER, INTENT(OUT) :: ierr
REAL(dp), PARAMETER :: half = 0.5e0_dp, one = 1.e0_dp, &
quarter = 0.25e0_dp, zero = 0.e0_dp
INTEGER :: i, n
REAL(dp) :: bm1, bp2n, c(30), cn, coef, d(30), dj, &
j, l, lnx, n2, nu, q, r, s, sum, t, &
t2, tol, u, v, z
bm1 = (b - half) - half
nu = a + half*bm1
IF (y > 0.375e0_dp) THEN
lnx = LOG(x)
ELSE
lnx = alnrel(-y)
END IF
z = -nu*lnx
IF (b*z == zero) THEN
! THE EXPANSION CANNOT BE COMPUTED
ierr = 1
RETURN
END IF
! COMPUTATION OF THE EXPANSION
! SET R = EXP(-Z)*Z**B/GAMMA(B)
r = b*(one + gam1(b))*EXP(b*LOG(z))
r = r*EXP(a*lnx)*EXP(half*bm1*lnx)
u = algdiv(b, a) + b*LOG(nu)
u = r*EXP(-u)
IF (u == zero) THEN
! THE EXPANSION CANNOT BE COMPUTED
ierr = 1
RETURN
END IF
CALL grat1(b, z, r, q=q, eps=eps)
tol = 15.0e0_dp*eps
v = quarter*(one/nu)**2
t2 = quarter*lnx*lnx
l = w/u
j = q/r
sum = j
t = one
cn = one
n2 = zero
DO n = 1, 30
bp2n = b + n2
j = (bp2n*(bp2n + one)*j + (z + bp2n + one)*t)*v
n2 = n2 + 2.0e0_dp
t = t*t2
cn = cn/(n2*(n2 + one))
c(n) = cn
s = zero
IF (.NOT. (n == 1)) THEN
coef = b - n
DO i = 1, n - 1
s = s + coef*c(i)*d(n - i)
coef = coef + b
END DO
END IF
d(n) = bm1*cn + s/n
dj = d(n)*j
sum = sum + dj
IF (sum <= zero) THEN
! THE EXPANSION CANNOT BE COMPUTED
ierr = 1
RETURN
END IF
IF (ABS(dj) <= tol*(sum + l)) EXIT
END DO
! ADD THE RESULTS TO W
ierr = 0
w = w + u*sum
END SUBROUTINE bgrat
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \param x ...
!> \param r ...
!> \param p ...
!> \param q ...
!> \param eps ...
! **************************************************************************************************
SUBROUTINE grat1(a, x, r, p, q, eps)
!-----------------------------------------------------------------------
! EVALUATION OF P(A,X) AND Q(A,X) WHERE A <= 1 AND
! THE INPUT ARGUMENT R HAS THE VALUE E**(-X)*X**A/GAMMA(A)
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a, x, r
REAL(dp), INTENT(OUT), OPTIONAL :: p, q
REAL(dp), INTENT(IN) :: eps
REAL(dp), PARAMETER :: half = 0.5e0_dp, one = 1.e0_dp, &
quarter = 0.25e0_dp, three = 3.e0_dp, &
two = 2.e0_dp, zero = 0.e0_dp
REAL(dp) :: a2n, a2nm1, an, b2n, b2nm1, c, g, h, j, &
l, pp, qq, sum, t, tol, w, z
IF (a*x == zero) THEN
IF (x <= a) THEN
pp = zero
qq = one
ELSE
pp = one
qq = zero
END IF
IF (PRESENT(p)) p = pp
IF (PRESENT(q)) q = qq
RETURN
END IF
IF (a == half) THEN
IF (x < quarter) THEN
pp = ERF(SQRT(x))
qq = half + (half - pp)
ELSE
qq = ERFC(SQRT(x))
pp = half + (half - qq)
END IF
IF (PRESENT(p)) p = pp
IF (PRESENT(q)) q = qq
RETURN
END IF
IF (x < 1.1e0_dp) THEN
! TAYLOR SERIES FOR P(A,X)/X**A
an = three
c = x
sum = x/(a + three)
tol = three*eps/(a + one)
an = an + one
c = -c*(x/an)
t = c/(a + an)
sum = sum + t
DO WHILE (ABS(t) > tol)
an = an + one
c = -c*(x/an)
t = c/(a + an)
sum = sum + t
END DO
j = a*x*((sum/6.0e0_dp - half/(a + two))*x + one/(a + one))
z = a*LOG(x)
h = gam1(a)
g = one + h
IF ((x < quarter .AND. z > -.13394e0_dp) .OR. a < x/2.59e0_dp) THEN
l = rexp(z)
qq = ((half + (half + l))*j - l)*g - h
IF (qq <= zero) THEN
pp = one
qq = zero
ELSE
pp = half + (half - qq)
END IF
ELSE
w = EXP(z)
pp = w*g*(half + (half - j))
qq = half + (half - pp)
END IF
ELSE
! CONTINUED FRACTION EXPANSION
tol = 8.0e0_dp*eps
a2nm1 = one
a2n = one
b2nm1 = x
b2n = x + (one - a)
c = one
DO
a2nm1 = x*a2n + c*a2nm1
b2nm1 = x*b2n + c*b2nm1
c = c + one
a2n = a2nm1 + (c - a)*a2n
b2n = b2nm1 + (c - a)*b2n
a2nm1 = a2nm1/b2n
b2nm1 = b2nm1/b2n
a2n = a2n/b2n
b2n = one
IF (ABS(a2n - a2nm1/b2nm1) < tol*a2n) EXIT
END DO
qq = r*a2n
pp = half + (half - qq)
END IF
IF (PRESENT(p)) p = pp
IF (PRESENT(q)) q = qq
END SUBROUTINE grat1
! **************************************************************************************************
!> \brief ...
!> \param mu ...
!> \param x ...
!> \return ...
! **************************************************************************************************
FUNCTION esum(mu, x) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF EXP(MU + X)
!-----------------------------------------------------------------------
INTEGER, INTENT(IN) :: mu
REAL(dp), INTENT(IN) :: x
REAL(dp) :: fn_val
REAL(dp) :: w
IF (x > 0.0e0_dp) THEN
IF (mu > 0 .OR. mu + x < 0.0_dp) THEN
w = mu
fn_val = EXP(w)*EXP(x)
ELSE
w = mu + x
fn_val = EXP(w)
END IF
ELSE
IF (mu < 0 .OR. mu + x < 0.0_dp) THEN
w = mu
fn_val = EXP(w)*EXP(x)
ELSE
w = mu + x
fn_val = EXP(w)
END IF
END IF
END FUNCTION esum
! **************************************************************************************************
!> \brief ...
!> \param x ...
!> \return ...
! **************************************************************************************************
FUNCTION rlog1(x) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF THE FUNCTION X - LN(1 + X)
!-----------------------------------------------------------------------
! A = RLOG (0.7)
! B = RLOG (4/3)
!------------------------
REAL(dp), INTENT(IN) :: x
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: a = .566749439387324e-01_dp, b = .456512608815524e-01_dp, &
p0 = .333333333333333e+00_dp, p1 = -.224696413112536e+00_dp, &
p2 = .620886815375787e-02_dp, q1 = -.127408923933623e+01_dp, q2 = .354508718369557e+00_dp
REAL(dp) :: r, t, u, up2, w, w1
IF (x < -0.39e0_dp .OR. x > 0.57e0_dp) THEN
w = (x + 0.5e0_dp) + 0.5e0_dp
fn_val = x - LOG(w)
RETURN
END IF
! ARGUMENT REDUCTION
IF (x < -0.18e0_dp) THEN
u = (x + 0.3e0_dp)/0.7e0_dp
up2 = u + 2.0e0_dp
w1 = a - u*0.3e0_dp
ELSEIF (x > 0.18e0_dp) THEN
t = 0.75e0_dp*x
u = t - 0.25e0_dp
up2 = t + 1.75e0_dp
w1 = b + u/3.0e0_dp
ELSE
u = x
up2 = u + 2.0e0_dp
w1 = 0.0e0_dp
END IF
! SERIES EXPANSION
r = u/up2
t = r*r
w = ((p2*t + p1)*t + p0)/((q2*t + q1)*t + 1.0e0_dp)
fn_val = r*(u - 2.0e0_dp*t*w) + w1
END FUNCTION rlog1
! **************************************************************************************************
!> \brief ...
!> \param a ...
!> \return ...
! **************************************************************************************************
FUNCTION gamln1(a) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF LN(GAMMA(1 + A)) FOR -0.2 .LE. A .LE. 1.25
!-----------------------------------------------------------------------
REAL(dp), INTENT(IN) :: a
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: p0 = .577215664901533e+00_dp, p1 = .844203922187225e+00_dp, &
p2 = -.168860593646662e+00_dp, p3 = -.780427615533591e+00_dp, &
p4 = -.402055799310489e+00_dp, p5 = -.673562214325671e-01_dp, &
p6 = -.271935708322958e-02_dp, q1 = .288743195473681e+01_dp, &
q2 = .312755088914843e+01_dp, q3 = .156875193295039e+01_dp, q4 = .361951990101499e+00_dp, &
q5 = .325038868253937e-01_dp, q6 = .667465618796164e-03_dp, r0 = .422784335098467e+00_dp, &
r1 = .848044614534529e+00_dp, r2 = .565221050691933e+00_dp, r3 = .156513060486551e+00_dp, &
r4 = .170502484022650e-01_dp, r5 = .497958207639485e-03_dp
REAL(dp), PARAMETER :: s1 = .124313399877507e+01_dp, s2 = .548042109832463e+00_dp, &
s3 = .101552187439830e+00_dp, s4 = .713309612391000e-02_dp, s5 = .116165475989616e-03_dp
REAL(dp) :: w, x
IF (a < 0.6e0_dp) THEN
w = ((((((p6*a + p5)*a + p4)*a + p3)*a + p2)*a + p1)*a + p0)/ &
((((((q6*a + q5)*a + q4)*a + q3)*a + q2)*a + q1)*a + 1.0e0_dp)
fn_val = -a*w
ELSE
x = (a - 0.5e0_dp) - 0.5e0_dp
w = (((((r5*x + r4)*x + r3)*x + r2)*x + r1)*x + r0)/ &
(((((s5*x + s4)*x + s3)*x + s2)*x + s1)*x + 1.0e0_dp)
fn_val = x*w
END IF
END FUNCTION gamln1
! **************************************************************************************************
!> \brief ...
!> \param xx ...
!> \return ...
! **************************************************************************************************
FUNCTION psi(xx) RESULT(fn_val)
!---------------------------------------------------------------------
! EVALUATION OF THE DIGAMMA FUNCTION
! -----------
! PSI(XX) IS ASSIGNED THE VALUE 0 WHEN THE DIGAMMA FUNCTION CANNOT
! BE COMPUTED.
! THE MAIN COMPUTATION INVOLVES EVALUATION OF RATIONAL CHEBYSHEV
! APPROXIMATIONS PUBLISHED IN MATH. COMP. 27, 123-127(1973) BY
! CODY, STRECOK AND THACHER.
!---------------------------------------------------------------------
! PSI WAS WRITTEN AT ARGONNE NATIONAL LABORATORY FOR THE FUNPACK
! PACKAGE OF SPECIAL FUNCTION SUBROUTINES. PSI WAS MODIFIED BY
! A.H. MORRIS (NSWC).
!---------------------------------------------------------------------
REAL(dp), INTENT(IN) :: xx
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: dx0 = 1.461632144968362341262659542325721325e0_dp, p1(7) = (/ &
.895385022981970e-02_dp, .477762828042627e+01_dp, .142441585084029e+03_dp, &
.118645200713425e+04_dp, .363351846806499e+04_dp, .413810161269013e+04_dp, &
.130560269827897e+04_dp/), p2(4) = (/-.212940445131011e+01_dp, -.701677227766759e+01_dp, &
-.448616543918019e+01_dp, -.648157123766197e+00_dp/), piov4 = .785398163397448e0_dp, q1(6)&
= (/.448452573429826e+02_dp, .520752771467162e+03_dp, .221000799247830e+04_dp, &
.364127349079381e+04_dp, .190831076596300e+04_dp, .691091682714533e-05_dp/)
REAL(dp), PARAMETER :: q2(4) = (/.322703493791143e+02_dp, .892920700481861e+02_dp, &
.546117738103215e+02_dp, .777788548522962e+01_dp/)
INTEGER :: i, m, n, nq
REAL(dp) :: aug, den, sgn, upper, w, x, xmax1, xmx0, &
xsmall, z
!---------------------------------------------------------------------
! PIOV4 = PI/4
! DX0 = ZERO OF PSI TO EXTENDED PRECISION
!---------------------------------------------------------------------
!---------------------------------------------------------------------
! COEFFICIENTS FOR RATIONAL APPROXIMATION OF
! PSI(X) / (X - X0), 0.5 <= X <= 3.0
!---------------------------------------------------------------------
!---------------------------------------------------------------------
! COEFFICIENTS FOR RATIONAL APPROXIMATION OF
! PSI(X) - LN(X) + 1 / (2*X), X > 3.0
!---------------------------------------------------------------------
!---------------------------------------------------------------------
! MACHINE DEPENDENT CONSTANTS ...
! XMAX1 = THE SMALLEST POSITIVE FLOATING POINT CONSTANT
! WITH ENTIRELY INTEGER REPRESENTATION. ALSO USED
! AS NEGATIVE OF LOWER BOUND ON ACCEPTABLE NEGATIVE
! ARGUMENTS AND AS THE POSITIVE ARGUMENT BEYOND WHICH
! PSI MAY BE REPRESENTED AS ALOG(X).
! XSMALL = ABSOLUTE ARGUMENT BELOW WHICH PI*COTAN(PI*X)
! MAY BE REPRESENTED BY 1/X.
!---------------------------------------------------------------------
xmax1 = ipmpar(3)
xmax1 = MIN(xmax1, 1.0e0_dp/dpmpar(1))
xsmall = 1.e-9_dp
!---------------------------------------------------------------------
x = xx
aug = 0.0e0_dp
IF (x < 0.5e0_dp) THEN
!---------------------------------------------------------------------
! X .LT. 0.5, USE REFLECTION FORMULA
! PSI(1-X) = PSI(X) + PI * COTAN(PI*X)
!---------------------------------------------------------------------
IF (ABS(x) <= xsmall) THEN
IF (x == 0.0e0_dp) THEN
! ERROR RETURN
fn_val = 0.0e0_dp
RETURN
END IF
!---------------------------------------------------------------------
! 0 .LT. ABS(X) .LE. XSMALL. USE 1/X AS A SUBSTITUTE
! FOR PI*COTAN(PI*X)
!---------------------------------------------------------------------
aug = -1.0e0_dp/x
x = 1.0e0_dp - x
ELSE
!---------------------------------------------------------------------
! REDUCTION OF ARGUMENT FOR COTAN
!---------------------------------------------------------------------
w = -x
sgn = piov4
IF (w <= 0.0e0_dp) THEN
w = -w
sgn = -sgn
END IF
!---------------------------------------------------------------------
! MAKE AN ERROR EXIT IF X .LE. -XMAX1
!---------------------------------------------------------------------
IF (w >= xmax1) THEN
! ERROR RETURN
fn_val = 0.0e0_dp
RETURN
END IF
nq = INT(w)
w = w - nq
nq = INT(w*4.0e0_dp)
w = 4.0e0_dp*(w - nq*.25e0_dp)
!---------------------------------------------------------------------
! W IS NOW RELATED TO THE FRACTIONAL PART OF 4.0 * X.
! ADJUST ARGUMENT TO CORRESPOND TO VALUES IN FIRST
! QUADRANT AND DETERMINE SIGN
!---------------------------------------------------------------------
n = nq/2
IF ((n + n) /= nq) w = 1.0e0_dp - w
z = piov4*w
m = n/2
IF ((m + m) /= n) sgn = -sgn
!---------------------------------------------------------------------
! DETERMINE FINAL VALUE FOR -PI*COTAN(PI*X)
!---------------------------------------------------------------------
n = (nq + 1)/2
m = n/2
m = m + m
IF (m /= n) THEN
aug = sgn*((SIN(z)/COS(z))*4.0e0_dp)
ELSE
!---------------------------------------------------------------------
! CHECK FOR SINGULARITY
!---------------------------------------------------------------------
IF (z == 0.0e0_dp) THEN
! ERROR RETURN
fn_val = 0.0e0_dp
RETURN
END IF
!---------------------------------------------------------------------
! USE COS/SIN AS A SUBSTITUTE FOR COTAN, AND
! SIN/COS AS A SUBSTITUTE FOR TAN
!---------------------------------------------------------------------
aug = sgn*((COS(z)/SIN(z))*4.0e0_dp)
END IF
x = 1.0e0_dp - x
END IF
END IF
IF (x <= 3.0e0_dp) THEN
!---------------------------------------------------------------------
! 0.5 .LE. X .LE. 3.0
!---------------------------------------------------------------------
den = x
upper = p1(1)*x
DO i = 1, 5
den = (den + q1(i))*x
upper = (upper + p1(i + 1))*x
END DO
den = (upper + p1(7))/(den + q1(6))
xmx0 = x - dx0
fn_val = den*xmx0 + aug
RETURN
END IF
!---------------------------------------------------------------------
! IF X .GE. XMAX1, PSI = LN(X)
!---------------------------------------------------------------------
IF (x < xmax1) THEN
!---------------------------------------------------------------------
! 3.0 .LT. X .LT. XMAX1
!---------------------------------------------------------------------
w = 1.0e0_dp/(x*x)
den = w
upper = p2(1)*w
DO i = 1, 3
den = (den + q2(i))*w
upper = (upper + p2(i + 1))*w
END DO
aug = upper/(den + q2(4)) - 0.5e0_dp/x + aug
END IF
fn_val = aug + LOG(x)
END FUNCTION psi
! **************************************************************************************************
!> \brief ...
!> \param a0 ...
!> \param b0 ...
!> \return ...
! **************************************************************************************************
FUNCTION betaln(a0, b0) RESULT(fn_val)
!-----------------------------------------------------------------------
! EVALUATION OF THE LOGARITHM OF THE BETA FUNCTION
!-----------------------------------------------------------------------
! E = 0.5*LN(2*PI)
!--------------------------
REAL(dp), INTENT(IN) :: a0, b0
REAL(dp) :: fn_val
REAL(dp), PARAMETER :: e = .918938533204673e0_dp
INTEGER :: i, n
REAL(dp) :: a, b, c, h, u, v, w, z
!--------------------------
a = MIN(a0, b0)
b = MAX(a0, b0)
!-----------------------------------------------------------------------
! PROCEDURE WHEN A .GE. 8
!-----------------------------------------------------------------------
IF (a >= 8.0e0_dp) THEN
w = bcorr(a, b)
h = a/b
c = h/(1.0e0_dp + h)
u = -(a - 0.5e0_dp)*LOG(c)
v = b*alnrel(h)
IF (u > v) THEN
fn_val = (((-0.5e0_dp*LOG(b) + e) + w) - v) - u
ELSE
fn_val = (((-0.5e0_dp*LOG(b) + e) + w) - u) - v
END IF
!-----------------------------------------------------------------------
! PROCEDURE WHEN A .LT. 1
!-----------------------------------------------------------------------
ELSEIF (a < 1.0e0_dp) THEN
IF (b < 8.0e0_dp) THEN
fn_val = LOG_GAMMA(a) + (LOG_GAMMA(b) - LOG_GAMMA(a + b))
ELSE
fn_val = LOG_GAMMA(a) + algdiv(a, b)
END IF
!-----------------------------------------------------------------------
! PROCEDURE WHEN 1 .LE. A .LT. 8
!-----------------------------------------------------------------------
ELSEIF (a <= 2.0e0_dp) THEN
IF (b <= 2.0e0_dp) THEN
fn_val = LOG_GAMMA(a) + LOG_GAMMA(b) - gsumln(a, b)
RETURN
END IF
w = 0.0e0_dp
IF (b < 8.0e0_dp) THEN
! REDUCTION OF B WHEN B .LT. 8
n = INT(b - 1.0e0_dp)
z = 1.0e0_dp
DO i = 1, n
b = b - 1.0e0_dp
z = z*(b/(a + b))
END DO
fn_val = w + LOG(z) + (LOG_GAMMA(a) + (LOG_GAMMA(b) - gsumln(a, b)))
RETURN
END IF
fn_val = LOG_GAMMA(a) + algdiv(a, b)
! REDUCTION OF A WHEN B .LE. 1000
ELSEIF (b <= 1000.0e0_dp) THEN
n = INT(a - 1.0e0_dp)
w = 1.0e0_dp
DO i = 1, n
a = a - 1.0e0_dp
h = a/b
w = w*(h/(1.0e0_dp + h))
END DO
w = LOG(w)
IF (b >= 8.0e0_dp) THEN
fn_val = w + LOG_GAMMA(a) + algdiv(a, b)
RETURN
END IF
! REDUCTION OF B WHEN B .LT. 8
n = INT(b - 1.0e0_dp)
z = 1.0e0_dp
DO i = 1, n
b = b - 1.0e0_dp
z = z*(b/(a + b))
END DO
fn_val = w + LOG(z) + (LOG_GAMMA(a) + (LOG_GAMMA(b) - gsumln(a, b)))
ELSE
! REDUCTION OF A WHEN B .GT. 1000
n = INT(a - 1.0e0_dp)
w = 1.0e0_dp
DO i = 1, n
a = a - 1.0e0_dp
w = w*(a/(1.0e0_dp + a/b))
END DO
fn_val = (LOG(w) - n*LOG(b)) + (LOG_GAMMA(a) + algdiv(a, b))
END IF