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mod_bigquad.F90
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module bigquad
use precision
implicit none
private
real(kind=real2), save :: faca(21)
real(kind=real2), parameter :: tol = epsilon(faca(1))
public :: gauss_Legendre,gaussl_nodes,gaussl_weights,stieltjes_c
public :: baratella_dleg_approx
contains
!**********************************************************************
!> @breif Calculates Gauss-Legendre nodes for Large N
!> @par This calculates the Gauss-Legendre nodes for large n based on
!! Townsend & Hale (2013) and Bogeart et al. (2013). Recomended for N < 100.
!! Although Golub-Welsch is still acceptable until N~=1000
!> @param[in] n order (number of points)
!> @param[out] x nodes, preallocated for speed
!> @param[out] t theta of nodes such that x = cos(t), preallocated for speed
!**********************************************************************
subroutine gaussl_nodes(n,x,t)
use precision
use maths, only : pi,factorialArrayReal
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(out) :: x(n),t(n)
integer(kind=int1) :: n2,k
n2 = floor(real(n,kind=real2)/2d0)
faca = factorialArrayReal(20)
do k=1,n2
t(k) = theta0(n,k)
enddo
call gaussl_node_adjust(n,n2,t)
do k=1,n2
t(n-k+1) = pi - t(k)
x(k) = cos(t(k))
x(n-k+1) = -cos(t(k))
enddo
if(mod(n,2) .ne. 0) x(n2 + 1) = 0d0
return
end subroutine gaussl_nodes
!**********************************************************************
!> @breif Calculates Gauss-Legendre nodes for Large N
!> @par This calculates the Gauss-Legendre nodes for large n based on
!! Townsend & Hale (2013) and Bogeart et al. (2013). Recomended for N < 100.
!! Although Golub-Welsch is still acceptable until N~=1000
!> @param[in] n order (number of points)
!> @param[out] x nodes, preallocated for speed
!> @param[out] t theta of nodes such that x = cos(t), preallocated for speed
!**********************************************************************
subroutine gaussl_weights(n,w)
use precision
use maths, only : pi,factorialArrayReal
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(out) :: w(n)
integer(kind=int1) :: n2,k
real(kind=real2) :: t(n),dpn
n2 = floor(real(n,kind=real2)/2d0)
faca = factorialArrayReal(20)
do k=1,n2
t(k) = theta0(n,k)
enddo
call gaussl_node_adjust(n,n2,t,w)
do k=1,n2
w(n-k+1) = w(k)
enddo
if(mod(n,2) .ne. 0) then
dpn = baratella_dleg_approx(n,pi/2d0)
w(n2 + 1) = 2d0/(dpn*dpn)
endif
return
end subroutine gaussl_weights
!**********************************************************************
!> @breif Calculates Gauss-Legendre quadrature for Large N
!> @par This calculates the Gauss-Legendre nodes adn weights for large n based on
!! Townsend & Hale (2013) and Bogeart et al. (2013). Recomended for N < 100.
!! Although Golub-Welsch is still acceptable until N~=1000
!> @param[in] n order (number of points)
!> @param[out] x nodes
!> @param[out] w weigths
!**********************************************************************
subroutine gauss_Legendre(n,x,w)
use precision
use maths, only : pi,factorialArrayReal
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(out) :: x(n),w(n)
integer(kind=int1) :: n2,k
real(kind=real2) :: t(n),dpn
n2 = floor(real(n,kind=real2)/2d0)
faca = factorialArrayReal(20)
do k=1,n2
t(k) = theta0(n,k)
enddo
call gaussl_node_adjust(n,n2,t,w)
do k=1,n2
x(k) = cos(t(k))
x(n-k+1) = -cos(t(k))
w(n-k+1) = w(k)
enddo
if(mod(n,2) .ne. 0) then
x(n2 + 1) = 0d0
dpn = baratella_dleg_approx(n,pi/2d0)
w(n2 + 1) = 2d0/(dpn*dpn)
endif
return
end subroutine gauss_Legendre
!**********************************************************************
!> @brief Adjust inital guess of node
!> @par Adjust the initial guess of thenodes location using Newtons method
!! together with suitable methods for estimating the value of the legendre
!! polynomial and its gradient.
!> @param[in] n order
!> @param[in] n2 floor(order/2)
!> @param[in] t0 initial guess of node theta
!> @param[out] t refined vaule of theta
!**********************************************************************
subroutine gaussl_node_adjust(n,n2,t,w)
use precision
use maths, only : pi
implicit none
integer(kind=int1), intent(in) :: n,n2
real(kind=real2), intent(inout) :: t(n)
real(kind=real2), optional, intent(out) :: w(n)
integer(kind=int1) :: k,lim
real(kind=real2) :: cn
lim = max(10,floor(n/2000d0))
lim = 10
cn = stieltjes_c(n)
if(present(w))then
do k=1,n2
if(k .gt. lim) then
call gaussl_newton_central(n,k,cn,t(k),w(k))
else
call gaussl_newton_boundary(n,k,t(k),w(k))
endif
enddo
else
do k=1,n2
if(k .gt. lim) then
call gaussl_newton_central(n,k,cn,t(k))
else
call gaussl_newton_boundary(n,k,t(k))
endif
enddo
endif
return
end subroutine gaussl_node_adjust
!**********************************************************************
subroutine gaussl_newton_central(n,k,cn,t,w)
use precision
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2), intent(in) :: cn
real(kind=real2), intent(inout) :: t
real(kind=real2), optional, intent(out) :: w
integer(kind=int1) :: j
integer(kind=int1), parameter :: jmax = 20
real(kind=real2) :: delt,pn,dpn,ttol
delt = 1d0
j = 0
ttol = t*tol
do while(abs(delt) .gt. ttol)
j = j + 1
call stieltjes_leg_approx(n,k,cn,t,pn,dpn)
delt = pn/(dpn)
t = t + delt
if(j .gt. jmax)then
print *,'ERROR: BIGQ CONV FAIL, CENTRAL',pn,k
exit
endif
enddo
! For more accurate weights uncomment this line
! WARNING, this will be much slower
!dpn = baratella_dleg_approx(n,t)
if(present(w)) w = 2d0/(dpn*dpn)
return
end subroutine gaussl_newton_central
!**********************************************************************
subroutine gaussl_newton_boundary(n,k,t,w)
use precision
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2), intent(inout) :: t
real(kind=real2), optional, intent(out) :: w
integer(kind=int1), parameter :: jmax = 20
integer(kind=int1) :: j
real(kind=real2) :: delt,pn,dpn,ttol
delt = 1d0
j = 0
ttol = t*tol
do while(abs(delt) .gt. ttol)
j = j + 1
pn = bogaert_leg_approx(n,t)
dpn = baratella_dleg_approx(n,t)
delt = pn/(dpn)
t = t + delt
if(j .gt. jmax)then
print *,'ERROR: BIGQ CONV FAIL, BOUNDARY',pn,k
exit
endif
enddo
! For more accurate weights uncomment this line
! WARNING, this will be much slower
!dpn = baratella_dleg_approx(n,t)
if(present(w)) w = 2d0/(dpn*dpn)
return
end subroutine gaussl_newton_boundary
!**********************************************************************
subroutine stieltjes_leg_approx(n,k,cn,t,pn,dpn) ! best for |x|< 0.866 (=cos(pi/6))
use precision
use maths, only : pi
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2), intent(in) :: cn,t
real(kind=real2), intent(out) :: pn,dpn
integer(kind=int1) :: nm,m
real(kind=real2), parameter :: pisixth = pi/6d0
real(kind=real2) :: rn,rm,beta,del,hnm,z
real(kind=real2) :: sina,cosa,sinz,cosz,cott,r2sin,r2sinm
rn = real(n,kind=real2)
z = (rn + 0.5d0)*t - 0.25d0*pi
del = z - real(k,kind=real2)*pi
sinz = taylor_sin_kpi(k,del)
cosz = taylor_cos_kpi(k,del)
cott = 1d0/tan(t)
r2sin = 1d0/(2d0*sin(t))
r2sinm = sqrt(r2sin)
pn = 0d0
dpn = 0d0
nm = 15
if(t .gt. pisixth) nm = 7
do m=0,nm-1
rm = real(m,kind=real2)
beta = rm*(t - 0.5d0*pi)
sina = sinz*cos(beta) + cosz*sin(beta)
cosa = cosz*cos(beta) - sinz*sin(beta)
hnm = stieltjes_h(n,m)
pn = pn + hnm*cosa*r2sinm
dpn = dpn + hnm*((rm - 0.5d0)*cosa*cott + (rn + rm - 0.5d0)*sina)*r2sinm
r2sinm = r2sinm*r2sin
enddo
pn = cn*pn
dpn = cn*dpn
return
end subroutine stieltjes_leg_approx
!**********************************************************************
function taylor_sin_kpi(k,del) result(z)
use precision
use maths, only : pi
implicit none
integer(kind=int1), intent(in) :: k
real(kind=real2), intent(in) :: del
integer(kind=int1), parameter :: nm = 10
integer(kind=int1) :: i
real(kind=real2) :: z,c,m1,delz
m1 = 1d0
if(mod(k,2) .eq. 0) m1 = -1d0
c = 1d0/del
z = sin(real(k)*pi)
do i=1,nm
m1 = -m1
c = c*del*del
delz = m1*c/faca(2*i)
z = z + delz
enddo
return
end function taylor_sin_kpi
!**********************************************************************
function taylor_cos_kpi(k,del) result(z)
use precision
implicit none
integer(kind=int1), intent(in) :: k
real(kind=real2), intent(in) :: del
integer(kind=int1), parameter :: nm = 10
integer(kind=int1) :: i
real(kind=real2) :: z,c,m1,delz
m1 = -1d0
if(mod(k,2) .eq. 0) m1 = 1d0
c = 1d0
z = m1
do i=1,nm
m1 = -m1
c = c*del*del
delz = m1*c/faca(2*i+1)
z = z + delz
enddo
return
end function taylor_cos_kpi
!**********************************************************************
function stieltjes_c(n) result(cn)
use precision
use maths, only : ex,pi
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2) :: cn
real(kind=real2) :: rn
rn = real(n,kind=real2)
cn = quotentent_large_indice(n,0.5d0)
cn = cn*sqrt(rn/(rn + 0.5d0))*sqrt(ex/rn)
cn = cn*stirling_series(rn)/stirling_series(rn + 0.5d0)
cn = cn*sqrt(4d0/pi)
return
end function stieltjes_c
!**********************************************************************
function stirling_series(x) result(s)
use precision
implicit none
real(kind=real2), intent(in) :: x
real(kind=real2) :: s
! First 10 terms in taylor expansion for stirlings series
! cn is normally only calculated onceper quad so 10 is not expensive
real(kind=real2), parameter :: si(10) = [ &
8.3333333333333329E-002, &
3.4722222222222220E-003, &
-2.6813271604938273E-003, &
-2.2947209362139917E-004, &
7.8403922172006662E-004, &
6.9728137583658571E-005, &
-5.9216643735369393E-004, &
-5.1717909082605919E-005, &
8.3949872067208726E-004, &
7.2048954160200109E-005]
integer(kind=int1) :: i
real(kind=real2) :: rx,rxn
rx = 1d0/x
rxn = rx
s = 1d0
do i=1,10
s = s + si(i)*rxn
rxn = rx*rxn
enddo
return
end function stirling_series
!**********************************************************************
function quotentent_large_indice(n,a) result(z)
! calculates (n/(n+a))**(n+a) for large values of n
use precision
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(in) :: a
real(kind=real2) :: z
integer(kind=int1) :: j
real(kind=real2) :: s,rn,ds
s = 0d0
rn = real(n,kind=real2)
ds = 1d0
j = 0
do while(abs(ds) .gt. tol)
j = j + 1
ds = ((-a/rn)**real(j,kind=real2))/real(j,kind=real2)
s = s + ds
enddo
z = (rn + a)*s
z = exp(z)
return
end function quotentent_large_indice
!**********************************************************************
function stieltjes_h(n,m) result(h)
use precision
implicit none
integer(kind=int1), intent(in) :: n,m
real(kind=real2) :: h
integer(kind=int1) :: j
real(kind=real2) :: rn,rj
h = 1d0
if(m .gt. 0)then
rn = real(n,kind=real2)
do j=1,m
rj = real(j,kind=real2)
h = h*(rj - 0.5d0)*(rj - 0.5d0)
h = h/(rj*(rn + rj + 0.5d0))
enddo
endif
return
end function stieltjes_h
!**********************************************************************
function bogaert_leg_approx(n,t) result(pn) ! best for cos(pi/6) <= |x| <= 1 as expensive
use precision
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(in) :: t
real(kind=real2) :: pn
integer(kind=int1) :: i
real(kind=real2) :: v,y,fn(7),c,rv2
v = real(n,kind=real2) + 0.5d0
rv2 = 1/v/v
y = t*v
fn = bogaert_f(y)
c = 1d0
pn = 0d0
do i=0,6
pn = pn + fn(i+1)*c
c = c*rv2
enddo
return
end function bogaert_leg_approx
!**********************************************************************
function bogaert_f(y) result(fn)
use precision
implicit none
real(kind=real2), intent(in) :: y
real(kind=real2) :: fn(7)
! Bessel function approxiamtion parameters from Bogaert, Michiels and Fostier
real(kind=real2), parameter :: c21 = 1d0/8d0
real(kind=real2), parameter :: c22 = -1d0/12d0
real(kind=real2), parameter :: c42 = 11d0/384d0
real(kind=real2), parameter :: c43 = -7d0/160d0
real(kind=real2), parameter :: c44 = 1d0/160d0
real(kind=real2), parameter :: c63 = 173d0/15360d0
real(kind=real2), parameter :: c64 = -101d0/3584d0
real(kind=real2), parameter :: c65 = 671d0/80640d0
real(kind=real2), parameter :: c66 = -61d0/120960d0
real(kind=real2), parameter :: c84 = 22931d0/3440640d0
real(kind=real2), parameter :: c85 = -90497d0/3870720d0
real(kind=real2), parameter :: c86 = 217d0/20480d0
real(kind=real2), parameter :: c87 = -1261d0/967680d0
real(kind=real2), parameter :: c88 = 1261d0/29030400d0
real(kind=real2), parameter :: cX5 = 1319183d0/247726080d0
real(kind=real2), parameter :: cX6 = -10918993/454164480d0
real(kind=real2), parameter :: cX7 = 1676287d0/113541120d0
real(kind=real2), parameter :: cX8 = -7034857d0/2554675200d0
real(kind=real2), parameter :: cX9 = 1501d0/8110080d0
real(kind=real2), parameter :: cXX = -79d0/20275200d0
real(kind=real2), parameter :: cZ6 = 233526463d0/43599790080d0
real(kind=real2), parameter :: cZ7 = -1396004969d0/47233105920d0
real(kind=real2), parameter :: cZ8 = 2323237523d0/101213798400d0
real(kind=real2), parameter :: cZ9 = -72836747d0/12651724800d0
real(kind=real2), parameter :: cZX = 3135577d0/5367398400d0
real(kind=real2), parameter :: cZY = -1532789d0/61993451520d0
real(kind=real2), parameter :: cZZ = 66643d0/185980354560d0
integer(kind=int1) :: i
real(kind=real2) :: h(13),c
c = 1d0
do i=1,13
h(i) = bessel_jn(i-1,y)*c
c = c*y
enddo
fn(1) = h(1)
fn(2) = c21*h(2) + c22*h(3)
fn(3) = c42*h(3) + c43*h(4) + c44*h(5)
fn(4) = c63*h(4) + c64*h(5) + c65*h(6) + c66*h(7)
fn(5) = c84*h(5) + c85*h(6) + c86*h(7) + c87*h(8) + c88*h(9)
fn(6) = cX5*h(6) + cX6*h(7) + cX7*h(8) + cX8*h(9) + cX9*h(10) + cXX*h(11)
fn(7) = cZ6*h(7) + cZ7*h(8) + cZ8*h(9) + cZ9*h(10) + cZX*h(11) + &
cZY*h(12) + cZZ*h(13)
return
end function bogaert_f
!**********************************************************************
function baratella_dleg_approx(n,t) result(dpn) ! best for cos(pi/6) <= |x| <=1
use precision
implicit none
integer(kind=int1), intent(in) :: n
real(kind=real2), intent(in) :: t
real(kind=real2) :: dpn
integer(kind=int1), parameter :: lmax = 15
integer(kind=int1) :: l
real(kind=real2) :: delj,rt,mt,mtl,rfacl
rt = (real(n,kind=real2)+ 0.5d0)*t
rfacl = 1d0
mt = -t
mtl = 1d0
dpn = 0d0
l = 0
delj = 1d0
do l=1,lmax
rfacl = rfacl/real(l,kind=real2)
delj = bessel0d(l,rt)*mtl*rfacl
dpn = dpn + delj
mtl = mtl*mt
enddo
dpn = (t*dpn + (cos(t) - 1d0)*bessel_jn(0,rt))/sin(t)
dpn = dpn*sqrt(t/sin(t))
dpn = -real(n,kind=real2)*dpn
return
end function baratella_dleg_approx
!**********************************************************************
function bessel0d(l,z) result(jl)
use precision
implicit none
integer(kind=int1), intent(in) :: l
real(kind=real2), intent(in) :: z
real(kind=real2) :: jl
integer(kind=int1) :: j
real(kind=real2) :: m1,c
jl = 0d0
m1 = -1d0
c = 1d0
do j=0,l
m1 = -m1
jl = jl + bessel_jn((-l + 2*j),z)*m1*c
c = c*real(l-j,kind=real2)/real(j+1,kind=real2)
enddo
jl = jl*(2d0**(-l))
return
end function bessel0d
!**********************************************************************
function theta0(n,k) result(tk)
use precision
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2) :: tk
real(kind=real2) :: xk
xk = tricomi_approx(n,k)
if(abs(xk) .gt. 0.5d0) xk = olver_approx(n,k)
tk = acos(xk)
return
end function theta0
!**********************************************************************
function tricomi_approx(n,k) result(xk) ! best for |xk| < 0.5
use precision
use maths, only : pi
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2) :: xk
real(kind=real2) :: phi,rn
rn = real(n,kind=real2)
phi = (real(k,kind=real2) - 0.25d0)*pi/(rn + 0.5d0)
xk = 39d0 - 28d0/(sin(phi)**2d0)
xk = -xk/(384d0*rn**4d0)
xk = xk - (rn - 1d0)/(8d0*rn**3d0)
xk = 1d0 + xk
xk = xk*cos(phi)
return
end function tricomi_approx
!**********************************************************************
function olver_approx(n,k) result(xk) ! best for 0.5 <= |xk| <=1
use precision
implicit none
integer(kind=int1), intent(in) :: n,k
real(kind=real2) :: xk
real(kind=real2) :: psi,nph
nph = real(n,kind=real2) + 0.5d0
psi = bessel0_root(k)/nph
xk = psi/tan(psi) - 1d0
xk = xk/(8d0*psi*nph*nph)
xk = xk + psi
xk = cos(xk)
return
end function olver_approx
!**********************************************************************
function bessel0_root(k) result(j0)
use precision
use maths, only : pi
implicit none
integer(kind=int1), intent(in) :: k
real(kind=real2) :: j0
! Approximation parameters from JCP 42, pp. 403-405 (1981)
real(kind=real2), parameter :: a0 = 0.682894897349453d-1
real(kind=real2), parameter :: a1 = 0.131420807470708d+0
real(kind=real2), parameter :: a2 = 0.245988241803681d-1
real(kind=real2), parameter :: a3 = 0.813005721543268d-3
real(kind=real2), parameter :: b0 = 1d0
real(kind=real2), parameter :: b1 = 0.116837242570470d+1
real(kind=real2), parameter :: b2 = 0.200991122197811d+0
real(kind=real2), parameter :: b3 = 0.650404577261471d-2
real(kind=real2) :: beta
beta = (real(k,kind=real2) - 0.25d0)*pi
j0 = a0 + a1*beta**2 + a2*beta**4 + a3*beta**6
j0 = j0/(beta*(b0 + b1*beta**2 + b2*beta**4 + b3*beta**6))
j0 = beta + j0
return
end function bessel0_root
!**********************************************************************
end module bigquad