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makequad.f90
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makequad.f90
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!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!
! This module contains routines for constructing generalized Gaussian quadrature rules.
! They are wrappers around the codes in chebyshev_quad.f90 and gaussian_quad.f90.
!
! The following subroutines are publicly callable:
!
! ggquad - construct a generalized Gaussian quadrature rule for a collection of
! input functions given on an interval (a,b) and specified via an external
! subroutine
!
! ggquad_prod - construct a generalized Gaussian quadrature ruke for a collection of
! functions of the form
!
! f_i(x) g_j(x) i=1,...,n1, j=1,...,m1,
!
! where the f_i and g_j are specified by the user via external subroutines
!
! ggquad_interp - use Kirill Serkh's procedure to construct an n-point quadrature
! which integrates a collection of n smooth (or, very mildly singular) functions
! and whose nodes are stable interpolation points for the collection
!
! IMPORTANT NOTE: THE FUNCTION f(x) = 1 MUST BE AMONG THE INPUT FUNCTIONS OR
! THE RESULTING QUADRATURE MIGHT NOT ACCURATELY INTEGRATE THE INPUT FUNCTIONS
!
! ggquad_interp_sing - use Kirill Serkh's procedure to construct an n-point
! "interpolation quadrature" for a collection of n functions given on the interval
! (-1,1) and which are singular at 0
!
! IMPORTANT NOTE: THE FUNCTION f(x) = 1 MUST BE AMONG THE INPUT FUNCTIONS OR
! THE RESULTING QUADRATURE MIGHT NOT ACCURATELY INTEGRATE THE INPUT FUNCTIONS
! ggquad_sing - construct a generalized Gaussian quadrature rule for a collection
! of functions given on the interval (-1,1) that are strongly singular at 0
! (e.g., |x|^alpha for alpha close to -1)
!
! ggquad_singprod - construct a generalized Gaussian quadrature rule for a collection
! of functions of the form
!
! f_i(x) g_j(x) i=1,...,n1, j=1,...,m1
!
! where the f_i are given on the interval (-1,1) and are strongly singular at 0,
! and the g_j(x) are nonsingular.
!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
module makequad
use utils
use legendre
use legepw
use linalg
use chebyshev_quad
use gaussian_quad
use iso_c_binding
! set this to one to suppress output from these routines
integer, private, parameter :: ifsuppress = 0
interface
subroutine ggfun(nfuns,n,xs,vals,userptr)
!
! Return the (n,nfuns) matrix vals whose jth column gives the
! values of the jth input function at a collection of points
! specified by the xs array. The parameter userptr is a "void *"
! pointer which is supplied by the user to the quadrature routines
! and passed on to this routine.
!
import c_ptr
double precision :: xs(n)
double precision :: vals(:,:)
type(c_ptr) :: userptr
end subroutine
end interface
type ggwrapper_data
type(c_ptr) :: userptr
procedure(ggfun), pointer, nopass :: userfun
integer :: ifscale
integer :: ifodd
integer :: ifeven
double precision :: dsub
end type ggwrapper_data
contains
subroutine ggquad(ifsing,a,b,nfuns,fun,userptr,nquad,xs,whts)
implicit double precision (a-h,o-z)
procedure(ggfun) :: fun
double precision, allocatable, intent(out) :: xs(:), whts(:)
type(c_ptr) :: userptr
!
! Construct a generalized Gaussian quadrature for a collection of functions
! of the form
!
! f_i(x), i=1,...,n,
!
! where the f_i are supplied by the user via an external subroutine.
!
! Input parameters:
! ifsing - an integer parameter which, when set to 1, indicates that the
! input functions are given on an interval of the form (a,b) with
! a <= 0 <= b, and that they have mild singularities at 0
!
! (a,b) - the interval on which the input functions are given
! nfuns - the number of input functions
! fun - an external subroutine conforming to the ggfun interface which
! supplies the values of the input functions
! userptr - a user-supplied "void *" pointer which is passed to the subroutine
! fun
!
! Output parameters:
! (nquad,xs,whts) - the resulting quadrature rule
!
type(legepw_disc) :: disc
type(c_ptr) :: wrapptr
type(ggwrapper_data), pointer :: wrapdata
double precision, allocatable :: ab0(:,:)
double precision, allocatable :: vals0(:,:), vals(:,:), r(:,:)
double precision, allocatable :: xs0(:), whts0(:)
double precision, allocatable :: us(:), uwhts(:), rnorms(:)
integer, allocatable :: ipivs(:)
eps0 = epsilon(0.0d0)
epsdisc = 1.0d-13
epsqr = 1.0d-13
epsnewt = 1.0d-7
if (eps0 .lt. 1.0d-17) then
epsdisc = 1.0d-15
epsqr = 1.0d-15
epsnewt = 1.0d-9
endif
if (eps0 .lt. 1.0d-30) then
epsdisc = 1.0d-30
epsqr = 1.0d-30
epsnewt = 1.0d-15
endif
nlege = 30
dsub = 1
a0 = a
b0 = b
if (ifsing .eq. 1) then
dsub = 3
a0 =-abs(a)**(1.0d0/dsub)
b0 = b**(1.0d0/dsub)
if (a .lt. 0 .AND. b .gt. 0) then
nints0 = 2
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = 0.0d0
ab0(1,2) = 0.0d0
ab0(2,2) = b0
elseif (b .gt. 0) then
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = 0.0d0
ab0(2,1) = b0
else
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = 0.0d0
endif
else
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = b0
endif
!
! Discretize the user-supplied collection of functions
!
allocate(wrapdata)
wrapdata%userptr = userptr
wrapdata%userfun => fun
wrapdata%ifscale = 1
wrapdata%dsub = dsub
wrapptr = c_loc(wrapdata)
call legepw_init(disc,nlege,nints0,ab0)
call legepw_adap(epsdisc,nfuns,ggwrapper,disc,wrapptr)
call legepw_quad(disc,nquad0,xs0,whts0)
if (ifsuppress .ne. 1) then
call prini("in ggquad, nints = ",disc%nints)
call prini("in ggquad, nquad0 = ",nquad0)
endif
!call prin2("in ggquad, ab = ",disc%ab)
!
! Orthonormalize the input functions
!
allocate(vals0(nquad0,nfuns))
call ggwrapper(nfuns,nquad0,xs0,whts0,vals0,wrapptr)
call qrdecomp(epsqr,nquad0,nfuns,vals0,krank,ipivs,vals,r)
allocate(rnorms(krank))
do i=1,krank
rnorms(i) = abs(r(i,i))
end do
if (ifsuppress .ne. 1) then
call prini("in ggquad, nfuns = ",nfuns)
call prini("in ggquad, krank = ",krank)
call prin2("in ggquad, rnorms = ",rnorms)
endif
!
! Construct the oversampled "Chebyshev" quadrature
!
call chebquad(disc,krank,vals,nquad,us,uwhts)
if (ifsuppress .ne. 1) then
call prin2("in ggquad, chebxs = ",us)
call prin2("in ggquad, chebwhts = ",uwhts)
endif
!
! Perform Newton iterations
!
iorder=1
call gaussquad(epsnewt,iorder,disc,krank,vals,nquad,us,uwhts)
!
! Apply the substitution
!
allocate(xs(nquad), whts(nquad))
xs = us**dsub
whts = uwhts * dsub*us**(dsub-1)
if (ifsuppress .ne. 1) then
call prini("in ggquad, nfuns = ",nfuns)
call prini("in ggquad, krank = ",krank)
call prini("in ggquad, final nquad = ",nquad)
call prin2("in ggquad, xs = ",xs)
call prin2("in ggquad, whts = ",whts)
endif
end subroutine
subroutine ggquad_prod(ifsing,a,b,nfuns1,nfuns2,fun1,fun2,userptr1,userptr2, &
nquad,xs,whts)
implicit double precision (a-h,o-z)
procedure(ggfun) :: fun1, fun2
double precision, allocatable, intent(out) :: xs(:), whts(:)
type(c_ptr) :: userptr1, userptr2
!
! Construct a generalized Gaussian quadrature for a collection of functions
! of the form
!
! f_i(x) g_j(x) i=1,...,n, j=1,..,m
!
! where the f_i and g_j are supplied by the user via an external subroutines.
!
! Input parameters:
! ifsing - an integer parameter which, when set to 1, indicates that the input
! functions are given on an interval of the form (a,b) with a <= 0 <= b
! and that the f_i(x) might have a mild sinuglarities at 0.
!
! (a,b) - the interval on which the input functions are given
! nfuns1 - the number of f_i's
! nfuns2 - the number of g_j's
! fun1 - an external subroutine conforming to the ggfun interface which
! supplies the values of the f_i's
! fun2 - an external subroutien conforming to the ggfun interface which
! supplies the values of the g_j's
! userptr1 - a user-supplied "void *" pointer which is passed to the subroutines
! fun1
! userptr2 - a user-supplied "void *" pointer which is passed to the subroutines
! fun2
!
! Output parameters:
! (nquad,xs,whts) - the resulting quadrature rule
!
type(legepw_disc) :: disc
type(c_ptr) :: wrapptr
type(ggwrapper_data), pointer :: wrapdata
double precision, allocatable :: vals10(:,:), vals1(:,:), rnorms1(:)
double precision, allocatable :: vals20(:,:), vals2(:,:), rnorms2(:)
double precision, allocatable :: prods0(:,:), prods(:,:), rnorms(:), r(:,:)
double precision, allocatable :: ab0(:,:)
double precision, allocatable :: xs0(:), whts0(:)
double precision, allocatable :: us0(:), uwhts0(:)
double precision, allocatable :: us(:), uwhts(:)
integer, allocatable :: ipivs(:)
eps0 = epsilon(0.0d0)
epsdisc = 1.0d-13
epsqr = 1.0d-13
epsnewt = 1.0d-7
if (eps0 .lt. 1.0d-17) then
epsdisc = 1.0d-15
epsqr = 1.0d-15
epsnewt = 1.0d-7
endif
if (eps0 .lt. 1.0d-30) then
epsdisc = 1.0d-30
epsqr = 1.0d-30
epsnewt = 1.0d-15
endif
nlege = 30
dsub = 1
a0 = a
b0 = b
if (ifsing .eq. 1) then
dsub = 5
a0 = sign(1.0d0,a)*abs(a)**(1.0d0/dsub)
b0 = b**(1.0d0/dsub)
if (a .lt. 0 .AND. b .gt. 0) then
nints0 = 2
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = 0.0d0
ab0(1,2) = 0.0d0
ab0(2,2) = b0
elseif (b .gt. 0) then
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = 0.0d0
ab0(2,1) = b0
else
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = 0.0d0
endif
else
nints0 = 1
allocate(ab0(2,nints0))
ab0(1,1) = a0
ab0(2,1) = b0
endif
!
! Discretize the two user-supplied collections of functions
!
call legepw_init(disc,nlege,nints0,ab0)
allocate(wrapdata)
wrapdata%dsub = dsub
wrapptr = c_loc(wrapdata)
wrapdata%ifscale = 0
wrapdata%userfun => fun2
wrapdata%userptr = userptr2
call legepw_adap(epsdisc,nfuns2,ggwrapper,disc,wrapptr)
call prini("in ggquad_prod, nints = ",disc%nints)
wrapdata%ifscale = 1
wrapdata%userfun => fun1
wrapdata%userptr = userptr1
call legepw_adap(epsdisc,nfuns1,ggwrapper,disc,wrapptr)
call prini("in ggquad_prod, nints = ",disc%nints)
call legepw_order(disc,nlege*2)
call legepw_quad(disc,nquad0,xs0,whts0)
call prini("in ggquad_prod, nquad0 = ",nquad0)
!call prin2("in ggquad_prod, ab = ",disc%ab)
!
! Orthonormalize each collection of input functions separately
!
allocate(vals10(nquad0,nfuns1), vals20(nquad0,nfuns2) )
wrapdata%userfun => fun1
wrapdata%userptr = userptr1
wrapdata%ifscale = 1
call ggwrapper(nfuns1,nquad0,xs0,whts0,vals10,wrapptr)
wrapdata%userfun => fun2
wrapdata%userptr = userptr2
wrapdata%ifscale = 0
call ggwrapper(nfuns2,nquad0,xs0,whts0,vals20,wrapptr)
call qrdecomp(epsqr,nquad0,nfuns1,vals10,krank1,ipivs,vals1,r)
allocate(rnorms1(krank1))
do i=1,krank1
rnorms1(i) = abs(r(i,i))
end do
if (ifsuppress .ne. 1) then
call prini("in ggquad_prod, nfuns1 = ",nfuns1)
call prini("in ggquad_prod, krank1 = ",krank1)
call prin2("in ggquad_prod, rnorms1 = ",rnorms1)
endif
call qrdecomp(epsqr,nquad0,nfuns2,vals20,krank2,ipivs,vals2,r)
allocate(rnorms2(krank2))
do i=1,krank2
rnorms2(i) = abs(r(i,i))
end do
if (ifsuppress .ne. 1) then
call prini("in ggquad_prod, nfuns2 = ",nfuns2)
call prini("in ggquad_prod, krank2 = ",krank2)
call prin2("in ggquad_prod, rnorms2 = ",rnorms2)
endif
!
! Scale each set of functions by their "sinuglar values"
!
do j=1,krank1
vals1(:,j) = vals1(:,j) * rnorms1(j)
end do
do j=1,krank2
vals2(:,j) = vals2(:,j) * rnorms2(j)
end do
!
! Construct the products and orthonormalize 'em
!
nprods = krank1*krank2
allocate(prods0(nquad0,nprods) )
do i=1,krank1
do j=1,krank2
idx = (i-1)*krank2
prods0(:,idx+j) = vals1(:,i)*vals2(:,j)*1/sqrt(whts0)
end do
end do
call qrdecomp(epsqr,nquad0,nprods,prods0,krank,ipivs,prods,r)
allocate(rnorms(krank))
do i=1,krank
rnorms(i) = abs(r(i,i))
end do
if (ifsuppress .ne. 1) then
call prini("in ggquad_prod, krank = ",krank)
call prin2("in ggquad_prod, rnorms = ",rnorms)
endif
!
! Construct the oversampled "Chebyshev" quadrature
!
call chebquad(disc,krank,prods,nquad,us,uwhts)
!
! Perform Newton iterations
!
iorder=1
call gaussquad(epsnewt,iorder,disc,krank,prods,nquad,us,uwhts)
!
! Apply the substitution (if any)
!
allocate(xs(nquad), whts(nquad))
xs = us**dsub
whts = uwhts * dsub*us**(dsub-1)
if (ifsuppress .ne. 1) then
call prini("in ggquad_prod, nprods = ",nprods)
call prini("in ggquad_prod, krank = ",krank)
call prini("in ggquad_prod, nquad = ",nquad)
call prin2("in ggquad_prod, xs = ",xs)
call prin2("in ggquad_prod, whts = ",whts)
call prina("")
endif
end subroutine
subroutine ggquad_interp(a,b,nfuns,fun,userptr,nquad,xs,whts,rcond)
implicit double precision (a-h,o-z)
procedure(ggfun) :: fun
double precision, allocatable, intent(out) :: xs(:), whts(:)
type(c_ptr) :: userptr
!
! Use Kirill Serkh's procedure to construct am interpolation quadrature for a
! collection of user-supplied smooth (or very mildly singular) functions
!
! Input parameters:
! (a,b) - the domain of definition of the functions
! nfuns - the number of input functions
! fun - an external subroutine conforming to the ggfun interface which
! supplies the values of the input functions
! userptr - a user-supplied "void *" pointer which is passed to the subroutines
! fun
!
! Output parameters:
! (nquad,xs,whts) - the resulting quadrature rule
! rcond - an upper bound on the
!
type(legepw_disc) :: disc
type(c_ptr) :: wrapptr
type(ggwrapper_data), pointer :: wrapdata
double precision, allocatable :: ab0(:,:)
double precision, allocatable :: xs0(:), whts0(:)
double precision, allocatable :: vals0(:,:), vals(:,:), rnorms(:), r(:,:)
double precision, allocatable :: prods0(:,:), prods(:,:), rnormsprods(:)
integer, allocatable :: ipivs(:), idxs(:)
double precision, allocatable :: amatr(:,:)
eps0 = epsilon(0.0d0)
epsdisc = 1.0d-14
epsqr = 1.0d-14
epsnewt = 1.0d-7
if (eps0 .lt. 1.0d-17) then
epsdisc = 1.0d-15
epsqr = 1.0d-15
epsnewt = 1.0d-9
endif
if (eps0 .lt. 1.0d-30) then
epsdisc = 1.0d-30
epsqr = 1.0d-30
epsnewt = 1.0d-15
endif
nlege = 30
dsub = 1
call legepw_init(disc,nlege,a,b)
allocate(wrapdata)
wrapdata%dsub = dsub
wrapptr = c_loc(wrapdata)
wrapdata%ifscale = 0
wrapdata%userfun => fun
wrapdata%userptr = userptr
call legepw_adap(epsdisc,nfuns,ggwrapper,disc,wrapptr)
call prini("in ggquad_interp, nints = ",disc%nints)
!call legepw_order(disc,nlege*2)
call legepw_quad(disc,nquad0,xs0,whts0)
call prini("in ggquad_interp, nquad0 = ",nquad0)
!call prin2("in ggquad_interp, ab = ",disc%ab)
allocate(vals0(nquad0,nfuns))
call ggwrapper(nfuns,nquad0,xs0,whts0,vals0,wrapptr)
call qrdecomp(epsqr,nquad0,nfuns,vals0,krank,ipivs,vals,r)
allocate(rnorms(krank))
do i=1,krank
rnorms(i) = abs(r(i,i))
end do
do j=1,krank
dd = rnorms(j)
if (dd .lt. epsnewt) then
krank = j-1
exit
endif
vals(:,j) = vals(:,j) * sqrt(dd)
end do
call prini("in ggquad_interp, nfuns = ",nfuns)
call prini("in ggquad_interp, krank = ",krank)
call prin2("in ggquad_interp, rnorms = ",rnorms(1:krank))
!
! Construct the products and orthonormalize them
!
nprods = krank*krank
allocate(prods0(nquad0,nprods) )
do i=1,krank
do j=1,krank
idx = (i-1)*krank
prods0(:,idx+j) = vals(:,i)*vals(:,j)/sqrt(whts0)
end do
end do
eps = epsnewt/10
call qrdecomp(eps,nquad0,nprods,prods0,krankprods,ipivs,prods,r)
allocate(rnormsprods(krankprods))
do i=1,krankprods
dd = abs(r(i,i))
rnormsprods(i) = dd
if (dd .lt. epsnewt) then
krankprods = i-1
exit
endif
end do
!krankprods = min(2*krank,krankprods)
call prini("in ggquad_interp, krankprods = ",krankprods)
call prin2("in ggquad_interp, rnormsprods = ",rnormsprods(1:krankprods))
!
! Construct the oversampled "Chebyshev" quadrature
!
call chebquad(disc,krankprods,prods(:,1:krankprods),nquad,xs,whts)
call prin2("in ggquad_interp, xs = ",xs)
call prin2("in ggquad_interp, whts = ",whts)
!
! Perform Newton iterations to reduce the quadrature rule
!
iorder=1
call gaussquad(epsnewt,iorder,disc,krankprods,prods(:,1:krankprods),nquad,xs,whts)
!
! Apply the substitution
!
call prini("in ggquad_interp, krank = ",krank)
call prini("in ggquad_interp, krankprods = ",krankprods)
call prini("in ggquad_interp, nquad = ",nquad)
call prin2("in ggquad_interp, xs = ",xs)
call prin2("in ggquad_interp, whts = ",whts)
!
! Bound the condition number of interpolation
!
do j=1,krank
vals(:,j) = vals(:,j)/sqrt(rnorms(j))
end do
allocate(amatr(nquad,krank))
do i=1,nquad
x = xs(i)
wht = whts(i)
call legepw_interp(disc,vals,x,amatr(i,:))
amatr(i,:) = amatr(i,:) *sqrt(wht)
end do
dd = norm2(matmul(transpose(amatr),amatr)-eye(krank))
rcond = sqrt( (1+dd**2) /(1-dd**2) )
call prind("in ggquad_interp, ||A*A - I||_2 = ",dd)
call prind("in ggquad_interp, rcond = ",rcond)
call prina("")
deallocate(amatr)
end subroutine
subroutine ggquad_interp_sing(nfuns,fun,userptr,nquad,xs,whts,rcond)
implicit double precision (a-h,o-z)
procedure(ggfun) :: fun
double precision, allocatable, intent(out) :: xs(:), whts(:)
type(c_ptr) :: userptr
!
! Use Kirill Serkh's procedure to construct am interpolation quadrature for a
! collection of user-supplied functions given on the interval (-1,1) and which
! can have singularities at 0
!
! Input parameters:
! nfuns - the number of input functions
! fun - an external subroutine conforming to the ggfun interface which
! supplies the values of the input functions
! userptr - a user-supplied "void *" pointer which is passed to the subroutines
! fun
!
! Output parameters:
! (nquad,xs,whts) - the resulting quadrature rule
! rcond - an upper bound on the "condition number of the basis"
!
type(legepw_disc) :: disc
type(c_ptr) :: wrapptr
type(ggwrapper_data), pointer :: wrapdata
double precision, allocatable :: ab0(:,:)
double precision, allocatable :: xs0(:), whts0(:)
double precision, allocatable :: vals0(:,:), vals(:,:), rnorms(:), r(:,:)
double precision, allocatable :: prods0(:,:), prods(:,:), rnormsprods(:)
double precision, allocatable :: us(:), uwhts(:)
integer, allocatable :: ipivs(:), idxs(:)
double precision, allocatable :: amatr(:,:), rints(:), rints0(:)
eps0 = epsilon(0.0d0)
epsdisc = 1.0d-14
epsqr = 1.0d-14
epsnewt = 1.0d-10
if (eps0 .lt. 1.0d-17) then
epsdisc = 1.0d-17
epsqr = 1.0d-17
epsnewt = 1.0d-10
endif
if (eps0 .lt. 1.0d-30) then
epsdisc = 1.0d-30
epsqr = 1.0d-30
epsnewt = 1.0d-20
endif
nlege = 30
!
! First construct a quadratue rule for the
!
nints0 = 4
allocate(ab0(2,nints0))
ab0(1,1) = -70.0d0
ab0(2,1) = -1.0d0
ab0(1,2) = -1.0d0
ab0(2,2) = 0.0d0
ab0(1,3) = 0.0d0
ab0(2,3) = 1.0d0
ab0(1,4) = 1.0d0
ab0(2,4) = 70.0d0
call legepw_init(disc,nlege,nints0,ab0)
allocate(wrapdata)
wrapdata%dsub = dsub
wrapptr = c_loc(wrapdata)
wrapdata%userfun => fun
wrapdata%userptr = userptr
call legepw_adap(epsdisc,nfuns,ggwrapper3,disc,wrapptr)
call prini("in ggquad_interp_sing, nints = ",disc%nints)
! promoting the order is unnecessary
!call legepw_order(disc,nlege*2)
call legepw_quad(disc,nquad0,xs0,whts0)
call prini("in ggquad_interp_sing, nquad0 = ",nquad0)
call prin2("in ggquad_interp_sing, ab = ",disc%ab)
allocate(vals0(nquad0,nfuns))
call ggwrapper3(nfuns,nquad0,xs0,whts0,vals0,wrapptr)
call qrdecomp(epsqr,nquad0,nfuns,vals0,krank,ipivs,vals,r)
allocate(rnorms(krank))
do i=1,krank
rnorms(i) = abs(r(i,i))
end do
do j=1,krank
dd = rnorms(j)
if (dd .lt. epsnewt) then
krank = j-1
exit
endif
vals(:,j) = vals(:,j) * sqrt(dd)
end do
call prini("in ggquad_interp_sing, nfuns = ",nfuns)
call prini("in ggquad_interp_sing, krank = ",krank)
call prin2("in ggquad_interp_sing, rnorms = ",rnorms(1:krank))
!
! Construct the products and orthonormalize them
!
nprods = krank*krank
allocate(prods0(nquad0,nprods) )
do i=1,krank
do j=1,krank
idx = (i-1)*krank
prods0(:,idx+j) = vals(:,i)*vals(:,j)/sqrt(whts0)
end do
end do
eps = epsnewt/100
call qrdecomp(eps,nquad0,nprods,prods0,krankprods,ipivs,prods,r)
allocate(rnormsprods(krankprods))
do i=1,krankprods
dd = abs(r(i,i))
rnormsprods(i) = dd
if (dd .lt. epnewt) then
krankprods = i-1
exit
endif
end do
call prini("in ggquad_interp_sing, krankprods = ",krankprods)
call prin2("in ggquad_interp_sing, rnormsprods = ",rnormsprods(1:krankprods))
!
! Construct the oversampled "Chebyshev" quadrature
!
call chebquad(disc,krankprods,prods(:,1:krankprods),nquad,us,uwhts)
!
! Perform Newton iterations to reduce the quadrature rule
!
iorder=1
call gaussquad(epsnewt,iorder,disc,krankprods,prods(:,1:krankprods),nquad,us,uwhts)
!
! Apply the substitution and sort the nodes
!
allocate(xs(nquad), whts(nquad), idxs(nquad))
xs = exp(-abs(us))* sign(1.0d0,us)
whts = exp(-abs(us))*uwhts
call insort2(nquad,xs,idxs)
whts = whts(idxs)
call prini("in ggquad_interp_sing, krank = ",krank)
call prini("in ggquad_interp_sing, krankprods = ",krankprods)
call prini("in ggquad_interp_sing, nquad = ",nquad)
call prin2("in ggquad_interp_sing, xs = ",xs)
call prin2("in ggquad_interp_sing, whts = ",whts)
!
! Bound the condition number of interpolation
!
do j=1,krank
vals(:,j) = vals(:,j)/sqrt(rnorms(j))
end do
allocate(amatr(nquad,krank))
do i=1,nquad
x = us(i)
wht = uwhts(i)
call legepw_interp(disc,vals,x,amatr(i,:))
amatr(i,:) = amatr(i,:) *sqrt(wht)
end do
dd = norm2(matmul(transpose(amatr),amatr)-eye(krank))
rcond = sqrt(1+dd**2)/sqrt(1-dd**2)
deallocate(amatr)
call prind("in ggquad_interp_sing, rcond = ",rcond)
end subroutine
subroutine ggquad_sing(nfuns,fun,userptr,nquad,xs,whts)
implicit double precision (a-h,o-z)
procedure(ggfun) :: fun
double precision, allocatable, intent(out) :: xs(:), whts(:)
type(c_ptr) :: userptr
!
! Construct a generalized Gaussian quadrature on the interval (-1,1) for
! a collection of the functions which are strongly sinuglar at 0.
!
! Input parameters:
! nfuns - the number of input functions
! fun - an external subroutine conforming to the ggfun interface which
! supplies the values of the input functions
! userptr - a user-supplied "void *" pointer which is passed to the subroutine
! fun
!
! Output parameters:
! (nquad,xs,whts) - the resulting quadrature rule
!
type(legepw_disc) :: disc
type(c_ptr) :: wrapptr
type(ggwrapper_data), pointer :: wrapdata
double precision, allocatable :: vals0(:,:), vals(:,:), r(:,:), ab0(:,:)
double precision, allocatable :: xs0(:), whts0(:)
double precision, allocatable :: us(:), uwhts(:), rnorms(:)
integer, allocatable :: ipivs(:), idxs(:)
eps0 = epsilon(0.0d0)
epsdisc = 1.0d-14
epsqr = 1.0d-14
epsnewt = 1.0d-7
if (eps0 .lt. 1.0d-17) then
epsdisc = 1.0d-17
epsqr = 1.0d-17
epsnewt = 1.0d-12
endif
if (eps0 .lt. 1.0d-30) then
epsdisc = 1.0d-30
epsqr = 1.0d-30
epsnewt = 1.0d-15
endif
nlege = 30
nints0 = 4
allocate(ab0(2,nints0))
ab0(1,1) = -200.0d0
ab0(2,1) = -1.0d0
ab0(1,2) = -1.0d0
ab0(2,2) = 0.0d0
ab0(1,3) = 0.0d0
ab0(2,3) = 1.0d0
ab0(1,4) = 1.0d0
ab0(2,4) = 200.0d0
! Discretize the user-supplied collection of functions
!
allocate(wrapdata)
wrapdata%ifscale = 1
wrapdata%userptr = userptr
wrapdata%userfun => fun
wrapptr = c_loc(wrapdata)
call legepw_init(disc,nlege,nints0,ab0)
call legepw_adap(epsdisc,nfuns,ggwrapper2,disc,wrapptr)
call legepw_quad(disc,nquad0,xs0,whts0)
call prini("in ggquad_sing, nints = ",disc%nints)
call prini("in ggquad_sing, nquad0 = ",nquad0)
!
! Orthonormalize the input functions
!
allocate(vals0(nquad0,nfuns))
call ggwrapper2(nfuns,nquad0,xs0,whts0,vals0,wrapptr)
call qrdecomp(epsqr,nquad0,nfuns,vals0,krank,ipivs,vals,r)
allocate(rnorms(krank))
do i=1,krank
rnorms(i) = abs(r(i,i))
end do
call prini("in ggquad_sing, krank = ",krank)
call prin2("in ggquad_sing, rnorms = ",rnorms)
!
! Construct the oversampled "Chebyshev" quadrature
!
call chebquad(disc,krank,vals,nquad,us,uwhts)
!
! Perform Newton iterations
!
iorder=1
call gaussquad(epsnewt,iorder,disc,krank,vals,nquad,us,uwhts)
!