quadpack_generic Module

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

key for choice of local integration rule. a gauss-kronrod pair is used with:

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork limit determines the maximum number of subintervals in the partition of the given integration interval (a,b), limit>=1. if limit<1, the routine will end with ier = 6.

dimensioning parameter for work lenw must be at least limit*4. if lenw<limit*4, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least limit, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit3+iwork(1)),... , work(limit3+iwork(k)) form a decreasing sequence with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

vector of dimension at least lenw on return work(1), ..., work(last) contain the left end points of the subintervals in the partition of (a,b), work(limit+1), ..., work(limit+last) contain the right end points, work(limit*2+1), ..., work(limit*2+last) contain the integral approximations over the subintervals, work(limit3+1), ..., work(limit3+last) contain the error estimates.

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

key for choice of local integration rule a gauss-kronrod pair is used with

gives an upperbound on the number of subintervals in the partition of (a,b), limit>=1.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least limit, the first k elements of which are pointers to the error estimates over the subintervals, such that elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

number of subintervals actually produced in the subdivision process

function subprogram defining the integrand function f(x).

finite bound of integration range (has no meaning if interval is doubly-infinite)

indicating the kind of integration range involved:

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork limit determines the maximum number of subintervals in the partition of the given integration interval (a,b), limit>=1. if limit<1, the routine will end with ier = 6.

dimensioning parameter for work lenw must be at least limit*4. if lenw<limit*4, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least limit, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit*3+iwork(1)),...,work(limit*3+iwork(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

vector of dimension at least lenw on return: * work(1), ..., work(last) contain the left end points of the subintervals in the partition of (a,b), * work(limit+1), ..., work(limit+last) contain the right end points, * work(limit*2+1), ...,work(limit*2+last) contain the integral approximations over the subintervals, * work(limit*3+1), ..., work(limit*3) contain the error estimates.

function subprogram defining the integrand function f(x).

finite bound of integration range (has no meaning if interval is doubly-infinite)

indicating the kind of integration range involved * inf = 1 corresponds to (bound,+infinity) * inf = -1 corresponds to (-infinity,bound) * inf = 2 corresponds to (-infinity,+infinity)

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upper bound on the number of subintervals in the partition of (a,b), limit>=1

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the transformed integration range (0,1).

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the transformed integration range (0,1).

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension limit, the first k elements of which are pointers to the error estimates over the subintervals, such that elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

number of subintervals actually produced in the subdivision process

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

number equal to two more than the number of user-supplied break points within the integration range, npts>=2. if npts2<2, the routine will end with ier = 6.

vector of dimension npts2, the first (npts2-2) elements of which are the user provided break points. if these points do not constitute an ascending sequence there will be an automatic sorting.

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork. leniw determines limit = (leniw-npts2)/2, which is the maximum number of subintervals in the partition of the given integration interval (a,b), leniw>=(3*npts2-2). if leniw<(3*npts2-2), the routine will end with ier = 6.

dimensioning parameter for work. lenw must be at least leniw*2-npts2. if lenw<leniw*2-npts2, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least leniw. on return, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit*3+iwork(1)),...,work(limit*3+iwork(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise iwork(limit+1), ...,iwork(limit+last) contain the subdivision levels of the subintervals, i.e. if (aa,bb) is a subinterval of (p1,p2) where p1 as well as p2 is a user-provided break point or integration limit, then (aa,bb) has level l if abs(bb-aa) = abs(p2-p1)*2**(-l), iwork(limit*2+1), ..., iwork(limit*2+npts2) have no significance for the user, note that limit = (leniw-npts2)/2.

vector of dimension at least lenw. on return:

number equal to two more than the number of user-supplied break points within the integration range, npts2>=2. if npts2<2, the routine will end with ier = 6.

vector of dimension npts2, the first (npts2-2) elements of which are the user provided break points. if these points do not constitute an ascending sequence there will be an automatic sorting.

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upper bound on the number of subintervals in the partition of (a,b), limit>=npts2 if limit<npts2, the routine will end with ier = 6.

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least npts2, containing the integration limits and the break points of the interval in ascending sequence.

vector of dimension at least limit, the first k elements of which are pointers to the error estimates over the subintervals, such that elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

vector of dimension at least limit, containing the subdivision levels of the subinterval, i.e. if (aa,bb) is a subinterval of (p1,p2) where p1 as well as p2 is a user-provided break point or integration limit, then (aa,bb) has level l if abs(bb-aa) = abs(p2-p1)*2**(-l).

vector of dimension at least npts2, after first integration over the intervals (pts(i)),pts(i+1), i = 0,1, ..., npts2-2, the error estimates over some of the intervals may have been increased artificially, in order to put their subdivision forward. if this happens for the subinterval numbered k, ndin(k) is put to 1, otherwise ndin(k) = 0.

number of subintervals actually produced in the subdivisions process

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork. limit determines the maximum number of subintervals in the partition of the given integration interval (a,b), limit>=1. if limit<1, the routine will end with ier = 6.

dimensioning parameter for work. lenw must be at least limit*4. if lenw<limit*4, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, determines the number of significant elements actually in the work arrays.

vector of dimension at least limit, the first k elements of which contain pointers to the error estimates over the subintervals such that work(limit*3+iwork(1)),...,work(limit*3+iwork(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

vector of dimension at least lenw. on return:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upperbound on the number of subintervals in the partition of (a,b)

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages: * ier = 1 maximum number of subdivisions allowed has been achieved. one can allow more sub- divisions by increasing the value of limit (and taking the according dimension adjustments into account). however, if this yields no improvement it is advised to analyze the integrand in order to determine the integration difficulties. if the position of a local difficulty can be determined (e.g. singularity, discontinuity within the interval) one will probably gain from splitting up the interval at this point and calling the integrator on the subranges. if possible, an appropriate special-purpose integrator should be used, which is designed for handling the type of difficulty involved. * ier = 2 the occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved. the error may be under-estimated. * ier = 3 extremely bad integrand behaviour occurs at some points of the integration interval. * ier = 4 the algorithm does not converge. roundoff error is detected in the extrapolation table. it is presumed that the requested tolerance cannot be achieved, and that the returned result is the best which can be obtained. * ier = 5 the integral is probably divergent, or slowly convergent. it must be noted that divergence can occur with any other value of ier. * ier = 6 the input is invalid, because epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28). result, abserr, neval, last, rlist(1), iord(1) and elist(1) are set to zero. alist(1) and blist(1) are set to a and b respectively.

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least limit, the first k elements of which are pointers to the error estimates over the subintervals, such that elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

number of subintervals actually produced in the subdivision process

function subprogram defining the integrand function f(x).

under limit of integration

upper limit of integration

parameter in the weight function, c/=a, c/=b. if c = a or c = b, the routine will end with ier = 6 .

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate or the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages: * ier = 1 maximum number of subdivisions allowed has been achieved. one can allow more sub- divisions by increasing the value of limit (and taking the according dimension adjustments into account). however, if this yields no improvement it is advised to analyze the integrand in order to determine the integration difficulties. if the position of a local difficulty can be determined (e.g. singularity, discontinuity within the interval) one will probably gain from splitting up the interval at this point and calling appropriate integrators on the subranges. * ier = 2 the occurrence of roundoff error is detected, which prevents the requested tolerance from being achieved. * ier = 3 extremely bad integrand behaviour occurs at some points of the integration interval. * ier = 6 the input is invalid, because c = a or c = b or (epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28)) or limit<1 or lenw<limit*4. esult, abserr, neval, last are set to zero. except when lenw or limit is invalid, iwork(1), work(limit*2+1) and work(limit*3+1) are set to zero, work(1) is set to a and work(limit+1) to b.

dimensioning parameter for iwork. limit determines the maximum number of subintervals in the partition of the given integration interval (a,b), limit>=1. if limit<1, the routine will end with ier = 6.

dimensioning parameter for work. lenw must be at least limit*4. if lenw<limit*4, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least limit, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit*3+iwork(1)),...,work(limit*3+iwork(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

vector of dimension at least lenw. on return:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upper bound on the number of subintervals in the partition of (a,b), limit>=1

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least limit, the first k elements of which are pointers to the error estimates over the subintervals, so that elist(iord(1)), ..., elist(iord(k)) with k = last if last<=(limit/2+2), and k = limit+1-last otherwise, form a decreasing sequence

number of subintervals actually produced in the subdivision process

function subprogram defining the integrand function f(x).

lower limit of integration

parameter in the integrand weight function

indicates which of the weight functions is used:

absolute accuracy requested, epsabs>0. if epsabs<=0, the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

limlst gives an upper bound on the number of cycles, limlst>=3. if limlst<3, the routine will end with ier = 6.

on return, lst indicates the number of cycles actually needed for the integration. if omega = 0, then lst is set to 1.

dimensioning parameter for iwork. on entry, (leniw-limlst)/2 equals the maximum number of subintervals allowed in the partition of each cycle, leniw>=(limlst+2). if leniw<(limlst+2), the routine will end with ier = 6.

maxp1 gives an upper bound on the number of chebyshev moments which can be stored, i.e. for the intervals of lengths abs(b-a)*2**(-l), l = 0,1, ..., maxp1-2, maxp1>=1. if maxp1<1, the routine will end with ier = 6.

dimensioning parameter for work. lenw must be at least leniw*2+maxp1*25. if lenw<(leniw*2+maxp1*25), the routine will end with ier = 6.

vector of dimension at least leniw on return, iwork(k) for k = 1, 2, ..., lst contain the error flags on the cycles.

vector of dimension at least lenw on return:

function subprogram defining the integrand function f(x).

lower limit of integration

parameter in the weight function

indicates which weight function is used:

absolute accuracy requested, epsabs>0 if epsabs<=0, the routine will end with ier = 6.

limlst gives an upper bound on the number of cycles, limlst>=1. if limlst<3, the routine will end with ier = 6.

gives an upper bound on the number of subintervals allowed in the partition of each cycle, limit>=1 each cycle, limit>=1.

gives an upper bound on the number of chebyshev moments which can be stored, i.e. for the intervals of lengths abs(b-a)*2**(-l),l=0,1, ..., maxp1-2, maxp1>=1``

approximation to the integral x

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

vector of dimension at least limlst rslst(k) contains the integral contribution over the interval (a+(k-1)c,a+kc) where c = (2*int(abs(omega))+1)*pi/abs(omega), k = 1, 2, ..., lst. note that, if omega = 0, rslst(1) contains the value of the integral over (a,infinity).

vector of dimension at least limlst erlst(k) contains the error estimate corresponding with rslst(k).

vector of dimension at least limlst ierlst(k) contains the error flag corresponding with rslst(k). for the meaning of the local error flags see description of output parameter ier.

number of subintervals needed for the integration if omega = 0 then lst is set to 1.

vector of dimension at least limit

vector of dimension at least limit

vector of dimension at least limit

vector of dimension at least limit

vector of dimension at least limit, providing space for the quantities needed in the subdivision process of each cycle

vector of dimension at least limit, providing space for the quantities needed in the subdivision process of each cycle

array of dimension at least (maxp1,25), providing space for the chebyshev moments needed within the cycles (see also routine dqc25f)

function subprogram defining the function f(x).

lower limit of integration

upper limit of integration

parameter in the integrand weight function

indicates which of the weight functions is used

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork. leniw/2 equals the maximum number of subintervals allowed in the partition of the given integration interval (a,b), leniw>=2. if leniw<2, the routine will end with ier = 6.

gives an upper bound on the number of chebyshev moments which can be stored, i.e. for the intervals of lengths abs(b-a)*2**(-l), l=0,1, ..., maxp1-2, maxp1>=1 if maxp1<1, the routine will end with ier = 6.

dimensioning parameter for work lenw must be at least leniw*2+maxp1*25. if lenw<(leniw*2+maxp1*25), the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least leniw on return, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit*3+iwork(1)), .., work(limit*3+iwork(k)) form a decreasing sequence, with limit = lenw/2 , and k = last if last<=(limit/2+2), and k = limit+1-last otherwise. furthermore, iwork(limit+1), ..., iwork(limit+last) indicate the subdivision levels of the subintervals, such that iwork(limit+i) = l means that the subinterval numbered i is of length abs(b-a)*2**(1-l).

vector of dimension at least lenw. on return:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

parameter in the integrand weight function

indicates which of the weight functions is to be used:

absolute accuracy requested

relative accuracy requested. if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upper bound on the number of subdivisions in the partition of (a,b), limit>=1.

if dqawoe is to be used only once, icall must be set to 1. assume that during this call, the chebyshev moments (for clenshaw-curtis integration of degree 24) have been computed for intervals of lengths (abs(b-a))*2**(-l), l=0,1,2,...momcom-1. if icall>1 this means that dqawoe has been called twice or more on intervals of the same length abs(b-a). the chebyshev moments already computed are then re-used in subsequent calls. if icall<1, the routine will end with ier = 6.

gives an upper bound on the number of chebyshev moments which can be stored, i.e. for the intervals of lengths abs(b-a)*2**(-l), l=0,1, ..., maxp1-2, maxp1>=1. if maxp1<1, the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

on return, last equals the number of subintervals produces in the subdivision process, which determines the number of significant elements actually in the work arrays.

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least limit, the first k elements of which are pointers to the error estimates over the subintervals, such that elist(iord(1)), ..., elist(iord(k)) form a decreasing sequence, with k = last if last<=(limit/2+2), and k = limit+1-last otherwise.

vector of dimension at least limit, containing the subdivision levels of the subintervals, i.e. iwork(i) = l means that the subinterval numbered i is of length abs(b-a)*2**(1-l)

indicating that the chebyshev moments have been computed for intervals of lengths (abs(b-a))*2**(-l), l=0,1,2, ..., momcom-1, momcom<maxp1

array of dimension (maxp1,25) containing the chebyshev moments

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration, b>a if b<=a, the routine will end with ier = 6.

parameter in the integrand function, alfa>(-1) if alfa<=(-1), the routine will end with ier = 6.

parameter in the integrand function, beta>(-1) if beta<=(-1), the routine will end with ier = 6.

indicates which weight function is to be used:

absolute accuracy requested

relative accuracy requested. if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

dimensioning parameter for iwork limit determines the maximum number of subintervals in the partition of the given integration interval (a,b), limit>=2. if limit<2, the routine will end with ier = 6.

dimensioning parameter for work lenw must be at least limit*4. if lenw<limit*4, the routine will end with ier = 6.

on return, last equals the number of subintervals produced in the subdivision process, which determines the significant number of elements actually in the work arrays.

vector of dimension limit, the first k elements of which contain pointers to the error estimates over the subintervals, such that work(limit*3+iwork(1)), ..., work(limit*3+iwork(k)) form a decreasing sequence with k = last if last<=(limit/2+2), and k = limit+1-last otherwise

on return:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration, b>a. if b<=a, the routine will end with ier = 6.

parameter in the weight function, alfa>(-1) if alfa<=(-1), the routine will end with ier = 6.

parameter in the weight function, beta>(-1) if beta<=(-1), the routine will end with ier = 6.

indicates which weight function is to be used:

absolute accuracy requested

relative accuracy requested. if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

gives an upper bound on the number of subintervals in the partition of (a,b), limit>=2 if limit<2, the routine will end with ier = 6.

approximation to the integral

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

vector of dimension at least limit, the first last elements of which are the left end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit, the first last elements of which are the right end points of the subintervals in the partition of the given integration range (a,b)

vector of dimension at least limit,the first last elements of which are the integral approximations on the subintervals

vector of dimension at least limit, the first last elements of which are the moduli of the absolute error estimates on the subintervals

vector of dimension at least limit, the first k of which are pointers to the error estimates over the subintervals, so that elist(iord(1)), ..., elist(iord(k)) with k = last if last<=(limit/2+2), and k = limit+1-last otherwise form a decreasing sequence

number of subintervals actually produced in the subdivision process

function subprogram defining the integrand function f(x).

left end point of the integration interval

right end point of the integration interval, b>a

parameter in the weight function

approximation to the integral. result is computed by using a generalized clenshaw-curtis method if c lies within ten percent of the integration interval. in the other case the 15-point kronrod rule obtained by optimal addition of abscissae to the 7-point gauss rule, is applied.

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

key which is decreased by 1 if the 15-point gauss-kronrod scheme has been used

number of integrand evaluations

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

parameter in the weight function

indicates which weight function is to be used

the length of interval (a,b) is equal to the length of the original integration interval divided by 2**nrmom (we suppose that the routine is used in an adaptive integration process, otherwise set nrmom = 0). nrmom must be zero at the first call.

gives an upper bound on the number of chebyshev moments which can be stored, i.e. for the intervals of lengths abs(bb-aa)*2**(-l), l = 0,1,2, ..., maxp1-2.

key which is one when the moments for the current interval have been computed

approximation to the integral i

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

approximation to the integral j

approximation to the integral of abs(f-i/(b-a))

for each interval length we need to compute the chebyshev moments. momcom counts the number of intervals for which these moments have already been computed. if nrmom<momcom or ksave = 1, the chebyshev moments for the interval (a,b) have already been computed and stored, otherwise we compute them and we increase momcom.

array of dimension at least (maxp1,25) containing the modified chebyshev moments for the first momcom momcom interval lengths

function subprogram defining the integrand f(x).

left end point of the original interval

right end point of the original interval, b>a

lower limit of integration, bl>=a

upper limit of integration, br<=b

parameter in the weight function

parameter in the weight function

modified chebyshev moments for the application of the generalized clenshaw-curtis method (computed in subroutine dqmomo)

modified chebyshev moments for the application of the generalized clenshaw-curtis method (computed in subroutine dqmomo)

modified chebyshev moments for the application of the generalized clenshaw-curtis method (computed in subroutine dqmomo)

modified chebyshev moments for the application of the generalized clenshaw-curtis method (computed in subroutine dqmomo)

approximation to the integral result is computed by using a generalized clenshaw-curtis method if b1 = a or br = b. in all other cases the 15-point kronrod rule is applied, obtained by optimal addition of abscissae to the 7-point gauss rule.

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

approximation to the integral of abs(f*w-i/(b-a))

which determines the weight function * = 1 w(x) = (x-a)**alfa*(b-x)**beta * = 2 w(x) = (x-a)**alfa*(b-x)**beta*log(x-a) * = 3 w(x) = (x-a)**alfa*(b-x)**beta*log(b-x) * = 4 w(x) = (x-a)**alfa*(b-x)**beta*log(x-a)*log(b-x)

number of integrand evaluations

vector of dimension 11 containing the values cos(k*pi/24), k = 1, ..., 11

vector of dimension 25 containing the function values at the points (b+a+(b-a)*cos(k*pi/24))/2, k = 0, ...,24, where (a,b) is the approximation interval. fval(1) and fval(25) are divided by two (these values are destroyed at output).

vector of dimension 13 containing the chebyshev coefficients for degree 12

vector of dimension 25 containing the chebyshev coefficients for degree 24

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 15-point kronrod rule (resk) obtained by optimal addition of abscissae to the7-point gauss rule(resg).

estimate of the modulus of the absolute error, which should not exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a)) over (a,b)

function subprogram defining the integrand function f(x).

finite bound of original integration range (set to zero if inf = +2)

the integral is computed as the sum of two integrals, one over (-infinity,0) and one over (0,+infinity).

lower limit for integration over subrange of (0,1)

upper limit for integration over subrange of (0,1)

approximation to the integral i. result is computed by applying the 15-point kronrod rule(resk) obtained by optimal addition of abscissae to the 7-point gauss rule(resg).

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs((transformed integrand)-i/(b-a)) over (a,b)

function subprogram defining the integrand function f(x).

function subprogram defining the integrand weight function w(x).

parameter in the weight function

parameter in the weight function

parameter in the weight function

parameter in the weight function

key for indicating the type of weight function

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 15-point kronrod rule (resk) obtained by optimal addition of abscissae to the 7-point gauss rule (resg).

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

approximation to the integral of abs(f)

approximation to the integral of abs(f-i/(b-a))

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 21-point kronrod rule (resk) obtained by optimal addition of abscissae to the 10-point gauss rule (resg).

estimate of the modulus of the absolute error, which should not exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a)) over (a,b)

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 31-point gauss-kronrod rule (resk), obtained by optimal addition of abscissae to the 15-point gauss rule (resg).

estimate of the modulus of the modulus, which should not exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a)) over (a,b)

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 41-point gauss-kronrod rule (resk) obtained by optimal addition of abscissae to the 20-point gauss rule (resg).

estimate of the modulus of the absolute error, which should not exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a)) over (a,b)

function subroutine defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i. result is computed by applying the 51-point kronrod rule (resk) obtained by optimal addition of abscissae to the 25-point gauss rule (resg).

estimate of the modulus of the absolute error, which should not exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a)) over (a,b)

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

approximation to the integral i result is computed by applying the 61-point kronrod rule (resk) obtained by optimal addition of abscissae to the 30-point gauss rule (resg).

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

approximation to the integral j

approximation to the integral of abs(f-i/(b-a))

parameter in the weight function w(x), alfa>(-1)

parameter in the weight function w(x), beta>(-1)

i(k) is the integral over (-1,1) of (1+x)**alfa*t(k-1,x), k = 1, ..., 25.

rj(k) is the integral over (-1,1) of (1-x)**beta*t(k-1,x), k = 1, ..., 25.

rg(k) is the integral over (-1,1) of (1+x)**alfa*log((1+x)/2)*t(k-1,x), k = 1, ..., 25.

rh(k) is the integral over (-1,1) of (1-x)**beta*log((1-x)/2)*t(k-1,x), k = 1, ..., 25.

input parameter indicating the modified moments to be computed:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration

absolute accuracy requested

relative accuracy requested if epsabs<=0 and epsrel<max(50*rel.mach.acc.,0.5e-28), the routine will end with ier = 6.

approximation to the integral i result is obtained by applying the 21-point gauss-kronrod rule (res21) obtained by optimal addition of abscissae to the 10-point gauss rule (res10), or by applying the 43-point rule (res43) obtained by optimal addition of abscissae to the 21-point gauss-kronrod rule, or by applying the 87-point rule (res87) obtained by optimal addition of abscissae to the 43-point rule.

estimate of the modulus of the absolute error, which should equal or exceed abs(i-result)

number of integrand evaluations

error messages:

function subprogram defining the integrand function f(x).

lower limit of integration.

upper limit of integration.

the value of the integral to the specified relative accuracy.

relative accuracy required. when the relative difference of two successive formulae does not exceed epsil the last formula computed is taken as the result.

number integrand evaluations.

on exit normally icheck=0. however if convergence to the accuracy requested is not achieved icheck=1 on exit.

array of abscissas, which must be in increasing order.

array of function values. i.e., y(i)=func(x(i))

The integer number of function values supplied. N >= 2 unless XLO = XUP.

lower limit of integration

upper limit of integration. Must have XLO <= XUP

computed approximate value of integral

A status code:

function subprogram defining the integrand function f(x).

lower limit of integration

upper limit of integration (may be less than A)

a requested error tolerance. Normally, pick a value 0 < ERR < 1.0e-8.

computed value of the integral. Hopefully, ANS is accurate to within ERR * integral of ABS(FUN(X)).

a status code:

the number of function evaluations actually used to do the integration. A value of K > 1000 indicates a difficult problem; other programs may be more efficient. DQNC79 will gracefully give up if K exceeds 5000.

function subprogram defining the integrand function f(x).

lower bound of the integration

upper bound of the integration

is a requested pseudorelative error tolerance. normally pick a value of abs(error_tol) so that dtol < abs(error_tol) <= 1.0e-3 where dtol is the larger of 1.0e-18and the real unit roundoff d1mach(4). ans will normally have no more error than abs(error_tol) times the integral of the absolute value of f(x). usually, smaller values of error_tol yield more accuracy and require more function evaluations.

computed value of integral

status code:

an estimate of the absolute error in ans. the estimated error is solely for information to the user and should not be used as a correction to the computed integral.

function subprogram defining the integrand function f(x).

lower bound of the integration

upper bound of the integration

relative error tolerance

computed value of integral

status code:

function subprogram defining the integrand function f(x).

lower bound of the integration

upper bound of the integration

relative error tolerance

computed value of integral

status code: