pm_quadPack::setNodeWeightGK Interface Reference

Return the Kronrod \(2n + 1\) nodes and weights of the extension to the \(n\)-point Gauss-Legendre quadrature, as well as the \(n\)-point Gauss-Legendre weights. More...

Detailed Description

Return the Kronrod \(2n + 1\) nodes and weights of the extension to the \(n\)-point Gauss-Legendre quadrature, as well as the \(n\)-point Gauss-Legendre weights.

The abscissas and weights for both the Gauss and Kronrod rules are computed for integration over the interval \([-1,+1]\).
Also, only the nonnegative abscissas are computed because the quadrature formulae is symmetric with respect to the origin.
See the documentation of pm_quadPack for more details on the Gauss-Kronrod quadrature.

Parameters

[in]	order	: The input scalar integer of default kind IK containing the Number of Points in the Gauss quadrature rule. (optional. If present, then the rest of the (output vector) arguments must be all allocatable.)
[out]	nodeK	: The output contiguous or allocatable vector of size (1 : n + 1) of type real of kind any supported by the processor (e.g., RK, RK32, RK64, or RK128), containing the Gauss-Kronrod nodes (abscissas). If the input argument order is present, then Node must be an allocatable vector that will be reallocated to order+1 on output. If the input argument order is missing, then Node must be a contiguous vector of size order + 1, where order is the desired number of points in the Gauss rule.
[out]	weightK	: The output contiguous or allocatable vector of the same type, kind, and size as nodeK containing the Gauss-Kronrod optimal extension weights. If the input argument order is present, then weightK must be an allocatable vector that will be reallocated to order+1 on output. If the input argument order is missing, then weightK must be a contiguous vector of size order + 1, where order is the desired number of points in the Gauss rule.
[out]	weightG	: The output contiguous or allocatable vector of size (size(nodeK) / 2), of the same type and kind as nodeK containing the Gauss-Legendre weights. By definition, the elements of weightG(:) correspond to the node elements nodeK(2::2). If the input argument order is present, then weightG must be an allocatable vector that will be reallocated to (order+1)/2 on output. If the input argument order is missing, then weightG must be a contiguous vector of size order + 1, where order is the desired number of points in the Gauss rule.

Possible calling interfaces ⛓

use pm_quadPack, only: setNodeWeightGK
 
call setNodeWeightGK(nodeK, weightK, weightG)

Warning

The number of points \(n\) in the Gauss quadrature rule is determined from the size of the specified nodeK argument as n = size(nodeK) - 1.
As such, the sizes of weightK and weightG must be size(nodeK) - 1 and size(nodeK) / 2 respectively.
These conditions are verified only if the library is built with the preprocessor macro CHECK_ENABLED=1.

Remarks

The procedures under discussion are impure.

See also

Robert Piessens, Maria Branders, 1974, A Note on the Optimal Addition of Abscissas to Quadrature Formulas of Gauss and Lobatto, Mathematics of Computation, 28, 125, 135-139.
Laurie, 1997, Calculation of gauss-kronrod quadrature rules.
Kronrod, 1965, Nodes and weights of quadrature formulas.
FORTRAN90 version by John Burkardt.

Example usage ⛓

program example
 
    use pm_kind, only: SK, IK, RKG => RKH ! testing with highest real precision available. all other real kinds are also supported.
    use pm_quadpack, only: nodeK15, weightK15, weightG7
    use pm_quadpack, only: nodeK21, weightK21, weightG10
    use pm_quadpack, only: nodeK31, weightK31, weightG15
    use pm_quadpack, only: nodeK41, weightK41, weightG20
    use pm_quadpack, only: nodeK51, weightK51, weightG25
    use pm_quadpack, only: nodeK61, weightK61, weightG30
    use pm_quadpack, only: setNodeWeightGK
    use pm_io, only: display_type
 
    implicit none
 
    integer(IK) , parameter     :: MAX_NPG = 30_IK  !   MAXimum Number of Points in the Gauss quadrature rule considered in this example.
    real(RKG)                   :: nodeK(MAX_NPG+1), weightK(MAX_NPG+1), weightG(MAX_NPG+1)
    real(RKG)   , allocatable   :: refNodeK(:), refWeightK(:), refWeightG(:)
 
    type(display_type) :: disp
    disp = display_type(file = "main.out.F90")
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 7-15 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG7, RKG)
    refWeightK = real(weightK15, RKG)
    refNodeK = real(nodeK15, RKG)
    call setNodeWeight(npg = 7)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 10-21 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG10, RKG)
    refWeightK = real(weightK21, RKG)
    refNodeK = real(nodeK21, RKG)
    call setNodeWeight(npg = 10)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 15-31 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG15, RKG)
    refWeightK = real(weightK31, RKG)
    refNodeK = real(nodeK31, RKG)
    call setNodeWeight(npg = 15)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 20-41 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG20, RKG)
    refWeightK = real(weightK41, RKG)
    refNodeK = real(nodeK41, RKG)
    call setNodeWeight(npg = 20)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 25-51 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG25, RKG)
    refWeightK = real(weightK51, RKG)
    refNodeK = real(nodeK51, RKG)
    call setNodeWeight(npg = 25)
 
    call disp%skip()
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%show("! Compute the 30-61 Gauss-Kronrod nodes and weights.")
    call disp%show("!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%")
    call disp%skip()
 
    refWeightG = real(weightG30, RKG)
    refWeightK = real(weightK61, RKG)
    refNodeK = real(nodeK61, RKG)
    call setNodeWeight(npg = 30)
 
contains
 
    subroutine setNodeWeight(npg)
        use pm_val2str, only: getStr
        integer(IK) , intent(in)    :: npg  !   number of points in the Gauss quadrature rule.
        integer(IK)                 :: npk  !   number of extra points by the Kronrod extension rule ( = # points in the Gauss rule + 1).
        integer(IK)                 :: i,j
        call disp%skip()
        call disp%show("npk = "//getStr(npg)//" + 1")
                        npk = npg + 1_IK
        call disp%show("call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))")
                        call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
        call disp%show("[nodeK, weightK, weightG]")
        do i = 1, npk
            if (mod(i, 2_IK) == 0_IK) then
                call disp%show( [nodeK(i), weightK(i), weightG(i/2)] )
            else
                call disp%show( [nodeK(i), weightK(i)] )
            end if
        end do
        call disp%skip()
        call disp%show("[nodeK - exact, weightK - exact, weightG - exact]")
        do i = 1, npk
            j = npk - i + 1
            if (mod(i, 2_IK) == 0_IK) then
                call disp%show( [nodeK(i) - refNodeK(j), weightK(i) - refWeightK(j), weightG(i/2) - refWeightG(size(refWeightG) - i/2 + 1)] )
            else
                call disp%show( [nodeK(i) - refNodeK(j), weightK(i) - refWeightK(j)] )
            end if
        end do
        call disp%skip()
        deallocate(refNodeK, refWeightG, refWeightK)
    end subroutine
 
end program example

Example Unix compile command via Intel ifort compiler ⛓

#!/usr/bin/env sh
rm main.exe
ifort -fpp -standard-semantics -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example Windows Batch compile command via Intel ifort compiler ⛓

del main.exe
set PATH=..\..\..\lib;%PATH%
ifort /fpp /standard-semantics /O3 /I:..\..\..\include main.F90 ..\..\..\lib\libparamonte*.lib /exe:main.exe
main.exe

Example Unix / MinGW compile command via GNU gfortran compiler ⛓

#!/usr/bin/env sh
rm main.exe
gfortran -cpp -ffree-line-length-none -O3 -Wl,-rpath,../../../lib -I../../../inc main.F90 ../../../lib/libparamonte* -o main.exe
./main.exe

Example output ⛓

 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 7-15 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 7 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.991455371120812639206854697526328534, +0.229353220105292249637320080589695566E-1
+0.949107912342758524526189684047851219, +0.630920926299785532907006631892035206E-1, +0.129484966168869693270611432679081161
+0.864864423359769072789712788640926167, +0.104790010322250183839876322541518041
+0.741531185599394439863864773280788426, +0.140653259715525918745189590510237726, +0.279705391489276667901467771423779330
+0.586087235467691130294144838258729596, +0.169004726639267902826583426598550208
+0.405845151377397166906606412076961454, +0.190350578064785409913256402421013653, +0.381830050505118944950369775488975074
+0.207784955007898467600689403773244930, +0.204432940075298892414161999234648970
+0.00000000000000000000000000000000000, +0.209482141084727828012999174891714262, +0.417959183673469387755102040816326510
 
[nodeK - exact, weightK - exact, weightG - exact]
+0.00000000000000000000000000000000000, -0.361111864572606722447995864234673872E-34
+0.00000000000000000000000000000000000, -0.770371977754894341222391177033970927E-33, -0.866668474974256133875190074163217293E-33
+0.00000000000000000000000000000000000, +0.240741243048404481631997242823115915E-34
+0.00000000000000000000000000000000000, -0.192592994438723585305597794258492732E-33, -0.240741243048404481631997242823115915E-33
+0.00000000000000000000000000000000000, -0.722223729145213444895991728469347744E-34
+0.00000000000000000000000000000000000, -0.240741243048404481631997242823115915E-34, -0.481482486096808963263994485646231830E-34
+0.240741243048404481631997242823115915E-34, -0.120370621524202240815998621411557957E-33
+0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 10-21 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 10 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.995657163025808080735527280689002850, +0.116946388673718742780643960621920455E-1
+0.973906528517171720077964012084452104, +0.325581623079647274788189724593892905E-1, +0.666713443086881375935688098933313063E-1
+0.930157491355708226001207180059508367, +0.547558965743519960313813002445801872E-1
+0.865063366688984510732096688423493068, +0.750396748109199527670431409161898610E-1, +0.149451349150580593145776339657697151
+0.780817726586416897063717578345042409, +0.931254545836976055350654650833663122E-1
+0.679409568299024406234327365114873548, +0.109387158802297641899210590325804986, +0.219086362515982043995534934228163250
+0.562757134668604683339000099272694155, +0.123491976262065851077958109831074157
+0.433395394129247190799265943165784184, +0.134709217311473325928054001771706780, +0.269266719309996355091226921569469300
+0.294392862701460198131126603103865508, +0.142775938577060080797094273138717120
+0.148874338981631210884826001129719980, +0.147739104901338491374841515972067985, +0.295524224714752870173892994651338306
+0.00000000000000000000000000000000000, +0.149445554002916905664936468389821279
 
[nodeK - exact, weightK - exact, weightG - exact]
+0.00000000000000000000000000000000000, -0.300926553810505602039996553528894894E-35
+0.962964972193617926527988971292463659E-34, -0.469445423944388739182394623505076034E-33, -0.481482486096808963263994485646231830E-33
+0.00000000000000000000000000000000000, +0.120370621524202240815998621411557957E-34
+0.00000000000000000000000000000000000, -0.144444745829042688979198345693869549E-33, -0.192592994438723585305597794258492732E-33
+0.00000000000000000000000000000000000, -0.361111864572606722447995864234673872E-34
+0.00000000000000000000000000000000000, +0.240741243048404481631997242823115915E-34, +0.481482486096808963263994485646231830E-34
+0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
+0.00000000000000000000000000000000000, -0.481482486096808963263994485646231830E-34, -0.481482486096808963263994485646231830E-34
-0.481482486096808963263994485646231830E-34, +0.481482486096808963263994485646231830E-34
+0.00000000000000000000000000000000000, -0.722223729145213444895991728469347744E-34, +0.00000000000000000000000000000000000
+0.00000000000000000000000000000000000, +0.722223729145213444895991728469347744E-34
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 15-31 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 15 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.998002298693397060285172840152271212, +0.537747987292334898779205143012766868E-2
+0.987992518020485428489565718586612614, +0.150079473293161225383747630758065355E-1, +0.307532419961172683546283935772036556E-1
+0.967739075679139134257347978784337170, +0.254608473267153201868740010196533869E-1
+0.937273392400705904307758947710209522, +0.353463607913758462220379484783596456E-1, +0.703660474881081247092674164506669230E-1
+0.897264532344081900882509656454495847, +0.445897513247648766082272993732796876E-1
+0.848206583410427216200648320774216849, +0.534815246909280872653431472394303324E-1, +0.107159220467171935011869546685869334
+0.790418501442465932967649294817947367, +0.620095678006706402851392309608029347E-1
+0.724417731360170047416186054613938033, +0.698541213187282587095200770991473795E-1, +0.139570677926154314447804794511028232
+0.650996741297416970533735895313274663, +0.768496807577203788944327774826590358E-1
+0.570972172608538847537226737253910681, +0.830805028231330210382892472861039900E-1, +0.166269205816993933553200860481209062
+0.485081863640239680693655740232350637, +0.885644430562117706472754436937743237E-1
+0.394151347077563369897207370981045485, +0.931265981708253212254868727473457423E-1, +0.186161000015562211026800561866422892
+0.299180007153168812166780024266388908, +0.966427269836236785051799076275893564E-1
+0.201194093997434522300628303394596212, +0.991735987217919593323931734846031084E-1, +0.198431485327111576456118326443839338
+0.101142066918717499027074231447392276, +0.100769845523875595044946662617569744
+0.00000000000000000000000000000000000, +0.101330007014791549017374792767492326, +0.202578241925561272880620199967519315
 
[nodeK - exact, weightK - exact, weightG - exact]
+0.00000000000000000000000000000000000, +0.188079096131566001274997845955559308E-34
+0.00000000000000000000000000000000000, -0.732756158528581140967391607842859066E-33, -0.761344181140579173161191280428104081E-33
-0.962964972193617926527988971292463659E-34, +0.270833898429455041835996898176005404E-34
+0.962964972193617926527988971292463659E-34, -0.403241582106077506733595381728719157E-33, -0.421297175334707842855995174940452851E-33
+0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
+0.00000000000000000000000000000000000, +0.361111864572606722447995864234673872E-34, +0.361111864572606722447995864234673872E-34
+0.00000000000000000000000000000000000, +0.00000000000000000000000000000000000
+0.00000000000000000000000000000000000, -0.962964972193617926527988971292463659E-34, -0.962964972193617926527988971292463659E-34
+0.00000000000000000000000000000000000, +0.240741243048404481631997242823115915E-34
+0.00000000000000000000000000000000000, +0.204630056591143809387197656399648528E-33, +0.240741243048404481631997242823115915E-33
+0.00000000000000000000000000000000000, +0.240741243048404481631997242823115915E-34
+0.00000000000000000000000000000000000, +0.240741243048404481631997242823115915E-34, +0.722223729145213444895991728469347744E-34
-0.481482486096808963263994485646231830E-34, +0.240741243048404481631997242823115915E-34
+0.00000000000000000000000000000000000, -0.240741243048404481631997242823115915E-34, +0.240741243048404481631997242823115915E-34
-0.601853107621011204079993107057789787E-34, +0.240741243048404481631997242823115915E-34
+0.00000000000000000000000000000000000, -0.216667118743564033468797518540804323E-33, +0.00000000000000000000000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 20-41 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 20 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.998859031588277663838315576545863027, +0.307358371852053150121829324603096036E-2
+0.993128599185094924786122388471320311, +0.860026985564294219866178795010156739E-2, +0.176140071391521183118619623518520174E-1
+0.981507877450250259193342994720216953, +0.146261692569712529837879603088683438E-1
+0.963971927277913791267666131197277245, +0.203883734612665235980102314327544473E-1, +0.406014298003869413310399522749318474E-1
+0.940822633831754753519982722212443384, +0.258821336049511588345050670961531257E-1
+0.912234428251325905867752441203298155, +0.312873067770327989585431193238004526E-1, +0.626720483341090635695065351870413037E-1
+0.878276811252281976077442995113078476, +0.366001697582007980305572407072109909E-1
+0.839116971822218823394529061701520649, +0.416688733279736862637883059368949448E-1, +0.832767415767047487247581432220464148E-1
+0.795041428837551198350638833272787951, +0.464348218674976747202318809261075123E-1
+0.746331906460150792614305070355641587, +0.509445739237286919327076700503450663E-1, +0.101930119817240435036750135480350030
+0.693237656334751384805490711845931522, +0.551951053482859947448323724197773487E-1
+0.636053680726515025452836696226285966, +0.591114008806395723749672206485942313E-1, +0.118194531961518417312377377711382301
+0.575140446819710315342946036586425083, +0.626532375547811680258701221742549883E-1
+0.510867001950827098004364050955250976, +0.658345971336184221115635569693980091E-1, +0.131688638449176626898494499748163180
+0.443593175238725103199992213492640121, +0.686486729285216193456234118853677711E-1
+0.373706088715419560672548177024927261, +0.710544235534440683057903617232102466E-1, +0.142096109318382051329298325067165001
+0.301627868114913004320555356858592298, +0.730306903327866674951894176589131234E-1
+0.227785851141645078080496195368574615, +0.745828754004991889865814183624874938E-1, +0.149172986472603746787828737001969411
+0.152605465240922675505220241022677516, +0.757044976845566746595427753766165649E-1
+0.765265211334973337546404093988382039E-1, +0.763778676720807367055028350380610118E-1, +0.152753387130725850698084331955097591
+0.00000000000000000000000000000000000, +0.766007119179996564450499015301016947E-1
 
[nodeK - exact, weightK - exact, weightG - exact]
+0.00000000000000000000000000000000000, -0.270833898429455041835996898176005404E-34
+0.00000000000000000000000000000000000, -0.779399774369209509283591073639837774E-33, -0.800464633135944901426390832386860417E-33
+0.00000000000000000000000000000000000, -0.120370621524202240815998621411557957E-34
+0.00000000000000000000000000000000000, -0.258796836277034817754397036034849608E-33, -0.264815367353244929795196967105427506E-33
+0.00000000000000000000000000000000000, -0.180555932286303361223997932117336936E-34
+0.00000000000000000000000000000000000, -0.282870960581875265917596760317161200E-33, -0.300926553810505602039996553528894894E-33
+0.00000000000000000000000000000000000, -0.180555932286303361223997932117336936E-34
+0.00000000000000000000000000000000000, +0.204630056591143809387197656399648528E-33, +0.204630056591143809387197656399648528E-33
+0.00000000000000000000000000000000000, -0.601853107621011204079993107057789787E-35
+0.00000000000000000000000000000000000, +0.120370621524202240815998621411557957E-33, +0.156481807981462913060798207835025345E-33
+0.00000000000000000000000000000000000, +0.180555932286303361223997932117336936E-34
+0.00000000000000000000000000000000000, +0.120370621524202240815998621411557957E-34, +0.120370621524202240815998621411557957E-34
-0.962964972193617926527988971292463659E-34, +0.120370621524202240815998621411557957E-34
+0.00000000000000000000000000000000000, +0.601853107621011204079993107057789787E-34, +0.481482486096808963263994485646231830E-34
+0.00000000000000000000000000000000000, -0.361111864572606722447995864234673872E-34
+0.00000000000000000000000000000000000, +0.842594350669415685711990349880905702E-34, +0.722223729145213444895991728469347744E-34
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-0.120370621524202240815998621411557957E-34, +0.120370621524202240815998621411557957E-34, +0.00000000000000000000000000000000000
+0.00000000000000000000000000000000000, -0.481482486096808963263994485646231830E-34
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 25-51 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 25 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.999262104992609834193457486540340575, +0.198738389233031592650785188284342961E-2
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+0.00000000000000000000000000000000000, +0.615808180678329350787598242400646323E-1, +0.123176053726715451203902873079050139
 
[nodeK - exact, weightK - exact, weightG - exact]
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-0.962964972193617926527988971292463659E-34, -0.180555932286303361223997932117336936E-34
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-0.481482486096808963263994485646231830E-34, -0.180555932286303361223997932117336936E-34
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-0.240741243048404481631997242823115915E-34, +0.601853107621011204079993107057789787E-35
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+0.00000000000000000000000000000000000, +0.782409039907314565303991039175126723E-34, +0.00000000000000000000000000000000000
 
 
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
! Compute the 30-61 Gauss-Kronrod nodes and weights.
!%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 
 
npk = 30 + 1
call setNodeWeightGK(nodeK(1:npk), weightK(1:npk), weightG(1:npk/2))
[nodeK, weightK, weightG]
+0.999484410050490637571325895705810795, +0.138901369867700762455159122675971330E-2
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+0.492480467861778574993693061207708789, +0.448148001331626631923555516167232373E-1
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+0.400401254830394392535476211542660634, +0.471855465692991539452614781810994885E-1
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+0.254636926167889846439805129817805114, +0.497956834270742063578115693799423231E-1, +0.995934205867952670627802821035694751E-1
+0.204525116682309891438957671002024697, +0.504059214027823468408930856535850308E-1
+0.153869913608583546963794672743255924, +0.508817958987496064922974730498046820E-1, +0.101762389748405504596428952168553985
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+0.00000000000000000000000000000000000, +0.514947294294515675583404336470992344E-1
 
[nodeK - exact, weightK - exact, weightG - exact]
+0.00000000000000000000000000000000000, +0.135416949214727520917998449088002702E-34
+0.962964972193617926527988971292463659E-34, -0.179728384263324470818387941595132475E-32, -0.184467977485839934050517887313212570E-32
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-0.240741243048404481631997242823115915E-34, -0.180555932286303361223997932117336936E-34
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+0.00000000000000000000000000000000000, -0.722223729145213444895991728469347744E-34
 
 

Test:

test_pm_quadPack

Final Remarks ⛓

---

If you believe this algorithm or its documentation can be improved, we appreciate your contribution and help to edit this page's documentation and source file on GitHub.
For details on the naming abbreviations, see this page.
For details on the naming conventions, see this page.
This software is distributed under the MIT license with additional terms outlined below.

1. If you use any parts or concepts from this library to any extent, please acknowledge the usage by citing the relevant publications of the ParaMonte library.
   
2. If you regenerate any parts/ideas from this library in a programming environment other than those currently supported by this ParaMonte library (i.e., other than C, C++, Fortran, MATLAB, Python, R), please also ask the end users to cite this original ParaMonte library.
   

This software is available to the public under a highly permissive license.
Help us justify its continued development and maintenance by acknowledging its benefit to society, distributing it, and contributing to it.

Copyright

Computational Data Science Lab

Author:

Amir Shahmoradi, Oct 16, 2009, 11:14 AM, Michigan

Definition at line 2036 of file pm_quadPack.F90.

---

The documentation for this interface was generated from the following file:

• src/fortran/main/pm_quadPack.F90