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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
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ParaMonte Fortran 2.0.0
Parallel Monte Carlo and Machine Learning Library
See the latest version documentation. |
This module contains classes and procedures for non-adaptive and adaptive global numerical quadrature and Cauchy Principal Value of 1D functions with various types of singularities and points of difficulties via the Gauss-Kronrod and Clenshaw-Curtis quadrature formulae.
More...
Data Types | |
| interface | getQuadErr |
Compute the 1D integral of the input scalar (potentially singular) integrand getFunc on a finite or semi/fully-infinite interval (a, b) and estimate its absolute error via the requested adaptive global Gauss-Kronrod (GK) extension rules. More... | |
| interface | getQuadGK |
Compute the 1D integral of the input scalar integrand getFunc on the finite or infinite interval [a, b] and estimate its absolute error via the requested non-adaptive Gauss-Kronrod (GK) extension rule. More... | |
| type | GK15_type |
| This is the indicator type for generating instances of objects that indicate the use of 7-point Gauss-Legendre quadrature with 15-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| type | GK21_type |
| This is the indicator type for generating instances of objects that indicate the use of 10-point Gauss-Legendre quadrature with 11-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| type | GK31_type |
| This is the indicator type for generating instances of objects that indicate the use of 15-point Gauss-Legendre quadrature with 31-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| type | GK41_type |
| This is the indicator type for generating instances of objects that indicate the use of 20-point Gauss-Legendre quadrature with 41-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| type | GK51_type |
| This is the indicator type for generating instances of objects that indicate the use of 25-point Gauss-Legendre quadrature with 51-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| type | GK61_type |
| This is the indicator type for generating instances of objects that indicate the use of 30-point Gauss-Legendre quadrature with 61-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| interface | isFailedQuad |
Compute the 1D integral of the input scalar (potentially singular) integrand getFunc on a finite or semi/fully-infinite interval (a, b) and estimate its absolute error via the requested adaptive global quadrature rule. More... | |
| interface | setChebExpan |
Compute and return the series expansion of the input function values via the Chebyshev polynomials of the first kind of degrees 12 and 24 using Fast Fourier Transform method. More... | |
| interface | setErrSorted |
| Sort the input local error estimates list (resulting from the interval subdivision process of the Adaptive Global quadrature procedures of this module) in descending order. More... | |
| interface | setNodeWeightGK |
| 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... | |
| interface | setSeqLimEps |
| Return the limit of a given sequence of approximations via the Epsilon method of Wynn (1961). More... | |
| interface | wcauchy_type |
This is the derived type for constructing objects that signify the computation of the Cauchy Principal Value of arbitrary function whose Cauchy singularities is stored in the cs components of the objects of this type.More... | |
| interface | wcos_type |
| This is the derived type for constructing objects that signify the computation of the integral an arbitrary Cose-weighted function. More... | |
| type | weps_type |
| This is the indicator type for generating instances of objects that indicate the use of Epsilon extrapolation method of Wynn (1961) to accelerate the convergence of integrations of functions with integrable singularities and infinities using the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadErr. More... | |
| interface | wsin_type |
| This is the derived type for constructing objects that signify the computation of the integral an arbitrary Sine-weighted function. More... | |
Variables | |
| character(*, SK), parameter | MODULE_NAME = "@pm_quadPack" |
| type(GK15_type), parameter | GK15 = GK15_type() |
| The scalar constant object of type GK15_type that indicates the use of 7-point Gauss-Legendre quadrature with 15-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(4), parameter | nodeG7 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 4.0584515137739716690660641207696146334738201409937012638704325179466381322612565532831268972774658776528675866604802e-01_RKB , 7.4153118559939443986386477328078840707414764714139026011995535196742987467218051379282683236686324705969251809311201e-01_RKB , 9.4910791234275852452618968404785126240077093767061778354876910391306333035484014080573077002792572414430073966699522e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 7-points Gauss quadrature rule. More... | |
| real(RKB), dimension(4), parameter | weightG7 = [ 4.1795918367346938775510204081632653061224489795918367346938775510204081632653061224489795918367346938775510204081633e-01_RKB , 3.8183005050511894495036977548897513387836508353386273475108345103070554643412970834868465934404480145031467176458536e-01_RKB , 2.7970539148927666790146777142377958248692506522659876453701403269361881043056267681324094290119761876632337521337205e-01_RKB , 1.2948496616886969327061143267908201832858740225994666397720863872465523497204230871562541816292084508948440200163443e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 7-points Gauss quadrature rule. More... | |
| real(RKB), dimension(8), parameter | nodeK15 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 2.0778495500789846760068940377324491347978440714517064971384573461986693844943520226910343227183698530560857645062738e-01_RKB , 4.0584515137739716690660641207696146334738201409937012638704325179466381322612565532831268972774658776528675866604802e-01_RKB , 5.8608723546769113029414483825872959843678075060436095130499289319880373607444407464511674498935942098956811555121368e-01_RKB , 7.4153118559939443986386477328078840707414764714139026011995535196742987467218051379282683236686324705969251809311201e-01_RKB , 8.6486442335976907278971278864092620121097230707408814860145771276706770813259572103585847859604590541475281326027862e-01_RKB , 9.4910791234275852452618968404785126240077093767061778354876910391306333035484014080573077002792572414430073966699522e-01_RKB , 9.9145537112081263920685469752632851664204433837033470129108741357244173934653407235924503509626841760744349505339308e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 15-points Kronrod extension to the 7-points Gauss quadrature rule. More... | |
| real(RKB), dimension(8), parameter | weightK15 = [ 2.0948214108472782801299917489171426369776208022370431671299800656137515132325648616816908211675949102392971459688215e-01_RKB , 2.0443294007529889241416199923464908471651760418071835742447095312045467698546598879348374292009347554167803659293064e-01_RKB , 1.9035057806478540991325640242101368282607807545535835588544088036744058072410212679605964605106377593834568683551139e-01_RKB , 1.6900472663926790282658342659855028410624490030294424149734006755695680921619029112936702403855359908156070095656537e-01_RKB , 1.4065325971552591874518959051023792039988975724799857556174546893312708093090950408097379122415555910759700350860143e-01_RKB , 1.0479001032225018383987632254151801744375665421383061189339065133963746321576289524167571627509311333949422518201492e-01_RKB , 6.3092092629978553290700663189204286665071157211550707113605545146983997477964874928199170264504441995865872491871943e-02_RKB , 2.2935322010529224963732008058969591993560811275746992267507430254711815787976075946156368168156289483493617134063245e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 15-points Kronrod extension to the 7-points Gauss quadrature rule. More... | |
| type(GK21_type), parameter | GK21 = GK21_type() |
| The scalar constant object of type GK21_type that indicates the use of 10-point Gauss-Legendre quadrature with 21-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(5), parameter | nodeG10 = [ 1.4887433898163121088482600112971998461756485942069169570798925351590361735566852137117762979946369123003116080525534e-01_RKB , 4.3339539412924719079926594316578416220007183765624649650270151314376698907770350122510275795011772122368293504099894e-01_RKB , 6.7940956829902440623432736511487357576929471183480946766481718895255857539507492461507857357048037949983390204739932e-01_RKB , 8.6506336668898451073209668842349304852754301496533045252195973184537475513805556135679072894604577069440463108641177e-01_RKB , 9.7390652851717172007796401208445205342826994669238211923121206669659520323463615962572356495626855625823304251877421e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 10-points Gauss quadrature rule. More... | |
| real(RKB), dimension(5), parameter | weightG10 = [ 2.9552422471475287017389299465133832942104671702685360135430802975599593821715232927035659579375421672271716440125256e-01_RKB , 2.6926671930999635509122692156946935285975993846088379580056327624215343231917927676422663670925276075559581145036870e-01_RKB , 2.1908636251598204399553493422816319245877187052267708988095654363519991065295128124268399317720219278659121687281289e-01_RKB , 1.4945134915058059314577633965769733240255663966942736783547726875323865472663001094594726463473195191400575256104544e-01_RKB , 6.6671344308688137593568809893331792857864834320158145128694881613412064084087101776785509685058877821090054714520419e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 10-points Gauss quadrature rule. More... | |
| real(RKB), dimension(11), parameter | nodeK21 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.4887433898163121088482600112971998461756485942069169570798925351590361735566852137117762979946369123003116080525534e-01_RKB , 2.9439286270146019813112660310386556616268662515695791864888229172724611166332737888445523178268237359119185139299872e-01_RKB , 4.3339539412924719079926594316578416220007183765624649650270151314376698907770350122510275795011772122368293504099894e-01_RKB , 5.6275713466860468333900009927269414084301388194196695886034621458779266353216327549712087854169992422106448211158815e-01_RKB , 6.7940956829902440623432736511487357576929471183480946766481718895255857539507492461507857357048037949983390204739932e-01_RKB , 7.8081772658641689706371757834504237716340752029815717974694859999505607982761420654526977234238996241110129779403362e-01_RKB , 8.6506336668898451073209668842349304852754301496533045252195973184537475513805556135679072894604577069440463108641177e-01_RKB , 9.3015749135570822600120718005950834622516790998193924230349406866828415983091673055011194572851007884702013619684320e-01_RKB , 9.7390652851717172007796401208445205342826994669238211923121206669659520323463615962572356495626855625823304251877421e-01_RKB , 9.9565716302580808073552728068900284792126058721947892436337916111757023046774867357152325996912076724298149077812671e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 21-points Kronrod extension to the 10-points Gauss quadrature rule. More... | |
| real(RKB), dimension(11), parameter | weightK21 = [ 1.4944555400291690566493646838982120374523631668747280383560851873698964478511841925721030705689540264726493367634340e-01_RKB , 1.4773910490133849137484151597206804552373162548520660451819195439885993016735696405732703959182882254268727823258502e-01_RKB , 1.4277593857706008079709427313871706088597905653190555560741004743970770449909340027811131706283756428281146832304737e-01_RKB , 1.3470921731147332592805400177170683276099191300855971406636668491320291400121282036676953159488271772384389604997640e-01_RKB , 1.2349197626206585107795810983107415951230034952864832764467994120974054238975454689681538622363738230836484113389878e-01_RKB , 1.0938715880229764189921059032580496027181329983434522007819675829826550372891432168683899432674553842507906611591517e-01_RKB , 9.3125454583697605535065465083366344390018828880760031970085038760177735672200775237414123061615827474831165614953012e-02_RKB , 7.5039674810919952767043140916190009395219382000910088173697048048430404342858495178813808730646554086856929327903059e-02_RKB , 5.4755896574351996031381300244580176373721114058333557524432615804784098927818975325116301569003298086458722055550981e-02_RKB , 3.2558162307964727478818972459389760617388939845662609571537504232714121820165498692381607605384626494546068817765276e-02_RKB , 1.1694638867371874278064396062192048396217332481931888927598147525622222058064992651806736704969967250888097490233242e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 21-points Kronrod extension to the 10-points Gauss quadrature rule. More... | |
| type(GK31_type), parameter | GK31 = GK31_type() |
| The scalar constant object of type GK31_type that indicates the use of 15-point Gauss-Legendre quadrature with 31-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(8), parameter | nodeG15 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 2.0119409399743452230062830339459620781283645446263767961594972460994823900302018760183625806752105908967902257386509e-01_RKB , 3.9415134707756336989720737098104546836275277615869825503116534395160895778696141797549711416165976202589352169635648e-01_RKB , 5.7097217260853884753722673725391064123838639628274960485326541705419537986975857948341462856982614477912646497026257e-01_RKB , 7.2441773136017004741618605461393800963089929458410256355142342070412378167792521899610109760313432626923598549381925e-01_RKB , 8.4820658341042721620064832077421685136625617473699263409572755876067507517414548519760771975082148085090373835713340e-01_RKB , 9.3727339240070590430775894771020947124399627351530445790136307635020297379704552795054758617426808659746824044603157e-01_RKB , 9.8799251802048542848956571858661258114697281712376148999999751558738843736901942471272205036831914497667516843990079e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 15-points Gauss quadrature rule. More... | |
| real(RKB), dimension(8), parameter | weightG15 = [ 2.0257824192556127288062019996751931483866215800947735679670411605143539875474607409339344071278803213535148267082999e-01_RKB , 1.9843148532711157645611832644383932481869255995754199348473792792912479753343426813331499916481782320766020854889310e-01_RKB , 1.8616100001556221102680056186642282450622601227792840281549572731001325550269916061894976888609932360539977709001384e-01_RKB , 1.6626920581699393355320086048120881113090018009841290732186519056355356321227851771070517429241553621484461540657185e-01_RKB , 1.3957067792615431444780479451102832252085027531551124320239112863108844454190781168076825736357133363814908889327664e-01_RKB , 1.0715922046717193501186954668586930341554371575810198068702238912187799485231579972568585713760862404439808767837506e-01_RKB , 7.0366047488108124709267416450667338466708032754330719825907292914387055512874237044840452066693939219355489858595041e-02_RKB , 3.0753241996117268354628393577204417721748144833434074264228285504237189467117168039038770732399404002516991188859473e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 15-points Gauss quadrature rule. More... | |
| real(RKB), dimension(16), parameter | nodeK31 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.0114206691871749902707423144739233878745105740164180495800189504151097862454083050931321451540380998341273193681967e-01_RKB , 2.0119409399743452230062830339459620781283645446263767961594972460994823900302018760183625806752105908967902257386509e-01_RKB , 2.9918000715316881216678002426638896266160338274382080184125545738918081102513884467602322020157243563662094470221235e-01_RKB , 3.9415134707756336989720737098104546836275277615869825503116534395160895778696141797549711416165976202589352169635648e-01_RKB , 4.8508186364023968069365574023235061286633893089407312129367943604080239955167155974371848690848595275551258416303565e-01_RKB , 5.7097217260853884753722673725391064123838639628274960485326541705419537986975857948341462856982614477912646497026257e-01_RKB , 6.5099674129741697053373589531327469254694822609259966708966160576093305841043840794460394747228060367236079289132544e-01_RKB , 7.2441773136017004741618605461393800963089929458410256355142342070412378167792521899610109760313432626923598549381925e-01_RKB , 7.9041850144246593296764929481794734686214051995697617332365280643308302974631807059994738664225445530963711137343440e-01_RKB , 8.4820658341042721620064832077421685136625617473699263409572755876067507517414548519760771975082148085090373835713340e-01_RKB , 8.9726453234408190088250965645449588283177871149442786763972687601078537721473771221195399661919716123038835639691946e-01_RKB , 9.3727339240070590430775894771020947124399627351530445790136307635020297379704552795054758617426808659746824044603157e-01_RKB , 9.6773907567913913425734797878433722528335733730013163797468062226335804249452174804319385048203118506304424717089291e-01_RKB , 9.8799251802048542848956571858661258114697281712376148999999751558738843736901942471272205036831914497667516843990079e-01_RKB , 9.9800229869339706028517284015227120907340644231555723034839427970683348682837134566648979907760125278631896777136104e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 31-points Kronrod extension to the 15-points Gauss quadrature rule. More... | |
| real(RKB), dimension(16), parameter | weightK31 = [ 1.0133000701479154901737479276749254677092627259659629246734858372174107615774696665932418050683956749891773195816338e-01_RKB , 1.0076984552387559504494666261756972191634838013536373069278929029488122760822761077475060185965408326901925180106227e-01_RKB , 9.9173598721791959332393173484603131059567260816713281734860095693651563064308745717056680128223790739026832596087552e-02_RKB , 9.6642726983623678505179907627589335136656568630495198973407668882934392359962841826511402504664592185391687490319950e-02_RKB , 9.3126598170825321225486872747345718561927881321317330560285879189052002874531855060114908990458716740695847509343865e-02_RKB , 8.8564443056211770647275443693774303212266732690655967817996052574877144544749814260718837576325109922207832119243346e-02_RKB , 8.3080502823133021038289247286103789601554188253368717607281604875233630643885056057630789228337088859687986285569521e-02_RKB , 7.6849680757720378894432777482659006722109101167947000584089097112470821092034084418224731527690291913686588446455555e-02_RKB , 6.9854121318728258709520077099147475786045435140671549698798093177992675624987998849748628778570667518643649536771245e-02_RKB , 6.2009567800670640285139230960802932190400004210329723569147829395618376206272317333030584268303808639229575334680414e-02_RKB , 5.3481524690928087265343147239430296771554760947116739813222888752727413616259625439714812475198987513183153639571249e-02_RKB , 4.4589751324764876608227299373279690223256649667921096570980823211805450700059906366455036418897149593261561551176267e-02_RKB , 3.5346360791375846222037948478360048122630678992420820868148023340902501837247680978434662724296810081131106317333086e-02_RKB , 2.5460847326715320186874001019653359397271745046864640508377984982400903447009185267605205778819712848080691366407461e-02_RKB , 1.5007947329316122538374763075807268094639436437387634979291759700896494746154334398961710227490402528151677469993935e-02_RKB , 5.3774798729233489877920514301276498183080402431284197876486169536848635554354599213793172596490038991436925569025913e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 31-points Kronrod extension to the 15-points Gauss quadrature rule. More... | |
| type(GK41_type), parameter | GK41 = GK41_type() |
| The scalar constant object of type GK41_type that indicates the use of 20-point Gauss-Legendre quadrature with 41-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(10), parameter | nodeG20 = [ 7.6526521133497333754640409398838211004796266813497500804795244384256342048336978241545114181556215606998505646364133e-02_RKB , 2.2778585114164507808049619536857462474308893768292747231463573920717134186355582779495212519096870803177373131560430e-01_RKB , 3.7370608871541956067254817702492723739574632170568271182794861351564576437305952789589568363453337894476772208852815e-01_RKB , 5.1086700195082709800436405095525099842549132920242683347234861989473497039076572814403168305086777919832943068843526e-01_RKB , 6.3605368072651502545283669622628593674338911679936846393944662254654126258543013255870319549576130658211710937772596e-01_RKB , 7.4633190646015079261430507035564159031073067956917644413954590606853535503815506468110411362064752061238490065167656e-01_RKB , 8.3911697182221882339452906170152068532962936506563737325249272553286109399932480991922934056595764922060422035306914e-01_RKB , 9.1223442825132590586775244120329811304918479742369177479588221915807089120871907893644472619292138737876039175464603e-01_RKB , 9.6397192727791379126766613119727722191206032780618885606353759389204158078438305698001812525596471563131043491596423e-01_RKB , 9.9312859918509492478612238847132027822264713090165589614818413121798471762775378083944940249657220927472894034724419e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 21-points Gauss quadrature rule. More... | |
| real(RKB), dimension(10), parameter | weightG20 = [ 1.5275338713072585069808433195509759349194864511237859727470104981759745316273778153557248783650390593544001842813788e-01_RKB , 1.4917298647260374678782873700196943669267990408136831649621121780984442259558678069396132603521048105170913854567338e-01_RKB , 1.4209610931838205132929832506716493303451541339202030333736708298382808749793436761694922428320058260133068573666201e-01_RKB , 1.3168863844917662689849449974816313491611051114698352699643649370885435642948093314355797518397262924510598005463625e-01_RKB , 1.1819453196151841731237737771138228700504121954896877544688995202017474835051151630572868782581901744606267543092317e-01_RKB , 1.0193011981724043503675013548034987616669165602339255626197161619685232202539434647534931576947985821375859035525483e-01_RKB , 8.3276741576704748724758143222046206100177828583163290744882060785693082894079419471375190843790839349096116111932764e-02_RKB , 6.2672048334109063569506535187041606351601076578436364099584345437974811033665678644563766056832203512603253399592073e-02_RKB , 4.0601429800386941331039952274932109879090639989951536817606854561832296750987328295538920623044384976189825709675075e-02_RKB , 1.7614007139152118311861962351852816362143105543336732524349326677348419259621847817403105542146097668703716227512570e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 20-points Gauss quadrature rule. More... | |
| real(RKB), dimension(21), parameter | nodeK41 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 7.6526521133497333754640409398838211004796266813497500804795244384256342048336978241545114181556215606998505646364133e-02_RKB , 1.5260546524092267550522024102267752791167622481841730660174156703809133685751696356987995886397049724808931527012542e-01_RKB , 2.2778585114164507808049619536857462474308893768292747231463573920717134186355582779495212519096870803177373131560430e-01_RKB , 3.0162786811491300432055535685859226061539650501373092456926374427956957435978384116066498234762220215751079886015902e-01_RKB , 3.7370608871541956067254817702492723739574632170568271182794861351564576437305952789589568363453337894476772208852815e-01_RKB , 4.4359317523872510319999221349264010784010101082300309613315028346299543059315258601993479156987847429893626854030516e-01_RKB , 5.1086700195082709800436405095525099842549132920242683347234861989473497039076572814403168305086777919832943068843526e-01_RKB , 5.7514044681971031534294603658642513281381264014771682537415885495717468074720062012357788489049470208285175093670561e-01_RKB , 6.3605368072651502545283669622628593674338911679936846393944662254654126258543013255870319549576130658211710937772596e-01_RKB , 6.9323765633475138480549071184593153338642585141021417904687378454301191710739219011546672416325022748282227809465165e-01_RKB , 7.4633190646015079261430507035564159031073067956917644413954590606853535503815506468110411362064752061238490065167656e-01_RKB , 7.9504142883755119835063883327278794295938959911578029703855163894322697871710382866701777890251824617748545658564370e-01_RKB , 8.3911697182221882339452906170152068532962936506563737325249272553286109399932480991922934056595764922060422035306914e-01_RKB , 8.7827681125228197607744299511307846671124526828251164853898086998248145904743220740840261624245683876748360309079747e-01_RKB , 9.1223442825132590586775244120329811304918479742369177479588221915807089120871907893644472619292138737876039175464603e-01_RKB , 9.4082263383175475351998272221244338027429557377965291059536839973186796006557571220888218676776618448841584569497535e-01_RKB , 9.6397192727791379126766613119727722191206032780618885606353759389204158078438305698001812525596471563131043491596423e-01_RKB , 9.8150787745025025919334299472021694456725093981023759869077533318793098857465723460898060491887511355706497739384103e-01_RKB , 9.9312859918509492478612238847132027822264713090165589614818413121798471762775378083944940249657220927472894034724419e-01_RKB , 9.9885903158827766383831557654586300999957020432629666866666860339324411793311982967839129772854179884971700274369367e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 41-points Kronrod extension to the 20-points Gauss quadrature rule. More... | |
| real(RKB), dimension(21), parameter | weightK41 = [ 7.6600711917999656445049901530101740827932500628670118055485349620314721456712029449597396569857880493210849110825276e-02_RKB , 7.6377867672080736705502835038061001800801036764945996714946431116936745542061941050008345047482501253320401746334511e-02_RKB , 7.5704497684556674659542775376616558263363155900414326194855223272348838596099414841886740468379707283366777797425290e-02_RKB , 7.4582875400499188986581418362487528616116493572092273080047040726969899567887364227664202642942357104526915332274625e-02_RKB , 7.3030690332786667495189417658913112760626845234552742380174250771849743831660040966804802312464527721645765620253776e-02_RKB , 7.1054423553444068305790361723210167412912159322210143921628270586407381879789525901086146473278095159807542174985045e-02_RKB , 6.8648672928521619345623411885367801715489704958239860400434264173923806029589970941711224257967651039544669425313433e-02_RKB , 6.5834597133618422111563556969397943147223506343381443709751749639944420314384296347503523810096842402960802728781816e-02_RKB , 6.2653237554781168025870122174254980585819744698897886186553324157100424088919284503451596742588386343548162830898103e-02_RKB , 5.9111400880639572374967220648594217136419365977042191748388047204015262840407696611508732839851952697839735487615776e-02_RKB , 5.5195105348285994744832372419777329194753456228153116909812131213177827707884692917845453999535518818940813085110223e-02_RKB , 5.0944573923728691932707670050344948664836365809262579747517140086119113476866735641054822574173198900379392130050979e-02_RKB , 4.6434821867497674720231880926107516842127071007077929289994127933243222585938804392953931185146446072587020288747981e-02_RKB , 4.1668873327973686263788305936894738043960843153010324860966353235271889596379726462208702081068715463576895020003842e-02_RKB , 3.6600169758200798030557240707211008487453496747498001651070009441973280061489266074044986901436324295513243878212345e-02_RKB , 3.1287306777032798958543119323800737887769280362813337359554598005322423266047996771926031069705049476071896145456496e-02_RKB , 2.5882133604951158834505067096153142999479118048674944526997797755374306421629440393392427198869345793286369198147609e-02_RKB , 2.0388373461266523598010231432754705122838627940185929365371868214433006532030353671253640300679157504987977281782909e-02_RKB , 1.4626169256971252983787960308868356163881050162249770342103474631076960029748751959380482484308382288261238476948520e-02_RKB , 8.6002698556429421986617879501023472521289227667077976622450602031426535362696437838448828009554532025301579670206091e-03_RKB , 3.0735837185205315012182932460309874880335046882543449198461628212114333665590378156706265241414469306987988292234740e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 41-points Kronrod extension to the 20-points Gauss quadrature rule. More... | |
| type(GK51_type), parameter | GK51 = GK51_type() |
| The scalar constant object of type GK51_type that indicates the use of 25-point Gauss-Legendre quadrature with 51-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(13), parameter | nodeG25 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.2286469261071039638735981880803680553220534604978373842389353789270883496885841582643884994633105537597765980412320e-01_RKB , 2.4386688372098843204519036279745158640563315632598447642113565325038747278585595067977636776325034060327548499765742e-01_RKB , 3.6117230580938783773582173012764066742207834704337506979457877784674538239569654860329531506093761400789294612122812e-01_RKB , 4.7300273144571496052218211500919204133181773846162729090723082769560327584128603010315684778279363544192787010704498e-01_RKB , 5.7766293024122296772368984161265406739573503929151825664548350776102301275263202227671659646579649084013116066120581e-01_RKB , 6.7356636847346836448512063324762217588341672807274931705965696177828773684928421158196368568030932194044282149314388e-01_RKB , 7.5925926303735763057728286520436097638752201889833412091838973544501862882026240760763679724185230331463919586229073e-01_RKB , 8.3344262876083400142102110869356956946096411382352078602086471546171813247709012525322973947759168107133491065937347e-01_RKB , 8.9499199787827536885104200678280495417455484975358390306170168295917151090119945137118600693039178162093726882638296e-01_RKB , 9.4297457122897433941401116965847053190520157060899014192745249713729532254404926130890521815127348327109666786665572e-01_RKB , 9.7666392145951751149831538647959406774537055531440674467098742731616386753588055389644670948300617866819865983054648e-01_RKB , 9.9555696979049809790878494689390161725756264940480817121080493113293348134372793448728802635294700756868258870429256e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 25-points Gauss quadrature rule. More... | |
| real(RKB), dimension(13), parameter | weightG25 = [ 1.2317605372671545120390287307905014243823362751815166539135219731691200794926142128460112517504958377310054583945994e-01_RKB , 1.2224244299031004168895951894585150583505924756305904090758008223203896721918010243033540891078906637115620156845304e-01_RKB , 1.1945576353578477222817812651290104739017670141372642551958788133518409022018773502442869720975271321374348568426235e-01_RKB , 1.1485825914571164833932554586955580864093619166818014959151499003148279667112542256534429898558156273250513652351744e-01_RKB , 1.0851962447426365311609395705011661934007758798672201615649430734883929279360844269339768350029654172135832773427565e-01_RKB , 1.0053594906705064420220689039268582698846609452814190706986904199941294815904602968195565620373258211755226681206658e-01_RKB , 9.1028261982963649811497220702891653380992558959334310970483768967017384678410526902484398142953718885872521590850372e-02_RKB , 8.0140700335001018013234959669111302290225732853675893716201462973612828934801289559457377714225318048243957479325813e-02_RKB , 6.8038333812356917207187185656707968554709494354636562615071226410003654051711473106651522969481873733098761760660898e-02_RKB , 5.4904695975835191925936891540473324160109985553111349048508498244593774678436511895711924079433444763756746828817613e-02_RKB , 4.0939156701306312655623487711645953660845783364104346504698414899297432880215512770478971055110424130123527015425511e-02_RKB , 2.6354986615032137261901815295299144935963281703322468755366165783870934008879499371529821528172928890350362464605104e-02_RKB , 1.1393798501026287947902964113234773603320526292909696448948061116189891729766743355923677112945033505688431618009664e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 25-points Gauss quadrature rule. More... | |
| real(RKB), dimension(26), parameter | nodeK51 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 6.1544483005685078886546392366796631281724348039823545274305431751687279361558658545141048781022691067898008423227288e-02_RKB , 1.2286469261071039638735981880803680553220534604978373842389353789270883496885841582643884994633105537597765980412320e-01_RKB , 1.8371893942104889201596988875952841578528447834990555215034512653236752851109815617651867160645591242103823539931527e-01_RKB , 2.4386688372098843204519036279745158640563315632598447642113565325038747278585595067977636776325034060327548499765742e-01_RKB , 3.0308953893110783016747890998033932920041937876655194685731578452573120372337209717349617882111662416355753711853559e-01_RKB , 3.6117230580938783773582173012764066742207834704337506979457877784674538239569654860329531506093761400789294612122812e-01_RKB , 4.1788538219303774885181439459457248709336998140069528034955785068796932076966599548717224205109797297615032607570119e-01_RKB , 4.7300273144571496052218211500919204133181773846162729090723082769560327584128603010315684778279363544192787010704498e-01_RKB , 5.2632528433471918259962377815801017803683252320191114313002425180471455022502695302371008520604638341970901082293650e-01_RKB , 5.7766293024122296772368984161265406739573503929151825664548350776102301275263202227671659646579649084013116066120581e-01_RKB , 6.2681009901031741278812268162451788101954628995068510806525222008437260184181183053045236423845198752346149030569920e-01_RKB , 6.7356636847346836448512063324762217588341672807274931705965696177828773684928421158196368568030932194044282149314388e-01_RKB , 7.1776640681308438818665407977329778059771167555515582423493486823991612820974965089522905953765860328116692570706602e-01_RKB , 7.5925926303735763057728286520436097638752201889833412091838973544501862882026240760763679724185230331463919586229073e-01_RKB , 7.9787379799850005941041090499430656940863230009338267661706934499488650817643824077118950314443984031474353711531825e-01_RKB , 8.3344262876083400142102110869356956946096411382352078602086471546171813247709012525322973947759168107133491065937347e-01_RKB , 8.6584706529327559544899696958834008820284409402823690293965213246691432948180280120756708738064779055576005302835351e-01_RKB , 8.9499199787827536885104200678280495417455484975358390306170168295917151090119945137118600693039178162093726882638296e-01_RKB , 9.2074711528170156174634608454633063157457035996277199700642836501131385042631212407808952281702820179915510491592339e-01_RKB , 9.4297457122897433941401116965847053190520157060899014192745249713729532254404926130890521815127348327109666786665572e-01_RKB , 9.6161498642584251241813003366016724169212642963709676666624520141292893281185666917636407790823210892689040877316178e-01_RKB , 9.7666392145951751149831538647959406774537055531440674467098742731616386753588055389644670948300617866819865983054648e-01_RKB , 9.8803579453407724763733101457740622707248415209160748131449972199405186821347293686245404742032360498210710718706868e-01_RKB , 9.9555696979049809790878494689390161725756264940480817121080493113293348134372793448728802635294700756868258870429256e-01_RKB , 9.9926210499260983419345748654034059370452496042279618586228697762904524428167719073818746102238075978747461480736921e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 51-points Kronrod extension to the 25-points Gauss quadrature rule. More... | |
| real(RKB), dimension(26), parameter | weightK51 = [ 6.1580818067832935078759824240064553190436936903140808056908996403358367244202623293256774502185186717703954810463664e-02_RKB , 6.1471189871425316661544131965264177586537962876885022711111683500151700796198726558483367566537422877227096643444043e-02_RKB , 6.1128509717053048305859030416292711922678552321960938357322028070390133769952032831204895569347757809858568165047769e-02_RKB , 6.0539455376045862945360267517565427162312365710457079923487043144554747810689514408013582515489930908693681447570811e-02_RKB , 5.9720340324174059979099291932561853835363045476189975483372207816149988460708299020779612375010639778624011960832019e-02_RKB , 5.8689680022394207961974175856787764139795646254828315293243700305012569486054157617049685031506591863121580010947248e-02_RKB , 5.7437116361567832853582693939506471994832856823896682976509412313367495727224381199978598247737089593472710899482737e-02_RKB , 5.5950811220412317308240686382747346820271035112771802428932791066115158268338607019365831655460314732208940609352540e-02_RKB , 5.4251129888545490144543370459875606826076838441263383072163293312936923476650934130242315028422047795830492882862973e-02_RKB , 5.2362885806407475864366712137872714887351550723707596350905793656046659248541276597504566497990926306481919129870507e-02_RKB , 5.0277679080715671963325259433440084440587630604775975142050968279743014641141402310302584542633557037153607386127936e-02_RKB , 4.7982537138836713906392255756914754983592207423271169651235865196757913880334117810235517477328110033499422471098658e-02_RKB , 4.5502913049921788909870584752660393043707768935695327316724254392794299567957035458208970599641697203261236226745020e-02_RKB , 4.2872845020170049476895792439495161101999504199883328877919242515738957655253932048951366960802592343905647433925806e-02_RKB , 4.0083825504032382074839284467075646401410549266591308713115878386835777315058451955614116158949614066927183232852042e-02_RKB , 3.7116271483415543560330625367619875995997802688047764805628702762773009669395760582294525748583875707140577080663373e-02_RKB , 3.4002130274329337836748795229551203225670528250050443083264193121524339063344855010257660547708022429300203676502386e-02_RKB , 3.0792300167387488891109020215228585600877162393292487644544830559965388047996492709248618249084851477787538356572832e-02_RKB , 2.7475317587851737802948455517811078614796013288710603199613621069727810352835469926107822047433566792405123805901196e-02_RKB , 2.4009945606953216220092489164881081392931528209659330290734972342536012282191913069778658241972047765300060007037359e-02_RKB , 2.0435371145882835456568292235938973678758006097668937220074531550163622566841885855957623103354443247806459277197725e-02_RKB , 1.6847817709128298231516667536336315840402654624706139411175769276842182270078960078544597372646532637619276509222462e-02_RKB , 1.3236229195571674813656405846976238077578084997863654732213860488560614587634395544002156258192582265590155862296710e-02_RKB , 9.4739733861741516072077105236553238716453268483726334971394029603529306140359023187904705754719643032594360138998941e-03_RKB , 5.5619321353567137580402369010655220701769295496290984052961210793810038857581724171021610100708799763006942755331129e-03_RKB , 1.9873838923303159265078518828434098894299804282505973837653346298985629336820118753523093675303476883723992297810124e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 51-points Kronrod extension to the 25-points Gauss quadrature rule. More... | |
| type(GK61_type), parameter | GK61 = GK61_type() |
| The scalar constant object of type GK61_type that indicates the use of 30-point Gauss-Legendre quadrature with 61-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK. More... | |
| real(RKB), dimension(15), parameter | nodeG30 = [ 5.1471842555317695833025213166722573749141453666569564255160843987964755210427109055870090707285485841217089963590678e-02_RKB , 1.5386991360858354696379467274325592041855197124433846171896298291578714851081610139692310651074078557990111754952062e-01_RKB , 2.5463692616788984643980512981780510788278930330251842616428597508896353156907880290636628138423620257595521678255758e-01_RKB , 3.5270472553087811347103720708937386065363100802142562659418446890026941623319107866436039675211352945165817827083104e-01_RKB , 4.4703376953808917678060990032285400016240759386142440975447738172761535172858420700400688872124189834257262048739699e-01_RKB , 5.3662414814201989926416979331107279416417800693029710545274348291201490861897837863114116009718990258091585830703557e-01_RKB , 6.2052618298924286114047755643118929920736469282952813259505117012433531497488911774115258445532782106478789996137481e-01_RKB , 6.9785049479331579693229238802664006838235380065395465637972284673997672124315996069538163644008904690545069439941341e-01_RKB , 7.6777743210482619491797734097450313169488361723290845320649438736515857017299504505260960258623968420224697596501719e-01_RKB , 8.2956576238276839744289811973250191643906869617034167880695298345365650658958163508295244350814016004371545455777732e-01_RKB , 8.8256053579205268154311646253022559005668914714648423206832605312161626269519165572921583828573210485349058106849548e-01_RKB , 9.2620004742927432587932427708047400408647453682532906091103713367942299565110232681677288015055886244486106298320068e-01_RKB , 9.6002186496830751221687102558179766293035921740392339948566167242493995770706842922718944370380002378239172677454384e-01_RKB , 9.8366812327974720997003258160566280194031785470971136351718001015114429536479104370207597166035471368057762560137209e-01_RKB , 9.9689348407464954027163005091869528334088203811775079010809429780238769521016374081588201955806171741257405095963817e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 30-points Gauss quadrature rule. More... | |
| real(RKB), dimension(15), parameter | weightG30 = [ 1.0285265289355884034128563670541504386837555706492822258631898667601623865660942939262884632188870916503815852709086e-01_RKB , 1.0176238974840550459642895216855404463270628948712684086426094541964251360531767494547599781978391198881693385887696e-01_RKB , 9.9593420586795267062780282103569476529869263666704277221365146183946660389908809018092299289324184705373523229592037e-02_RKB , 9.6368737174644259639468626351809865096406461430160245912994275732837534742003123724951247818104195363343093583583429e-02_RKB , 9.2122522237786128717632707087618767196913234418234107527675047001973047070094168298464052916811907158954949394100501e-02_RKB , 8.6899787201082979802387530715125702576753328743545344012222129882153582254261494247955033509639105330215477601953921e-02_RKB , 8.0755895229420215354694938460529730875892803708439299890258593706051180567026345604212402769217808080749416147400962e-02_RKB , 7.3755974737705206268243850022190734153770526037049438941269182374599399314635211710401352716638183270192254236882630e-02_RKB , 6.5974229882180495128128515115962361237442953656660378967031516042143672466094179365819913911598737439478205808271237e-02_RKB , 5.7493156217619066481721689402056128797120670721763134548715799003232147409954376925211999650950125355559974348279846e-02_RKB , 4.8402672830594052902938140422807517815271809197372736345191936791805425677102152797767439563562263454374645955072007e-02_RKB , 3.8799192569627049596801936446347692033200976766395352107732789705946970952769793919055026279035105656340228558382274e-02_RKB , 2.8784707883323369349719179611292043639588894546287496474180122608145988940013933101730206711484171554940392262251283e-02_RKB , 1.8466468311090959142302131912047269096206533968181403371298365514585599521307973654080519029675417955638095832046164e-02_RKB , 7.9681924961666056154658834746736224504806965871517212294851633569200384329013332941536616922861735209846506562158817e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 30-points Gauss quadrature rule. More... | |
| real(RKB), dimension(31), parameter | nodeK61 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 5.1471842555317695833025213166722573749141453666569564255160843987964755210427109055870090707285485841217089963590678e-02_RKB , 1.0280693796673703014709675131800059247190133296515840552101946914632788253917872738234797140786490207720254922664913e-01_RKB , 1.5386991360858354696379467274325592041855197124433846171896298291578714851081610139692310651074078557990111754952062e-01_RKB , 2.0452511668230989143895767100202470952410426459556377447604465028350321894663245495592565235317147819577892124850607e-01_RKB , 2.5463692616788984643980512981780510788278930330251842616428597508896353156907880290636628138423620257595521678255758e-01_RKB , 3.0407320227362507737267710719925655353115778980946272844421536998312150442387767304001423699909778588529370119457430e-01_RKB , 3.5270472553087811347103720708937386065363100802142562659418446890026941623319107866436039675211352945165817827083104e-01_RKB , 4.0040125483039439253547621154266063361104593297078395983186610656429170689311759061175527015710247383961903284673474e-01_RKB , 4.4703376953808917678060990032285400016240759386142440975447738172761535172858420700400688872124189834257262048739699e-01_RKB , 4.9248046786177857499369306120770879564426564096318697026073340982988422546396352776837047452262025983265531109327026e-01_RKB , 5.3662414814201989926416979331107279416417800693029710545274348291201490861897837863114116009718990258091585830703557e-01_RKB , 5.7934523582636169175602493217254049590705158881215289208126016612312833567812241903809970751783808208940322061083509e-01_RKB , 6.2052618298924286114047755643118929920736469282952813259505117012433531497488911774115258445532782106478789996137481e-01_RKB , 6.6006106412662696137005366814927075303835037480883390955067197339904937499734522076788020517029688190998858739703079e-01_RKB , 6.9785049479331579693229238802664006838235380065395465637972284673997672124315996069538163644008904690545069439941341e-01_RKB , 7.3379006245322680472617113136952764566938172775468549208701399518300016463613325382024664531597318795933262446521430e-01_RKB , 7.6777743210482619491797734097450313169488361723290845320649438736515857017299504505260960258623968420224697596501719e-01_RKB , 7.9972783582183908301366894232268324073569842937778450923647349548686662567326007229195202524185356472023967927713548e-01_RKB , 8.2956576238276839744289811973250191643906869617034167880695298345365650658958163508295244350814016004371545455777732e-01_RKB , 8.5720523354606109895865851065894385682080017062359612850504551739119887225712932688031120704657195642614071367390794e-01_RKB , 8.8256053579205268154311646253022559005668914714648423206832605312161626269519165572921583828573210485349058106849548e-01_RKB , 9.0557330769990779854652255892595831956897536366222841356404766397803760239449631913585074426842574155323901785046522e-01_RKB , 9.2620004742927432587932427708047400408647453682532906091103713367942299565110232681677288015055886244486106298320068e-01_RKB , 9.4437444474855997941583132403743912158564371496498093181748940139520917000657342753448871376849848523800667868447591e-01_RKB , 9.6002186496830751221687102558179766293035921740392339948566167242493995770706842922718944370380002378239172677454384e-01_RKB , 9.7311632250112626837469386842370688488763796428343933853755850185624118958166838288308561708261486365954975485787212e-01_RKB , 9.8366812327974720997003258160566280194031785470971136351718001015114429536479104370207597166035471368057762560137209e-01_RKB , 9.9163099687040459485862836610948572485050033374616325510019923349807489603260796605556191495843575227494654783755353e-01_RKB , 9.9689348407464954027163005091869528334088203811775079010809429780238769521016374081588201955806171741257405095963817e-01_RKB , 9.9948441005049063757132589570581081946887394701850801923632642830748016674843587830656468823145435723317885056396548e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 61-points Kronrod extension to the 30-points Gauss quadrature rule. More... | |
| real(RKB), dimension(31), parameter | weightK61 = [ 5.1494729429451567558340433647099307532736880396464168074637323362474083844397567724480716864880173808112573901197920e-02_RKB , 5.1426128537459025933862879215781259829552034862395987263855824172761589259406892072066110681184224608133314131500422e-02_RKB , 5.1221547849258772170656282604944208251146952425246327553509056805511015401279553971190412722969308620984161625812560e-02_RKB , 5.0881795898749606492297473049804691853384914260919239920771942080972542646780575571132056254070929858650733836163479e-02_RKB , 5.0405921402782346840893085653585028902197018251622233664243959211066713308635283713447747907973700791599900911248852e-02_RKB , 4.9795683427074206357811569379942328539209602813696108951047392842948482646220377655098341924089250200477846596263918e-02_RKB , 4.9055434555029778887528165367238173605887405295296569579490717901328215644590555247522873065246297467067324397612445e-02_RKB , 4.8185861757087129140779492298304592605799236108429800057373350872433793583969368428942672063270298939865425225579922e-02_RKB , 4.7185546569299153945261478181099486482884807300628457194141861551725533289490897029020276525603515502104799540544222e-02_RKB , 4.6059238271006988116271735559373580594692875571824924004732379492293604006446052672252973438978639166425766841417488e-02_RKB , 4.4814800133162663192355551616723243757431392796373009889680201194063503947907899189061064792111919040540351834527742e-02_RKB , 4.3452539701356069316831728117073258074603308631703168064888805495738640839573863333942084117196541456054957383622173e-02_RKB , 4.1969810215164246147147541285969757790088656718992374820388720323852655511200365790379948462006156953358103259681948e-02_RKB , 4.0374538951535959111995279752468114216126062126030255633998289613810846761059740961836828802959573901107306640876603e-02_RKB , 3.8678945624727592950348651532281050250923629821553846790376130679337402056620700554139109487533759557982632153728099e-02_RKB , 3.6882364651821229223911065617135967736955164781030337670005198584196134970154169862584193360751243227989492571664973e-02_RKB , 3.4979338028060024137499670731467875097226912794818719972208457232177786702008744219498470603846784465175225933802357e-02_RKB , 3.2981447057483726031814191016853927510599291213858385714519347641452316582381008804994515341969205985818543200837577e-02_RKB , 3.0907257562387762472884252943092272635270458523807153426840486964022086189874056947717446328187131273807982629114591e-02_RKB , 2.8754048765041292843978785354334211144679160542074930035102280759132174815469834227854660515366003136772757344886331e-02_RKB , 2.6509954882333101610601709335075414366517579522748565770867438338472138903658077617652522759934474895733739329287706e-02_RKB , 2.4191162078080601365686370725232026760391377828182462432228943562944885267501070688006470962871743661192935455117297e-02_RKB , 2.1828035821609192297167485738338993401507296056834912773630422358720439403382559079356058602393879803560534375378340e-02_RKB , 1.9414141193942381173408951050128455851421014191431525770276066536497179079025540486072726114628763606440143557769099e-02_RKB , 1.6920889189053272627572289420322092368566703783835191139883410840546679978551861043620089451681146020853650713611444e-02_RKB , 1.4369729507045804812451432443580010195841899895001505873565899403000198662495821906144274682894222591414503342336172e-02_RKB , 1.1823015253496341742232898853250592896264406250607818326302431548265365155855182739401700032519141448997853772603766e-02_RKB , 9.2732796595177634284411468920243604212700249381931076964956469143626665557434385492325784596343112153704094886248672e-03_RKB , 6.6307039159312921733198263697501681336283882177812585973955597357837568277731921327731815844512598157843672104469554e-03_RKB , 3.8904611270998840512672018445155032785151429848864649214200101281144733676455451061226273655941038347210163533085954e-03_RKB , 1.3890136986770076245515912267596996810488412919632724534411055332301367130989865366956251556423820479579333920310978e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 61-points Kronrod extension to the 30-points Gauss quadrature rule. More... | |
| integer(IK), parameter | MAXLEN_EPSTAB = 50_IK |
The scalar integer of default kind IK, containing the maximum size of the Table of Epsilon algorithm. More... | |
| type(weps_type), parameter | weps = weps_type() |
| The scalar constant object of type weps_type that indicates the use of Epsilon extrapolation method of Wynn (1961) to accelerate the convergence of integrations of functions with integrable singularities and infinities using the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadErr. More... | |
This module contains classes and procedures for non-adaptive and adaptive global numerical quadrature and Cauchy Principal Value of 1D functions with various types of singularities and points of difficulties via the Gauss-Kronrod and Clenshaw-Curtis quadrature formulae.
The algorithms of this module significantly extend the original routines of the venerable QuadPack library in FORTRAN77.
Question
Is there a hassle-free simple nimble reliable numerical integrator that I can use as the first try?
Answer:
YES, as described below.
Gaussian Quadrature is an approximation of the definite integral of a function, usually stated as a weighted sum of function values at specified points within the integration domain.
An \(n\)-point Gaussian quadrature rule, named after Carl Friedrich Gauss, is a quadrature rule constructed to yield an exact result for polynomials of degree \(2n − 1\) or less by a suitable choice of the nodes \(x_i\) and weights \(w_i\) for \(i = \{1, \ldots, n\}\).
The modern formulation using orthogonal polynomials was developed by Carl Gustav Jacobi in 1826.
The most common domain of integration for such a rule is taken as \([−1, 1]\), so the rule is stated as
\begin{equation} \large \int_{-1}^{+1} f(x) \mathrm{d}x \approx \sum_{i = 1}^{n} w_i f(x_i) ~, \end{equation}
which is exact for a polynomial of degree \(2n − 1\) or less. This exact rule is known as the Gauss-Legendre quadrature rule.
The integration of polynomials is trivial and is rarely done via numerical methods.
However, transcendental and numerically defined functions are frequently integrated via Gaussian quadrature, and the defining problem becomes how to estimate the remainder.
The Gauss-Legendre quadrature rule will only be an accurate approximation to the integral above if \(f(x)\) is well-approximated by a polynomial of degree \(2n − 1\) or less on \([−1, 1]\).
The Gauss-Legendre quadrature rule is typically not used for integrable functions with endpoint singularities.
The key to the Gaussian quadrature is the derivation of the nodes and weights for which established formulae exist (e.g., setNodeWeightGK).
It is possible to compute a Gaussian quadrature of order \(n\) and another of order (e.g., \(2n+1\)) and use the difference as an error estimate.
However, this is not optimal, as the zeros of the Legendre polynomials (i.e., the nodes of Gaussian quadrature) are never the same for different orders.
This requires \(3n + 1\) function evaluations to be performed.
The error estimation challenge mentioned above limits the direct application of the Gauss-Legendre quadrature to integration problems.
Higher order Gauss-Legendre quadrature (corresponding to higher numbers of nodes and weights) does not necessarily overlap with the lower-order nodes.
This means that the lower-order calculations of \(f(x)\) cannot be used in subsequent higher-order Gauss-Legendre quadrature, which has two consequences,
The Gauss-Kronrod quadrature is an extension of Gaussian quadrature, enabling the use of lower-order nodes and computations in higher-order quadrature.
Kronrod (1964) considered the problem of how to interleave nodes into a Gauss-Legendre quadrature in such a way that all previous function evaluations can be reused, while increasing the order of polynomials that can be integrated exactly.
Kronrod discovered that one could integrate polynomials of degree \(3n + 1\) by adding \(n + 1\) nodes (computed from the zeros of the Legendre-Stieltjes polynomials) to a Gaussian quadrature of order \(n\).
Later, Patterson (1968, 1969) showed how to compute continued extensions of this kind (e.g., setNodeWeightGK).
This allows an a posteriori error estimate to be provided while still preserving exponential convergence.
This error estimate is, however, frequently pessimistic. This is particularly true when the estimated error is smaller than \(1.25 \times 10^{-6}\) meaning that the integration result is generally much more accurate than what is understood from the error estimate.
The Gauss-Kronrod is a powerful driver in Adaptive Global Quadrature methods (e.g., as implemented in getQuadGK).
The Clenshaw-Curtis quadrature (Clenshaw and Curtis (1960)) handles the numerical computation of the integral of a function \(f(x)\) over a finite interval, with weight function \(w = 1\).
The idea of Clenshaw-Curtis integration is to approximate \(f(x)\) by a truncated Chebyshev polynomials expansion, which can be integrated exactly.
For functions with a slowly convergent Chebyshev series expansion, use of the Clenshaw-Curtis method is not advisable.
However, there is a possibility to retain the advantage of the computational ease of the Clenshaw-Curtis method while extending its practical applicability to functions with slowly convergent Chebyshev expansions.
This is accomplished by transferring the difficult part of the integrand, which causes the slow rate of convergence of the Chebyshev series (e.g. a rapidly varying factor), into a weight function.
Even certain singular weight functions, becoming infinite somewhere in the integration interval, can be treated successfully by this technique.
This is makes the Clenshaw-Curtis quadrature rules particularly suitable for computing the Cauchy Principal Value of arbitrary functions when combined with the Gauss-Kronrod rules.
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.
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.
The scalar constant object of type GK15_type that indicates the use of 7-point Gauss-Legendre quadrature with 15-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 674 of file pm_quadPack.F90.
The scalar constant object of type GK21_type that indicates the use of 10-point Gauss-Legendre quadrature with 21-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 870 of file pm_quadPack.F90.
The scalar constant object of type GK31_type that indicates the use of 15-point Gauss-Legendre quadrature with 31-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 1074 of file pm_quadPack.F90.
The scalar constant object of type GK41_type that indicates the use of 20-point Gauss-Legendre quadrature with 41-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 1294 of file pm_quadPack.F90.
The scalar constant object of type GK51_type that indicates the use of 25-point Gauss-Legendre quadrature with 51-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 1528 of file pm_quadPack.F90.
The scalar constant object of type GK61_type that indicates the use of 30-point Gauss-Legendre quadrature with 61-point Kronrod extension formulae as the drivers in the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadGK.
Example usage ⛓
ifort compiler ⛓ ifort compiler ⛓ gfortran compiler ⛓
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.
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.
Definition at line 1778 of file pm_quadPack.F90.
| integer(IK), parameter pm_quadPack::MAXLEN_EPSTAB = 50_IK |
The scalar integer of default kind IK, containing the maximum size of the Table of Epsilon algorithm.
This parameter is used by the various quadrature procedures of this module to perform integration extrapolations.
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.
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.
Definition at line 4501 of file pm_quadPack.F90.
| character(*, SK), parameter pm_quadPack::MODULE_NAME = "@pm_quadPack" |
Definition at line 149 of file pm_quadPack.F90.
| real(RKB), dimension(5), parameter pm_quadPack::nodeG10 = [ 1.4887433898163121088482600112971998461756485942069169570798925351590361735566852137117762979946369123003116080525534e-01_RKB , 4.3339539412924719079926594316578416220007183765624649650270151314376698907770350122510275795011772122368293504099894e-01_RKB , 6.7940956829902440623432736511487357576929471183480946766481718895255857539507492461507857357048037949983390204739932e-01_RKB , 8.6506336668898451073209668842349304852754301496533045252195973184537475513805556135679072894604577069440463108641177e-01_RKB , 9.7390652851717172007796401208445205342826994669238211923121206669659520323463615962572356495626855625823304251877421e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 10-points Gauss quadrature rule.
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.
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.
Definition at line 894 of file pm_quadPack.F90.
| real(RKB), dimension(8), parameter pm_quadPack::nodeG15 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 2.0119409399743452230062830339459620781283645446263767961594972460994823900302018760183625806752105908967902257386509e-01_RKB , 3.9415134707756336989720737098104546836275277615869825503116534395160895778696141797549711416165976202589352169635648e-01_RKB , 5.7097217260853884753722673725391064123838639628274960485326541705419537986975857948341462856982614477912646497026257e-01_RKB , 7.2441773136017004741618605461393800963089929458410256355142342070412378167792521899610109760313432626923598549381925e-01_RKB , 8.4820658341042721620064832077421685136625617473699263409572755876067507517414548519760771975082148085090373835713340e-01_RKB , 9.3727339240070590430775894771020947124399627351530445790136307635020297379704552795054758617426808659746824044603157e-01_RKB , 9.8799251802048542848956571858661258114697281712376148999999751558738843736901942471272205036831914497667516843990079e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 15-points Gauss quadrature rule.
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.
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.
Definition at line 1098 of file pm_quadPack.F90.
| real(RKB), dimension(10), parameter pm_quadPack::nodeG20 = [ 7.6526521133497333754640409398838211004796266813497500804795244384256342048336978241545114181556215606998505646364133e-02_RKB , 2.2778585114164507808049619536857462474308893768292747231463573920717134186355582779495212519096870803177373131560430e-01_RKB , 3.7370608871541956067254817702492723739574632170568271182794861351564576437305952789589568363453337894476772208852815e-01_RKB , 5.1086700195082709800436405095525099842549132920242683347234861989473497039076572814403168305086777919832943068843526e-01_RKB , 6.3605368072651502545283669622628593674338911679936846393944662254654126258543013255870319549576130658211710937772596e-01_RKB , 7.4633190646015079261430507035564159031073067956917644413954590606853535503815506468110411362064752061238490065167656e-01_RKB , 8.3911697182221882339452906170152068532962936506563737325249272553286109399932480991922934056595764922060422035306914e-01_RKB , 9.1223442825132590586775244120329811304918479742369177479588221915807089120871907893644472619292138737876039175464603e-01_RKB , 9.6397192727791379126766613119727722191206032780618885606353759389204158078438305698001812525596471563131043491596423e-01_RKB , 9.9312859918509492478612238847132027822264713090165589614818413121798471762775378083944940249657220927472894034724419e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 21-points Gauss quadrature rule.
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.
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.
Definition at line 1318 of file pm_quadPack.F90.
| real(RKB), dimension(13), parameter pm_quadPack::nodeG25 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.2286469261071039638735981880803680553220534604978373842389353789270883496885841582643884994633105537597765980412320e-01_RKB , 2.4386688372098843204519036279745158640563315632598447642113565325038747278585595067977636776325034060327548499765742e-01_RKB , 3.6117230580938783773582173012764066742207834704337506979457877784674538239569654860329531506093761400789294612122812e-01_RKB , 4.7300273144571496052218211500919204133181773846162729090723082769560327584128603010315684778279363544192787010704498e-01_RKB , 5.7766293024122296772368984161265406739573503929151825664548350776102301275263202227671659646579649084013116066120581e-01_RKB , 6.7356636847346836448512063324762217588341672807274931705965696177828773684928421158196368568030932194044282149314388e-01_RKB , 7.5925926303735763057728286520436097638752201889833412091838973544501862882026240760763679724185230331463919586229073e-01_RKB , 8.3344262876083400142102110869356956946096411382352078602086471546171813247709012525322973947759168107133491065937347e-01_RKB , 8.9499199787827536885104200678280495417455484975358390306170168295917151090119945137118600693039178162093726882638296e-01_RKB , 9.4297457122897433941401116965847053190520157060899014192745249713729532254404926130890521815127348327109666786665572e-01_RKB , 9.7666392145951751149831538647959406774537055531440674467098742731616386753588055389644670948300617866819865983054648e-01_RKB , 9.9555696979049809790878494689390161725756264940480817121080493113293348134372793448728802635294700756868258870429256e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 25-points Gauss quadrature rule.
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.
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.
Definition at line 1552 of file pm_quadPack.F90.
| real(RKB), dimension(15), parameter pm_quadPack::nodeG30 = [ 5.1471842555317695833025213166722573749141453666569564255160843987964755210427109055870090707285485841217089963590678e-02_RKB , 1.5386991360858354696379467274325592041855197124433846171896298291578714851081610139692310651074078557990111754952062e-01_RKB , 2.5463692616788984643980512981780510788278930330251842616428597508896353156907880290636628138423620257595521678255758e-01_RKB , 3.5270472553087811347103720708937386065363100802142562659418446890026941623319107866436039675211352945165817827083104e-01_RKB , 4.4703376953808917678060990032285400016240759386142440975447738172761535172858420700400688872124189834257262048739699e-01_RKB , 5.3662414814201989926416979331107279416417800693029710545274348291201490861897837863114116009718990258091585830703557e-01_RKB , 6.2052618298924286114047755643118929920736469282952813259505117012433531497488911774115258445532782106478789996137481e-01_RKB , 6.9785049479331579693229238802664006838235380065395465637972284673997672124315996069538163644008904690545069439941341e-01_RKB , 7.6777743210482619491797734097450313169488361723290845320649438736515857017299504505260960258623968420224697596501719e-01_RKB , 8.2956576238276839744289811973250191643906869617034167880695298345365650658958163508295244350814016004371545455777732e-01_RKB , 8.8256053579205268154311646253022559005668914714648423206832605312161626269519165572921583828573210485349058106849548e-01_RKB , 9.2620004742927432587932427708047400408647453682532906091103713367942299565110232681677288015055886244486106298320068e-01_RKB , 9.6002186496830751221687102558179766293035921740392339948566167242493995770706842922718944370380002378239172677454384e-01_RKB , 9.8366812327974720997003258160566280194031785470971136351718001015114429536479104370207597166035471368057762560137209e-01_RKB , 9.9689348407464954027163005091869528334088203811775079010809429780238769521016374081588201955806171741257405095963817e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 30-points Gauss quadrature rule.
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.
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.
Definition at line 1802 of file pm_quadPack.F90.
| real(RKB), dimension(4), parameter pm_quadPack::nodeG7 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 4.0584515137739716690660641207696146334738201409937012638704325179466381322612565532831268972774658776528675866604802e-01_RKB , 7.4153118559939443986386477328078840707414764714139026011995535196742987467218051379282683236686324705969251809311201e-01_RKB , 9.4910791234275852452618968404785126240077093767061778354876910391306333035484014080573077002792572414430073966699522e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 7-points Gauss quadrature rule.
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.
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.
Definition at line 698 of file pm_quadPack.F90.
| real(RKB), dimension(8), parameter pm_quadPack::nodeK15 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 2.0778495500789846760068940377324491347978440714517064971384573461986693844943520226910343227183698530560857645062738e-01_RKB , 4.0584515137739716690660641207696146334738201409937012638704325179466381322612565532831268972774658776528675866604802e-01_RKB , 5.8608723546769113029414483825872959843678075060436095130499289319880373607444407464511674498935942098956811555121368e-01_RKB , 7.4153118559939443986386477328078840707414764714139026011995535196742987467218051379282683236686324705969251809311201e-01_RKB , 8.6486442335976907278971278864092620121097230707408814860145771276706770813259572103585847859604590541475281326027862e-01_RKB , 9.4910791234275852452618968404785126240077093767061778354876910391306333035484014080573077002792572414430073966699522e-01_RKB , 9.9145537112081263920685469752632851664204433837033470129108741357244173934653407235924503509626841760744349505339308e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 15-points Kronrod extension to the 7-points Gauss quadrature rule.
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.
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.
Definition at line 754 of file pm_quadPack.F90.
| real(RKB), dimension(11), parameter pm_quadPack::nodeK21 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.4887433898163121088482600112971998461756485942069169570798925351590361735566852137117762979946369123003116080525534e-01_RKB , 2.9439286270146019813112660310386556616268662515695791864888229172724611166332737888445523178268237359119185139299872e-01_RKB , 4.3339539412924719079926594316578416220007183765624649650270151314376698907770350122510275795011772122368293504099894e-01_RKB , 5.6275713466860468333900009927269414084301388194196695886034621458779266353216327549712087854169992422106448211158815e-01_RKB , 6.7940956829902440623432736511487357576929471183480946766481718895255857539507492461507857357048037949983390204739932e-01_RKB , 7.8081772658641689706371757834504237716340752029815717974694859999505607982761420654526977234238996241110129779403362e-01_RKB , 8.6506336668898451073209668842349304852754301496533045252195973184537475513805556135679072894604577069440463108641177e-01_RKB , 9.3015749135570822600120718005950834622516790998193924230349406866828415983091673055011194572851007884702013619684320e-01_RKB , 9.7390652851717172007796401208445205342826994669238211923121206669659520323463615962572356495626855625823304251877421e-01_RKB , 9.9565716302580808073552728068900284792126058721947892436337916111757023046774867357152325996912076724298149077812671e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 21-points Kronrod extension to the 10-points Gauss quadrature rule.
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.
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.
Definition at line 952 of file pm_quadPack.F90.
| real(RKB), dimension(16), parameter pm_quadPack::nodeK31 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 1.0114206691871749902707423144739233878745105740164180495800189504151097862454083050931321451540380998341273193681967e-01_RKB , 2.0119409399743452230062830339459620781283645446263767961594972460994823900302018760183625806752105908967902257386509e-01_RKB , 2.9918000715316881216678002426638896266160338274382080184125545738918081102513884467602322020157243563662094470221235e-01_RKB , 3.9415134707756336989720737098104546836275277615869825503116534395160895778696141797549711416165976202589352169635648e-01_RKB , 4.8508186364023968069365574023235061286633893089407312129367943604080239955167155974371848690848595275551258416303565e-01_RKB , 5.7097217260853884753722673725391064123838639628274960485326541705419537986975857948341462856982614477912646497026257e-01_RKB , 6.5099674129741697053373589531327469254694822609259966708966160576093305841043840794460394747228060367236079289132544e-01_RKB , 7.2441773136017004741618605461393800963089929458410256355142342070412378167792521899610109760313432626923598549381925e-01_RKB , 7.9041850144246593296764929481794734686214051995697617332365280643308302974631807059994738664225445530963711137343440e-01_RKB , 8.4820658341042721620064832077421685136625617473699263409572755876067507517414548519760771975082148085090373835713340e-01_RKB , 8.9726453234408190088250965645449588283177871149442786763972687601078537721473771221195399661919716123038835639691946e-01_RKB , 9.3727339240070590430775894771020947124399627351530445790136307635020297379704552795054758617426808659746824044603157e-01_RKB , 9.6773907567913913425734797878433722528335733730013163797468062226335804249452174804319385048203118506304424717089291e-01_RKB , 9.8799251802048542848956571858661258114697281712376148999999751558738843736901942471272205036831914497667516843990079e-01_RKB , 9.9800229869339706028517284015227120907340644231555723034839427970683348682837134566648979907760125278631896777136104e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 31-points Kronrod extension to the 15-points Gauss quadrature rule.
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.
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.
Definition at line 1162 of file pm_quadPack.F90.
| real(RKB), dimension(21), parameter pm_quadPack::nodeK41 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 7.6526521133497333754640409398838211004796266813497500804795244384256342048336978241545114181556215606998505646364133e-02_RKB , 1.5260546524092267550522024102267752791167622481841730660174156703809133685751696356987995886397049724808931527012542e-01_RKB , 2.2778585114164507808049619536857462474308893768292747231463573920717134186355582779495212519096870803177373131560430e-01_RKB , 3.0162786811491300432055535685859226061539650501373092456926374427956957435978384116066498234762220215751079886015902e-01_RKB , 3.7370608871541956067254817702492723739574632170568271182794861351564576437305952789589568363453337894476772208852815e-01_RKB , 4.4359317523872510319999221349264010784010101082300309613315028346299543059315258601993479156987847429893626854030516e-01_RKB , 5.1086700195082709800436405095525099842549132920242683347234861989473497039076572814403168305086777919832943068843526e-01_RKB , 5.7514044681971031534294603658642513281381264014771682537415885495717468074720062012357788489049470208285175093670561e-01_RKB , 6.3605368072651502545283669622628593674338911679936846393944662254654126258543013255870319549576130658211710937772596e-01_RKB , 6.9323765633475138480549071184593153338642585141021417904687378454301191710739219011546672416325022748282227809465165e-01_RKB , 7.4633190646015079261430507035564159031073067956917644413954590606853535503815506468110411362064752061238490065167656e-01_RKB , 7.9504142883755119835063883327278794295938959911578029703855163894322697871710382866701777890251824617748545658564370e-01_RKB , 8.3911697182221882339452906170152068532962936506563737325249272553286109399932480991922934056595764922060422035306914e-01_RKB , 8.7827681125228197607744299511307846671124526828251164853898086998248145904743220740840261624245683876748360309079747e-01_RKB , 9.1223442825132590586775244120329811304918479742369177479588221915807089120871907893644472619292138737876039175464603e-01_RKB , 9.4082263383175475351998272221244338027429557377965291059536839973186796006557571220888218676776618448841584569497535e-01_RKB , 9.6397192727791379126766613119727722191206032780618885606353759389204158078438305698001812525596471563131043491596423e-01_RKB , 9.8150787745025025919334299472021694456725093981023759869077533318793098857465723460898060491887511355706497739384103e-01_RKB , 9.9312859918509492478612238847132027822264713090165589614818413121798471762775378083944940249657220927472894034724419e-01_RKB , 9.9885903158827766383831557654586300999957020432629666866666860339324411793311982967839129772854179884971700274369367e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 41-points Kronrod extension to the 20-points Gauss quadrature rule.
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.
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.
Definition at line 1386 of file pm_quadPack.F90.
| real(RKB), dimension(26), parameter pm_quadPack::nodeK51 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 6.1544483005685078886546392366796631281724348039823545274305431751687279361558658545141048781022691067898008423227288e-02_RKB , 1.2286469261071039638735981880803680553220534604978373842389353789270883496885841582643884994633105537597765980412320e-01_RKB , 1.8371893942104889201596988875952841578528447834990555215034512653236752851109815617651867160645591242103823539931527e-01_RKB , 2.4386688372098843204519036279745158640563315632598447642113565325038747278585595067977636776325034060327548499765742e-01_RKB , 3.0308953893110783016747890998033932920041937876655194685731578452573120372337209717349617882111662416355753711853559e-01_RKB , 3.6117230580938783773582173012764066742207834704337506979457877784674538239569654860329531506093761400789294612122812e-01_RKB , 4.1788538219303774885181439459457248709336998140069528034955785068796932076966599548717224205109797297615032607570119e-01_RKB , 4.7300273144571496052218211500919204133181773846162729090723082769560327584128603010315684778279363544192787010704498e-01_RKB , 5.2632528433471918259962377815801017803683252320191114313002425180471455022502695302371008520604638341970901082293650e-01_RKB , 5.7766293024122296772368984161265406739573503929151825664548350776102301275263202227671659646579649084013116066120581e-01_RKB , 6.2681009901031741278812268162451788101954628995068510806525222008437260184181183053045236423845198752346149030569920e-01_RKB , 6.7356636847346836448512063324762217588341672807274931705965696177828773684928421158196368568030932194044282149314388e-01_RKB , 7.1776640681308438818665407977329778059771167555515582423493486823991612820974965089522905953765860328116692570706602e-01_RKB , 7.5925926303735763057728286520436097638752201889833412091838973544501862882026240760763679724185230331463919586229073e-01_RKB , 7.9787379799850005941041090499430656940863230009338267661706934499488650817643824077118950314443984031474353711531825e-01_RKB , 8.3344262876083400142102110869356956946096411382352078602086471546171813247709012525322973947759168107133491065937347e-01_RKB , 8.6584706529327559544899696958834008820284409402823690293965213246691432948180280120756708738064779055576005302835351e-01_RKB , 8.9499199787827536885104200678280495417455484975358390306170168295917151090119945137118600693039178162093726882638296e-01_RKB , 9.2074711528170156174634608454633063157457035996277199700642836501131385042631212407808952281702820179915510491592339e-01_RKB , 9.4297457122897433941401116965847053190520157060899014192745249713729532254404926130890521815127348327109666786665572e-01_RKB , 9.6161498642584251241813003366016724169212642963709676666624520141292893281185666917636407790823210892689040877316178e-01_RKB , 9.7666392145951751149831538647959406774537055531440674467098742731616386753588055389644670948300617866819865983054648e-01_RKB , 9.8803579453407724763733101457740622707248415209160748131449972199405186821347293686245404742032360498210710718706868e-01_RKB , 9.9555696979049809790878494689390161725756264940480817121080493113293348134372793448728802635294700756868258870429256e-01_RKB , 9.9926210499260983419345748654034059370452496042279618586228697762904524428167719073818746102238075978747461480736921e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 51-points Kronrod extension to the 25-points Gauss quadrature rule.
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.
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.
Definition at line 1626 of file pm_quadPack.F90.
| real(RKB), dimension(31), parameter pm_quadPack::nodeK61 = [ 0.0000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000e+00_RKB , 5.1471842555317695833025213166722573749141453666569564255160843987964755210427109055870090707285485841217089963590678e-02_RKB , 1.0280693796673703014709675131800059247190133296515840552101946914632788253917872738234797140786490207720254922664913e-01_RKB , 1.5386991360858354696379467274325592041855197124433846171896298291578714851081610139692310651074078557990111754952062e-01_RKB , 2.0452511668230989143895767100202470952410426459556377447604465028350321894663245495592565235317147819577892124850607e-01_RKB , 2.5463692616788984643980512981780510788278930330251842616428597508896353156907880290636628138423620257595521678255758e-01_RKB , 3.0407320227362507737267710719925655353115778980946272844421536998312150442387767304001423699909778588529370119457430e-01_RKB , 3.5270472553087811347103720708937386065363100802142562659418446890026941623319107866436039675211352945165817827083104e-01_RKB , 4.0040125483039439253547621154266063361104593297078395983186610656429170689311759061175527015710247383961903284673474e-01_RKB , 4.4703376953808917678060990032285400016240759386142440975447738172761535172858420700400688872124189834257262048739699e-01_RKB , 4.9248046786177857499369306120770879564426564096318697026073340982988422546396352776837047452262025983265531109327026e-01_RKB , 5.3662414814201989926416979331107279416417800693029710545274348291201490861897837863114116009718990258091585830703557e-01_RKB , 5.7934523582636169175602493217254049590705158881215289208126016612312833567812241903809970751783808208940322061083509e-01_RKB , 6.2052618298924286114047755643118929920736469282952813259505117012433531497488911774115258445532782106478789996137481e-01_RKB , 6.6006106412662696137005366814927075303835037480883390955067197339904937499734522076788020517029688190998858739703079e-01_RKB , 6.9785049479331579693229238802664006838235380065395465637972284673997672124315996069538163644008904690545069439941341e-01_RKB , 7.3379006245322680472617113136952764566938172775468549208701399518300016463613325382024664531597318795933262446521430e-01_RKB , 7.6777743210482619491797734097450313169488361723290845320649438736515857017299504505260960258623968420224697596501719e-01_RKB , 7.9972783582183908301366894232268324073569842937778450923647349548686662567326007229195202524185356472023967927713548e-01_RKB , 8.2956576238276839744289811973250191643906869617034167880695298345365650658958163508295244350814016004371545455777732e-01_RKB , 8.5720523354606109895865851065894385682080017062359612850504551739119887225712932688031120704657195642614071367390794e-01_RKB , 8.8256053579205268154311646253022559005668914714648423206832605312161626269519165572921583828573210485349058106849548e-01_RKB , 9.0557330769990779854652255892595831956897536366222841356404766397803760239449631913585074426842574155323901785046522e-01_RKB , 9.2620004742927432587932427708047400408647453682532906091103713367942299565110232681677288015055886244486106298320068e-01_RKB , 9.4437444474855997941583132403743912158564371496498093181748940139520917000657342753448871376849848523800667868447591e-01_RKB , 9.6002186496830751221687102558179766293035921740392339948566167242493995770706842922718944370380002378239172677454384e-01_RKB , 9.7311632250112626837469386842370688488763796428343933853755850185624118958166838288308561708261486365954975485787212e-01_RKB , 9.8366812327974720997003258160566280194031785470971136351718001015114429536479104370207597166035471368057762560137209e-01_RKB , 9.9163099687040459485862836610948572485050033374616325510019923349807489603260796605556191495843575227494654783755353e-01_RKB , 9.9689348407464954027163005091869528334088203811775079010809429780238769521016374081588201955806171741257405095963817e-01_RKB , 9.9948441005049063757132589570581081946887394701850801923632642830748016674843587830656468823145435723317885056396548e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the abscissas of the 61-points Kronrod extension to the 30-points Gauss quadrature rule.
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.
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.
Definition at line 1880 of file pm_quadPack.F90.
| real(RKB), dimension(5), parameter pm_quadPack::weightG10 = [ 2.9552422471475287017389299465133832942104671702685360135430802975599593821715232927035659579375421672271716440125256e-01_RKB , 2.6926671930999635509122692156946935285975993846088379580056327624215343231917927676422663670925276075559581145036870e-01_RKB , 2.1908636251598204399553493422816319245877187052267708988095654363519991065295128124268399317720219278659121687281289e-01_RKB , 1.4945134915058059314577633965769733240255663966942736783547726875323865472663001094594726463473195191400575256104544e-01_RKB , 6.6671344308688137593568809893331792857864834320158145128694881613412064084087101776785509685058877821090054714520419e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 10-points Gauss quadrature rule.
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.
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.
Definition at line 923 of file pm_quadPack.F90.
| real(RKB), dimension(8), parameter pm_quadPack::weightG15 = [ 2.0257824192556127288062019996751931483866215800947735679670411605143539875474607409339344071278803213535148267082999e-01_RKB , 1.9843148532711157645611832644383932481869255995754199348473792792912479753343426813331499916481782320766020854889310e-01_RKB , 1.8616100001556221102680056186642282450622601227792840281549572731001325550269916061894976888609932360539977709001384e-01_RKB , 1.6626920581699393355320086048120881113090018009841290732186519056355356321227851771070517429241553621484461540657185e-01_RKB , 1.3957067792615431444780479451102832252085027531551124320239112863108844454190781168076825736357133363814908889327664e-01_RKB , 1.0715922046717193501186954668586930341554371575810198068702238912187799485231579972568585713760862404439808767837506e-01_RKB , 7.0366047488108124709267416450667338466708032754330719825907292914387055512874237044840452066693939219355489858595041e-02_RKB , 3.0753241996117268354628393577204417721748144833434074264228285504237189467117168039038770732399404002516991188859473e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 15-points Gauss quadrature rule.
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.
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.
Definition at line 1130 of file pm_quadPack.F90.
| real(RKB), dimension(10), parameter pm_quadPack::weightG20 = [ 1.5275338713072585069808433195509759349194864511237859727470104981759745316273778153557248783650390593544001842813788e-01_RKB , 1.4917298647260374678782873700196943669267990408136831649621121780984442259558678069396132603521048105170913854567338e-01_RKB , 1.4209610931838205132929832506716493303451541339202030333736708298382808749793436761694922428320058260133068573666201e-01_RKB , 1.3168863844917662689849449974816313491611051114698352699643649370885435642948093314355797518397262924510598005463625e-01_RKB , 1.1819453196151841731237737771138228700504121954896877544688995202017474835051151630572868782581901744606267543092317e-01_RKB , 1.0193011981724043503675013548034987616669165602339255626197161619685232202539434647534931576947985821375859035525483e-01_RKB , 8.3276741576704748724758143222046206100177828583163290744882060785693082894079419471375190843790839349096116111932764e-02_RKB , 6.2672048334109063569506535187041606351601076578436364099584345437974811033665678644563766056832203512603253399592073e-02_RKB , 4.0601429800386941331039952274932109879090639989951536817606854561832296750987328295538920623044384976189825709675075e-02_RKB , 1.7614007139152118311861962351852816362143105543336732524349326677348419259621847817403105542146097668703716227512570e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 20-points Gauss quadrature rule.
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.
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.
Definition at line 1352 of file pm_quadPack.F90.
| real(RKB), dimension(13), parameter pm_quadPack::weightG25 = [ 1.2317605372671545120390287307905014243823362751815166539135219731691200794926142128460112517504958377310054583945994e-01_RKB , 1.2224244299031004168895951894585150583505924756305904090758008223203896721918010243033540891078906637115620156845304e-01_RKB , 1.1945576353578477222817812651290104739017670141372642551958788133518409022018773502442869720975271321374348568426235e-01_RKB , 1.1485825914571164833932554586955580864093619166818014959151499003148279667112542256534429898558156273250513652351744e-01_RKB , 1.0851962447426365311609395705011661934007758798672201615649430734883929279360844269339768350029654172135832773427565e-01_RKB , 1.0053594906705064420220689039268582698846609452814190706986904199941294815904602968195565620373258211755226681206658e-01_RKB , 9.1028261982963649811497220702891653380992558959334310970483768967017384678410526902484398142953718885872521590850372e-02_RKB , 8.0140700335001018013234959669111302290225732853675893716201462973612828934801289559457377714225318048243957479325813e-02_RKB , 6.8038333812356917207187185656707968554709494354636562615071226410003654051711473106651522969481873733098761760660898e-02_RKB , 5.4904695975835191925936891540473324160109985553111349048508498244593774678436511895711924079433444763756746828817613e-02_RKB , 4.0939156701306312655623487711645953660845783364104346504698414899297432880215512770478971055110424130123527015425511e-02_RKB , 2.6354986615032137261901815295299144935963281703322468755366165783870934008879499371529821528172928890350362464605104e-02_RKB , 1.1393798501026287947902964113234773603320526292909696448948061116189891729766743355923677112945033505688431618009664e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 25-points Gauss quadrature rule.
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.
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.
Definition at line 1589 of file pm_quadPack.F90.
| real(RKB), dimension(15), parameter pm_quadPack::weightG30 = [ 1.0285265289355884034128563670541504386837555706492822258631898667601623865660942939262884632188870916503815852709086e-01_RKB , 1.0176238974840550459642895216855404463270628948712684086426094541964251360531767494547599781978391198881693385887696e-01_RKB , 9.9593420586795267062780282103569476529869263666704277221365146183946660389908809018092299289324184705373523229592037e-02_RKB , 9.6368737174644259639468626351809865096406461430160245912994275732837534742003123724951247818104195363343093583583429e-02_RKB , 9.2122522237786128717632707087618767196913234418234107527675047001973047070094168298464052916811907158954949394100501e-02_RKB , 8.6899787201082979802387530715125702576753328743545344012222129882153582254261494247955033509639105330215477601953921e-02_RKB , 8.0755895229420215354694938460529730875892803708439299890258593706051180567026345604212402769217808080749416147400962e-02_RKB , 7.3755974737705206268243850022190734153770526037049438941269182374599399314635211710401352716638183270192254236882630e-02_RKB , 6.5974229882180495128128515115962361237442953656660378967031516042143672466094179365819913911598737439478205808271237e-02_RKB , 5.7493156217619066481721689402056128797120670721763134548715799003232147409954376925211999650950125355559974348279846e-02_RKB , 4.8402672830594052902938140422807517815271809197372736345191936791805425677102152797767439563562263454374645955072007e-02_RKB , 3.8799192569627049596801936446347692033200976766395352107732789705946970952769793919055026279035105656340228558382274e-02_RKB , 2.8784707883323369349719179611292043639588894546287496474180122608145988940013933101730206711484171554940392262251283e-02_RKB , 1.8466468311090959142302131912047269096206533968181403371298365514585599521307973654080519029675417955638095832046164e-02_RKB , 7.9681924961666056154658834746736224504806965871517212294851633569200384329013332941536616922861735209846506562158817e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 30-points Gauss quadrature rule.
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.
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.
Definition at line 1841 of file pm_quadPack.F90.
| real(RKB), dimension(4), parameter pm_quadPack::weightG7 = [ 4.1795918367346938775510204081632653061224489795918367346938775510204081632653061224489795918367346938775510204081633e-01_RKB , 3.8183005050511894495036977548897513387836508353386273475108345103070554643412970834868465934404480145031467176458536e-01_RKB , 2.7970539148927666790146777142377958248692506522659876453701403269361881043056267681324094290119761876632337521337205e-01_RKB , 1.2948496616886969327061143267908201832858740225994666397720863872465523497204230871562541816292084508948440200163443e-01_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 7-points Gauss quadrature rule.
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.
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.
Definition at line 726 of file pm_quadPack.F90.
| real(RKB), dimension(8), parameter pm_quadPack::weightK15 = [ 2.0948214108472782801299917489171426369776208022370431671299800656137515132325648616816908211675949102392971459688215e-01_RKB , 2.0443294007529889241416199923464908471651760418071835742447095312045467698546598879348374292009347554167803659293064e-01_RKB , 1.9035057806478540991325640242101368282607807545535835588544088036744058072410212679605964605106377593834568683551139e-01_RKB , 1.6900472663926790282658342659855028410624490030294424149734006755695680921619029112936702403855359908156070095656537e-01_RKB , 1.4065325971552591874518959051023792039988975724799857556174546893312708093090950408097379122415555910759700350860143e-01_RKB , 1.0479001032225018383987632254151801744375665421383061189339065133963746321576289524167571627509311333949422518201492e-01_RKB , 6.3092092629978553290700663189204286665071157211550707113605545146983997477964874928199170264504441995865872491871943e-02_RKB , 2.2935322010529224963732008058969591993560811275746992267507430254711815787976075946156368168156289483493617134063245e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 15-points Kronrod extension to the 7-points Gauss quadrature rule.
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.
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.
Definition at line 786 of file pm_quadPack.F90.
| real(RKB), dimension(11), parameter pm_quadPack::weightK21 = [ 1.4944555400291690566493646838982120374523631668747280383560851873698964478511841925721030705689540264726493367634340e-01_RKB , 1.4773910490133849137484151597206804552373162548520660451819195439885993016735696405732703959182882254268727823258502e-01_RKB , 1.4277593857706008079709427313871706088597905653190555560741004743970770449909340027811131706283756428281146832304737e-01_RKB , 1.3470921731147332592805400177170683276099191300855971406636668491320291400121282036676953159488271772384389604997640e-01_RKB , 1.2349197626206585107795810983107415951230034952864832764467994120974054238975454689681538622363738230836484113389878e-01_RKB , 1.0938715880229764189921059032580496027181329983434522007819675829826550372891432168683899432674553842507906611591517e-01_RKB , 9.3125454583697605535065465083366344390018828880760031970085038760177735672200775237414123061615827474831165614953012e-02_RKB , 7.5039674810919952767043140916190009395219382000910088173697048048430404342858495178813808730646554086856929327903059e-02_RKB , 5.4755896574351996031381300244580176373721114058333557524432615804784098927818975325116301569003298086458722055550981e-02_RKB , 3.2558162307964727478818972459389760617388939845662609571537504232714121820165498692381607605384626494546068817765276e-02_RKB , 1.1694638867371874278064396062192048396217332481931888927598147525622222058064992651806736704969967250888097490233242e-02_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 21-points Kronrod extension to the 10-points Gauss quadrature rule.
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.
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.
Definition at line 987 of file pm_quadPack.F90.
| real(RKB), dimension(16), parameter pm_quadPack::weightK31 = [ 1.0133000701479154901737479276749254677092627259659629246734858372174107615774696665932418050683956749891773195816338e-01_RKB , 1.0076984552387559504494666261756972191634838013536373069278929029488122760822761077475060185965408326901925180106227e-01_RKB , 9.9173598721791959332393173484603131059567260816713281734860095693651563064308745717056680128223790739026832596087552e-02_RKB , 9.6642726983623678505179907627589335136656568630495198973407668882934392359962841826511402504664592185391687490319950e-02_RKB , 9.3126598170825321225486872747345718561927881321317330560285879189052002874531855060114908990458716740695847509343865e-02_RKB , 8.8564443056211770647275443693774303212266732690655967817996052574877144544749814260718837576325109922207832119243346e-02_RKB , 8.3080502823133021038289247286103789601554188253368717607281604875233630643885056057630789228337088859687986285569521e-02_RKB , 7.6849680757720378894432777482659006722109101167947000584089097112470821092034084418224731527690291913686588446455555e-02_RKB , 6.9854121318728258709520077099147475786045435140671549698798093177992675624987998849748628778570667518643649536771245e-02_RKB , 6.2009567800670640285139230960802932190400004210329723569147829395618376206272317333030584268303808639229575334680414e-02_RKB , 5.3481524690928087265343147239430296771554760947116739813222888752727413616259625439714812475198987513183153639571249e-02_RKB , 4.4589751324764876608227299373279690223256649667921096570980823211805450700059906366455036418897149593261561551176267e-02_RKB , 3.5346360791375846222037948478360048122630678992420820868148023340902501837247680978434662724296810081131106317333086e-02_RKB , 2.5460847326715320186874001019653359397271745046864640508377984982400903447009185267605205778819712848080691366407461e-02_RKB , 1.5007947329316122538374763075807268094639436437387634979291759700896494746154334398961710227490402528151677469993935e-02_RKB , 5.3774798729233489877920514301276498183080402431284197876486169536848635554354599213793172596490038991436925569025913e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 31-points Kronrod extension to the 15-points Gauss quadrature rule.
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.
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.
Definition at line 1202 of file pm_quadPack.F90.
| real(RKB), dimension(21), parameter pm_quadPack::weightK41 = [ 7.6600711917999656445049901530101740827932500628670118055485349620314721456712029449597396569857880493210849110825276e-02_RKB , 7.6377867672080736705502835038061001800801036764945996714946431116936745542061941050008345047482501253320401746334511e-02_RKB , 7.5704497684556674659542775376616558263363155900414326194855223272348838596099414841886740468379707283366777797425290e-02_RKB , 7.4582875400499188986581418362487528616116493572092273080047040726969899567887364227664202642942357104526915332274625e-02_RKB , 7.3030690332786667495189417658913112760626845234552742380174250771849743831660040966804802312464527721645765620253776e-02_RKB , 7.1054423553444068305790361723210167412912159322210143921628270586407381879789525901086146473278095159807542174985045e-02_RKB , 6.8648672928521619345623411885367801715489704958239860400434264173923806029589970941711224257967651039544669425313433e-02_RKB , 6.5834597133618422111563556969397943147223506343381443709751749639944420314384296347503523810096842402960802728781816e-02_RKB , 6.2653237554781168025870122174254980585819744698897886186553324157100424088919284503451596742588386343548162830898103e-02_RKB , 5.9111400880639572374967220648594217136419365977042191748388047204015262840407696611508732839851952697839735487615776e-02_RKB , 5.5195105348285994744832372419777329194753456228153116909812131213177827707884692917845453999535518818940813085110223e-02_RKB , 5.0944573923728691932707670050344948664836365809262579747517140086119113476866735641054822574173198900379392130050979e-02_RKB , 4.6434821867497674720231880926107516842127071007077929289994127933243222585938804392953931185146446072587020288747981e-02_RKB , 4.1668873327973686263788305936894738043960843153010324860966353235271889596379726462208702081068715463576895020003842e-02_RKB , 3.6600169758200798030557240707211008487453496747498001651070009441973280061489266074044986901436324295513243878212345e-02_RKB , 3.1287306777032798958543119323800737887769280362813337359554598005322423266047996771926031069705049476071896145456496e-02_RKB , 2.5882133604951158834505067096153142999479118048674944526997797755374306421629440393392427198869345793286369198147609e-02_RKB , 2.0388373461266523598010231432754705122838627940185929365371868214433006532030353671253640300679157504987977281782909e-02_RKB , 1.4626169256971252983787960308868356163881050162249770342103474631076960029748751959380482484308382288261238476948520e-02_RKB , 8.6002698556429421986617879501023472521289227667077976622450602031426535362696437838448828009554532025301579670206091e-03_RKB , 3.0735837185205315012182932460309874880335046882543449198461628212114333665590378156706265241414469306987988292234740e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 41-points Kronrod extension to the 20-points Gauss quadrature rule.
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.
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.
Definition at line 1431 of file pm_quadPack.F90.
| real(RKB), dimension(26), parameter pm_quadPack::weightK51 = [ 6.1580818067832935078759824240064553190436936903140808056908996403358367244202623293256774502185186717703954810463664e-02_RKB , 6.1471189871425316661544131965264177586537962876885022711111683500151700796198726558483367566537422877227096643444043e-02_RKB , 6.1128509717053048305859030416292711922678552321960938357322028070390133769952032831204895569347757809858568165047769e-02_RKB , 6.0539455376045862945360267517565427162312365710457079923487043144554747810689514408013582515489930908693681447570811e-02_RKB , 5.9720340324174059979099291932561853835363045476189975483372207816149988460708299020779612375010639778624011960832019e-02_RKB , 5.8689680022394207961974175856787764139795646254828315293243700305012569486054157617049685031506591863121580010947248e-02_RKB , 5.7437116361567832853582693939506471994832856823896682976509412313367495727224381199978598247737089593472710899482737e-02_RKB , 5.5950811220412317308240686382747346820271035112771802428932791066115158268338607019365831655460314732208940609352540e-02_RKB , 5.4251129888545490144543370459875606826076838441263383072163293312936923476650934130242315028422047795830492882862973e-02_RKB , 5.2362885806407475864366712137872714887351550723707596350905793656046659248541276597504566497990926306481919129870507e-02_RKB , 5.0277679080715671963325259433440084440587630604775975142050968279743014641141402310302584542633557037153607386127936e-02_RKB , 4.7982537138836713906392255756914754983592207423271169651235865196757913880334117810235517477328110033499422471098658e-02_RKB , 4.5502913049921788909870584752660393043707768935695327316724254392794299567957035458208970599641697203261236226745020e-02_RKB , 4.2872845020170049476895792439495161101999504199883328877919242515738957655253932048951366960802592343905647433925806e-02_RKB , 4.0083825504032382074839284467075646401410549266591308713115878386835777315058451955614116158949614066927183232852042e-02_RKB , 3.7116271483415543560330625367619875995997802688047764805628702762773009669395760582294525748583875707140577080663373e-02_RKB , 3.4002130274329337836748795229551203225670528250050443083264193121524339063344855010257660547708022429300203676502386e-02_RKB , 3.0792300167387488891109020215228585600877162393292487644544830559965388047996492709248618249084851477787538356572832e-02_RKB , 2.7475317587851737802948455517811078614796013288710603199613621069727810352835469926107822047433566792405123805901196e-02_RKB , 2.4009945606953216220092489164881081392931528209659330290734972342536012282191913069778658241972047765300060007037359e-02_RKB , 2.0435371145882835456568292235938973678758006097668937220074531550163622566841885855957623103354443247806459277197725e-02_RKB , 1.6847817709128298231516667536336315840402654624706139411175769276842182270078960078544597372646532637619276509222462e-02_RKB , 1.3236229195571674813656405846976238077578084997863654732213860488560614587634395544002156258192582265590155862296710e-02_RKB , 9.4739733861741516072077105236553238716453268483726334971394029603529306140359023187904705754719643032594360138998941e-03_RKB , 5.5619321353567137580402369010655220701769295496290984052961210793810038857581724171021610100708799763006942755331129e-03_RKB , 1.9873838923303159265078518828434098894299804282505973837653346298985629336820118753523093675303476883723992297810124e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 51-points Kronrod extension to the 25-points Gauss quadrature rule.
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.
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.
Definition at line 1676 of file pm_quadPack.F90.
| real(RKB), dimension(31), parameter pm_quadPack::weightK61 = [ 5.1494729429451567558340433647099307532736880396464168074637323362474083844397567724480716864880173808112573901197920e-02_RKB , 5.1426128537459025933862879215781259829552034862395987263855824172761589259406892072066110681184224608133314131500422e-02_RKB , 5.1221547849258772170656282604944208251146952425246327553509056805511015401279553971190412722969308620984161625812560e-02_RKB , 5.0881795898749606492297473049804691853384914260919239920771942080972542646780575571132056254070929858650733836163479e-02_RKB , 5.0405921402782346840893085653585028902197018251622233664243959211066713308635283713447747907973700791599900911248852e-02_RKB , 4.9795683427074206357811569379942328539209602813696108951047392842948482646220377655098341924089250200477846596263918e-02_RKB , 4.9055434555029778887528165367238173605887405295296569579490717901328215644590555247522873065246297467067324397612445e-02_RKB , 4.8185861757087129140779492298304592605799236108429800057373350872433793583969368428942672063270298939865425225579922e-02_RKB , 4.7185546569299153945261478181099486482884807300628457194141861551725533289490897029020276525603515502104799540544222e-02_RKB , 4.6059238271006988116271735559373580594692875571824924004732379492293604006446052672252973438978639166425766841417488e-02_RKB , 4.4814800133162663192355551616723243757431392796373009889680201194063503947907899189061064792111919040540351834527742e-02_RKB , 4.3452539701356069316831728117073258074603308631703168064888805495738640839573863333942084117196541456054957383622173e-02_RKB , 4.1969810215164246147147541285969757790088656718992374820388720323852655511200365790379948462006156953358103259681948e-02_RKB , 4.0374538951535959111995279752468114216126062126030255633998289613810846761059740961836828802959573901107306640876603e-02_RKB , 3.8678945624727592950348651532281050250923629821553846790376130679337402056620700554139109487533759557982632153728099e-02_RKB , 3.6882364651821229223911065617135967736955164781030337670005198584196134970154169862584193360751243227989492571664973e-02_RKB , 3.4979338028060024137499670731467875097226912794818719972208457232177786702008744219498470603846784465175225933802357e-02_RKB , 3.2981447057483726031814191016853927510599291213858385714519347641452316582381008804994515341969205985818543200837577e-02_RKB , 3.0907257562387762472884252943092272635270458523807153426840486964022086189874056947717446328187131273807982629114591e-02_RKB , 2.8754048765041292843978785354334211144679160542074930035102280759132174815469834227854660515366003136772757344886331e-02_RKB , 2.6509954882333101610601709335075414366517579522748565770867438338472138903658077617652522759934474895733739329287706e-02_RKB , 2.4191162078080601365686370725232026760391377828182462432228943562944885267501070688006470962871743661192935455117297e-02_RKB , 2.1828035821609192297167485738338993401507296056834912773630422358720439403382559079356058602393879803560534375378340e-02_RKB , 1.9414141193942381173408951050128455851421014191431525770276066536497179079025540486072726114628763606440143557769099e-02_RKB , 1.6920889189053272627572289420322092368566703783835191139883410840546679978551861043620089451681146020853650713611444e-02_RKB , 1.4369729507045804812451432443580010195841899895001505873565899403000198662495821906144274682894222591414503342336172e-02_RKB , 1.1823015253496341742232898853250592896264406250607818326302431548265365155855182739401700032519141448997853772603766e-02_RKB , 9.2732796595177634284411468920243604212700249381931076964956469143626665557434385492325784596343112153704094886248672e-03_RKB , 6.6307039159312921733198263697501681336283882177812585973955597357837568277731921327731815844512598157843672104469554e-03_RKB , 3.8904611270998840512672018445155032785151429848864649214200101281144733676455451061226273655941038347210163533085954e-03_RKB , 1.3890136986770076245515912267596996810488412919632724534411055332301367130989865366956251556423820479579333920310978e-03_RKB ] |
A vector of type real of the highest-precision kind available by the processor, containing the weights of the 61-points Kronrod extension to the 30-points Gauss quadrature rule.
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.
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.
Definition at line 1935 of file pm_quadPack.F90.
The scalar constant object of type weps_type that indicates the use of Epsilon extrapolation method of Wynn (1961) to accelerate the convergence of integrations of functions with integrable singularities and infinities using the (Adaptive) Global Gauss-Kronrod Quadrature of getQuadErr.
See the documentation of getQuadErr for example usage.
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.
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.
Definition at line 4551 of file pm_quadPack.F90.
1.9.3