/****** Numerical Quadrature: Nodes and Weights **********

Provided by: Rob Dittmar     dittmar@stls.frb.org
Research Department
Federal Reserve Bank of St. Louis
Date: Nov 96
Provided without guarantees for public non-commercial use.

Comments: Before you consider using numerical quadrature, you should be aware
of how expensive it will be to perform a quadruple integral.  Any quadrature rule
that evaluates the integrand at N nodes, will require N*N*N*N evaluations of
the integrand for a 4-dimensional integral.  This would probably take a
prohibative amount of time if it were necessary to evaluate a large number
of integrals.

On the other hand, if you only need to do it a handful of times, the usual
approach taken with integrals over the whole space or a half space is
Gaussian-Hermite or Gaussian-Laguerre quadrature.  Again this would depend
on the form of your integral.  If it is taken over the whole of 4-th
dimensional space you will probably want to use Hermite quadrature unless
the integrand has "fat" tails.  If the integral is take over half-spaces or
the integrand has fat tails, use Laguerre quadrature.

You will have to integrate variable by variable.  Starting with a set of
N*N*N*N function values, hold the first three fixed and use quadrature to
integrate out the fourth.  You will then have N*N*N values and you can
continue the process.

I have attached two text files that contain the nodes and weights for both
Laguerre and Hermite quadrature.  I recommend a book like "Introduction to
Numerical Analysis" by Atkinson for references to issues involved in
Gaussian quadrature.
***********************************************************************/
@Hermite quadrature:@

_hermint2 =
{-7.071067811865476e-1   8.86226925452758e-1,
7.071067811865476e-1    8.86226925452758e-1};

_hermint4 =
{-1.650680123885785e0    8.13128354472452e-2,
-5.246476232752904e-1   8.04914090005513e-1,
5.246476232752904e-1    8.04914090005513e-1,
1.650680123885785e0     8.13128354472452e-2};

_hermint8 =
{-2.930637420257244e0    1.996040722113676e-4,
-1.981656756695843e0    1.707798300741347e-2,
-1.15719371244678e0     2.078023258148919e-1,
-3.811869902073221e-1   6.611470125582414e-1,
3.811869902073221e-1    6.611470125582414e-1,
1.15719371244678e0      2.078023258148919e-1,
1.981656756695843e0     1.707798300741347e-2,
2.930637420257244e0     1.996040722113676e-4};

_hermint16 =
{-4.688738939305818e0    2.654807474011183e-10,
-3.869447904860123e0    2.320980844865211e-7,
-3.176999161979956e0    2.711860092537881e-5,
-2.546202157847481e0    9.32284008624181e-4,
-1.951787990916254e0    1.288031153550997e-2,
-1.380258539198881e0    8.38100413989858e-2,
-8.22951449144656e-1    2.806474585285337e-1,
-2.734810461381524e-1   5.079294790166138e-1,
2.734810461381524e-1    5.079294790166138e-1,
8.22951449144656e-1     2.806474585285337e-1,
1.380258539198881e0     8.38100413989858e-2,
1.951787990916254e0     1.288031153550997e-2,
2.546202157847481e0     9.32284008624181e-4,
3.176999161979956e0     2.711860092537881e-5,
3.869447904860123e0     2.320980844865211e-7,
4.688738939305818e0     2.654807474011183e-10};

_hermint32 =
{-7.125813909830728e0    7.310676427384165e-23,
-6.409498149269661e0    9.23173653651829e-19,
-5.812225949515914e0    1.197344017092849e-15,
-5.275550986515881e0    4.215010211326448e-13,
-4.777164503502596e0    5.933291463396639e-11,
-4.305547953351199e0    4.098832164770897e-9,
-3.853755485471445e0    1.574167792545594e-7,
-3.417167492818571e0    3.650585129562376e-6,
-2.992490825002374e0    5.416584061819983e-5,
-2.577249537732317e0    5.36268365527972e-4,
-2.169499183606112e0    3.654890326654428e-3,
-1.767654109463201e0    1.755342883157343e-2,
-1.370376410952872e0    6.045813095591262e-2,
-9.76500463589683e-1    1.512697340766425e-1,
-5.849787654359325e-1   2.774581423025299e-1,
-1.948407415693993e-1   3.752383525928024e-1,
1.948407415693993e-1    3.752383525928024e-1,
5.849787654359325e-1    2.774581423025299e-1,
9.76500463589683e-1     1.512697340766425e-1,
1.370376410952872e0     6.045813095591262e-2,
1.767654109463201e0     1.755342883157343e-2,
2.169499183606112e0     3.654890326654428e-3,
2.577249537732317e0     5.36268365527972e-4,
2.992490825002374e0     5.416584061819983e-5,
3.417167492818571e0     3.650585129562376e-6,
3.853755485471445e0     1.574167792545594e-7,
4.305547953351199e0     4.098832164770897e-9,
4.777164503502596e0     5.933291463396639e-11,
5.275550986515881e0     4.215010211326448e-13,
5.812225949515914e0     1.197344017092849e-15,
6.409498149269661e0     9.23173653651829e-19,
7.125813909830728e0     7.310676427384165e-23};

_hermint64 =
{-1.052612316796055e1    5.535706535856943e-49,
-9.89528758682954e0     1.67974799010816e-43,
-9.37315954964672e0     3.421138011255741e-39,
-8.90724909996477e0     1.557390624629764e-35,
-8.47752908337986e0     2.549660899113e-32,
-8.07368728501023e0     1.929103595464967e-29,
-7.689540164040497e0    7.861797788925911e-27,
-7.32101303278095e0     1.911706883300643e-24,
-6.965241120551108e0    2.982862784279851e-22,
-6.620112262636028e0    3.152254566503782e-20,
-6.284011228774828e0    2.351884710675819e-18,
-5.955666326799486e0    1.280093391322438e-16,
-5.634052164349973e0    5.218623726590847e-15,
-5.318325224633271e0    1.628340730709721e-13,
-5.007779602198769e0    3.959177766947724e-12,
-4.7018156474075e0      7.615217250145451e-11,
-4.399917168228138e0    1.173616742321549e-9,
-4.101634474566657e0    1.465125316476109e-8,
-3.80657151394536e0     1.495532936727247e-7,
-3.514375935740906e0    1.258340251031185e-6,
-3.224731291992036e0    8.78849923085036e-6,
-2.937350823004622e0    5.125929135786274e-5,
-2.651972435430635e0    2.509836985130625e-4,
-2.368354588632402e0    1.036329099507578e-3,
-2.086272879881762e0    3.622586978534459e-3,
-1.805517171465545e0    1.075604050987914e-2,
-1.525889140209863e0    2.720312895368892e-2,
-1.247200156943118e0    5.873998196409943e-2,
-9.69269423071178e-1    1.084983493061868e-1,
-6.919223058100446e-1   1.716858423490837e-1,
-4.149888241210788e-1   2.32994786062678e-1,
-1.383022449870097e-1   2.71377424941304e-1,
1.383022449870097e-1    2.71377424941304e-1,
4.149888241210788e-1    2.32994786062678e-1,
6.919223058100446e-1    1.716858423490837e-1,
9.69269423071178e-1     1.084983493061868e-1,
1.247200156943118e0     5.873998196409943e-2,
1.525889140209863e0     2.720312895368892e-2,
1.805517171465545e0     1.075604050987914e-2,
2.086272879881762e0     3.622586978534459e-3,
2.368354588632402e0     1.036329099507578e-3,
2.651972435430635e0     2.509836985130625e-4,
2.937350823004622e0     5.125929135786274e-5,
3.224731291992036e0     8.78849923085036e-6,
3.514375935740906e0     1.258340251031185e-6,
3.80657151394536e0      1.495532936727247e-7,
4.101634474566657e0     1.465125316476109e-8,
4.399917168228138e0     1.173616742321549e-9,
4.7018156474075e0       7.615217250145451e-11,
5.007779602198769e0     3.959177766947724e-12,
5.318325224633271e0     1.628340730709721e-13,
5.634052164349973e0     5.218623726590847e-15,
5.955666326799486e0     1.280093391322438e-16,
6.284011228774828e0     2.351884710675819e-18,
6.620112262636028e0     3.152254566503782e-20,
6.965241120551108e0     2.982862784279851e-22,
7.32101303278095e0      1.911706883300643e-24,
7.689540164040497e0     7.861797788925911e-27,
8.07368728501023e0      1.929103595464967e-29,
8.47752908337986e0      2.549660899113e-32,
8.90724909996477e0      1.557390624629764e-35,
9.37315954964672e0      3.421138011255741e-39,
9.89528758682954e0      1.67974799010816e-43,
1.052612316796055e1     5.535706535856943e-49};

_hermint128 =
{-1.529181976688274e1    1.799065980109285e-102,
-1.47338424735893e1     2.608172402409111e-95,
-1.427398130478783e1    1.404689771315089e-89,
-1.386520698447624e1    1.261249483338538e-84,
-1.348955641262314e1    3.401230086936636e-80,
-1.313777478802765e1    3.751215868804724e-76,
-1.280431208206713e1    2.041585797243985e-72,
-1.248551258534945e1    6.214244161830314e-69,
-1.217880861983124e1    1.156155164096375e-65,
-1.188231011887831e1    1.404467257740487e-62,
-1.159457505474145e1    1.17197850121298e-59,
-1.131447164428998e1    6.993072924051958e-57,
-1.104109097601963e1    3.082077383339299e-54,
-1.077368911516144e1    1.030486252055695e-51,
-1.051164732991487e1    2.672743751736068e-49,
-1.025444392847093e1    5.480217028978797e-47,
-1.000163379893012e1    9.02804013878664e-45,
-9.75283321343917e0     1.211779534130592e-42,
-9.50770832327906e0     1.341497481764369e-40,
-9.26596630029618e0     1.238085557976368e-38,
-9.02734841339479e0     9.6167080679675e-37,
-8.79162454488869e0     6.339913526366491e-35,
-8.55858879506451e0     3.574378895879421e-33,
-8.32805592079015e0     1.735103020282061e-31,
-8.0998584215079e0      7.296545006768406e-30,
-7.873844133535435e0    2.672923620058073e-28,
-7.649874227681007e0    8.57283048376936e-27,
-7.427821529952302e0    2.418403459647665e-25,
-7.207569103387334e0    6.025984032006456e-24,
-6.989009042644775e0    1.33136785903359e-22,
-6.772041443255929e0    2.617457583934811e-21,
-6.556573515264483e0    4.594007677329723e-20,
-6.34251881700178e0     7.220107316928292e-19,
-6.129796589422163e0    1.018933230423292e-17,
-5.918331175085812e0    1.294548159339371e-16,
-5.708051508768087e0    1.484223837513856e-15,
-5.49889066897391e0     1.539049730353545e-14,
-5.29078548147718e0     1.446347321190417e-13,
-5.083676167489341e0    1.234214486600556e-12,
-4.877506030264814e0    9.58031650873586e-12,
-4.672221174932639e0    6.775780487774555e-11,
-4.467770257148584e0    4.373186659848404e-10,
-4.264104256825519e0    2.579397229426394e-9,
-4.061176273749273e0    1.392190715293518e-8,
-3.858941342344282e0    6.88458112215435e-8,
-3.657356263235308e0    3.122872986178903e-7,
-3.456379449571737e0    1.300747003238199e-6,
-3.255970786350659e0    4.979924532590987e-6,
-3.05609150120268e0     1.75404858480939e-5,
-2.856704045297405e0    5.688743760040243e-5,
-2.657771983189484e0    1.700140882628094e-4,
-2.459259890565739e0    4.685515378084114e-4,
-2.261133258973062e0    1.191563814457167e-3,
-2.063358406708566e0    2.79783940160579e-3,
-1.865902395140599e0    6.068862406925888e-3,
-1.668732949803723e0    1.216691886446934e-2,
-1.471818385674486e0    2.255431016782442e-2,
-1.275127536089158e0    3.867395481063691e-2,
-1.078629684810909e0    6.136072100449007e-2,
-8.82294500792981e-1    9.01086783764489e-2,
-6.860919752173349e-1   1.225032731641357e-1,
-4.899923604154589e-1   1.542104352983544e-1,
-2.939661103002957e-1   1.797730839077993e-1,
-9.7983821955819e-2     1.940976118640877e-1,
9.7983821955819e-2      1.940976118640877e-1,
2.939661103002957e-1    1.797730839077993e-1,
4.899923604154589e-1    1.542104352983544e-1,
6.860919752173349e-1    1.225032731641357e-1,
8.82294500792981e-1     9.01086783764489e-2,
1.078629684810909e0     6.136072100449007e-2,
1.275127536089158e0     3.867395481063691e-2,
1.471818385674486e0     2.255431016782442e-2,
1.668732949803723e0     1.216691886446934e-2,
1.865902395140599e0     6.068862406925888e-3,
2.063358406708566e0     2.79783940160579e-3,
2.261133258973062e0     1.191563814457167e-3,
2.459259890565739e0     4.685515378084114e-4,
2.657771983189484e0     1.700140882628094e-4,
2.856704045297405e0     5.688743760040243e-5,
3.05609150120268e0      1.75404858480939e-5,
3.255970786350659e0     4.979924532590987e-6,
3.456379449571737e0     1.300747003238199e-6,
3.657356263235308e0     3.122872986178903e-7,
3.858941342344282e0     6.88458112215435e-8,
4.061176273749273e0     1.392190715293518e-8,
4.264104256825519e0     2.579397229426394e-9,
4.467770257148584e0     4.373186659848404e-10,
4.672221174932639e0     6.775780487774555e-11,
4.877506030264814e0     9.58031650873586e-12,
5.083676167489341e0     1.234214486600556e-12,
5.29078548147718e0      1.446347321190417e-13,
5.49889066897391e0      1.539049730353545e-14,
5.708051508768087e0     1.484223837513856e-15,
5.918331175085812e0     1.294548159339371e-16,
6.129796589422163e0     1.018933230423292e-17,
6.34251881700178e0      7.220107316928292e-19,
6.556573515264483e0     4.594007677329723e-20,
6.772041443255929e0     2.617457583934811e-21,
6.989009042644775e0     1.33136785903359e-22,
7.207569103387334e0     6.025984032006456e-24,
7.427821529952302e0     2.418403459647665e-25,
7.649874227681007e0     8.57283048376936e-27,
7.873844133535435e0     2.672923620058073e-28,
8.0998584215079e0       7.296545006768406e-30,
8.32805592079015e0      1.735103020282061e-31,
8.55858879506451e0      3.574378895879421e-33,
8.79162454488869e0      6.339913526366491e-35,
9.02734841339479e0      9.6167080679675e-37,
9.26596630029618e0      1.238085557976368e-38,
9.50770832327906e0      1.341497481764369e-40,
9.75283321343917e0      1.211779534130592e-42,
1.000163379893012e1     9.02804013878664e-45,
1.025444392847093e1     5.480217028978797e-47,
1.051164732991487e1     2.672743751736068e-49,
1.077368911516144e1     1.030486252055695e-51,
1.104109097601963e1     3.082077383339299e-54,
1.131447164428998e1     6.993072924051958e-57,
1.159457505474145e1     1.17197850121298e-59,
1.188231011887831e1     1.404467257740487e-62,
1.217880861983124e1     1.156155164096375e-65,
1.248551258534945e1     6.214244161830314e-69,
1.280431208206713e1     2.041585797243985e-72,
1.313777478802765e1     3.751215868804724e-76,
1.348955641262314e1     3.401230086936636e-80,
1.386520698447624e1     1.261249483338538e-84,
1.427398130478783e1     1.404689771315089e-89,
1.47338424735893e1      2.608172402409111e-95,
1.529181976688274e1     1.799065980109285e-102};


@Laguerre quadrature:@
_lagint2 =
{5.85786437626905e-1     8.53553390593274e-1,
3.414213562373095e0     1.464466094067262e-1};

_lagint4 =
{3.225476896193923e-1    6.031541043416337e-1,
1.745761101158346e0     3.574186924377996e-1,
4.536620296921128e0     3.888790851500538e-2,
9.39507091230113e0      5.392947055613275e-4};

_lagint8 =
{1.70279632305101e-1     3.691885893416375e-1,
9.0370177679938e-1      4.18786780814343e-1,
2.251086629866131e0     1.757949866371718e-1,
4.266700170287659e0     3.334349226121565e-2,
7.045905402393466e0     2.794536235225673e-3,
1.0758516010181e1       9.07650877335821e-5,
1.5740678641278e1       8.48574671627253e-7,
2.286313173688927e1     1.04800117487151e-9};

_lagint16 =
{8.76494104789278e-2     2.06151714957801e-1,
4.626963289150809e-1    3.310578549508842e-1,
1.141057774831227e0     2.657957776442141e-1,
2.12928364509838e0      1.362969342963775e-1,
3.437086633893207e0     4.732892869412522e-2,
5.078018614549768e0     1.129990008033945e-2,
7.070338535048234e0     1.849070943526311e-3,
9.43831433639194e0      2.042719153082785e-4,
1.221422336886616e1     1.48445868739813e-5,
1.544152736878162e1     6.8283193308712e-7,
1.918015685675314e1     1.881024841079673e-8,
2.351590569399191e1     2.862350242973882e-10,
2.857872974288214e1     2.127079033224103e-12,
3.458339870228662e1     6.297967002517869e-15,
4.194045264768834e1     5.050473700035513e-18,
5.170116033954333e1     4.161462370372856e-22};

_lagint32 =
{4.448936583326703e-2    1.09218341952385e-1,
2.345261095196185e-1    2.104431079388132e-1,
5.768846293018865e-1    2.35213229669848e-1,
1.072448753817818e0     1.95903335972881e-1,
1.722408776444645e0     1.299837862860718e-1,
2.528336706425795e0     7.057862386571744e-2,
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