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EoFit NIST Tests
Back to EoFit Nonlinear Least-Squares.
EoFit does an excellent job solving the NIST nonlinear test cases. The 27 test cases along with EoFit results are listed below in order of their level of difficulty.
Note that NIST results are provided with 10 digits of precision. EoFit default stopping tolerances produce slightly less resolution for robust fitting of a wide array of problems. The stopping tolerances may be adjusted depending on the application to provide higher resolution when needed. The EoFit results below use the default settings.
Higher level of difficulty
NIST Data Set: MGH09.dat
Equation: y = b1*(x^ 2+x*b2) / (x^ 2+x*b3+b4);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.9280693458e-001 1.14353122e-002 1.9280693319e-001 1.14353124e-002
b2: 1.9128232873e-001 1.96332209e-001 1.9128231670e-001 1.96332168e-001
b3: 1.2305650693e-001 8.08420312e-002 1.2305648396e-001 8.08420174e-002
b4: 1.3606233068e-001 9.00255423e-002 1.3606232570e-001 9.00255430e-002
Residual Sum Of Squares: Nist: 3.0750560385e-004 EoFit: 3.0750560385e-004
Residual StdDev: Nist: 6.6279236551e-003 EoFit: 6.6279236551e-003
NIST Data Set: Thurber.dat
Equation: y = (b1 + b2*x + b3*x^ 2 + b4*x^ 3) /
(1 + b5*x + b6*x^ 2 + b7*x^ 3);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.2881396800e+003 4.66479633e+000 1.2881396802e+003 4.66479638e+000
b2: 1.4910792535e+003 3.95711561e+001 1.4910792569e+003 3.95711638e+001
b3: 5.8323836877e+002 2.86986961e+001 5.8323837109e+002 2.86987021e+001
b4: 7.5416644291e+001 5.56753703e+000 7.5416644756e+001 5.56753828e+000
b5: 9.6629502864e-001 3.13333407e-002 9.6629503063e-001 3.13333470e-002
b6: 3.9797285797e-001 1.49849282e-002 3.9797285925e-001 1.49849325e-002
b7: 4.9727297349e-002 6.58423446e-003 4.9727298151e-002 6.58423477e-003
Residual Sum Of Squares: Nist: 5.6427082397e+003 EoFit: 5.6427082397e+003
Residual StdDev: Nist: 1.3714600784e+001 EoFit: 1.3714600784e+001
NIST Data Set: BoxBOD.dat
Equation: y = b1*(1-exp(-b2*x));
Param NIST Certified StdDev EoFit Result StdDev
b1: 2.1380940889e+002 1.23545152e+001 2.1380940909e+002 1.23545152e+001
b2: 5.4723748542e-001 1.04559932e-001 5.4723748355e-001 1.04559932e-001
Residual Sum Of Squares: Nist: 1.1680088766e+003 EoFit: 1.1680088766e+003
Residual StdDev: Nist: 1.7088072423e+001 EoFit: 1.7088072423e+001
NIST Data Set: Rat42.dat
Equation: y = b1 / (1+exp(b2-b3*x));
Param NIST Certified StdDev EoFit Result StdDev
b1: 7.2462237576e+001 1.73402834e+000 7.2462237601e+001 1.73402830e+000
b2: 2.6180768402e+000 8.82952175e-002 2.6180768430e+000 8.82952227e-002
b3: 6.7359200066e-002 3.44656634e-003 6.7359200059e-002 3.44656642e-003
Residual Sum Of Squares: Nist: 8.0565229338e+000 EoFit: 8.0565229338e+000
Residual StdDev: Nist: 1.1587725499e+000 EoFit: 1.1587725499e+000
NIST Data Set: Rat43.dat
Equation: y = b1 / ((1+exp(b2-b3*x))^(1/b4));
Param NIST Certified StdDev EoFit Result StdDev
b1: 6.9964151270e+002 1.63022978e+001 6.9964151704e+002 1.63022972e+001
b2: 5.2771253025e+000 2.08287358e+000 5.2771240962e+000 2.08287375e+000
b3: 7.5962938329e-001 1.95661235e-001 7.5962927693e-001 1.95661249e-001
b4: 1.2792483859e+000 6.87619364e-001 1.2792479713e+000 6.87619304e-001
Residual Sum Of Squares: Nist: 8.7864049080e+003 EoFit: 8.7864049080e+003
Residual StdDev: Nist: 2.8262414662e+001 EoFit: 2.8262414662e+001
NIST Data Set: MGH10.dat
Equation: y = b1 * exp(b2/(x+b3));
Param NIST Certified StdDev EoFit Result StdDev
b1: 5.6096364710e-003 1.56878925e-004 5.6096364235e-003 1.57168227e-004
b2: 6.1813463463e+003 2.33090211e+001 6.1813463533e+003 2.33674943e+001
b3: 3.4522363462e+002 7.84861035e-001 3.4522363486e+002 7.89072185e-001
Residual Sum Of Squares: Nist: 8.7945855171e+001 EoFit: 8.7945855171e+001
Residual StdDev: Nist: 2.6009740065e+000 EoFit: 2.6009740065e+000
NIST Data Set: Eckerle4.dat
Equation: y = (b1/b2) * exp(-0.5*((x-b3)/b2)^2);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.5543827178e+000 1.54080512e-002 1.5543827182e+000 1.54080513e-002
b2: 4.0888321754e+000 4.68030208e-002 4.0888321776e+000 4.68030225e-002
b3: 4.5154121844e+002 4.68005188e-002 4.5154121844e+002 4.68005189e-002
Residual Sum Of Squares: Nist: 1.4635887487e-003 EoFit: 1.4635887487e-003
Residual StdDev: Nist: 6.7629245447e-003 EoFit: 6.7629245447e-003
NIST Data Set: Bennett5.dat
Equation: y = b1 * (b2+x)^(-1/b3);
Param NIST Certified StdDev EoFit Result StdDev
b1: -2.5235058043e+003 2.97151754e+002 -2.5235361977e+003 2.97817081e+002
b2: 4.6736564644e+001 1.24488719e+000 4.6736691971e+001 1.24806619e+000
b3: 9.3218483193e-001 2.02722994e-002 9.3218275847e-001 2.03137517e-002
Residual Sum Of Squares: Nist: 5.2404744073e-004 EoFit: 5.2404744076e-004
Residual StdDev: Nist: 1.8629312528e-003 EoFit: 1.8629312528e-003
Average level of difficulty
NIST Data Set: Kirby2.dat
Equation: y = (b1 + b2*x + b3*x^2) /
(1 + b4*x + b5*x^2);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.6745063063e+000 8.79896343e-002 1.6745061730e+000 8.79896337e-002
b2: -1.3927397867e-001 4.11820414e-003 -1.3927397327e-001 4.11820418e-003
b3: 2.5961181191e-003 4.18565205e-005 2.5961180721e-003 4.18565222e-005
b4: -1.7241811870e-003 5.89318974e-005 -1.7241812270e-003 5.89319046e-005
b5: 2.1664802578e-005 2.01297619e-007 2.1664802325e-005 2.01297617e-007
Residual Sum Of Squares: Nist: 3.9050739624e+000 EoFit: 3.9050739624e+000
Residual StdDev: Nist: 1.6354535131e-001 EoFit: 1.6354535131e-001
NIST Data Set: Hahn1.dat
Equation: y = (b1 + b2*x + b3*x^2 + b4*x^3) /
(1 + b5*x + b6*x^2 + b7*x^3);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.0776351733e+000 1.70701547e-001 1.0776344815e+000 2.48225424e-001
b2: -1.2269296921e-001 1.20002892e-002 -1.2269292205e-001 1.97840232e-002
b3: 4.0863750610e-003 2.25083149e-004 4.0863742407e-003 4.16316371e-004
b4: -1.4262662514e-006 2.75780377e-007 -1.4262655712e-006 2.82352673e-008
b5: -5.7609940901e-003 2.47128882e-004 -5.7609938072e-003 9.79206737e-004
b6: 2.4053735503e-004 1.04493738e-005 2.4053731549e-004 9.24237037e-006
b7: -1.2314450199e-007 1.30273353e-008 -1.2314446641e-007 8.38009787e-008
Residual Sum Of Squares: Nist: 1.5324382854e+000 EoFit: 1.5324382854e+000
Residual StdDev: Nist: 8.1803852243e-002 EoFit: 8.1803852243e-002
NIST Data Set: Nelson.dat
Equation: y = b1 - b2*x1 * exp(-b3*x2);
Param NIST Certified StdDev EoFit Result StdDev
b1: 2.5906836021e+000 1.91499964e-002 2.5906836225e+000 1.91501112e-002
b2: 5.6177717026e-009 6.11240965e-009 5.6177862088e-009 6.11678481e-009
b3: -5.7701013174e-002 3.95723665e-003 -5.7701003783e-002 3.96320126e-003
Residual Sum Of Squares: Nist: 3.7976833176e+000 EoFit: 3.7976833176e+000
Residual StdDev: Nist: 1.7430280130e-001 EoFit: 1.7430280130e-001
NIST Data Set: MGH17.dat
Equation: y = b1 + b2*exp(-x*b4) + b3*exp(-x*b5);
Param NIST Certified StdDev EoFit Result StdDev
b1: 3.7541005211e-001 2.07231536e-003 3.7541004571e-001 2.07231673e-003
b2: 1.9358469127e+000 2.20316692e-001 1.9358461063e+000 2.20316582e-001
b3: -1.4646871366e+000 2.21757077e-001 -1.4646863259e+000 2.21756968e-001
b4: 1.2867534640e-002 4.48613581e-004 1.2867533033e-002 4.48613732e-004
b5: 2.2122699662e-002 8.94719966e-004 2.2122702982e-002 8.94720732e-004
Residual Sum Of Squares: Nist: 5.4648946975e-005 EoFit: 5.4648946975e-005
Residual StdDev: Nist: 1.3970497866e-003 EoFit: 1.3970497866e-003
NIST Data Set: Lanczos1.dat
Equation: y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x);
Param NIST Certified StdDev EoFit Result StdDev
b1: 9.5100000027e-002 5.33473042e-011 9.5100000028e-002 5.33888787e-011
b2: 1.0000000001e+000 2.74730382e-010 1.0000000001e+000 2.74944247e-010
b3: 8.6070000013e-001 1.35760622e-010 8.6070000014e-001 1.35867357e-010
b4: 3.0000000002e+000 3.33082531e-010 3.0000000002e+000 3.33343123e-010
b5: 1.5575999998e+000 1.88157314e-010 1.5575999998e+000 1.88304929e-010
b6: 5.0000000001e+000 1.10575005e-010 5.0000000001e+000 1.10661952e-010
Residual Sum Of Squares: Nist: 1.4307867721e-025 EoFit: 1.4329520047e-025
Residual StdDev: Nist: 8.9156129349e-014 EoFit: 8.9223564548e-014
Syntax: NIST
Reads NIST file, performs fit, and reports comparison
with NIST certified values.
NIST Data Set: Gauss3.dat
Equation: y = b1*exp( -b2*x ) + b3*exp( -(x-b4)^2 / b5^2 )
+ b6*exp( -(x-b7)^2 / b8^ 2 );
Param NIST Certified StdDev EoFit Result StdDev
b1: 9.8940368970e+001 5.30051928e-001 9.8940368994e+001 5.30051932e-001
b2: 1.0945879335e-002 1.25540589e-004 1.0945879336e-002 1.25540591e-004
b3: 1.0069553078e+002 8.12565873e-001 1.0069553077e+002 8.12565951e-001
b4: 1.1163619459e+002 3.53178598e-001 1.1163619455e+002 3.53178595e-001
b5: 2.3300500029e+001 3.65847830e-001 2.3300499970e+001 3.65847805e-001
b6: 7.3705031418e+001 1.20912391e+000 7.3705031578e+001 1.20912377e+000
b7: 1.4776164251e+002 4.04881834e-001 1.4776164246e+002 4.04881830e-001
b8: 1.9668221230e+001 3.78066343e-001 1.9668221261e+001 3.78066379e-001
Residual Sum Of Squares: Nist: 1.2444846360e+003 EoFit: 1.2444846360e+003
Residual StdDev: Nist: 2.2677077625e+000 EoFit: 2.2677077625e+000
NIST Data Set: Misra1c.dat
Equation: y = b1 * (1-(1+2*b2*x)^(-.5));
Param NIST Certified StdDev EoFit Result StdDev
b1: 6.3642725809e+002 4.66383266e+000 6.3642725276e+002 4.66383027e+000
b2: 2.0813627256e-004 1.77284232e-006 2.0813627459e-004 1.77284140e-006
Residual Sum Of Squares: Nist: 4.0966836971e-002 EoFit: 4.0966836971e-002
Residual StdDev: Nist: 5.8428615257e-002 EoFit: 5.8428615257e-002
NIST Data Set: Misra1d.dat
Equation: y = b1*b2*x*((1+b2*x)^(-1));
Param NIST Certified StdDev EoFit Result StdDev
b1: 4.3736970754e+002 3.64891743e+000 4.3736970786e+002 3.64891748e+000
b2: 3.0227324449e-004 2.93343545e-006 3.0227324424e-004 2.93343545e-006
Residual Sum Of Squares: Nist: 5.6419295283e-002 EoFit: 5.6419295283e-002
Residual StdDev: Nist: 6.8568272111e-002 EoFit: 6.8568272111e-002
NIST Data Set: Roszman1.dat
Equation: pi = 3.141592653589793238462643383279E0;
y = b1 - b2*x - atan(b3/(x-b4))/pi;
Param NIST Certified StdDev EoFit Result StdDev
b1: 2.0196866396e-001 1.91726660e-002 2.0196868395e-001 1.91738167e-002
b2: -6.1953516256e-006 3.20589317e-006 -6.1953549966e-006 3.20602532e-006
b3: 1.2044556708e+003 7.40509831e+001 1.2044556062e+003 7.40578355e+001
b4: -1.8134269537e+002 4.95735138e+001 -1.8134264174e+002 4.95774266e+001
Residual Sum Of Squares: Nist: 4.9484847331e-004 EoFit: 4.9484847331e-004
Residual StdDev: Nist: 4.8542984060e-003 EoFit: 4.8542984060e-003
NIST Data Set: ENSO.dat
Equation: pi = 3.141592653589793238462643383279E0;
y = b1 + b2*cos( 2*pi*x/12 ) + b3*sin( 2*pi*x/12 ) + b5*cos( 2*pi*x/b4 ) + b6*sin( 2*pi*x/b4 )
+ b8*cos( 2*pi*x/b7 ) + b9*sin( 2*pi*x/b7 );
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.0510749193e+001 1.74888325e-001 1.0510749103e+001 1.74888323e-001
b2: 3.0762128085e+000 2.43100521e-001 3.0762128168e+000 2.43100523e-001
b3: 5.3280138227e-001 2.43546866e-001 5.3280140877e-001 2.43546868e-001
b4: 4.4311088700e+001 9.44080260e-001 4.4311090985e+001 9.44080331e-001
b5: -1.6231428586e+000 2.80783696e-001 -1.6231425703e+000 2.80783783e-001
b6: 5.2554493756e-001 4.80737011e-001 5.2554590057e-001 4.80736937e-001
b7: 2.6887614440e+001 4.16129391e-001 2.6887614043e+001 4.16129378e-001
b8: 2.1232288488e-001 5.14600229e-001 2.1232262387e-001 5.14600221e-001
b9: 1.4966870418e+000 2.54344689e-001 1.4966870465e+000 2.54344648e-001
Residual Sum Of Squares: Nist: 7.8853978668e+002 EoFit: 7.8853978668e+002
Residual StdDev: Nist: 2.2269642403e+000 EoFit: 2.2269642403e+000
Lower level of difficulty
NIST Data Set: Misra1a.dat
Equation: y = b1*(1-exp(-b2*x));
Param NIST Certified StdDev EoFit Result StdDev
b1: 2.3894212918e+002 2.70700752e+000 2.3894212902e+002 2.70700702e+000
b2: 5.5015643181e-004 7.26686884e-006 5.5015643224e-004 7.26686752e-006
Residual Sum Of Squares: Nist: 1.2455138894e-001 EoFit: 1.2455138894e-001
Residual StdDev: Nist: 1.0187876330e-001 EoFit: 1.0187876330e-001
NIST Data Set: Misra1b.dat
Equation: y = b1 * (1-(1+b2*x/2)^(-2));
Param NIST Certified StdDev EoFit Result StdDev
b1: 3.3799746163e+002 3.16439502e+000 3.3799746100e+002 3.16439645e+000
b2: 3.9039091287e-004 4.25473218e-006 3.9039091372e-004 4.25473417e-006
Residual Sum Of Squares: Nist: 7.5464681533e-002 EoFit: 7.5464681533e-002
Residual StdDev: Nist: 7.9301471998e-002 EoFit: 7.9301471998e-002
NIST Data Set: Chwirut1.dat
Equation: y = exp(-b1*x)/(b2+b3*x);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.9027818370e-001 2.19385570e-002 1.9027818844e-001 2.19385584e-002
b2: 6.1314004477e-003 3.45000251e-004 6.1314005028e-003 3.45000274e-004
b3: 1.0530908399e-002 7.92818477e-004 1.0530908266e-002 7.92818540e-004
Residual Sum Of Squares: Nist: 2.3844771393e+003 EoFit: 2.3844771393e+003
Residual StdDev: Nist: 3.3616721320e+000 EoFit: 3.3616721320e+000
NIST Data Set: Chwirut2.dat
Equation: y = exp(-b1*x)/(b2+b3*x);
Param NIST Certified StdDev EoFit Result StdDev
b1: 1.6657666537e-001 3.83032868e-002 1.6657655489e-001 3.83032780e-002
b2: 5.1653291286e-003 6.66216051e-004 5.1653276025e-003 6.66216125e-004
b3: 1.2150007096e-002 1.53042348e-003 1.2150011107e-002 1.53042356e-003
Residual Sum Of Squares: Nist: 5.1304802941e+002 EoFit: 5.1304802941e+002
Residual StdDev: Nist: 3.1717133040e+000 EoFit: 3.1717133040e+000
NIST Data Set: Lanczos3.dat
Equation: y = b1*exp(-b2*x) + b3*exp(-b4*x) + b5*exp(-b6*x);
Param NIST Certified StdDev EoFit Result StdDev
b1: 8.6816414977e-002 1.71979089e-002 8.6816446894e-002 1.71983493e-002
b2: 9.5498101505e-001 9.70416245e-002 9.5498118637e-001 9.70440195e-002
b3: 8.4400777463e-001 4.14886633e-002 8.4400787779e-001 4.14898640e-002
b4: 2.9515951832e+000 1.07663125e-001 2.9515954129e+000 1.07666056e-001
b5: 1.5825685901e+000 5.83715763e-002 1.5825684551e+000 5.83732265e-002
b6: 4.9863565084e+000 3.44364030e-002 4.9863565923e+000 3.44373852e-002
Residual Sum Of Squares: Nist: 1.6117193594e-008 EoFit: 1.6117193594e-008
Residual StdDev: Nist: 2.9923229172e-005 EoFit: 2.9923229172e-005
NIST Data Set: Gauss1.dat
Equation: y = b1*exp( -b2*x ) + b3*exp( -(x-b4)^2 / b5^2 )
+ b6*exp( -(x-b7)^2 / b8^2 );
Param NIST Certified StdDev EoFit Result StdDev
b1: 9.8778210871e+001 5.75273127e-001 9.8778210645e+001 5.75273131e-001
b2: 1.0497276517e-002 1.14062890e-004 1.0497276494e-002 1.14062893e-004
b3: 1.0048990633e+002 5.88317758e-001 1.0048990622e+002 5.88317757e-001
b4: 6.7481111276e+001 1.04605934e-001 6.7481111230e+001 1.04605934e-001
b5: 2.3129773360e+001 1.74399511e-001 2.3129773436e+001 1.74399513e-001
b6: 7.1994503004e+001 6.26227939e-001 7.1994502984e+001 6.26227935e-001
b7: 1.7899805021e+002 1.24369882e-001 1.7899805021e+002 1.24369882e-001
b8: 1.8389389025e+001 2.01343128e-001 1.8389389015e+001 2.01343130e-001
Residual Sum Of Squares: Nist: 1.3158222432e+003 EoFit: 1.3158222432e+003
Residual StdDev: Nist: 2.3317980180e+000 EoFit: 2.3317980180e+000
NIST Data Set: Gauss2.dat
Equation: y = b1*exp( -b2*x ) + b3*exp( -(x-b4)^2 / b5^2 )
+ b6*exp( -(x-b7)^2 / b8^2 );
Param NIST Certified StdDev EoFit Result StdDev
b1: 9.9018328406e+001 5.37487669e-001 9.9018328424e+001 5.37487673e-001
b2: 1.0994945399e-002 1.33353068e-004 1.0994945406e-002 1.33353070e-004
b3: 1.0188022528e+002 5.92173158e-001 1.0188022529e+002 5.92173150e-001
b4: 1.0703095519e+002 1.50067983e-001 1.0703095517e+002 1.50067984e-001
b5: 2.3578584029e+001 2.26955951e-001 2.3578584018e+001 2.26955948e-001
b6: 7.2045589471e+001 6.17219659e-001 7.2045589403e+001 6.17219653e-001
b7: 1.5327010194e+002 1.94666743e-001 1.5327010195e+002 1.94666746e-001
b8: 1.9525972636e+001 2.64165494e-001 1.9525972696e+001 2.64165510e-001
Residual Sum Of Squares: Nist: 1.2475282092e+003 EoFit: 1.2475282092e+003
Residual StdDev: Nist: 2.2704790782e+000 EoFit: 2.2704790782e+000
NIST Data Set: DanWood.dat
Equation: y = b1*x^b2;
Param NIST Certified StdDev EoFit Result StdDev
b1: 7.6886226176e-001 1.82819739e-002 7.6886226068e-001 1.82819736e-002
b2: 3.8604055871e+000 5.17266109e-002 3.8604055900e+000 5.17266092e-002
Residual Sum Of Squares: Nist: 4.3173084083e-003 EoFit: 4.3173084083e-003
Residual StdDev: Nist: 3.2853114039e-002 EoFit: 3.2853114039e-002
Back to EoFit Nonlinear Least-Squares.