MGH10: More, Gabrow and Hillstrom example 10
In NISTnls: Nonlinear least squares examples from NIST
Description
Format
Details
Source
Examples
Description
The MGH10 data frame has 16 rows and 2 columns.
Format
This data frame contains the following columns:
- y
-
A numeric vector of response values.
- x
-
A numeric vector of input values.
Details
This problem was found to be difficult for some very
good algorithms.
See More, J. J., Garbow, B. S., and Hillstrom, K. E.
(1981). Testing unconstrained optimization software.
ACM Transactions on Mathematical Software. 7(1):
pp. 17-41.
Source
Meyer, R. R. (1970).
Theoretical and computational aspects of nonlinear
regression. In Nonlinear Programming, Rosen,
Mangasarian and Ritter (Eds).
New York, NY: Academic Press, pp. 465-486.
Examples
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val
plot(y ~ x, data = MGH10)
## check plot on log scale for shape
plot(y ~ x, data = MGH10, log = "y")
## starting values for this run are ridiculous
Try(fm1 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b1 = 2, b2 = 400000, b3 = 25000)))
Try(fm1a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10,
trace = TRUE, alg = "port",
start = c(b1 = 2, b2 = 400000, b3 = 25000)))
Try(fm2 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b1 = 0.02, b2 = 4000, b3 = 250)))
Try(fm2a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10,
trace = TRUE, alg = "port",
start = c(b1 = 0.02, b2 = 4000, b3 = 250)))
Try(fm3 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b2 = 400000, b3 = 25000),
algorithm = "plinear"))
Try(fm4 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b2 = 4000, b3 = 250),
algorithm = "plinear"))
Example output
4.515243e+15 : 2 400000 25000
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, start = c(b1 = 2, :
singular gradient
0: 2.2576214e+15: 2.00000 400000. 25000.0
1: 1.0145664e+15: 1.73774 396720. 25205.1
2: 1.7298446e+14: 1.15869 390479. 25589.1
3: 8.1439367e+13: -0.503567 390674. 24863.6
4: 1.1718825e+13: -0.454618 382885. 25811.0
5: 1.2600229e+12: -0.254665 377162. 26423.5
6: 3.0302969e+10: -0.0384114 375182. 26626.2
7: 6.4621654e+09: -0.0336407 363276. 27938.0
8: 5.2630421e+09: 0.0782266 365089. 27885.9
9: 7.5497543e+08: 0.0187004 366335. 27756.9
10: 6.8994245e+08: 0.0236504 366689. 27720.5
11: 6.8969576e+08: 0.0208720 367994. 27585.5
12: 6.8941263e+08: 0.0186784 369294. 27449.8
13: 6.8915234e+08: 0.0166257 370612. 27311.2
14: 6.8855695e+08: 0.0129023 373436. 27011.4
15: 6.8839318e+08: 0.0123215 373991. 26951.4
16: 6.8827082e+08: 0.0116952 374551. 26890.7
17: 6.8777049e+08: 0.00948390 376781. 26647.8
18: 6.8760160e+08: 0.00881958 377529. 26565.1
19: 6.8737641e+08: 0.00756032 379032. 26398.0
20: 6.8698947e+08: 0.00650637 380556. 26227.0
21: 6.8663184e+08: 0.00554073 382133. 26048.0
22: 6.8625624e+08: 0.00465776 383799. 25856.9
23: 6.8587953e+08: 0.00384221 385597. 25648.2
24: 6.8538201e+08: 0.00317026 387377. 25439.0
25: 6.8492994e+08: 0.00259020 389187. 25223.6
26: 6.8448566e+08: 0.00208497 391072. 24996.1
27: 6.8409215e+08: 0.00163997 393089. 24749.5
28: 6.8343225e+08: 0.00129030 395081. 24502.1
29: 6.8294785e+08: 0.000999117 397114. 24245.9
30: 6.8254376e+08: 0.000755929 399250. 23972.4
31: 6.8243723e+08: 0.000551145 401563. 23671.3
32: 6.7998377e+08: 0.000488562 402702. 23520.2
33: 6.7969602e+08: 0.000409937 403898. 23360.2
34: 6.7919793e+08: 0.000341603 405169. 23188.5
35: 6.7875739e+08: 0.000278159 406549. 23000.0
36: 6.7813585e+08: 0.000226732 407916. 22811.2
37: 6.7758522e+08: 0.000182808 409306. 22616.9
38: 6.7704268e+08: 0.000145090 410754. 22412.1
39: 6.7656887e+08: 0.000112374 412301. 22190.6
40: 6.7575285e+08: 8.70784e-05 413830. 21968.6
41: 6.7517451e+08: 6.63164e-05 415390. 21738.9
42: 6.7468858e+08: 4.92956e-05 417028. 21494.4
43: 6.7457622e+08: 3.52515e-05 418797. 21226.2
44: 6.7148377e+08: 3.11041e-05 419669. 21091.8
45: 6.7116010e+08: 2.57689e-05 420588. 20948.9
46: 6.7054406e+08: 2.12118e-05 421568. 20795.2
47: 6.7002342e+08: 1.70179e-05 422634. 20626.3
48: 6.6925244e+08: 1.36755e-05 423691. 20457.2
49: 6.6858782e+08: 1.08582e-05 424765. 20283.4
50: 6.6793214e+08: 8.47942e-06 425883. 20100.5
50: 6.6793214e+08: 8.47942e-06 425883. 20100.5
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, alg = "port", :
Convergence failure: iteration limit reached without convergence (10)
1693607809 : 0.02 4000.00 250.00
1217987754 : 1.179854e-02 4.836962e+03 2.933311e+02
962449882 : 6.668662e-03 5.447176e+03 3.184697e+02
260923500 : 4.652186e-03 6.105943e+03 3.432606e+02
3907159 : 5.580519e-03 6.205900e+03 3.458634e+02
763.0635 : 5.606712e-03 6.181983e+03 3.452404e+02
87.94588 : 5.609632e-03 6.181347e+03 3.452237e+02
87.94586 : 5.609637e-03 6.181346e+03 3.452236e+02
Nonlinear regression model
model: y ~ b1 * exp(b2/(x + b3))
data: MGH10
b1 b2 b3
5.610e-03 6.181e+03 3.452e+02
residual sum-of-squares: 87.95
Number of iterations to convergence: 7
Achieved convergence tolerance: 2.101e-07
0: 8.4680390e+08: 0.0200000 4000.00 250.000
1: 3.5498521e+08: 0.0238126 4055.15 246.004
2: 3.4721146e+08: 0.0127067 4078.74 233.607
3: 42034709.: 0.0253330 4133.05 243.786
4: 8082518.2: 0.0354411 4258.53 258.712
5: 389985.73: 0.0457195 4424.43 276.338
6: 101057.51: 0.0502850 4440.01 280.104
7: 30372.903: 0.0484376 4514.08 284.724
8: 27230.622: 0.0480299 4521.08 285.005
9: 26959.036: 0.0475889 4527.70 285.266
10: 26536.602: 0.0468472 4538.84 285.704
11: 26085.189: 0.0461061 4550.03 286.135
12: 25857.253: 0.0458176 4554.76 286.327
13: 25671.969: 0.0455213 4559.42 286.509
14: 25243.127: 0.0443367 4578.06 287.239
15: 24513.347: 0.0431892 4596.87 287.973
16: 23802.106: 0.0420708 4615.73 288.707
17: 23122.002: 0.0409626 4634.69 289.434
18: 22419.238: 0.0398977 4653.72 290.172
19: 21802.562: 0.0387787 4674.28 290.969
20: 21061.081: 0.0377248 4694.29 291.742
21: 20646.974: 0.0364979 4718.21 292.665
22: 19791.958: 0.0353470 4741.60 293.564
23: 19051.373: 0.0342281 4765.11 294.466
24: 18888.594: 0.0329363 4793.14 295.539
25: 17761.400: 0.0317761 4819.58 296.548
26: 17011.253: 0.0306503 4846.16 297.559
27: 15610.381: 0.0300393 4861.38 298.136
28: 15195.312: 0.0294275 4876.61 298.713
29: 14804.820: 0.0289832 4887.95 299.142
30: 14480.382: 0.0283557 4904.17 299.758
31: 14050.008: 0.0277483 4920.25 300.365
32: 13635.929: 0.0271506 4936.40 300.972
33: 13224.660: 0.0265667 4952.61 301.581
34: 12837.069: 0.0259587 4969.88 302.230
35: 12423.948: 0.0253526 4987.54 302.892
36: 12123.434: 0.0246510 5008.49 303.676
37: 11666.696: 0.0239525 5030.07 304.482
38: 11185.211: 0.0232724 5051.75 305.290
39: 10974.680: 0.0225123 5076.70 306.218
40: 10398.753: 0.0217919 5101.32 307.131
41: 9906.4748: 0.0210919 5126.07 308.046
42: 8980.4444: 0.0207346 5139.36 308.536
43: 8728.7919: 0.0203945 5151.95 309.001
44: 8498.6396: 0.0201024 5162.96 309.406
45: 8282.6423: 0.0197724 5175.54 309.868
46: 8050.4333: 0.0194534 5187.95 310.325
47: 7824.7485: 0.0191396 5200.40 310.782
48: 7606.6599: 0.0187754 5215.04 311.318
49: 7348.7720: 0.0184266 5229.41 311.845
50: 7103.3202: 0.0180840 5243.84 312.373
50: 7103.3202: 0.0180840 5243.84 312.373
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, alg = "port", :
Convergence failure: iteration limit reached without convergence (10)
1366856107 : 4.000000e+05 2.500000e+04 1.494176e-03
Error in nls(y ~ exp(b2/(x + b3)), data = MGH10, trace = TRUE, start = c(b2 = 4e+05, :
step factor 0.000488281 reduced below 'minFactor' of 0.000976562
6977343 : 4.000000e+03 2.500000e+02 5.865021e-02
23094.19 : 5.139586e+03 3.089984e+02 2.108902e-02
13515.33 : 5.593556e+03 3.255459e+02 1.181885e-02
4941.471 : 6.139263e+03 3.442822e+02 6.007070e-03
88.21938 : 6.180663e+03 3.452042e+02 5.614985e-03
87.94586 : 6.181346e+03 3.452236e+02 5.609638e-03
87.94586 : 6.181346e+03 3.452236e+02 5.609637e-03
Nonlinear regression model
model: y ~ exp(b2/(x + b3))
data: MGH10
b2 b3 .lin
6.181e+03 3.452e+02 5.610e-03
residual sum-of-squares: 87.95
Number of iterations to convergence: 6
Achieved convergence tolerance: 6.406e-08
NISTnls documentation built on May 2, 2019, 2:37 a.m.
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Description Format Details Source Examples
Description
The MGH10 data frame has 16 rows and 2 columns.
Format
This data frame contains the following columns:
- y
-
A numeric vector of response values.
- x
-
A numeric vector of input values.
Details
This problem was found to be difficult for some very good algorithms.
See More, J. J., Garbow, B. S., and Hillstrom, K. E. (1981). Testing unconstrained optimization software. ACM Transactions on Mathematical Software. 7(1): pp. 17-41.
Source
Meyer, R. R. (1970). Theoretical and computational aspects of nonlinear regression. In Nonlinear Programming, Rosen, Mangasarian and Ritter (Eds). New York, NY: Academic Press, pp. 465-486.
Examples
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 | Try <- function(expr) if (!inherits(val <- try(expr), "try-error")) val
plot(y ~ x, data = MGH10)
## check plot on log scale for shape
plot(y ~ x, data = MGH10, log = "y")
## starting values for this run are ridiculous
Try(fm1 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b1 = 2, b2 = 400000, b3 = 25000)))
Try(fm1a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10,
trace = TRUE, alg = "port",
start = c(b1 = 2, b2 = 400000, b3 = 25000)))
Try(fm2 <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b1 = 0.02, b2 = 4000, b3 = 250)))
Try(fm2a <- nls(y ~ b1 * exp(b2/(x+b3)), data = MGH10,
trace = TRUE, alg = "port",
start = c(b1 = 0.02, b2 = 4000, b3 = 250)))
Try(fm3 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b2 = 400000, b3 = 25000),
algorithm = "plinear"))
Try(fm4 <- nls(y ~ exp(b2/(x+b3)), data = MGH10, trace = TRUE,
start = c(b2 = 4000, b3 = 250),
algorithm = "plinear"))
|
Example output
4.515243e+15 : 2 400000 25000
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, start = c(b1 = 2, :
singular gradient
0: 2.2576214e+15: 2.00000 400000. 25000.0
1: 1.0145664e+15: 1.73774 396720. 25205.1
2: 1.7298446e+14: 1.15869 390479. 25589.1
3: 8.1439367e+13: -0.503567 390674. 24863.6
4: 1.1718825e+13: -0.454618 382885. 25811.0
5: 1.2600229e+12: -0.254665 377162. 26423.5
6: 3.0302969e+10: -0.0384114 375182. 26626.2
7: 6.4621654e+09: -0.0336407 363276. 27938.0
8: 5.2630421e+09: 0.0782266 365089. 27885.9
9: 7.5497543e+08: 0.0187004 366335. 27756.9
10: 6.8994245e+08: 0.0236504 366689. 27720.5
11: 6.8969576e+08: 0.0208720 367994. 27585.5
12: 6.8941263e+08: 0.0186784 369294. 27449.8
13: 6.8915234e+08: 0.0166257 370612. 27311.2
14: 6.8855695e+08: 0.0129023 373436. 27011.4
15: 6.8839318e+08: 0.0123215 373991. 26951.4
16: 6.8827082e+08: 0.0116952 374551. 26890.7
17: 6.8777049e+08: 0.00948390 376781. 26647.8
18: 6.8760160e+08: 0.00881958 377529. 26565.1
19: 6.8737641e+08: 0.00756032 379032. 26398.0
20: 6.8698947e+08: 0.00650637 380556. 26227.0
21: 6.8663184e+08: 0.00554073 382133. 26048.0
22: 6.8625624e+08: 0.00465776 383799. 25856.9
23: 6.8587953e+08: 0.00384221 385597. 25648.2
24: 6.8538201e+08: 0.00317026 387377. 25439.0
25: 6.8492994e+08: 0.00259020 389187. 25223.6
26: 6.8448566e+08: 0.00208497 391072. 24996.1
27: 6.8409215e+08: 0.00163997 393089. 24749.5
28: 6.8343225e+08: 0.00129030 395081. 24502.1
29: 6.8294785e+08: 0.000999117 397114. 24245.9
30: 6.8254376e+08: 0.000755929 399250. 23972.4
31: 6.8243723e+08: 0.000551145 401563. 23671.3
32: 6.7998377e+08: 0.000488562 402702. 23520.2
33: 6.7969602e+08: 0.000409937 403898. 23360.2
34: 6.7919793e+08: 0.000341603 405169. 23188.5
35: 6.7875739e+08: 0.000278159 406549. 23000.0
36: 6.7813585e+08: 0.000226732 407916. 22811.2
37: 6.7758522e+08: 0.000182808 409306. 22616.9
38: 6.7704268e+08: 0.000145090 410754. 22412.1
39: 6.7656887e+08: 0.000112374 412301. 22190.6
40: 6.7575285e+08: 8.70784e-05 413830. 21968.6
41: 6.7517451e+08: 6.63164e-05 415390. 21738.9
42: 6.7468858e+08: 4.92956e-05 417028. 21494.4
43: 6.7457622e+08: 3.52515e-05 418797. 21226.2
44: 6.7148377e+08: 3.11041e-05 419669. 21091.8
45: 6.7116010e+08: 2.57689e-05 420588. 20948.9
46: 6.7054406e+08: 2.12118e-05 421568. 20795.2
47: 6.7002342e+08: 1.70179e-05 422634. 20626.3
48: 6.6925244e+08: 1.36755e-05 423691. 20457.2
49: 6.6858782e+08: 1.08582e-05 424765. 20283.4
50: 6.6793214e+08: 8.47942e-06 425883. 20100.5
50: 6.6793214e+08: 8.47942e-06 425883. 20100.5
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, alg = "port", :
Convergence failure: iteration limit reached without convergence (10)
1693607809 : 0.02 4000.00 250.00
1217987754 : 1.179854e-02 4.836962e+03 2.933311e+02
962449882 : 6.668662e-03 5.447176e+03 3.184697e+02
260923500 : 4.652186e-03 6.105943e+03 3.432606e+02
3907159 : 5.580519e-03 6.205900e+03 3.458634e+02
763.0635 : 5.606712e-03 6.181983e+03 3.452404e+02
87.94588 : 5.609632e-03 6.181347e+03 3.452237e+02
87.94586 : 5.609637e-03 6.181346e+03 3.452236e+02
Nonlinear regression model
model: y ~ b1 * exp(b2/(x + b3))
data: MGH10
b1 b2 b3
5.610e-03 6.181e+03 3.452e+02
residual sum-of-squares: 87.95
Number of iterations to convergence: 7
Achieved convergence tolerance: 2.101e-07
0: 8.4680390e+08: 0.0200000 4000.00 250.000
1: 3.5498521e+08: 0.0238126 4055.15 246.004
2: 3.4721146e+08: 0.0127067 4078.74 233.607
3: 42034709.: 0.0253330 4133.05 243.786
4: 8082518.2: 0.0354411 4258.53 258.712
5: 389985.73: 0.0457195 4424.43 276.338
6: 101057.51: 0.0502850 4440.01 280.104
7: 30372.903: 0.0484376 4514.08 284.724
8: 27230.622: 0.0480299 4521.08 285.005
9: 26959.036: 0.0475889 4527.70 285.266
10: 26536.602: 0.0468472 4538.84 285.704
11: 26085.189: 0.0461061 4550.03 286.135
12: 25857.253: 0.0458176 4554.76 286.327
13: 25671.969: 0.0455213 4559.42 286.509
14: 25243.127: 0.0443367 4578.06 287.239
15: 24513.347: 0.0431892 4596.87 287.973
16: 23802.106: 0.0420708 4615.73 288.707
17: 23122.002: 0.0409626 4634.69 289.434
18: 22419.238: 0.0398977 4653.72 290.172
19: 21802.562: 0.0387787 4674.28 290.969
20: 21061.081: 0.0377248 4694.29 291.742
21: 20646.974: 0.0364979 4718.21 292.665
22: 19791.958: 0.0353470 4741.60 293.564
23: 19051.373: 0.0342281 4765.11 294.466
24: 18888.594: 0.0329363 4793.14 295.539
25: 17761.400: 0.0317761 4819.58 296.548
26: 17011.253: 0.0306503 4846.16 297.559
27: 15610.381: 0.0300393 4861.38 298.136
28: 15195.312: 0.0294275 4876.61 298.713
29: 14804.820: 0.0289832 4887.95 299.142
30: 14480.382: 0.0283557 4904.17 299.758
31: 14050.008: 0.0277483 4920.25 300.365
32: 13635.929: 0.0271506 4936.40 300.972
33: 13224.660: 0.0265667 4952.61 301.581
34: 12837.069: 0.0259587 4969.88 302.230
35: 12423.948: 0.0253526 4987.54 302.892
36: 12123.434: 0.0246510 5008.49 303.676
37: 11666.696: 0.0239525 5030.07 304.482
38: 11185.211: 0.0232724 5051.75 305.290
39: 10974.680: 0.0225123 5076.70 306.218
40: 10398.753: 0.0217919 5101.32 307.131
41: 9906.4748: 0.0210919 5126.07 308.046
42: 8980.4444: 0.0207346 5139.36 308.536
43: 8728.7919: 0.0203945 5151.95 309.001
44: 8498.6396: 0.0201024 5162.96 309.406
45: 8282.6423: 0.0197724 5175.54 309.868
46: 8050.4333: 0.0194534 5187.95 310.325
47: 7824.7485: 0.0191396 5200.40 310.782
48: 7606.6599: 0.0187754 5215.04 311.318
49: 7348.7720: 0.0184266 5229.41 311.845
50: 7103.3202: 0.0180840 5243.84 312.373
50: 7103.3202: 0.0180840 5243.84 312.373
Error in nls(y ~ b1 * exp(b2/(x + b3)), data = MGH10, trace = TRUE, alg = "port", :
Convergence failure: iteration limit reached without convergence (10)
1366856107 : 4.000000e+05 2.500000e+04 1.494176e-03
Error in nls(y ~ exp(b2/(x + b3)), data = MGH10, trace = TRUE, start = c(b2 = 4e+05, :
step factor 0.000488281 reduced below 'minFactor' of 0.000976562
6977343 : 4.000000e+03 2.500000e+02 5.865021e-02
23094.19 : 5.139586e+03 3.089984e+02 2.108902e-02
13515.33 : 5.593556e+03 3.255459e+02 1.181885e-02
4941.471 : 6.139263e+03 3.442822e+02 6.007070e-03
88.21938 : 6.180663e+03 3.452042e+02 5.614985e-03
87.94586 : 6.181346e+03 3.452236e+02 5.609638e-03
87.94586 : 6.181346e+03 3.452236e+02 5.609637e-03
Nonlinear regression model
model: y ~ exp(b2/(x + b3))
data: MGH10
b2 b3 .lin
6.181e+03 3.452e+02 5.610e-03
residual sum-of-squares: 87.95
Number of iterations to convergence: 6
Achieved convergence tolerance: 6.406e-08
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