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From: leo nidas on 3 Oct 2009 20:45

Hi there,

I want to solve or b the : (X'X)*b=X'*y. I have some issues with the inverse of X'*X.
(vector b should estimate the vector [5 0.1048]. (regression coefficients))
MATLAB(R2007b).

In the beginning I used "inv" and got a warning:

Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 1.000436e-148.

b =

4.9346
0.0000 (Inspitethe warning the solution seems to approach the real values)

I also used pinv but I am not sure about the result
b =

1.0e-073 *

0.0000
0.8577 (Here the result looks weird, don't get a warning though)

I also used linfactor (Timothy A. Davis)

bnaive=linfactor (linfactor(Ximp'*Ximp),Ximp'*y')
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 3.483766e-074.
> In linfactor at 175
Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 3.483766e-074.
> In linfactor at 175

b =

4.9346
0.0000 (Again a warning but a good looking solution)

And finally ginv (Luis Frank, jan 2009.)

and got either [5.4417 0] or [1.0e-072 * 0 0.1171] without warnings.

So I am a little confused..Which is correct?...Below I give the data. Note that X=[ones(n,1) x]. where n=300.


x=
1.0e+073 *

6.0217
6.9355
6.9355
0.0000
6.9355
6.9355
0.0000
6.6810
4.5431
6.1384
0.0000
5.8016
6.2875
6.9355
4.5401
0.0000
6.9355
6.6662
6.9355
5.9197
6.9355
0.0000
6.9355
6.1304
5.6790
6.9355
6.9355
6.9355
6.9355
4.7936
0.0000
0.0000
6.6226
0.0000
0.0000
0.0000
6.9355
6.9355
4.5560
4.9384
6.9355
5.7786
0.0000
6.9355
0.0000
5.4005
6.2339
6.9355
0.0000
0.0000
5.1291
6.9355
6.9355
6.9355
6.9355
0.0000
0.0000
6.9355
6.9355
0.0000
5.1252
6.1170
0.0000
6.9355
6.9355
6.9355
6.9355
0.0000
5.0401
6.8300
4.8740
0.0000
6.6515
0.0000
5.0013
0.0000
6.9355
0.0000
4.5467
5.7678
6.9355
0.0000
6.5277
4.9925
6.9355
6.1809
5.0394
5.2346
0.0000
0.0000
6.8909
6.9355
5.6816
4.9700
0.0000
0.0000
5.0839
4.9574
0.0000
5.2523
5.1478
6.9355
6.9355
0.0000
0.0000
4.5564
5.8503
0.0000
6.9355
4.5315
4.6217
5.7917
6.9355
0.0000
6.9355
5.0533
6.9355
6.9355
0.0000
6.9355
0.0000
6.9355
6.3646
6.9355
0.0000
5.4656
0.0000
6.9355
6.9355
6.9355
6.9355
0.0000
0.0000
4.8783
6.9355
6.9355
0.0000
4.7015
4.8418
4.8581
6.9355
6.9355
6.9355
5.4954
0.0000
6.3505
6.9355
6.9355
6.9355
0.0000
6.9355
0.0000
6.9355
0.0000
5.9320
6.9355
4.8874
6.9355
6.4735
6.9355
6.9355
0.0000
5.6360
5.2314
6.9355
6.9355
5.5291
6.9355
6.8946
6.9355
6.9355
0.0000
6.7673
5.3281
0.0000
6.9355
6.9355
0.0000
6.9355
5.9098
6.9355
6.9355
0.0000
0.0000
6.9355
6.9355
0.0000
6.9355
0.0000
5.3169
5.4155
0.0000
6.9355
6.0202
0.0000
0.0000
6.9355
6.9355
6.9355
5.7784
5.9336
6.9355
5.6023
0.0000
6.9355
0.0000
6.9355
4.5746
6.9355
0.0000
6.9355
4.5628
0.0000
0.0000
4.6689
5.9038
0.0000
4.9554
4.9408
6.9355
4.5740
4.7817
6.9355
5.6149
6.9355
4.7791
4.5862
5.8192
0.0000
0.0000
4.5331
5.2861
6.9355
6.9355
5.4413
6.9355
5.3689
0.0000
4.6382
5.2439
0.0000
6.9355
0.0000
6.9355
6.9355
5.2391
6.9355
6.9355
6.9355
0.0000
6.9355
6.9355
5.6362
6.9355
0.0000
0.0000
0.0000
6.9355
6.9355
6.8841
0.0000
5.0142
6.9355
5.4128
0.0000
0.0000
0.0000
6.0928
6.5998
6.9355
0.0000
6.9355
4.7202
6.9355
4.7974
5.1762
0.0000
6.9355
6.5298
0.0000
0.0000
6.9355
6.1336
5.4558
0.0000
0.0000
5.6442
0.0000
5.2961
6.9355
5.2238
6.6430
6.9355
6.9355
6.9355
0.0000
6.9355
5.2214
6.9355
5.3165





y=

4.8111
5.2604
6.3000
4.1397
5.7850
8.0680
2.6153
7.1341
6.3813
5.1344
4.8746
6.0043
5.6167
5.6686
5.7879
4.9851
3.6736
5.5030
5.2088
4.8137
4.5450
5.3842
6.1297
5.1319
4.4700
4.7678
6.2697
5.3731
5.0480
6.1615
5.4038
4.4000
4.7029
6.3949
5.6621
4.0477
3.5602
4.6434
3.3360
6.5198
4.1286
6.4947
4.7885
7.4672
5.7104
6.4073
5.3284
5.6730
4.6356
4.8480
4.2965
5.5603
4.5348
3.5832
5.4171
3.9930
4.4879
4.5334
6.2194
5.2592
3.0510
5.2886
5.4761
5.2904
7.3873
5.0014
4.8700
6.0900
6.1828
4.2469
6.5676
6.3699
6.2629
4.0863
6.6842
5.3526
5.6865
4.5678
5.3424
5.1886
4.5782
4.1196
4.4998
3.3030
4.5550
6.1645
4.8040
6.3907
4.4269
4.4374
5.4680
5.7414
7.3257
6.1399
2.9633
4.5267
5.0329
5.7673
4.0117
5.6086
5.6665
4.2401
4.2609
5.3666
4.9009
5.3461
6.6157
5.3064
3.0179
5.4734
5.8616
2.4153
7.7829
3.8170
5.0442
5.2099
5.3179
4.8733
4.2587
6.2033
5.1428
5.3150
7.5840
5.6033
5.5261
6.4733
6.0306
4.9166
4.4991
4.8719
5.5547
6.5007
5.8086
4.2790
5.7095
5.4276
6.6512
5.2374
7.2476
4.2253
4.7902
7.4058
6.3936
5.5561
5.6328
7.4945
7.6635
5.7805
5.1112
2.7504
4.1025
6.9345
5.1199
4.9941
4.5647
5.4471
4.4141
6.7696
4.7769
5.2200
6.0108
3.3302
7.3792
5.5906
4.9745
5.3593
5.9843
5.7817
5.0402
6.0475
5.5631
5.7827
4.4960
5.1135
3.3456
4.6051
6.6434
5.9896
5.9918
5.3772
5.5311
4.5358
5.0157
4.4853
6.6768
7.7551
5.4937
6.8068
4.5231
4.2895
5.0643
4.9405
4.0419
5.7608
5.1294
5.0549
6.1442
3.8324
5.8736
6.0473
6.0278
5.8280
4.7676
5.5682
3.9522
4.2703
5.3895
8.2634
4.2123
4.6494
5.7650
5.0053
5.8436
4.5409
6.0050
7.0507
5.7757
5.6222
5.6440
5.8824
2.8991
4.4518
3.7672
5.0790
4.9698
4.8897
5.7037
6.3800
5.2329
5.6111
4.8948
6.2990
3.1272
4.5467
4.2359
4.1112
5.6829
4.6515
5.6744
4.5980
3.9540
5.2330
4.4072
6.5943
4.3242
4.0208
6.2833
6.7559
5.9868
4.6251
7.6454
6.0673
5.8195
6.1676
4.6909
2.9627
5.2211
5.4769
5.8216
4.6540
6.1510
5.9472
6.8202
5.9406
7.1331
4.8461
4.9920
7.4089
5.8690
6.6683
3.4887
3.5328
4.7221
5.7539
4.4846
5.2311
6.5396
6.0771
5.4340
4.1203
5.8677
6.3185
3.8214
6.7323
4.0473
2.8361
5.0833
4.1271
5.7618
3.8940
4.7097
4.9934
3.7886
4.9597
6.5558
5.7036
4.1438
5.5269
6.2652
7.1042
From: Petros Mpogiatzis on 3 Oct 2009 21:03
"leo nidas" <bleonidas25(a)yahoo.gr> wrote in message <ha8r6f$pf7$1(a)fred.mathworks.com>...
>
> Hi there,
>
> I want to solve or b the : (X'X)*b=X'*y. I have some issues with the inverse of X'*X.
> (vector b should estimate the vector [5 0.1048]. (regression coefficients))
> MATLAB(R2007b).
>
> In the beginning I used "inv" and got a warning:
>
> Warning: Matrix is close to singular or badly scaled.
> Results may be inaccurate. RCOND = 1.000436e-148.
>
> b =
>
> 4.9346
> 0.0000 (Inspitethe warning the solution seems to approach the real values)
>
> I also used pinv but I am not sure about the result
> b =
>
> 1.0e-073 *
>
> 0.0000
> 0.8577 (Here the result looks weird, don't get a warning though)
>
> I also used linfactor (Timothy A. Davis)
>
> bnaive=linfactor (linfactor(Ximp'*Ximp),Ximp'*y')
> Warning: Matrix is close to singular or badly scaled.
> Results may be inaccurate. RCOND = 3.483766e-074.
> > In linfactor at 175
> Warning: Matrix is close to singular or badly scaled.
> Results may be inaccurate. RCOND = 3.483766e-074.
> > In linfactor at 175
>
> b =
>
> 4.9346
> 0.0000 (Again a warning but a good looking solution)
>
> And finally ginv (Luis Frank, jan 2009.)
>
> and got either [5.4417 0] or [1.0e-072 * 0 0.1171] without warnings.
>
> So I am a little confused..Which is correct?...Below I give the data. Note that X=[ones(n,1) x]. where n=300.
>
>
> x=
> 1.0e+073 *
>
> 6.0217
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 6.9355
> 0.0000
> 6.6810
> 4.5431
> 6.1384
> 0.0000
> 5.8016
> 6.2875
> 6.9355
> 4.5401
> 0.0000
> 6.9355
> 6.6662
> 6.9355
> 5.9197
> 6.9355
> 0.0000
> 6.9355
> 6.1304
> 5.6790
> 6.9355
> 6.9355
> 6.9355
> 6.9355
> 4.7936
> 0.0000
> 0.0000
> 6.6226
> 0.0000
> 0.0000
> 0.0000
> 6.9355
> 6.9355
> 4.5560
> 4.9384
> 6.9355
> 5.7786
> 0.0000
> 6.9355
> 0.0000
> 5.4005
> 6.2339
> 6.9355
> 0.0000
> 0.0000
> 5.1291
> 6.9355
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 0.0000
> 6.9355
> 6.9355
> 0.0000
> 5.1252
> 6.1170
> 0.0000
> 6.9355
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 5.0401
> 6.8300
> 4.8740
> 0.0000
> 6.6515
> 0.0000
> 5.0013
> 0.0000
> 6.9355
> 0.0000
> 4.5467
> 5.7678
> 6.9355
> 0.0000
> 6.5277
> 4.9925
> 6.9355
> 6.1809
> 5.0394
> 5.2346
> 0.0000
> 0.0000
> 6.8909
> 6.9355
> 5.6816
> 4.9700
> 0.0000
> 0.0000
> 5.0839
> 4.9574
> 0.0000
> 5.2523
> 5.1478
> 6.9355
> 6.9355
> 0.0000
> 0.0000
> 4.5564
> 5.8503
> 0.0000
> 6.9355
> 4.5315
> 4.6217
> 5.7917
> 6.9355
> 0.0000
> 6.9355
> 5.0533
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 0.0000
> 6.9355
> 6.3646
> 6.9355
> 0.0000
> 5.4656
> 0.0000
> 6.9355
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 0.0000
> 4.8783
> 6.9355
> 6.9355
> 0.0000
> 4.7015
> 4.8418
> 4.8581
> 6.9355
> 6.9355
> 6.9355
> 5.4954
> 0.0000
> 6.3505
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 0.0000
> 6.9355
> 0.0000
> 5.9320
> 6.9355
> 4.8874
> 6.9355
> 6.4735
> 6.9355
> 6.9355
> 0.0000
> 5.6360
> 5.2314
> 6.9355
> 6.9355
> 5.5291
> 6.9355
> 6.8946
> 6.9355
> 6.9355
> 0.0000
> 6.7673
> 5.3281
> 0.0000
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 5.9098
> 6.9355
> 6.9355
> 0.0000
> 0.0000
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 0.0000
> 5.3169
> 5.4155
> 0.0000
> 6.9355
> 6.0202
> 0.0000
> 0.0000
> 6.9355
> 6.9355
> 6.9355
> 5.7784
> 5.9336
> 6.9355
> 5.6023
> 0.0000
> 6.9355
> 0.0000
> 6.9355
> 4.5746
> 6.9355
> 0.0000
> 6.9355
> 4.5628
> 0.0000
> 0.0000
> 4.6689
> 5.9038
> 0.0000
> 4.9554
> 4.9408
> 6.9355
> 4.5740
> 4.7817
> 6.9355
> 5.6149
> 6.9355
> 4.7791
> 4.5862
> 5.8192
> 0.0000
> 0.0000
> 4.5331
> 5.2861
> 6.9355
> 6.9355
> 5.4413
> 6.9355
> 5.3689
> 0.0000
> 4.6382
> 5.2439
> 0.0000
> 6.9355
> 0.0000
> 6.9355
> 6.9355
> 5.2391
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 6.9355
> 5.6362
> 6.9355
> 0.0000
> 0.0000
> 0.0000
> 6.9355
> 6.9355
> 6.8841
> 0.0000
> 5.0142
> 6.9355
> 5.4128
> 0.0000
> 0.0000
> 0.0000
> 6.0928
> 6.5998
> 6.9355
> 0.0000
> 6.9355
> 4.7202
> 6.9355
> 4.7974
> 5.1762
> 0.0000
> 6.9355
> 6.5298
> 0.0000
> 0.0000
> 6.9355
> 6.1336
> 5.4558
> 0.0000
> 0.0000
> 5.6442
> 0.0000
> 5.2961
> 6.9355
> 5.2238
> 6.6430
> 6.9355
> 6.9355
> 6.9355
> 0.0000
> 6.9355
> 5.2214
> 6.9355
> 5.3165
>
>
>
>
>
> y=
>
> 4.8111
> 5.2604
> 6.3000
> 4.1397
> 5.7850
> 8.0680
> 2.6153
> 7.1341
> 6.3813
> 5.1344
> 4.8746
> 6.0043
> 5.6167
> 5.6686
> 5.7879
> 4.9851
> 3.6736
> 5.5030
> 5.2088
> 4.8137
> 4.5450
> 5.3842
> 6.1297
> 5.1319
> 4.4700
> 4.7678
> 6.2697
> 5.3731
> 5.0480
> 6.1615
> 5.4038
> 4.4000
> 4.7029
> 6.3949
> 5.6621
> 4.0477
> 3.5602
> 4.6434
> 3.3360
> 6.5198
> 4.1286
> 6.4947
> 4.7885
> 7.4672
> 5.7104
> 6.4073
> 5.3284
> 5.6730
> 4.6356
> 4.8480
> 4.2965
> 5.5603
> 4.5348
> 3.5832
> 5.4171
> 3.9930
> 4.4879
> 4.5334
> 6.2194
> 5.2592
> 3.0510
> 5.2886
> 5.4761
> 5.2904
> 7.3873
> 5.0014
> 4.8700
> 6.0900
> 6.1828
> 4.2469
> 6.5676
> 6.3699
> 6.2629
> 4.0863
> 6.6842
> 5.3526
> 5.6865
> 4.5678
> 5.3424
> 5.1886
> 4.5782
> 4.1196
> 4.4998
> 3.3030
> 4.5550
> 6.1645
> 4.8040
> 6.3907
> 4.4269
> 4.4374
> 5.4680
> 5.7414
> 7.3257
> 6.1399
> 2.9633
> 4.5267
> 5.0329
> 5.7673
> 4.0117
> 5.6086
> 5.6665
> 4.2401
> 4.2609
> 5.3666
> 4.9009
> 5.3461
> 6.6157
> 5.3064
> 3.0179
> 5.4734
> 5.8616
> 2.4153
> 7.7829
> 3.8170
> 5.0442
> 5.2099
> 5.3179
> 4.8733
> 4.2587
> 6.2033
> 5.1428
> 5.3150
> 7.5840
> 5.6033
> 5.5261
> 6.4733
> 6.0306
> 4.9166
> 4.4991
> 4.8719
> 5.5547
> 6.5007
> 5.8086
> 4.2790
> 5.7095
> 5.4276
> 6.6512
> 5.2374
> 7.2476
> 4.2253
> 4.7902
> 7.4058
> 6.3936
> 5.5561
> 5.6328
> 7.4945
> 7.6635
> 5.7805
> 5.1112
> 2.7504
> 4.1025
> 6.9345
> 5.1199
> 4.9941
> 4.5647
> 5.4471
> 4.4141
> 6.7696
> 4.7769
> 5.2200
> 6.0108
> 3.3302
> 7.3792
> 5.5906
> 4.9745
> 5.3593
> 5.9843
> 5.7817
> 5.0402
> 6.0475
> 5.5631
> 5.7827
> 4.4960
> 5.1135
> 3.3456
> 4.6051
> 6.6434
> 5.9896
> 5.9918
> 5.3772
> 5.5311
> 4.5358
> 5.0157
> 4.4853
> 6.6768
> 7.7551
> 5.4937
> 6.8068
> 4.5231
> 4.2895
> 5.0643
> 4.9405
> 4.0419
> 5.7608
> 5.1294
> 5.0549
> 6.1442
> 3.8324
> 5.8736
> 6.0473
> 6.0278
> 5.8280
> 4.7676
> 5.5682
> 3.9522
> 4.2703
> 5.3895
> 8.2634
> 4.2123
> 4.6494
> 5.7650
> 5.0053
> 5.8436
> 4.5409
> 6.0050
> 7.0507
> 5.7757
> 5.6222
> 5.6440
> 5.8824
> 2.8991
> 4.4518
> 3.7672
> 5.0790
> 4.9698
> 4.8897
> 5.7037
> 6.3800
> 5.2329
> 5.6111
> 4.8948
> 6.2990
> 3.1272
> 4.5467
> 4.2359
> 4.1112
> 5.6829
> 4.6515
> 5.6744
> 4.5980
> 3.9540
> 5.2330
> 4.4072
> 6.5943
> 4.3242
> 4.0208
> 6.2833
> 6.7559
> 5.9868
> 4.6251
> 7.6454
> 6.0673
> 5.8195
> 6.1676
> 4.6909
> 2.9627
> 5.2211
> 5.4769
> 5.8216
> 4.6540
> 6.1510
> 5.9472
> 6.8202
> 5.9406
> 7.1331
> 4.8461
> 4.9920
> 7.4089
> 5.8690
> 6.6683
> 3.4887
> 3.5328
> 4.7221
> 5.7539
> 4.4846
> 5.2311
> 6.5396
> 6.0771
> 5.4340
> 4.1203
> 5.8677
> 6.3185
> 3.8214
> 6.7323
> 4.0473
> 2.8361
> 5.0833
> 4.1271
> 5.7618
> 3.8940
> 4.7097
> 4.9934
> 3.7886
> 4.9597
> 6.5558
> 5.7036
> 4.1438
> 5.5269
> 6.2652
> 7.1042


use the backslash operator:
b=(X'X)\X'y;

see ex. here:
http://blogs.mathworks.com/pick/2009/06/26/dont-let-that-inv-go-past-your-eyes-to-solve-that-system-factorize/

and here: http://blogs.mathworks.com/loren/2007/05/16/purpose-of-inv/
From: leo nidas on 4 Oct 2009 05:11

I forgot to mention that I already have tried the backslash operator and I get a warning :

Warning: Matrix is close to singular or badly scaled.
Results may be inaccurate. RCOND = 1.000436e-148.

ans =

4.9346
0.0000

Thanx for the answer! And for any other answers in advance.
From: Bruno Luong on 4 Oct 2009 05:32
"leo nidas" <bleonidas25(a)yahoo.gr> wrote in message <ha8r6f$pf7$1(a)fred.mathworks.com>...


Note that X=[ones(n,1) x]. where n=300.
>

When you do linear algebra, the first thing is to SCALE your variable correctly. Combine a vector of a magnitude 1 with x in the order of 1e73 is very bad.

I assume you want to find a linear polynomial coefficients that fit your data.

In this case work with xi := x/1e73 (xi will be between 0 and 1)

The fitting you want
y ~= b0 + b1*x becomes
y ~= [ones(n,1) x/1e73] * [b0 b1*1e73]'

solve for b2 = [b0 b1*1e73] by least square

xi = x/1e73
A2 = [ones(n,1) xi]
b2 = A2 \ y
b = [b2(1) b2(2)/1e73]

% Bruno

PS: A quick look of GINV indicates it's a not good quality code, I would say even amateurish one.
From: John D'Errico on 4 Oct 2009 05:59
"Petros Mpogiatzis" <painter(a)geo.auth.gr> wrote in message <ha8s85$4qh$1(a)fred.mathworks.com>...
> "leo nidas" <bleonidas25(a)yahoo.gr> wrote in message <ha8r6f$pf7$1(a)fred.mathworks.com>...
> >
> > Hi there,
> >
> > I want to solve or b the : (X'X)*b=X'*y. I have some issues with the inverse of X'*X.
> > (vector b should estimate the vector [5 0.1048]. (regression coefficients))
> > MATLAB(R2007b).
> >
> > In the beginning I used "inv" and got a warning:
> >
> > Warning: Matrix is close to singular or badly scaled.
> > Results may be inaccurate. RCOND = 1.000436e-148.
> >
> > b =
> >
> > 4.9346
> > 0.0000 (Inspitethe warning the solution seems to approach the real values)
> >
> > I also used pinv but I am not sure about the result
> > b =
> >
> > 1.0e-073 *
> >
> > 0.0000
> > 0.8577 (Here the result looks weird, don't get a warning though)
> >
> > I also used linfactor (Timothy A. Davis)
> >
> > bnaive=linfactor (linfactor(Ximp'*Ximp),Ximp'*y')
> > Warning: Matrix is close to singular or badly scaled.
> > Results may be inaccurate. RCOND = 3.483766e-074.
> > > In linfactor at 175
> > Warning: Matrix is close to singular or badly scaled.
> > Results may be inaccurate. RCOND = 3.483766e-074.
> > > In linfactor at 175
> >
> > b =
> >
> > 4.9346
> > 0.0000 (Again a warning but a good looking solution)
> >
> > And finally ginv (Luis Frank, jan 2009.)
> >
> > and got either [5.4417 0] or [1.0e-072 * 0 0.1171] without warnings.
> >
> > So I am a little confused..Which is correct?...Below I give the data. Note that X=[ones(n,1) x]. where n=300.
> >
> >
> > x=
> > 1.0e+073 *
> >
> > 6.0217

(snip)

> > 6.2652
> > 7.1042
>
>
> use the backslash operator:
> b=(X'X)\X'y;

NO!!!!!

Don't just blindly use backslash on the normal equations!!!!!
Once you form the normal equations, you have already
screwed yourself.

Sigh.

Don't use the normal equations to solve a linear regression
problem in the first place. Solve it as

b = X\y;

Or, solve it as

b = pinv(X)*y;

This is the correct way to solve the linear regression problem.
When you use the normal equations, as people seem to teach
each other, INCORRECTLY, to solve a linear regression problem,
you worsen the conditioning of any regression that you will
solve.

But BEFORE you do that, scale this data properly, as I show
how below.

So next, we must deal with that little, inconsequential factor
of 1e73 on your data! Look at the variable x! See the 1e73
that comes out front. It says

> > x=
> > 1.0e+073 *
> >
> > 6.0217

This means that this regression will generally return garbage,
unless you are careful.

Don't just throw such unscaled data into a linear algebra
problem, and worse, then throw the normal equations at it!
It is this that is causing the other part of your problem.

b = [ones(300,1),x/1e73]\y;
b =
4.9346
0.079934

Only now should you rescale b to compensate for the
prior scaling of x.

format long g
b(2) = b(2) * 1e73
b =
4.93457076016252
7.99340988034286e+71

See that no warning message was generated now.

John
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