S.O.S. Mathematics CyberBoard

Hello,

When calculating least-squares we use the form Xw=y
(where X is a matrix and y is a vector of appropriate dimensions).

This gives us an approximation, and it can happen that the resulting w will be such that Xw<y.

Is there a way how to make every entry in w big enough so that every Xw≥y?

And is it possible, at the same time, to make sure every entry in Xw is less that some other number?

Thank you
Best regards
hugo

Check out Fourier Motzkin elimination.

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Hello Shadow, thank you very much for your help. I tried what you suggest.

When calculating least-squares we use the form Xw=y.

I have created this (10x5)Matrix X:
2 4 1 3 3
3 5 2 2 3
6 6 1 1 4
0 7 3 7 5
2 8 4 10 2
1 0 5 3 3
1 6 0 0 7
2 7 7 5 0
3 8 8 4 7
4 9 0 3 8

and this (10x1) vector y1.....y10:
105
45
36
78
20
60
30
20
79
94

nvar = 5;
then, when I Fourier Motzkin elimination (fourmotz) in Matlab: fourmotz(X,y,nvar)

.....I get this Matrix M:
0 7 3 7 5
1 0 5 3 3
6 6 1 1 4
3 5 2 2 3
2 4 1 3 3
4 9 0 3 8
3 8 8 4 7
2 7 7 5 0
2 8 4 10 2
1 6 0 0 7

What shall I do then?

I tried to calculate least square from this M Matrix, giving me w1....w5:
5.5543
-5.8469
0.6521
6.3349
11.9513

and then create a table by multiplication of these w numbers with transposed M, like:
w1*Mn1,m1..........w1*Mn1,m10
.............................
.............................
.............................
w5*Mn5,m1..........w5*Mn5,m10

and do sum of its columns, thinking I would get values same or bigger than y1.....y10:
105 45 36 78 20 60 30 20 79 94 (transposed vector of y)

But I did not get it. I got these numbers:
65.12 63.67 53.03 37.25 43.23 84.21 84.10 6.41 54.19 54.13

So my algorithm is obviously wrong.
Do you know what have I done wrong please
Best regards
hugo