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## Re: least square

 From: heberf Subject: Re: least square Date: Thu, 5 Aug 1999 16:14:21 -0500 (CDT)

```I tried this myself and I think the Octave solution is the correct one.  But
maybe I misunderstand what you mean by a least square soluntion to the
overdetermined problem.  What I THINK you mean is find x to minimize the norm of

Ax - b

(which is of course the same as minimizing the sum of squared elements of the
above vector).  This problem has a closed form solution which is given by

x = inv(A'*A)*A'*b

You can check that for your problem this return the same thing as x = A\b.  Now
notice that the norm of Ax -b is 2.3748 while for the solution you suggest the
norm is bigger.  Also note that since one column of A is a vector of ones the
sum of Ax - b should be zero.  It is for the Octave solution.  For your
solution
it's considerably larger (I assumed that you miswrote -0.365 and 0.783 as -365
and 783 in your e-mail).

Does this help?  If not what do you mean by a least squares solution?

Heber Farnsworth

>
>        I ran the following standard example taken from IMSL(and also
>           checked with Matlab ) for least square solution of Ax = b,
>           where
>
> octave:4> A = [1,5,15,5;1, 4,17,4 ;1,7,14,7;1,3,18,3;1,1,15,1;\
>               1,8,11,8;1,3,9,3;1,4,10,0]
> A =
>
>    1   5  15   5
>    1   4  17   4
>    1   7  14   7
>    1   3  18   3
>    1   1  15   1
>    1   8  11   8
>    1   3   9   3
>    1   4  10   0
>
> octave:5> b = [30;31;35;29;18;35;20;22]
> b =
>
>   30
>   31
>   35
>   29
>   18
>   35
>   20
>   22
>
>  octave:6> x = A\b
> x =
>
>   1.48165
>   2.59948
>   1.01204
>   0.22104
>
> The actual LEAST SQUARE solution for the overdetermines system
>  ( which is ,as I assume, not same as the minimum norm soln for
>   singular matrices ) is:
>
>           X = .636, 2.845, 1.058, 000.
>  with residuals  r = -.733, .996, -365, 783, -1.353, -.036, 1.306, -.597
>
>  Does octave support a linear least square solving code or am I doing
>   something wrong? ( Mr. David Doolin suggested that I resubmit this ).
>
>                               Thanking you,
>
>                               sincerely,
>
>                               A. K. Mitra, address@hidden
>
>
>
>
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