Polynomial regression for multiple variables

I have explanation for you.
First of all we will observe two columns A and B. In column A we have numbers from 1 to 10 with step of 1 and on B from 10 to 100 with step of 10.

If you write A1:A10^{2;1;0;0} and with a standard method (choose a property range (4 col. and 10 rows) and ctr+shift+enter) execute this. You will get the matrix of 4 columns and 10 rows.
You will get something like this:

A^2  A       A^0  A^0
 1	1	1	1
4	2	1	1
9	3	1	1
16	4	1	1
25	5	1	1
36	6	1	1
49	7	1	1
64	8	1	1
81	9	1	1
100	10	1	1

And now you write B1:B10^{0;0;1;2} and with a standard method (choose a property range (4 col. and 10 rows) and ctr+shift+enter) execute this. You will get the matrix of 4 columns and 10 rows.
You will get something like this:

B^0	B^0	B^1	B^2
1	1	10	100
1	1	20	400
1	1	30	900
1	1	40	1600
1	1	50	2500
1	1	60	3600
1	1	70	4900
1	1	80	6400
1	1	90	8100
1	1	100	10000

And on last we have A1:A10^{2;1;0;0}*B1:B10^{0;0;1;2}.
And we will get this:

A^2	A	B	B^2
1	1	10	100
4	2	20	400
9	3	30	900
16	4	40	1600
25	5	50	2500
36	6	60	3600
49	7	70	4900
64	8	80	6400
81	9	90	8100
100	10	100	10000

Better explanation in the picture below.
aha111.jpg

Now the right question is WHY in this format and WHY is only 4 colums?
The trick is in part of {0;0} on the bouth side.
The first column of the first matrix is multiplied with the first column of the second matrix, than the second column of the first matrix is multiplied with the second column of the second matrix, etc.
Simple as that.
P.S.
You can not have 0 data in your matrix becose 0^0 is error.
But you can put instead of 0, 1^-9 .There will be no error and on the other hand you will have the number very close to zero.
ZTC

When I do these kind of regressions, I usually set them up like shg suggested in the linked thread -- build the matrix in the spreadsheet, then refer to the matrix in the LINEST() function. I guess that, with millions of cells at my disposal, I don't see any advantage to building complex arrays inside of the LINEST() function.

I sometimes think that the easiest way to understand what these arrays are doing is to actually put them in the spreadsheet and see. For example, understanding the * operator on arrays, I might set up two arrays, then add an array formula to an adjacent block of cells.

A,B,A*B
1,1,={A2:A16*B2:B16}
2,2,={A2:A16*B2:B16}
...

where column C2:C16 is array entered.

This spreadsheet will show you how I would begin to analyze that kind of array function.