{smcl}
{bf:Mata course:  Accuracy 2}
{hline}


{title:1.  Solvers vs. inversion}

{p 4 4 2}
Pretend we have a numeric method to solve 

		{it:A}{it:X} = {it:B}

{p 4 4 2}
for {it:X}.  Then that method can be used to obtain the inversion of 
A, because inv(A) = X such that 

		{it:A}{it:X} = {it:I}

{p 4 4 2}
Hence, we use our method, and substitute {it:B}={it:I}.



{title:2.  Is using a solver for inversion inaccurate?}

{p 4 4 2}
No.

{p 4 4 2}
The {it:B}={it:I} matrix contains 0s and 1s only:

{p 8 12 2}
1.  Multiplying by 0 introduces no loss of precision:  0*{it:z} = 0.

{p 8 12 2}
2.  Multiplying by 1 introduces no loss of precision:  1*{it:z} = z.

{p 8 12 2}
3.  Adding 0 introduces no loss of precision:  0+{it:z} = {it:z}.

{p 8 12 2}
4.  Adding 1 could introduce a loss of precision, but such terms 
    seldom (never) arise.


{title:3.  Is using a solver for inversion efficient?}

{p 4 4 2}
No.

{p 4 4 2}
We do lots of calculations, just to multiply them by 0, which we add to other 
nonzsero terms, and we are often multiplying or dividing results by 1.


{title:4.  Solving versus inverting}

{p 4 4 2}
We statistician's are often thought to think in terms of inverses, e.g., 

		{it:b} = ({it:X}'{it:X})^(-1){it:X}'{it:y}

{p 4 4 2}
Numerical types look down on us, because better is 


		Solve  ({it:X}'{it:X}){it:b} = {it:X}'{it:y}  for  {it:b}

{p 4 4 2}
That is because solving requires fewer operations (additions, importantly)
than inverting {it:X}'{it:X} and then multiplying the result by
{it:X}'{it:y}.

{p 4 4 2}
There are {it:k}^2 additions in that last multiplication.


{title:Inverses serve a purpose}

{p 4 4 2}
It is from inverses that we statisticians obtain our variances.


{title:The way Stata's -regress- works}

{p 4 8 2}
1.  Form {it:X}'{it:X} and {it:X}'{it:y}.{break}
    (Formed accurately, with means removed.)

{p 4 8 2}
2.  Invert {it:X}'{it:X}.{break}
    Use ({it:X}'{it:X})^(-1) for standard errors (variance matrix).

{p 4 8 2}
3.  Solve ({it:X}'{it:X}){it:b} = {it:X}'{it:y} to obtain {it:b}.{break}
    Do not use use ({it:X}'{it:X})^(-1) from step (2).

{p 4 4 2}
Concerning collinearity, it is identified at step (2) and then the reduced 
problem is solved in step (3).




{hline}
{bf:{view talk.smcl:Top}}