How to determine if a matrix is ​​“positive definite” through SQL?

Question

Is there any way, only in MSSQL, to determine if the next maxtrix will be evaluated as positive definite?

A C D G H I
A 1.00 0.68 0.24 0.62 0.90 0.00
C 0.68 1.00 0.25 0.46 0.61 0.00
D 0.24 0.25 1.00 0.60 0.08 0.00
G 0.62 0.46 0.60 1.00 0.46 0.00
H 0.90 0.61 0.08 0.46 1.00 0.00
I 0.00 0.00 0.00 0.00 0.00 1.00

We are currently using the third-party ExtremeNumerics application to handle definitions in a fairly convenient Blackbox way. If I had a SQL table in which I could input assets, correlated asset and value, would there be a way to do the math?

I poked around some, and I did not see anything in MSSQL that handles math math.

thank.

edit: Microsoft SQL 2008

Answer 1

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SQL Server 2008, table datatype

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Groundwork

, :

create type Matrix
as table ( Row int, Col int, Val float )
go

, , table SQL Server 2008.

-, ( ):

create function Determinant ( @matrix Matrix readonly )
returns float
as
begin
    -- Base case of recursion
    if ((select count(*) from @matrix) = 1) 
        return (select Val from @matrix)

, ( 1x1), . -, "" , 1..n

    -- canonicalize row and col numbers (doesn't affect answer)
    declare @rowMap table ( fr_row int, to_row int )
    declare @colMap table ( fr_col int, to_col int )

    insert @rowMap
    select row, row_number() over(order by row) from @matrix
    group by row

    insert @colMap
    select col, row_number() over(order by col) from @matrix
    group by col

    declare @canonicalMatrix Matrix
    insert @canonicalMatrix 
    select 
        to_row, to_col, Val
    from @matrix m
    inner join @rowMap rm on m.row = rm.fr_row
    inner join @colMap cm on m.col = cm.fr_col

, . - , , , ,

    -- apply laplace expansion on first row
    return
    (
        select sum(
            (case col % 2 
            when 1 then 1   -- odd col
            when 0 then -1  -- even col
            end
            )
                    * Val 
                    * dbo.DeterminantOfMinor ( @canonicalMatrix, 1, col ) 
            ) from @canonicalMatrix where row = 1
    )
end
go

, DeterminantOfMinor , table SQL Server:

create function dbo.DeterminantOfMinor ( 
    @matrix Matrix readonly
    , @drop_row int
    , @drop_col int 
)
returns float
as
begin

    declare @minor Matrix
    insert @minor select * from @matrix 
        where row <> @drop_row and col <> @drop_col
    return
        dbo.Determinant( @minor )

end
go

, .

, PD, . , () , , , , , ( ):

create function dbo.is_positive_definite ( @matrix Matrix readonly )
returns bit
as
begin
    -- base case of recursion
    -- a 1x1 matrix is PD iff the single value is positive
    if ((select count(*) from @matrix) = 1) 
        return (select case when Val > 0 then 1 else 0 end from @matrix)

, :

    declare @smallerMat Matrix
    insert @smallerMat
    select row, col, Val from @matrix
    where row < (select max(row) from @matrix)
    and col < (select max(col) from @matrix)

, , PD:

    -- for our input to be PD, its smaller version must be PD:
    return
        ( select case dbo.is_positive_definite( @smallerMat )
        when 1 then 
                (select case 
                     when dbo.Determinant ( @matrix ) > 0 
                     then 1 
                     else 0 
                     end)
        else 0
        end
        )

end
go

!

:

declare @test Matrix

insert @test values ( 1, 1, 1.00 )
insert @test values ( 1, 2, 0.68 )
insert @test values ( 1, 3, 0.24 )
/* snip */
insert @test values ( 6, 4, 0.00 )
insert @test values ( 6, 5, 0.00 )
insert @test values ( 6, 6, 1.00 )

select dbo.Determinant ( @test )
select dbo.is_positive_definite ( @test )


----------------------
0.0333962320000001

(1 row(s) affected)


-----
1

(1 row(s) affected)

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n   Time (s)
1   < 1
2   < 1
3   < 1
4   < 1
5   1
6   17

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