MNORM

Updated 2023-10-17 13:35:11.157000

Syntax

SELECT [westclintech].[wct].[MNORM] (
   <@Matrix, nvarchar(max),>
  ,<@NormType, nvarchar(4000),>)

Description

Use the scalar function MNORM to compute a matrix norm. The norm can be the one ('O') norm, the infinity ('I') norm, the Frobenius ('F') norm, the maximum modulus ('M') among elements of a matrix, or the '2'-norm.

Arguments

@Matrix

The string representation of the matrix with columns separated by commas and rows separated by semi-colons.

@NormType

The matrix norm to be calculated

Return Type

float

Remarks

If NormType IS NULL then NormType = '0'.

Set NormType = '0' or '1' for the 1-norm, the maximum absolute column sum.

Set NormType = 'I' for the infinity-norm, the maximum absolute row sum.

Set NormType = 'M' for the maximum modulus of all the matrix elements.

Set NormType = 'F' for the Frobenius norm, the square root of the sum of the square of the elements.

Set NormType = '2' for the 2-norm, the largest singular value of Matrix.

Examples

Example #1

--Matrix dimension

DECLARE @n as int = 9;

--Create the Hilbert Matrix

DECLARE @H as nvarchar(max) =

        (

            SELECT wct.NMATRIX2STRING(i, j, val)

            FROM

            (

                SELECT i.SeriesValue - 1 as i,

                       j.SeriesValue - 1 as j,

                       1 / (i.SeriesValue + j.SeriesValue - 1) as val

                FROM wct.SeriesFloat(1, @n, NULL, NULL, NULL) i

                    CROSS APPLY wct.SeriesInt(1, @n, NULL, NULL, NULL) j

            ) n

        );

--Calculate each norm

SELECT x.type,

       wct.MNORM(@H, x.type) as norm

FROM

(

    VALUES

        ('1'),

        ('F'),

        ('M'),

        ('I'),

        ('2')

) x (type)

ORDER BY 1;

This produces the following result.

{"columns":[{"field":"type"},{"field":"norm","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"type":"1","norm":"2.82896825396825"},{"type":"2","norm":"1.72588266090185"},{"type":"F","norm":"1.75587190971665"},{"type":"I","norm":"2.82896825396825"},{"type":"M","norm":"1"}]}