SQL Server GRAD Function

Updated 2024-03-05 17:03:47.060000

Description

Use the scalar function GRAD to numerically compute the gradient of a function

\nabla{F}\left(x,y,z\right)=\left(\frac{dF}{dx},\frac{dF}{dy},\frac{dF}{dz}\right)

using central finite difference.

Syntax

SELECT [westclintech].[wct].[GRAD] (
   <@Func, nvarchar(max),>
  ,<@VarNames, nvarchar(4000),>
  ,<@X, nvarchar(4000),>
  ,<@H, float,>)

Arguments

@Func

The function to be evaluated, as a string. The function must be in the form of a SELECT statement.

@VarNames

The names of the variables.

@X

The starting point for the minimization.

@H

Step size.

Return Type

nvarchar(max)

Remarks

If Func returns a NULL then NULL is returned.

If Func is not a valid SELECT statement then NULL is returned.

If no solution is found then NULL is returned.

If X is NULL then @X = ''.

If H is NULL then H = 0.

Examples

Example #1

Calculate the gradient of

\frac{x^3}{y^2}\times{z^2}+\frac{12xy}{2z}

at the point (1,1,1). Since the result is returned as a string, we will use the MATRIX function to unpack the results into a vector format.

SELECT RowNum,
       ItemValue
FROM wct.MATRIX(wct.GRAD('SELECT POWER(@x,3) / POWER(@y,2) * POWER(@z,2) + 12*@x*@y 
          / 2*@z', '@x,@y,@z', '1,1,1', NULL));

This produces the following result.

RowNum	ItemValue
0	8.99999999801187
1	3.99999999899416
2	7.99999999828166

Example #2

Using the same function from the previous formula we use the FDERIV function to demonstrate the calculation of the gradient.

SELECT wct.FDERIV(
                     REPLACE(
                                REPLACE('SELECT POWER(@x,3) / POWER(@y,2) * POWER(
                                          @z,2) + 12*@x*@y / 2*@z', '@y', 1),
                                '@z',
                                1
                            ),
                     '@x',
                     1,
                     1,
                     NULL,
                     'C'
                 ) as [dF/dx],
       wct.FDERIV(
                     REPLACE(
                                REPLACE('SELECT POWER(@x,3) / POWER(@y,2) * POWER(
                                          @z,2) + 12*@x*@y / 2*@z', '@x', 1),
                                '@z',
                                1
                            ),
                     '@y',
                     1,
                     1,
                     NULL,
                     'C'
                 ) as [dF/dy],
       wct.FDERIV(
                     REPLACE(
                                REPLACE('SELECT POWER(@x,3) / POWER(@y,2) * POWER(
                                          @z,2) + 12*@x*@y / 2*@z', '@x', 1),
                                '@y',
                                1
                            ),
                     '@z',
                     1,
                     1,
                     NULL,
                     'C'
                 ) as [dF/dz];