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XLeratorDB/math Documentation

SQL Server JACOBIAN function


JACOBIAN
Updated: 16 April 2018

Use the scalar function JACOBIAN to numerically compute the Jacobian matrix. The Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

Syntax
SELECT [wct].[JACOBIAN] (
   <@Func, nvarchar(max),>
  ,<@VarNames, nvarchar(4000),>
  ,<@X, nvarchar(4000),>
  ,<@H, float,>)
Arguments
Input NameDescription
@FuncThe function to be evaluated, as a string. The function must be in the form of a SELECT statement.
@VarNameThe name of the variable
@XThe point where the Jacobian is calculated
@HStep 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.
  • If @H <= 0 then @H =
  • Available in XLeratorDB / math 2008 only
Examples
Example #1

Calculate the Jacobian for the function

At the point (2.5,7/16*π,63/128* π)

Since the result is returned as a string, we will use the MATRIX function to unpack the results into a matrix format.

DECLARE @r as float = 2.5
DECLARE @theta as float = 7*PI()/16
DECLARE @phi as float = 63*PI()/128
DECLARE @func as varchar(max) = '
SELECT
     @r*SIN(@theta)*COS(@Phi) as x
    ,@r*SIN(@theta)*SIN(@Phi) as y
    ,@r*COS(@theta) as z'
DECLARE @varnames as varchar(max) = '@r,@theta,@phi'
DECLARE @X as varchar(max) = wct.MATRIX2STRING_q('SELECT 2.5, 7*PI()/16,63*PI()/128')
DECLARE @h as float = wct.POWER(2,-20)
 
SELECT
    *
FROM
    wct.MATRIX(
        wct.JACOBIAN(
             @func
            ,@varnames
            ,@X
            ,@h
            )
        )

This produces the following result.

See Also


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