BFGS

Updated 2024-03-01 22:07:59.877000

Syntax

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

Description

Use the scalar function BFGS to find the minimum of a functions using the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.

Arguments

@tol

Tolerance (for the solution).

@H

Step size.

@VarNames

The name of the variable.

@X

The starting point for the minimization.

@Func

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

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 = 2^(-52).

If tol is NULL then tol = 0.0001

Examples

Example #1

Calculate the minimum of

e^{x_1-1}+e^{-x_2+1}+\left(x_1-x_2\right)^2

Our initial estimate is the point (0,0). Since the result is returned as a string, we will use the MATRIX function to unpack the results into a vector format.

SELECT *

FROM wct.MATRIX(wct.BFGS('SELECT EXP(@x1-1) + EXP(-@x2+1) + POWER(@x1 - 

          @x2,2)', '@x1,@x2', '0,0', NULL, NULL));

This produces the following result.

{"columns":[{"field":"RowNum","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"ColNum","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"},{"field":"ItemValue","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"RowNum":"0","ColNum":"0","ItemValue":"0.796112971274252"},{"RowNum":"1","ColNum":"0","ItemValue":"1.20388757388411"}]}