QUADDE
Updated 2024-03-07 14:50:27.907000
Syntax
SELECT [westclintech].[wct].[QUADDE](
<@Func, nvarchar(max),>
,<@VarName, nvarchar(4000),>
,<@A, sql_variant,>
,<@B, sql_variant,>)
Description
Use the scalar function QUADDE to evaluate an infinite integral. QUADDE calculates the integral of the given function f(x) over the interval (-Infinity, Infinity) using Double Exponential Quadrature for non-Periodic functions.
Arguments
@A
the lower limit of integration.
@B
the upper limit of integration.
@VarName
the TSQL variable name. The variable name must start with '@'. @VarName must be of a type nvarchar or a type which implicitly converts to nvarchar.
@Func
the function to be integrated. @Func is a string containing any valid TSQL statement which includes a single variable that is the object of the integration. The variable name is defined in @VarName. @Func is of a type nvarchar or of any type which implicitly converts to nvarchar.
Return Type
float
Remarks
If @A is not '-Inf' and @B is not 'Inf' then NULL will be returned.
If @A is '-Inf' then the function will be integrate from -8 to @B.
If @A is not '-Inf' then the function will be integrated from @A to 8.
@A can be any floating point number or '-Inf'.
@B can be any floating point number or 'Inf'.
If @Func contains an undeclared SQL variable and it is not defined in @VarName a NULL will be returned.
For best results you should use only non-periodic functions. For periodic functions use QUADOSC instead.
Examples
In this example we want to evaluate the integral:
\int_1^\infty\frac{e^{-x}}{\sqrt{x}}
SELECT wct.QUADDE( 'SELECT EXP(-@x)/SQRT(@x)', --@Func
'@x', --@VarName
0, --@A
'Inf' --@B
) as Integral;
This produces the following result.
{"columns":[{"field":"Integral","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"Integral":"1.77245385090552"}]}
In this example we want to evaluate the integral:
\int_{-\infty}^0\frac{1}{1+x^2}
SELECT wct.QUADDE( 'SELECT 1/(1+POWER(@x,2))', --@Func
'@x', --@VarName
'-Inf', --@A
0 --@B
) as Integral;
This produces the following result.
{"columns":[{"field":"Integral","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"Integral":"1.5707963267949"}]}
In this example we want to evaluate the integral:
\int_{-\infty}^\infty\frac{x^2}{1+4x+3x^2-4x^3-2x^4+2x^5+x^6}
SELECT wct.QUADDE(
'SELECT POWER(@x,2)/(1+4*@x+3*POWER(@x,2)-4*POWER(@x,3)-2*POWER(@x,4)+2*POWER(@x,5)+POWER(@x,6))', --@Func
'@x', --@VarName
'-Inf', --@A
'Inf' --@B
) as Integral;
This produces the following result.
{"columns":[{"field":"Integral","headerClass":"ag-right-aligned-header","cellClass":"ag-right-aligned-cell"}],"rows":[{"Integral":"3.14159265358979"}]}
See Also
QUAD - Gauss-Kronrod 15-point quadrature
QUADGK - Gauss-Kronrod 21-point quadrature
QUADOSC - Double exponential quadrature for periodic functions