QUAD
Updated 2024-03-07 14:43:39.067000
Syntax
SELECT [westclintech][wctMath].[wct].[QUAD](
<@Func, nvarchar(max),>
,<@VarName, nvarchar(4000),>
,<@A, sql_variant,>
,<@B, sql_variant,>)
Description
Use the scalar function QUAD to evaluate an infinite integral. QUAD can be used to evaluate integrals in the regions -Infinity to a, a to Infinity, or -Infinity to Infinity, where -Infinity < a < Infinity. QUAD uses 15-point Gauss-Kronrod quadrature.
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 of 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.
Examples
In this example we want to evaluate the integral:
\int_1^\infty\frac{e^{-x}}{x}
SELECT wct.QUAD( 'SELECT EXP(-@x)/@x', --@Func
'@x', --@VarName
1, --@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":"0.21938393439552"}]}
In this example we want to evaluate the integral:
\int_{-\infty}^0\frac{\ln(1+x^2)}{x^2}
SELECT wct.QUAD( 'SELECT LOG(1+POWER(@x,2))/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":"3.14159265358684"}]}
In this example we want to evaluate the integral:
\int_{-\infty}^\infty1-\sqrt{2}\frac{\cosh{x}}{\sqrt{\cosh{2x}}
Note that COSH, the hyperbolic cosine function, is not a built-in SQL Server function, but it is part of the XLeratorDB library.
SELECT
wct.QUAD(
'SELECT 1 - SQRT(2)*wct.COSH(@x)/SQRT(wct.COSH(2*@x))', --@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":"-0.69314718056001"}]}
See Also
QUADDE - Double Exponential quadrature for non-periodic functions
QUADGK - Gauss-Kronrod 21-point quadrature
QUADOSC - Double exponential quadrature for periodic functions