Calculating Incomplete Beta Functions with IDL

 

Tolerance controls allow you to calculate the accuracy of the incomplete beta and gamma functions. See the Help Article #3182 for calculating Incomplete Gamma Functions.

Working With Tolerances in the Incomplete Beta Function: this example shows the difference in accuracy between the incomplete beta function computed with a low tolerance and the incomplete beta function computed with a high tolerance. The resulting surfaces show the relative errors of each. The relative error of the low tolerance ranges from 0 to 0.00002 percent. The relative error of the high tolerance ranges from 0 to 0.00000000008 percent.

More accuracy usually provides better results, but can cause slower computation speeds. If faster speeds are important, a less accurate calculation may be more desirable. This trade-off can be maintained through tolerances. Iteration controls allow you to expand the computation enough to converge to a result. Calculation of these functions may not converge to a result within the default number of iterations. If the number of iterations is increased, the calculation may converge.

 

Code Example: 

PRO usingIBETAwithEPS    

    ; Define an array of parametric exponents.
    parameterA = (DINDGEN(101)/100. + 1.D) # REPLICATE(1.D, 101)
    parameterB = REPLICATE(1.D, 101) # (DINDGEN(101)/10. + 1.D)
 
    ; Define the upper limits of integration.
    upperLimits = REPLICATE(0.1D, 101, 101)
 
    ; Compute the incomplete beta functions.
    betaFunctions = IBETA(parameterA, parameterB, $
       upperLimits)
 
    ; Compute the incomplete beta functions with a less
    ; accurate tolerance set.
    laBetaFunctions = IBETA(parameterA, parameterB, $
       upperLimits, EPS = 3.0e-4)
 
    ; Compute relative error.
    relativeError = 100.* $
       ABS((betaFunctions - laBetaFunctions)/betaFunctions)
 
    ; Display resulting relative error.
    WINDOW, 0, TITLE = 'Compare IBETA with Less Accurate EPS'
    SURFACE, relativeError, parameterA, parameterB, $
       /XSTYLE, /YSTYLE, TITLE = 'Relative Error', $
       XTITLE = 'Parameter A', YTITLE = 'Parameter B', $
       ZTITLE = 'Percent Error (%)', CHARSIZE = 1.5
 
    ; Compute the incomplete beta functions with a more
    ; accurate tolerance set.
    maBetaFunctions = IBETA(parameterA, parameterB, $
       upperLimits, EPS = 3.0e-10)
 
    ; Compute relative error.
    relativeError = 100.* $
       ABS((betaFunctions - maBetaFunctions)/betaFunctions)
 
    ; Display resulting relative error.
    WINDOW, 1, TITLE = 'Compare IBETA with More Accurate EPS'
    SURFACE, relativeError, parameterA, parameterB, $
       /XSTYLE, /YSTYLE, TITLE = 'Relative Error', $
       XTITLE = 'Parameter A', YTITLE = 'Parameter B', $
       ZTITLE = 'Percent Error (%)', CHARSIZE = 1.5

 
END

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Reviewed by BC on 09/05/2014