Module

for

Romberg Integration

   

Background for Romberg Integration.  To approximate the integral   [Graphics:Images/RombergMod_gr_1.gif]  by generating a table of approximations, and using  [Graphics:Images/RombergMod_gr_2.gif]  as the final answer.  
The approximations  [Graphics:Images/RombergMod_gr_3.gif]  are stored in a special lower triangular matrix.  The elements  [Graphics:Images/RombergMod_gr_4.gif]  of the first column are computed using the sequential trapezoidal rule
based on  [Graphics:Images/RombergMod_gr_5.gif]  subintervals of  [Graphics:Images/RombergMod_gr_6.gif];  then  [Graphics:Images/RombergMod_gr_7.gif] is computed using Romberg's rule.  

Elements of row  j  are   [Graphics:Images/RombergMod_gr_8.gif].

The algorithm is terminated when   [Graphics:Images/RombergMod_gr_9.gif][Graphics:Images/RombergMod_gr_10.gif].

Proof  Romberg Integration  Romberg Integration  

 

Animations (Romberg Integration  Romberg Integration).  

 

Computer Programs  Romberg Integration  Romberg Integration  

 

    The subroutine Romberg is "dynamic" in the following sense.  At the start, we initialize the array with the command  [Graphics:Images/RombergMod_gr_11.gif] and it contains one row and one element  [Graphics:Images/RombergMod_gr_12.gif],  in which we place one element  [Graphics:Images/RombergMod_gr_13.gif].  Next, the increment command,  [Graphics:Images/RombergMod_gr_14.gif]  is used to make  [Graphics:Images/RombergMod_gr_15.gif], and the Append command,  [Graphics:Images/RombergMod_gr_16.gif],   is invoked which adds a second row to  [Graphics:Images/RombergMod_gr_17.gif], which is initialized with zeros,  [Graphics:Images/RombergMod_gr_18.gif].  Then the TrapRule subroutine is called to perform the sequential trapezoidal rule and fill in the first entry  [Graphics:Images/RombergMod_gr_19.gif]  and Romberg's rule is used to fill in the second entry  [Graphics:Images/RombergMod_gr_20.gif].  And so it goes, the sequential trapezoidal rule is used to fill in the first entry in succeeding rows and  Romberg's rule fills in rest of the entries in that row.  The algorithm is terminated when  [Graphics:Images/RombergMod_gr_21.gif].  

Mathematica Subroutine (Romberg Integration).

[Graphics:Images/RombergMod_gr_22.gif]

Example 1  Investigate Romberg integration for approximating the integral  [Graphics:Images/RombergMod_gr_23.gif].  
Use the tolerances [Graphics:Images/RombergMod_gr_24.gif].  Compare with the analytic or "true value" of the integral.
Solution 1.

 

Example 2.  Use Romberg integration to compute a numerical approximation to the integral  [Graphics:Images/RombergMod_gr_74.gif].  
Use the tolerances [Graphics:Images/RombergMod_gr_75.gif].  Compare with the analytic or "true value" of the integral.
Solution 2.

 

Example 3.  Use Romberg integration to compute a numerical approximation to the integral  [Graphics:Images/RombergMod_gr_127.gif].  
Use the tolerances [Graphics:Images/RombergMod_gr_128.gif].  Compare with Mathematica's "numerical value" of the integral.
Solution 3.

 

Example 4.  Use Romberg integration to compute a numerical approximation to the integral  [Graphics:Images/RombergMod_gr_177.gif].  
Use the tolerances [Graphics:Images/RombergMod_gr_178.gif].  Compare with the analytic or "true value" of the integral.
Solution 4.

 

Example 5.  Use Romberg integration to compute a numerical approximation to the integral [Graphics:Images/RombergMod_gr_224.gif].  
Use the tolerances [Graphics:Images/RombergMod_gr_225.gif].  Compare with the analytic or "true value" of the integral.
Solution 5.

 

Various Scenarios and Animations for Romberg Integration.

Example 6.   Let  [Graphics:Images/RombergMod_gr_277.gif]  over  [Graphics:Images/RombergMod_gr_278.gif].  Use Romberg integration to approximate the value of the integral.  
Solution 6.

 

Animations (Romberg Integration  Romberg Integration).  

 

Old Lab Project (Romberg Integration  Romberg Integration).  Internet hyperlinks to an old lab project.  

 

Research Experience for Undergraduates

Romberg integration  Romberg integration  Internet hyperlinks to web sites and a bibliography of articles.  

 

Download this Mathematica Notebook Romberg Integration

 

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(c) John H. Mathews 2004