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Romberg Algorithm for Integration

Jay Pedersen

This estimates the integral value over interval [a,b] of input function f(x).

Routine: romberg Parameters: f - function to be integrated a, b - interval of integration N - number of columns to generated (zero-based) print_table - true or false, true to display generated table

<Text-field style="Heading 1" family="Arial" layout="Heading 1">Algorithm</Text-field>

romberg := proc(f::algebraic, a, b, N,print_table) local R,h,k,row,col; R := array(0..N,0..N); # Compute column 0, Trapezoid Rule approximations of # 1,2,4,8,..2^N subintervals h := evalf(b - a); R[0,0] := evalf(h/2 * (f(a)+f(b))); for row from 1 to N do; h := h/2; R[row,0] := evalf(0.5*R[row-1,0] + sum(h*f(a+(2*k-1)*h),k=1..2^(row-1))); # Compute [row,1]:[row,row], via Richardson extrapolation for col from 1 to row do; R[row,col] := ((4^col)*R[row,col-1] - R[row-1,col-1]) / (4^col - 1); end do; end do; # Display results if requested if (print_table) then for row from 0 to N do; for col from 0 to row do; printf("%12.10f ",R[row,col]); end do; printf("\134n"); end do; end if; return(R[N,N]); end proc:

<Text-field style="Heading 1" layout="Heading 1">Examples</Text-field>

# Estimate the integral of e^(-x^2)/2 from [0,1] f := x -> exp(-x^2/2); val := romberg(f,0,1,4,false);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLUkkZXhwRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMsJComIyIiIiIiI0Y1KUYoRjZGNSEiIkYlRiVGJQ==

NiM+SSR2YWxHNiIkIis9UkNjJikhIzU=

# Estimate integral of 2^x over [0,4], display table f := x -> 2^x; val := romberg(f,0,4,4,true);

NiM+SSJmRzYiZio2I0kieEdGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKSIiI0YoRiVGJUYl

34.0000000000 25.0000000000 22.0000000000 22.5000000000 21.6666666700 21.6444444500 21.8566017200 21.6421356300 21.6405002300 21.6404376300 21.6945506600 21.6405336400 21.6404268400 21.6404256800 21.6404256300

NiM+SSR2YWxHNiIkIitqRC9rQCEiKQ==

<Text-field style="Heading 1" layout="Heading 1">Restrictions</Text-field>

The input function must be Riemann integrable.

<Text-field style="Heading 1" layout="Heading 1">Notes</Text-field>

(1) Uses the trapezoid rule (with uniform spacing) to generate the first column. Uses properties of the error formula for the trapezoid rule integration (which is defined using the second derivative of the function). (2) To check if the method is working for a function; check table entries (set print_table = true); to see if the following holds:

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 \342\211\205 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

<Text-field style="Heading 1" layout="Heading 1">References</Text-field>

(1) Numerical Mathematics and Computing, 5th Edition Ward Cheney and David Kincaid. ISBN 0-534-38993-7. Section 5.3, Romberg Algorithm, Pages 223-224. (2) Dr. Steve From - University of Nebraska at Omaha mathematics professor; lecture notes from Numerical Methods course.

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