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Romberg Integranon 17
Truncation Error O(h!l O(h1) O(ht) O(hl)·· .
Step
~
0 1 2 3
Length hi
b-a=ho 0 To,o .-TTI.O VT,.o ~T3.0···
b-a
-2-==hj 1 TO,1 VTI.I V T,.,_ .,.
b-a =h) 2 T O,2 V T ,., _ ...
4 •
b-a
-8-~h3 3 ...
TO,J_
h=1 0.7500000oo 0.694444445 0.693174603 0.693147478 0.693147182
h=O.5 0.708333333 0.693253968 0.693147902 0.693147183
h =0.25 0.69702381 0.693154531 0.693147194
h-0.125 0.69412185 0.693147653 I(x) =.!. a=l
x b:;;,2
Figure 2.S.
Table of T values in the Romberg integration algorithm.
2.3.3. Romberg Integration Program. BASIC language Program B2-3 in
Appendix B implements (2.27), as iIlustrated in Figure 2.5. The only storage
required is in vector (single-subscript array) AU(·). The integrand function
should be coded by the user beginning in line 10,000; the values relurned by
the user's subroutine are expected to be labeled by the variable name "FC."
The table of values in the format of Figure 2.5 can be compared 10 the
program's computing sequence by adding lines to Program B2-3, as shown in
Table 2.1.
Run the example in the subroutine programmed in line 10,000 and after.
The integrand is Ilx, so that the integral is known in closed form, namely In x.
Input limits a= I to b=2, sO that the answer should be In2. The progress of
the Romberg algorilhm for this example is shown in Figure 2.5, and the
answer at termination is underlined. Parameter NO= I I in Appendix-B Program
B2-3 limits the algorithm to a maximum of 1025 evaluations of the
integrand function. The accuracy parameter EP = I.E - 5 usually produces at
Table 2.1.
Statements to Output tbe
Romberg Table
9062 PRINT"I1 AVOI)"
9064 PRINT.0; AV(I)
9282 PRINT
9284 PRINT.0; AV(I)
9352 PRINTII; AV(II)