By E. J. Redfern (auth.)
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Extra info for Introduction to Pascal for Computational Mathematics
The result of running a program will be either: 1. 2. 3. 4. A successful run, producing the required result from the program, or A successful run, producing an incorrect result, or An unsuccessful run, producing a list of runtime errors, or A successful run, producing what is thought to be the required result from the program. Again, error messages are displayed relating to particular lines but the actual error may have occurred much earlier in the program. For example, suppose we had a division operation in an assignment statement and at the point the statement is executed the divisor contains the value zero.
If we were to execute these statements, then on exit from the loop the identifier nextterm would contain an approximation to the limit of the sequence. 0001; nextterm := 1; REPEAT oldterm := nextterm; nextterm := 1 + nextterm/2; UNTIL ABS(nextterm/oldterm - 1) < epsilon; The accuracy of the approximation is determined by the value given to epsilon. The smaller the value the more accurate the answer, but the longer the calculation. From our mathematical knowledge of this sequence we know that it converges so no problem would arise in using this particular stopping rule in this case.
For each i. ) We could therefore calculate it using the following code: i : = 1; sum := 1; term := 1; REPEAT i := i + l; term := term * x I i; sum := sum + term; UNTIL ABS(term) < epsilon; where epsilon is some pre-set value reflecting the accuracy that we wanted to achieve. Alternatively, we could perform the summation over a fixed number of terms as follows: sum := 1; term := 1; FOR i := 1 TO n DO BEGIN term := term * x I i; sum := sum + term; END; Obviously as x increases, to produce the values of ex to the same order of accuracy as for small values of x, more terms need to be included in the series.