## Round Off Errors

Round off errors result from the fact that a computer can carry only a finite number of digits of any number. For instance, suppose that the database can store six digits and we wish to add $4.476276 + $3.388567 = $7.864843. In the machine, each of the numbers would be rounded to six significant digits:

$4.47628 + $3.38857 = $7.86485

The true answer, rounded off, would have a 4 in the sixth digit, but rounding error has caused this to be a 5.

In the Nortridge Loan System, interest accruals are calculated and stored to 10 decimal places. All transactions, and most on screen displays are rounded to the nearest cent.

Mathematicians Brower and Clemence (in 1961) derived a formula for the probable error after *n* calculations:

decimal places.

Example

The expected error after 100 calculations is:decimal places.

Each single calculation with a degree of precision of 10 decimal places has a degree of error of plus or minus .00000000005.

Differences will accumulate to whole cents when the number of decimal places in the Brower and Clemence calculation exceeds eight. This will occur when *n* exceeds 900,000. Therefore, the length of term before a loan would be expected to have lost a full cent of interest due to rounding errors is about 2,500 years.

The complete elimination of rounding errors in accruals over the lifetime of a 30-year loan would have required the eight digits of precision. Rounding errors would still exist; however, in the rounding to whole cents which is required in billing and transactions—different loan systems may handle rounding vs. truncation on the monthly payment amount differently—could cause a slight rounding difference when comparing between two loan systems. But, you will not have interest gained or lost due to rounding in NLS.

The preceding explanation of Rounding Errors is adapted from Fundamentals of Astrodynamics by Bate, Mueller, and White, 1971, pgs. 412 – 413 Section 9.5: Comments on Integration Schemes and Errors.