Pade' and reciprocal approximants and linear NOLOR functions

Converting a truncated infinite series to a rational fraction such as the Pade' Approximant (PA) can improve the accuracy of numerical calculations. The numerical integration of differential equations is an example. We have discovered a simpler method of reducing truncation errors and we call the new approximation the Reciprocal Approximant (RA). Although the RA is the easier to compute; nevertheless, for certain truncated infinite series the RA is more accurate than the PA. The NOLOR is the normal at any point of the curve of the natural logarithm of y'(dy/dx or rate) vs x (or time). It is like an ''instantaneous time constant'' which is not necessarily constant. The Nth order chemical reaction is an example of a linear NOLOR function; the derivative of NOLOR is a constant. Complex functions can often be adequately approximated with two or more segments of different linear NOLOR functions. For slab diffusion, the time derivative of NOLOR is approximately 2 before the process is 50% complete and approximately 0 after the process if 50% complete. The N value (for a chemical reaction) is the reciprocal of 1 minus the derivative of NOLOR with respect to x (or time). The inverse function of a linearmore » NOLOR function is also a linear NOLOR function. The first derivatives are reciprocals. The inverse NOLOR is the negative of the product of NOLOR multiplied by the first derivative. The sum of the derivative of NOLOR with respect to x plus the derivative of the inverse NOLOR with respect to y is equal to 1.« less