Analog computing implementing arbitrary non-linear functions using Chebyshev-polynomial-interpolation schemes and methods of use
The inventive disclosures described herein pertain to an improved physical analog computer that features the ability to evaluate arbitrary non-linear functions using an interpolation method based on Chebyshev polynomials. What has been developed is an improved method for non-linear-function generation in hybrid computing that relies on Chebyshev interpolation. The method requires an initial computation of the interpolation coefficients, which is to be carried out in the digital domain. These coefficients, along with the domain of definition of the non-linear function to be generated, are used during the programming of the analog domain to set multiplier and summer elements.
1. An improved physical electronic hybrid computer, comprising at least one non-linear-function generator, wherein with respect to a non-linear function of interest, and with respect to a desired level of accuracy, said at computer is adapted to perform the following steps:
with respect to Chebyschev polynomial interpolation, selecting a degree of interpolation;
digitally calculating interpolation coefficients for the function of interest to the selected degree of interpolation, thereby giving rise to an interpolant;
carrying out by means of digital calculation a comparison between the interpolant and the function of interest, by means of an error metric, arriving at an error value;
if the error value is consistent with the desired level of accuracy, then loading the interpolation coefficients into the function generator, and carrying out analog computation in which analog values are received as inputs to the function generator and analog values are yielded as outputs from the function generator; or
if the error value is inconsistent with the desired level of accuracy, then selecting a higher degree of interpolation and repeating the steps set forth above.
2. The improved physical electronic hybrid computer of claim 1 wherein the error metric is a root-means-squared measure.
3. The improved physical electronic hybrid computer of claim 1 , wherein said non-linear function of interest is selected from the group consisting of sine, cosine, logarithm and exponential.
4. A method for use with an improved physical electronic hybrid computer, the hybrid computer comprising at least one non-linear-function generator, the method carried out with respect to a non-linear function of interest, and with respect to a desired level of accuracy, the method comprising the steps of:
with respect to Chebyschev polynomial interpolation, selecting a degree of interpolation; digitally calculating interpolation coefficients for the function of interest to the selected degree of interpolation, thereby giving rise to an interpolant;
carrying out by means of digital calculation a comparison between the interpolant and the function of interest, by means of an error metric, arriving at an error value;
responsive to the error value being consistent with the desired level of accuracy, loading the interpolation coefficients into the function generator, and carrying out analog computation in which analog values are received as inputs to the function generator and analog values are yielded as outputs from the function generator.
5. A method for use with an improved physical electronic hybrid computer, the hybrid computer comprising at least one non-linear-function generator, the method carried out with respect to a non-linear function of interest, and with respect to a desired level of accuracy, the method comprising the steps of:
with respect to Chebyschev polynomial interpolation, selecting a first degree of interpolation; digitally calculating interpolation coefficients for the function of interest to the first degree of interpolation, thereby giving rise to an interpolant;
carrying out by means of digital calculation a comparison between the interpolant and the function of interest, by means of an error metric, arriving at an error value;
responsive to the error value not being consistent with the desired level of accuracy by selecting a second degree of interpolation, the second degree being higher than the first degree, digitally calculating interpolation coefficients for the function of interest to the second degree of interpolation, thereby giving rise to the interpolant;
carrying out by means of digital calculation a comparison between the interpolant and the function of interest, by means of an error metric, arriving at an error value;
responsive to the error value being consistent with the desired level of accuracy, loading the interpolation coefficients into the function generator, and carrying out analog computation in which analog values are received as inputs to the function generator and analog values are yielded as outputs from the function generator.