Difference between revisions of "Newton Polynomials"
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An '''interpolating polynomial''' is a polynomial that can be used to interpolate between a set of data points. Generally, an interpolating polynomial for an $$N$$-point data set will have $$N$$ coefficients and will therefore be a $$N-1$$st order polynomial. For example, to create an interpolating polynomial for five points, you would need to use a fourth-order polynomial. Higher-order interpolating polynomials suffer from overshoot/undershoot/oscillation issues in the presence of perturbances but lower-order interpolating polynomials can be useful in determining estimates of integrals and derivatives. | An '''interpolating polynomial''' is a polynomial that can be used to interpolate between a set of data points. Generally, an interpolating polynomial for an $$N$$-point data set will have $$N$$ coefficients and will therefore be a $$N-1$$st order polynomial. For example, to create an interpolating polynomial for five points, you would need to use a fourth-order polynomial. Higher-order interpolating polynomials suffer from overshoot/undershoot/oscillation issues in the presence of perturbances but lower-order interpolating polynomials can be useful in determining estimates of integrals and derivatives. | ||
| − | One issue with finding the coefficients of interpolating polynomials has to do with the linear algebra problem involved in finding the coefficients for interpolating polynomials. | + | One issue with finding the coefficients of interpolating polynomials has to do with the linear algebra problem involved in finding the coefficients for interpolating polynomials. The following example shows both problems noted above. The top graph demonstrates the numerical issues that can happen when the linear algebra has an ill-conditioned matrix while the lower graph - which uses shifted versions of the years - demonstrates the overshoot/undershoot/oscillation problem: |
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Revision as of 16:50, 12 November 2021
This page is a brief introduction to the utility of Newton Polynomials find finding equations to calculate estimates of integrals and derivatives using discrete data points. It is not meant to be an exhaustive explanation or proof of Newton Polynomials.
Introduction
An interpolating polynomial is a polynomial that can be used to interpolate between a set of data points. Generally, an interpolating polynomial for an $$N$$-point data set will have $$N$$ coefficients and will therefore be a $$N-1$$st order polynomial. For example, to create an interpolating polynomial for five points, you would need to use a fourth-order polynomial. Higher-order interpolating polynomials suffer from overshoot/undershoot/oscillation issues in the presence of perturbances but lower-order interpolating polynomials can be useful in determining estimates of integrals and derivatives.
One issue with finding the coefficients of interpolating polynomials has to do with the linear algebra problem involved in finding the coefficients for interpolating polynomials. The following example shows both problems noted above. The top graph demonstrates the numerical issues that can happen when the linear algebra has an ill-conditioned matrix while the lower graph - which uses shifted versions of the years - demonstrates the overshoot/undershoot/oscillation problem:
