Application of polynomial-smoothing formulas and related derivative-estimating formulas simplifies certain linear least-squares problems. These problems can then be solved by a computer method

Semtner, A. J.

English

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.
August 1970	 Brief 70-10078 
II 
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application. Tech Briefs are available on a subscription basis from the Clearinghouse for Federal Scientific and Technical 
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be directed to the Technology Utilization Division, NASA, Code UT, Washington, D.C. 20546. 
Polynomial-Smoothing and Derivative-Estimating Formulas 
for Functions of One or Two Independent Variables 
.
Application of polynomial-smoothing formulas 
and related derivative-estimating formulas can help 
to simplify certain linear least-squares problems. A 
method for solving linear least-squares problems by 
computer can then be employed. 
The report referenced in the note below provides 
tables of smoothing and derivative-estimating formu-
las for least-squares polynomial fitting at equally 
spaced abscissas. In the case of one independent 
variable, smoothing formulas are given for each odd 
number in of points up to 11, and for each degree of 
polynomial approximation such that I ii < ni. 
First and second derivative formulas for unit spacing 
are included. In the case of two independent variables, 
smoothing formulas are given for 3 x 3, 3 x 5, and 
5 x 5 planar arrays of points with linear and quadratic 
approximations. Derivative formulas are provided 
for all nonzero partial derivatives. 
The tables were calculated on an IBM 7094 with 
the help of a linear least-squares program. This pro-
gram can be used to develop smoothing formulas for 
more than two variables, or formulas based on other 
sets of functions.
Note: 
The following documentation may be obtained 
from:
Clearinghouse for Federal Scientific 
and Technical Information 
Springfield, Virginia 22151 
Single document price $3.00 
(or microfiche $0.65) 
Reference: 
NASA-CR-98657 (N69-14273), Poly-
nomial Smoothing Formulas and Deriva-
tive Formulas for One or Two Independ-
ent Variables 
Patent status: 
No patent action is contemplated by NASA. 
Source: Albert J. Semtnerof 
Caltech /J PL 
under contract to 
NASA Pasadena Office 
(NPO-1 1256) 
Category 09 
This document was prepared under the sponsorship of the National 
Aeronautics and Space Administration. Neither the United States 
Government nor any person acting on behalf of the United States
Governmtht assumes any liability resulting from the use of the 
information contained in this document, or warrants that such use 
will be free from privately ownedrights.
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Polynomial-smoothing and derivative-estimating formulas for functions of one or two independent variables

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