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decus/20-0026/dpecs.doc
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SUBROUTINE DPECS
PURPOSE
ECONOMIZATION OF A POLYNOMIAL FOR UNSYMMETRIC RANGE
USAGE
CALL DPECS(P,N,BOUND,EPS,TOL,WORK)
DESCRIPTION OF PARAMETERS
P - DOUBLE PRECISION COEFFICIENT VECTOR OF GIVEN
POLYNOMIAL
N - DIMENSION OF COEFFICIENT VECTOR P
BOUND - SINGLE PRECISION RIGHT HAND BOUNDARY OF INTERVAL
EPS - SINGLE PRECISION INITIAL ERROR BOUND
TOL - SINGLE PRECISION TOLERANCE FOR ERROR
WORK - DOUBLE PRECISION WORKING STORAGE OF DIMENSION N
REMARKS
THE INITIAL COEFFICIENT VECTOR P IS REPLACED BY THE
ECONOMIZED VECTOR.
THE INITIAL ERROR BOUND EPS IS REPLACED BY A FINAL
ERROR BOUND.
N IS REPLACED BY THE DIMENSION OF THE REDUCED POLYNOMIAL.
IN CASE OF AN ARBITRARY INTERVAL (XL,XR) IT IS NECESSARY
FIRST TO CALCULATE THE EXPANSION OF THE GIVEN POLYNOMIAL
WITH ARGUMENT X IN POWERS OF T = (X-XL).
THIS IS ACCOMPLISHED THROUGH SUBROUTINE DPCLD.
OPERATION IS BYPASSED IN CASE OF N LESS THAN 1.
SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED
NONE
METHOD
SUBROUTINE DPECS TAKES AN (N-1)ST DEGREE POLYNOMIAL
APPROXIMATION TO A FUNCTION F(X) VALID WITHIN A TOLERANCE
EPS OVER THE INTERVAL (0,BOUND) AND REDUCES IT IF POSSIBLE
TO A POLYNOMIAL OF LOWER DEGREE VALID WITHIN TOLERANCE
TOL.
THE COEFFICIENT VECTOR OF THE N-TH SHIFTED CHEBYSHEV
POLYNOMIAL IS CALCULATED FROM THE RECURSION FORMULA
A(K) = -A(K+1)*K*L*(2*K-1)/(2*(N+K-1)*(N-K+1)).
REFERENCE
K. A. BRONS, ALGORITHM 37, TELESCOPE 1, CACM VOL. 4, 1961,
NO. 3, PP. 151.