SLATEC Routines --- DPOLFT ---
*DECK DPOLFT
SUBROUTINE DPOLFT (N, X, Y, W, MAXDEG, NDEG, EPS, R, IERR, A)
C***BEGIN PROLOGUE DPOLFT
C***PURPOSE Fit discrete data in a least squares sense by polynomials
C in one variable.
C***LIBRARY SLATEC
C***CATEGORY K1A1A2
C***TYPE DOUBLE PRECISION (POLFIT-S, DPOLFT-D)
C***KEYWORDS CURVE FITTING, DATA FITTING, LEAST SQUARES, POLYNOMIAL FIT
C***AUTHOR Shampine, L. F., (SNLA)
C Davenport, S. M., (SNLA)
C Huddleston, R. E., (SNLL)
C***DESCRIPTION
C
C Abstract
C
C Given a collection of points X(I) and a set of values Y(I) which
C correspond to some function or measurement at each of the X(I),
C subroutine DPOLFT computes the weighted least-squares polynomial
C fits of all degrees up to some degree either specified by the user
C or determined by the routine. The fits thus obtained are in
C orthogonal polynomial form. Subroutine DP1VLU may then be
C called to evaluate the fitted polynomials and any of their
C derivatives at any point. The subroutine DPCOEF may be used to
C express the polynomial fits as powers of (X-C) for any specified
C point C.
C
C The parameters for DPOLFT are
C
C Input -- All TYPE REAL variables are DOUBLE PRECISION
C N - the number of data points. The arrays X, Y and W
C must be dimensioned at least N (N .GE. 1).
C X - array of values of the independent variable. These
C values may appear in any order and need not all be
C distinct.
C Y - array of corresponding function values.
C W - array of positive values to be used as weights. If
C W(1) is negative, DPOLFT will set all the weights
C to 1.0, which means unweighted least squares error
C will be minimized. To minimize relative error, the
C user should set the weights to: W(I) = 1.0/Y(I)**2,
C I = 1,...,N .
C MAXDEG - maximum degree to be allowed for polynomial fit.
C MAXDEG may be any non-negative integer less than N.
C Note -- MAXDEG cannot be equal to N-1 when a
C statistical test is to be used for degree selection,
C i.e., when input value of EPS is negative.
C EPS - specifies the criterion to be used in determining
C the degree of fit to be computed.
C (1) If EPS is input negative, DPOLFT chooses the
C degree based on a statistical F test of
C significance. One of three possible
C significance levels will be used: .01, .05 or
C .10. If EPS=-1.0 , the routine will
C automatically select one of these levels based
C on the number of data points and the maximum
C degree to be considered. If EPS is input as
C -.01, -.05, or -.10, a significance level of
C .01, .05, or .10, respectively, will be used.
C (2) If EPS is set to 0., DPOLFT computes the
C polynomials of degrees 0 through MAXDEG .
C (3) If EPS is input positive, EPS is the RMS
C error tolerance which must be satisfied by the
C fitted polynomial. DPOLFT will increase the
C degree of fit until this criterion is met or
C until the maximum degree is reached.
C
C Output -- All TYPE REAL variables are DOUBLE PRECISION
C NDEG - degree of the highest degree fit computed.
C EPS - RMS error of the polynomial of degree NDEG .
C R - vector of dimension at least NDEG containing values
C of the fit of degree NDEG at each of the X(I) .
C Except when the statistical test is used, these
C values are more accurate than results from subroutine
C DP1VLU normally are.
C IERR - error flag with the following possible values.
C 1 -- indicates normal execution, i.e., either
C (1) the input value of EPS was negative, and the
C computed polynomial fit of degree NDEG
C satisfies the specified F test, or
C (2) the input value of EPS was 0., and the fits of
C all degrees up to MAXDEG are complete, or
C (3) the input value of EPS was positive, and the
C polynomial of degree NDEG satisfies the RMS
C error requirement.
C 2 -- invalid input parameter. At least one of the input
C parameters has an illegal value and must be corrected
C before DPOLFT can proceed. Valid input results
C when the following restrictions are observed
C N .GE. 1
C 0 .LE. MAXDEG .LE. N-1 for EPS .GE. 0.
C 0 .LE. MAXDEG .LE. N-2 for EPS .LT. 0.
C W(1)=-1.0 or W(I) .GT. 0., I=1,...,N .
C 3 -- cannot satisfy the RMS error requirement with a
C polynomial of degree no greater than MAXDEG . Best
C fit found is of degree MAXDEG .
C 4 -- cannot satisfy the test for significance using
C current value of MAXDEG . Statistically, the
C best fit found is of order NORD . (In this case,
C NDEG will have one of the values: MAXDEG-2,
C MAXDEG-1, or MAXDEG). Using a higher value of
C MAXDEG may result in passing the test.
C A - work and output array having at least 3N+3MAXDEG+3
C locations
C
C Note - DPOLFT calculates all fits of degrees up to and including
C NDEG . Any or all of these fits can be evaluated or
C expressed as powers of (X-C) using DP1VLU and DPCOEF
C after just one call to DPOLFT .
C
C***REFERENCES L. F. Shampine, S. M. Davenport and R. E. Huddleston,
C Curve fitting by polynomials in one variable, Report
C SLA-74-0270, Sandia Laboratories, June 1974.
C***ROUTINES CALLED DP1VLU, XERMSG
C***REVISION HISTORY (YYMMDD)
C 740601 DATE WRITTEN
C 890531 Changed all specific intrinsics to generic. (WRB)
C 891006 Cosmetic changes to prologue. (WRB)
C 891006 REVISION DATE from Version 3.2
C 891214 Prologue converted to Version 4.0 format. (BAB)
C 900315 CALLs to XERROR changed to CALLs to XERMSG. (THJ)
C 900911 Added variable YP to DOUBLE PRECISION declaration. (WRB)
C 920501 Reformatted the REFERENCES section. (WRB)
C 920527 Corrected erroneous statements in DESCRIPTION. (WRB)
C***END PROLOGUE DPOLFT