*DECK POLFIT SUBROUTINE POLFIT (N, X, Y, W, MAXDEG, NDEG, EPS, R, IERR, A) C***BEGIN PROLOGUE POLFIT C***PURPOSE Fit discrete data in a least squares sense by polynomials C in one variable. C***LIBRARY SLATEC C***CATEGORY K1A1A2 C***TYPE SINGLE 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 POLFIT 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 PVALUE may then be C called to evaluate the fitted polynomials and any of their C derivatives at any point. The subroutine PCOEF 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 POLFIT are C C Input -- 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, POLFIT 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, POLFIT 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., POLFIT 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. POLFIT will increase the C degree of fit until this criterion is met or C until the maximum degree is reached. C C Output -- 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 PVALUE 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 POLFIT 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 - POLFIT 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 PVALUE and PCOEF C after just one call to POLFIT . 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 PVALUE, XERMSG C***REVISION HISTORY (YYMMDD) C 740601 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890531 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 920501 Reformatted the REFERENCES section. (WRB) C 920527 Corrected erroneous statements in DESCRIPTION. (WRB) C***END PROLOGUE POLFIT DOUBLE PRECISION TEMD1,TEMD2 DIMENSION X(*), Y(*), W(*), R(*), A(*) DIMENSION CO(4,3) SAVE CO DATA CO(1,1), CO(2,1), CO(3,1), CO(4,1), CO(1,2), CO(2,2), 1 CO(3,2), CO(4,2), CO(1,3), CO(2,3), CO(3,3), 2 CO(4,3)/-13.086850,-2.4648165,-3.3846535,-1.2973162, 3 -3.3381146,-1.7812271,-3.2578406,-1.6589279, 4 -1.6282703,-1.3152745,-3.2640179,-1.9829776/ C***FIRST EXECUTABLE STATEMENT POLFIT M = ABS(N) IF (M .EQ. 0) GO TO 30 IF (MAXDEG .LT. 0) GO TO 30 A(1) = MAXDEG MOP1 = MAXDEG + 1 IF (M .LT. MOP1) GO TO 30 IF (EPS .LT. 0.0 .AND. M .EQ. MOP1) GO TO 30 XM = M ETST = EPS*EPS*XM IF (W(1) .LT. 0.0) GO TO 2 DO 1 I = 1,M IF (W(I) .LE. 0.0) GO TO 30 1 CONTINUE GO TO 4 2 DO 3 I = 1,M 3 W(I) = 1.0 4 IF (EPS .GE. 0.0) GO TO 8 C C DETERMINE SIGNIFICANCE LEVEL INDEX TO BE USED IN STATISTICAL TEST FOR C CHOOSING DEGREE OF POLYNOMIAL FIT C IF (EPS .GT. (-.55)) GO TO 5 IDEGF = M - MAXDEG - 1 KSIG = 1 IF (IDEGF .LT. 10) KSIG = 2 IF (IDEGF .LT. 5) KSIG = 3 GO TO 8 5 KSIG = 1 IF (EPS .LT. (-.03)) KSIG = 2 IF (EPS .LT. (-.07)) KSIG = 3 C C INITIALIZE INDEXES AND COEFFICIENTS FOR FITTING C 8 K1 = MAXDEG + 1 K2 = K1 + MAXDEG K3 = K2 + MAXDEG + 2 K4 = K3 + M K5 = K4 + M DO 9 I = 2,K4 9 A(I) = 0.0 W11 = 0.0 IF (N .LT. 0) GO TO 11 C C UNCONSTRAINED CASE C DO 10 I = 1,M K4PI = K4 + I A(K4PI) = 1.0 10 W11 = W11 + W(I) GO TO 13 C C CONSTRAINED CASE C 11 DO 12 I = 1,M K4PI = K4 + I 12 W11 = W11 + W(I)*A(K4PI)**2 C C COMPUTE FIT OF DEGREE ZERO C 13 TEMD1 = 0.0D0 DO 14 I = 1,M K4PI = K4 + I TEMD1 = TEMD1 + DBLE(W(I))*DBLE(Y(I))*DBLE(A(K4PI)) 14 CONTINUE TEMD1 = TEMD1/DBLE(W11) A(K2+1) = TEMD1 SIGJ = 0.0 DO 15 I = 1,M K4PI = K4 + I K5PI = K5 + I TEMD2 = TEMD1*DBLE(A(K4PI)) R(I) = TEMD2 A(K5PI) = TEMD2 - DBLE(R(I)) 15 SIGJ = SIGJ + W(I)*((Y(I)-R(I)) - A(K5PI))**2 J = 0 C C SEE IF POLYNOMIAL OF DEGREE 0 SATISFIES THE DEGREE SELECTION CRITERION C IF (EPS) 24,26,27 C C INCREMENT DEGREE C 16 J = J + 1 JP1 = J + 1 K1PJ = K1 + J K2PJ = K2 + J SIGJM1 = SIGJ C C COMPUTE NEW B COEFFICIENT EXCEPT WHEN J = 1 C IF (J .GT. 1) A(K1PJ) = W11/W1 C C COMPUTE NEW A COEFFICIENT C TEMD1 = 0.0D0 DO 18 I = 1,M K4PI = K4 + I TEMD2 = A(K4PI) TEMD1 = TEMD1 + DBLE(X(I))*DBLE(W(I))*TEMD2*TEMD2 18 CONTINUE A(JP1) = TEMD1/DBLE(W11) C C EVALUATE ORTHOGONAL POLYNOMIAL AT DATA POINTS C W1 = W11 W11 = 0.0 DO 19 I = 1,M K3PI = K3 + I K4PI = K4 + I TEMP = A(K3PI) A(K3PI) = A(K4PI) A(K4PI) = (X(I)-A(JP1))*A(K3PI) - A(K1PJ)*TEMP 19 W11 = W11 + W(I)*A(K4PI)**2 C C GET NEW ORTHOGONAL POLYNOMIAL COEFFICIENT USING PARTIAL DOUBLE C PRECISION C TEMD1 = 0.0D0 DO 20 I = 1,M K4PI = K4 + I K5PI = K5 + I TEMD2 = DBLE(W(I))*DBLE((Y(I)-R(I))-A(K5PI))*DBLE(A(K4PI)) 20 TEMD1 = TEMD1 + TEMD2 TEMD1 = TEMD1/DBLE(W11) A(K2PJ+1) = TEMD1 C C UPDATE POLYNOMIAL EVALUATIONS AT EACH OF THE DATA POINTS, AND C ACCUMULATE SUM OF SQUARES OF ERRORS. THE POLYNOMIAL EVALUATIONS ARE C COMPUTED AND STORED IN EXTENDED PRECISION. FOR THE I-TH DATA POINT, C THE MOST SIGNIFICANT BITS ARE STORED IN R(I) , AND THE LEAST C SIGNIFICANT BITS ARE IN A(K5PI) . C SIGJ = 0.0 DO 21 I = 1,M K4PI = K4 + I K5PI = K5 + I TEMD2 = DBLE(R(I)) + DBLE(A(K5PI)) + TEMD1*DBLE(A(K4PI)) R(I) = TEMD2 A(K5PI) = TEMD2 - DBLE(R(I)) 21 SIGJ = SIGJ + W(I)*((Y(I)-R(I)) - A(K5PI))**2 C C SEE IF DEGREE SELECTION CRITERION HAS BEEN SATISFIED OR IF DEGREE C MAXDEG HAS BEEN REACHED C IF (EPS) 23,26,27 C C COMPUTE F STATISTICS (INPUT EPS .LT. 0.) C 23 IF (SIGJ .EQ. 0.0) GO TO 29 DEGF = M - J - 1 DEN = (CO(4,KSIG)*DEGF + 1.0)*DEGF FCRIT = (((CO(3,KSIG)*DEGF) + CO(2,KSIG))*DEGF + CO(1,KSIG))/DEN FCRIT = FCRIT*FCRIT F = (SIGJM1 - SIGJ)*DEGF/SIGJ IF (F .LT. FCRIT) GO TO 25 C C POLYNOMIAL OF DEGREE J SATISFIES F TEST C 24 SIGPAS = SIGJ JPAS = J NFAIL = 0 IF (MAXDEG .EQ. J) GO TO 32 GO TO 16 C C POLYNOMIAL OF DEGREE J FAILS F TEST. IF THERE HAVE BEEN THREE C SUCCESSIVE FAILURES, A STATISTICALLY BEST DEGREE HAS BEEN FOUND. C 25 NFAIL = NFAIL + 1 IF (NFAIL .GE. 3) GO TO 29 IF (MAXDEG .EQ. J) GO TO 32 GO TO 16 C C RAISE THE DEGREE IF DEGREE MAXDEG HAS NOT YET BEEN REACHED (INPUT C EPS = 0.) C 26 IF (MAXDEG .EQ. J) GO TO 28 GO TO 16 C C SEE IF RMS ERROR CRITERION IS SATISFIED (INPUT EPS .GT. 0.) C 27 IF (SIGJ .LE. ETST) GO TO 28 IF (MAXDEG .EQ. J) GO TO 31 GO TO 16 C C RETURNS C 28 IERR = 1 NDEG = J SIG = SIGJ GO TO 33 29 IERR = 1 NDEG = JPAS SIG = SIGPAS GO TO 33 30 IERR = 2 CALL XERMSG ('SLATEC', 'POLFIT', 'INVALID INPUT PARAMETER.', 2, + 1) GO TO 37 31 IERR = 3 NDEG = MAXDEG SIG = SIGJ GO TO 33 32 IERR = 4 NDEG = JPAS SIG = SIGPAS C 33 A(K3) = NDEG C C WHEN STATISTICAL TEST HAS BEEN USED, EVALUATE THE BEST POLYNOMIAL AT C ALL THE DATA POINTS IF R DOES NOT ALREADY CONTAIN THESE VALUES C IF(EPS .GE. 0.0 .OR. NDEG .EQ. MAXDEG) GO TO 36 NDER = 0 DO 35 I = 1,M CALL PVALUE (NDEG,NDER,X(I),R(I),YP,A) 35 CONTINUE 36 EPS = SQRT(SIG/XM) 37 RETURN END