PROGRAM GENERALFIT

IMPLICIT REAL*8 (A-H,O-Z)

C

C THIS PROGRAM IS FOR FITTING A SET OF DATE POINTS IN ONE VARIABLE,

C X, TO A POWER SERIES IN X, THEREBY FINDING THE DERIVATIVES WITH

C RESPECT TO X AT X = 0.

C

C IT USES THE ROMBERG SCHEME FOR PURGING HIGHER ORDER CONTAMINATION,

C AS IT WORKS THROUGH DERIVATIVES ORDER BY ORDER. THE ROMBERG SCHEME

C REQUIRES THAT THE DATA POINTS BE AT VALUES OF X THAT ARE IN THE

C RATIO OF 2, AS IN THE SET X = {0,3,6,12,...}. THE EVEN-TEMPERED

C ROMBERG SCHEME IN THIS PROGRAM REQUIRES ONLY THAT THE X VALUES

C HAVE A CONSISTENT RATIO FROM POINT TO POINT (I.E., RATIOS OTHER

C THAN 2 ARE ACCEPTABLE).

C

C THE PROGRAM DETERMINES THE RATIO OF X-VALUES BY TAKING AS

C INPUT DATA THE 2ND AND 3RD VALUES. THE FIRST X-VALUE IS 0.0.

C THE ASSOCIATED Y-VALUES MUST THEN BE ORDERED ACCORDING TO THE

C ORDER OF THE X-VALUES.

C

C A SPECIAL OPTION, CALLED "IEVEN", CAN BE USED TO HAVE THE FITTING

C DONE TO ELIMINATE ODD POWERS IN THE POWER SERIES EXPANSION. IN

C FACT, IT CAN BE USED FOR MORE SPECIALIZED FITS AS INDICATED BELOW.

C

C C. E. DYKSTRA - VERSION 1.1

C

DIMENSION E(40,40), X(40), P(40,40)

C

KLIMIT = 40

WRITE(6,800)

READ(5,*) KPTS, X(2), X(3), IEVEN

IF(KPTS. LE. KLIMIT) GO TO 10

WRITE(6,700) KPTS, KLIMIT

CALL EXIT

10 X(1) = 0.0D 0

C

C ANALYSIS CAN BE APPLIED TO EVEN POWER EXPANSIONS BY SIMPLY

C SQUARING THE COORDINATE VALUES

C

C IEVEN = 0 ALL POWERS OF X

C IEVEN = 1 POWERS OF X LIMITED TO 2, 4, 6, 8, ...

C IEVEN = 2 POWERS OF X LIMITED TO 4, 8, 12, ...

C

IF(IEVEN .NE. 0) X(2) = X(2) ** 2

IF(IEVEN .NE. 0) X(3) = X(3) ** 2

IF(IEVEN .EQ. 2) X(2) = X(2) ** 2

IF(IEVEN .EQ. 2) X(3) = X(3) ** 2

C = X(3) / X(2)

IF(C .LT. 1.0D 0) WRITE(6,710)

IF(KPTS .EQ. 3) GO TO 30

DO 20 K=4,KPTS

X(K) = X(K-1) * C

20 CONTINUE

30 REWIND 88

READ(88,*) (E(I,1),I=1,KPTS)

C

C FORM DIFFERENCES AS INITIAL DERIVATIVE VALUES

C

NORDER = KPTS

DO 50 N=2,NORDER

DO 40 I=N,KPTS

II = I - N + 1

E(I,N) = (E(I,N-1) - E(I-1,N-1)) / (X(I) - X(II))

40 CONTINUE

50 CONTINUE

C

C PURGE CONTAMINATION DUE TO HIGHER ORDER DERIVATIVES,

C DOING THIS DERIVATIVE BY DERIVATIVE

C

DO 90 N=2,NORDER

MAX = NORDER - N + 1

IF(MAX .LT. 2) GO TO 100

NDERIV = N - 1

NNNN = (IEVEN + 1) * NDERIV

WRITE(6,910) NNNN

N1 = N + 1

DO 60 I=N,KPTS

P(I,1) = E(I,N)

60 CONTINUE

WRITE(6,920)

WRITE(6,930) (I, P(I,1),I=N,KPTS)

KPTSN = KPTS

DO 80 ITER=2,MAX

ITPURGE = ITER + N - 2

KPTSN = KPTSN - 1

IF(KPTSN . LT. N1) GO TO 80

WRITE(6,940) ITPURGE

CQ = C**(ITER - 1)

CQ1 = CQ - 1.0D 0

DO 70 L=N1,KPTSN

P(L,ITER) = (CQ * P(L,ITER-1) - P(L+1,ITER-1)) / CQ1

70 CONTINUE

WRITE(6,930) (L,P(L,ITER),L=N1,KPTSN)

80 CONTINUE

90 CONTINUE

100 WRITE(6,950)

WRITE(6,960)

CALL EXIT

C

700 FORMAT(' >>> EXECUTION TERMINATED:',I6,' IS MORE THAN'

1 ' THE PROGRAM LIMIT OF'/

2 ' ',I6,' POINTS')

710 FORMAT(' >>> WARNING: RATIO OF VARIABLES IS LESS THAN 1'/)

800 FORMAT(' ENTER FORMAT-FREE (1) THE NUMBER OF DATA POINTS'/

1 ' (2) THE 2ND POINT-S VARIABLE VALUE'/

2 ' (3) THE 3RD POINT-S VARIABLE VALUE'/

3 ' (4) ZERO (NORMAL) OR ONE IF NO'/

4 ' ODD POWERS IN EXPANSION [N]'/)

910 FORMAT(/' RESULTS FOR DERIVATIVE'I5)

920 FORMAT(/' INITIAL VALUES BY POINTS')

930 FORMAT(I8,G16.8,I6,G16.8,I6,G16.8)

940 FORMAT(/' PURGING THROUGH'I5' VALUES BY POINTS:')

950 FORMAT(/' ... DERIVATIVE FITTING PROGRAM VERSION 1.1 '

1 ' BY C. E. DYKSTRA ... '/)

960 FORMAT(' VALUES GENERATED AFTER PURGING HIGHER ORDER'

3 ' CONTAMINATION ARE EXPANSION'/' COEFFICIENTS IN A SIMPLE'

4 ' POLYNOMIAL: F(X) = C0 + C1*X + C2*X*X + C3*X*X*X ...'

5 /' FACTORS OF 2, 6, 24, 120 AND SO ON SHOULD MULTIPLY'

6 ' THESE COEFFICIENTS'/' TO USE THEM AS '

7 ' DERIVATIVE VALUES IN A TAYLOR EXPANSION OF F(X).'/)

END