C DCAR 10
C ..................................................................DCAR 20
C DCAR 30
C SUBROUTINE DCAR DCAR 40
C DCAR 50
C PURPOSE DCAR 60
C TO COMPUTE, AT A GIVEN POINT X, AN APPROXIMATION Z TO THE DCAR 70
C DERIVATIVE OF AN ANALYTICALLY GIVEN FUNCTION FCT THAT IS 11- DCAR 80
C TIMES DIFFERENTIABLE IN A DOMAIN CONTAINING A CLOSED, 2-SIDED DCAR 90
C SYMMETRIC INTERVAL OF RADIUS ABSOLUTE H ABOUT X, USING FUNCTIONDCAR 100
C VALUES ONLY ON THAT CLOSED INTERVAL. DCAR 110
C DCAR 120
C USAGE DCAR 130
C CALL DCAR (X,H,IH,FCT,Z) DCAR 140
C PARAMETER FCT REQUIRES AN EXTERNAL STATEMENT DCAR 150
C DCAR 160
C DESCRIPTION OF PARAMETERS DCAR 170
C X - THE POINT AT WHICH THE DERIVATIVE IS TO BE COMPUTED DCAR 180
C H - THE NUMBER WHOSE ABSOLUTE VALUE DEFINES THE CLOSED, DCAR 190
C SYMMETRIC 2-SIDED INTERVAL ABOUT X (SEE PURPOSE) DCAR 200
C IH - INPUT PARAMETER (SEE REMARKS AND METHOD) DCAR 210
C IH NON-ZERO - THE SUBROUTINE GENERATES THE INTERNAL DCAR 220
C VALUE HH DCAR 230
C IH = 0 - THE INTERNAL VALUE HH IS SET TO ABSOLUTE H DCAR 240
C FCT - THE NAME OF THE EXTERNAL FUNCTION SUBPROGRAM THAT WILL DCAR 250
C GENERATE THE NECESSARY FUNCTION VALUES DCAR 260
C Z - RESULTING DERIVATIVE VALUE DCAR 270
C DCAR 280
C REMARKS DCAR 290
C (1) IF H = 0, THEN THERE IS NO COMPUTATION. DCAR 300
C (2) THE INTERNAL VALUE HH, WHICH IS DETERMINED ACCORDING TO DCAR 310
C IH, IS THE MAXIMUM STEP-SIZE USED IN THE COMPUTATION OF DCAR 320
C THE CENTRAL DIVIDED DIFFERENCES (SEE METHOD.) IF IH IS DCAR 330
C NON-ZERO, THEN THE SUBROUTINE GENERATES HH ACCORDING TO DCAR 340
C CRITERIA THAT BALANCE ROUND-OFF AND TRUNCATION ERROR. HH DCAR 350
C IS ALWAYS LESS THAN OR EQUAL TO ABSOLUTE H IN ABSOLUTE DCAR 360
C VALUE, SO THAT ALL COMPUTATION OCCURS WITHIN A RADIUS DCAR 370
C ABSOLUTE H OF X. DCAR 380
C DCAR 390
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED DCAR 400
C THE EXTERNAL FUNCTION SUBPROGRAM FCT(T) MUST BE FURNISHED BY DCAR 410
C THE USER. DCAR 420
C DCAR 430
C METHOD DCAR 440
C THE COMPUTATION OF Z IS BASED ON RICHARDSON'S AND ROMBERG'S DCAR 450
C EXTRAPOLATION METHOD AS APPLIED TO THE SEQUENCE OF CENTRAL DCAR 460
C DIVIDED DIFFERENCES ASSOCIATED WITH THE POINT PAIRS DCAR 470
C (X-(K*HH)/5,X+(K*HH)/5) K=1,...,5. (SEE FILLIPI, S. AND DCAR 480
C ENGELS, H., ALTES UND NEUES ZUR NUMERISCHEN DIFFERENTIATION, DCAR 490
C ELECTRONISCHE DATENVERARBEITUNG, ISS. 2 (1966), PP. 57-65.) DCAR 500
C DCAR 510
C ..................................................................DCAR 520
C DCAR 530
SUBROUTINE DCAR(X,H,IH,FCT,Z) DCAR 540
C DCAR 550
C DCAR 560
DIMENSION AUX(5) DCAR 570
C DCAR 580
C NO ACTION IN CASE OF ZERO INTERVAL LENGTH DCAR 590
IF(H)1,17,1 DCAR 600
C DCAR 610
C GENERATE STEPSIZE HH FOR DIVIDED DIFFERENCES DCAR 620
1 C=ABS(H) DCAR 630
IF(IH)2,9,2 DCAR 640
2 HH=.5 DCAR 650
IF(C-HH)3,4,4 DCAR 660
3 HH=C DCAR 670
4 A=FCT(X+HH) DCAR 680
B=FCT(X-HH) DCAR 690
Z=ABS((A-B)/(HH+HH)) DCAR 700
A=.5*ABS(A+B) DCAR 710
HH=.5 DCAR 720
IF(A-1.)6,6,5 DCAR 730
5 HH=HH*A DCAR 740
6 IF(Z-1.)8,8,7 DCAR 750
7 HH=HH/Z DCAR 760
8 IF(HH-C)10,10,9 DCAR 770
9 HH=C DCAR 780
C DCAR 790
C INITIALIZE DIFFERENTIATION LOOP DCAR 800
10 Z=(FCT(X+HH)-FCT(X-HH))/(HH+HH) DCAR 810
J=5 DCAR 820
JJ=J-1 DCAR 830
AUX(J)=Z DCAR 840
DH=HH/FLOAT(J) DCAR 850
DZ=1.7E38 DCAR 860
C DCAR 870
C START DIFFERENTIATION LOOP DCAR 880
11 J=J-1 DCAR 890
C=J DCAR 900
HH=C*DH DCAR 910
AUX(J)=(FCT(X+HH)-FCT(X-HH))/(HH+HH) DCAR 920
C DCAR 930
C INITIALIZE EXTRAPOLATION LOOP DCAR 940
D2=1.7E38 DCAR 950
B=0. DCAR 960
A=1./C DCAR 970
C DCAR 980
C START EXTRAPOLATION LOOP DCAR 990
DO 12 I=J,JJ DCAR1000
D1=D2 DCAR1010
B=B+A DCAR1020
HH=(AUX(I)-AUX(I+1))/(B*(2.+B)) DCAR1030
AUX(I+1)=AUX(I)+HH DCAR1040
C DCAR1050
C TEST ON OSCILLATING INCREMENTS DCAR1060
D2=ABS(HH) DCAR1070
IF(D2-D1)12,13,13 DCAR1080
12 CONTINUE DCAR1090
C END OF EXTRAPOLATION LOOP DCAR1100
C DCAR1110
C UPDATE RESULT VALUE Z DCAR1120
I=JJ+1 DCAR1130
GO TO 14 DCAR1140
13 D2=D1 DCAR1150
JJ=I DCAR1160
14 IF(D2-DZ)15,16,16 DCAR1170
15 DZ=D2 DCAR1180
Z=AUX(I) DCAR1190
16 IF(J-1)17,17,11 DCAR1200
C END OF DIFFERENTIATION LOOP DCAR1210
C DCAR1220
17 RETURN DCAR1230
END DCAR1240