C AHI 10
C ..................................................................AHI 20
C AHI 30
C SUBROUTINE AHI AHI 40
C AHI 50
C PURPOSE AHI 60
C TO INTERPOLATE FUNCTION VALUE Y FOR A GIVEN ARGUMENT VALUE AHI 70
C X USING A GIVEN TABLE (ARG,VAL) OF ARGUMENT, FUNCTION, AND AHI 80
C DERIVATIVE VALUES. AHI 90
C AHI 100
C USAGE AHI 110
C CALL AHI (X,ARG,VAL,Y,NDIM,EPS,IER) AHI 120
C AHI 130
C DESCRIPTION OF PARAMETERS AHI 140
C X - THE ARGUMENT VALUE SPECIFIED BY INPUT. AHI 150
C ARG - THE INPUT VECTOR (DIMENSION NDIM) OF ARGUMENT AHI 160
C VALUES OF THE TABLE (NOT DESTROYED). AHI 170
C VAL - THE INPUT VECTOR (DIMENSION 2*NDIM) OF FUNCTION AHI 180
C AND DERIVATIVE VALUES OF THE TABLE (DESTROYED). AHI 190
C FUNCTION AND DERIVATIVE VALUES MUST BE STORED IN AHI 200
C PAIRS, THAT MEANS BEGINNING WITH FUNCTION VALUE AT AHI 210
C POINT ARG(1) EVERY FUNCTION VALUE MUST BE FOLLOWED AHI 220
C BY THE VALUE OF DERIVATIVE AT THE SAME POINT. AHI 230
C Y - THE RESULTING INTERPOLATED FUNCTION VALUE. AHI 240
C NDIM - AN INPUT VALUE WHICH SPECIFIES THE NUMBER OF AHI 250
C POINTS IN TABLE (ARG,VAL). AHI 260
C EPS - AN INPUT CONSTANT WHICH IS USED AS UPPER BOUND AHI 270
C FOR THE ABSOLUTE ERROR. AHI 280
C IER - A RESULTING ERROR PARAMETER. AHI 290
C AHI 300
C REMARKS AHI 310
C (1) TABLE (ARG,VAL) SHOULD REPRESENT A SINGLE-VALUED AHI 320
C FUNCTION AND SHOULD BE STORED IN SUCH A WAY, THAT THE AHI 330
C DISTANCES ABS(ARG(I)-X) INCREASE WITH INCREASING AHI 340
C SUBSCRIPT I. TO GENERATE THIS ORDER IN TABLE (ARG,VAL), AHI 350
C SUBROUTINES ATSG, ATSM OR ATSE COULD BE USED IN A AHI 360
C PREVIOUS STAGE. AHI 370
C (2) NO ACTION BESIDES ERROR MESSAGE IN CASE NDIM LESS AHI 380
C THAN 1. AHI 390
C (3) INTERPOLATION IS TERMINATED EITHER IF THE DIFFERENCE AHI 400
C BETWEEN TWO SUCCESSIVE INTERPOLATED VALUES IS AHI 410
C ABSOLUTELY LESS THAN TOLERANCE EPS, OR IF THE ABSOLUTE AHI 420
C VALUE OF THIS DIFFERENCE STOPS DIMINISHING, OR AFTER AHI 430
C (2*NDIM-2) STEPS. FURTHER IT IS TERMINATED IF THE AHI 440
C PROCEDURE DISCOVERS TWO ARGUMENT VALUES IN VECTOR ARG AHI 450
C WHICH ARE IDENTICAL. DEPENDENT ON THESE FOUR CASES, AHI 460
C ERROR PARAMETER IER IS CODED IN THE FOLLOWING FORM AHI 470
C IER=0 - IT WAS POSSIBLE TO REACH THE REQUIRED AHI 480
C ACCURACY (NO ERROR). AHI 490
C IER=1 - IT WAS IMPOSSIBLE TO REACH THE REQUIRED AHI 500
C ACCURACY BECAUSE OF ROUNDING ERRORS. AHI 510
C IER=2 - IT WAS IMPOSSIBLE TO CHECK ACCURACY BECAUSE AHI 520
C NDIM IS LESS THAN 2, OR THE REQUIRED ACCURACY AHI 530
C COULD NOT BE REACHED BY MEANS OF THE GIVEN AHI 540
C TABLE. NDIM SHOULD BE INCREASED. AHI 550
C IER=3 - THE PROCEDURE DISCOVERED TWO ARGUMENT VALUES AHI 560
C IN VECTOR ARG WHICH ARE IDENTICAL. AHI 570
C AHI 580
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED AHI 590
C NONE AHI 600
C AHI 610
C METHOD AHI 620
C INTERPOLATION IS DONE BY MEANS OF AITKENS SCHEME OF AHI 630
C HERMITE INTERPOLATION. ON RETURN Y CONTAINS AN INTERPOLATED AHI 640
C FUNCTION VALUE AT POINT X, WHICH IS IN THE SENSE OF REMARK AHI 650
C (3) OPTIMAL WITH RESPECT TO GIVEN TABLE. FOR REFERENCE, SEE AHI 660
C F.B.HILDEBRAND, INTRODUCTION TO NUMERICAL ANALYSIS, AHI 670
C MCGRAW-HILL, NEW YORK/TORONTO/LONDON, 1956, PP.314-317, AND AHI 680
C GERSHINSKY/LEVINE, AITKEN-HERMITE INTERPOLATION, AHI 690
C JACM, VOL.11, ISS.3 (1964), PP.352-356. AHI 700
C AHI 710
C ..................................................................AHI 720
C AHI 730
SUBROUTINE AHI(X,ARG,VAL,Y,NDIM,EPS,IER) AHI 740
C AHI 750
C AHI 760
DIMENSION ARG(1),VAL(1) AHI 770
IER=2 AHI 780
H2=X-ARG(1) AHI 790
IF(NDIM-1)2,1,3 AHI 800
1 Y=VAL(1)+VAL(2)*H2 AHI 810
2 RETURN AHI 820
C AHI 830
C VECTOR ARG HAS MORE THAN 1 ELEMENT. AHI 840
C THE FIRST STEP PREPARES VECTOR VAL SUCH THAT AITKEN SCHEME CAN BE AHI 850
C USED. AHI 860
3 I=1 AHI 870
DO 5 J=2,NDIM AHI 880
H1=H2 AHI 890
H2=X-ARG(J) AHI 900
Y=VAL(I) AHI 910
VAL(I)=Y+VAL(I+1)*H1 AHI 920
H=H1-H2 AHI 930
IF(H)4,13,4 AHI 940
4 VAL(I+1)=Y+(VAL(I+2)-Y)*H1/H AHI 950
5 I=I+2 AHI 960
VAL(I)=VAL(I)+VAL(I+1)*H2 AHI 970
C END OF FIRST STEP AHI 980
C AHI 990
C PREPARE AITKEN SCHEME AHI 1000
DELT2=0. AHI 1010
IEND=I-1 AHI 1020
C AHI 1030
C START AITKEN-LOOP AHI 1040
DO 9 I=1,IEND AHI 1050
DELT1=DELT2 AHI 1060
Y=VAL(1) AHI 1070
M=(I+3)/2 AHI 1080
H1=ARG(M) AHI 1090
DO 6 J=1,I AHI 1100
K=I+1-J AHI 1110
L=(K+1)/2 AHI 1120
H=ARG(L)-H1 AHI 1130
IF(H)6,14,6 AHI 1140
6 VAL(K)=(VAL(K)*(X-H1)-VAL(K+1)*(X-ARG(L)))/H AHI 1150
DELT2=ABS(Y-VAL(1)) AHI 1160
IF(DELT2-EPS)11,11,7 AHI 1170
7 IF(I-5)9,8,8 AHI 1180
8 IF(DELT2-DELT1)9,12,12 AHI 1190
9 CONTINUE AHI 1200
C END OF AITKEN-LOOP AHI 1210
C AHI 1220
10 Y=VAL(1) AHI 1230
RETURN AHI 1240
C AHI 1250
C THERE IS SUFFICIENT ACCURACY WITHIN 2*NDIM-2 ITERATION STEPS AHI 1260
11 IER=0 AHI 1270
GOTO 10 AHI 1280
C AHI 1290
C TEST VALUE DELT2 STARTS OSCILLATING AHI 1300
12 IER=1 AHI 1310
RETURN AHI 1320
C AHI 1330
C THERE ARE TWO IDENTICAL ARGUMENT VALUES IN VECTOR ARG AHI 1340
13 Y=VAL(1) AHI 1350
14 IER=3 AHI 1360
RETURN AHI 1370
END AHI 1380