C VARM 10
C ..................................................................VARM 20
C VARM 30
C SUBROUTINE VARMX VARM 40
C VARM 50
C PURPOSE VARM 60
C PERFORM ORTHOGONAL ROTATIONS OF A FACTOR MATRIX. THIS VARM 70
C SUBROUTINE NORMALLY OCCURS IN A SEQUENCE OF CALLS TO SUB- VARM 80
C ROUTINES CORRE, EIGEN, TRACE, LOAD, VARMX IN THE PERFORMANCEVARM 90
C OF A FACTOR ANALYSIS. VARM 100
C VARM 110
C USAGE VARM 120
C CALL VARMX (M,K,A,NC,TV,H,F,D,IER) VARM 130
C VARM 140
C DESCRIPTION OF PARAMETERS VARM 150
C M - NUMBER OF VARIABLES AND NUMBER OF ROWS OF MATRIX A. VARM 160
C K - NUMBER OF FACTORS. VARM 170
C A - INPUT IS THE ORIGINAL FACTOR MATRIX, AND OUTPUT IS VARM 180
C THE ROTATED FACTOR MATRIX. THE ORDER OF MATRIX A VARM 190
C IS M X K. VARM 200
C NC - OUTPUT VARIABLE CONTAINING THE NUMBER OF ITERATION VARM 210
C CYCLES PERFORMED. VARM 220
C TV - OUTPUT VECTOR CONTAINING THE VARIANCE OF THE FACTOR VARM 230
C MATRIX FOR EACH ITERATION CYCLE. THE VARIANCE PRIORVARM 240
C TO THE FIRST ITERATION CYCLE IS ALSO CALCULATED. VARM 250
C THIS MEANS THAT NC+1 VARIANCES ARE STORED IN VECTOR VARM 260
C TV. MAXIMUM NUMBER OF ITERATION CYCLES ALLOWED IN VARM 270
C THIS SUBROUTINE IS 50. THEREFORE, THE LENGTH OF VARM 280
C VECTOR TV IS 51. VARM 290
C H - OUTPUT VECTOR OF LENGTH M CONTAINING THE ORIGINAL VARM 300
C COMMUNALITIES. VARM 310
C F - OUTPUT VECTOR OF LENGTH M CONTAINING THE FINAL VARM 320
C COMMUNALITIES. VARM 330
C D - OUTPUT VECTOR OF LENGTH M CONTAINING THE DIFFERENCESVARM 340
C BETWEEN THE ORIGINAL AND FINAL COMMUNALITIES. VARM 350
C IER - ERROR INDICATOR VARM 351
C IER=0 - NO ERROR VARM 352
C IER=1 - CONVERGENCE WAS NOT ACHIEVED IN 50 CYCLES VARM 353
C OF ROTATION VARM 354
C VARM 360
C REMARKS VARM 370
C IF VARIANCE COMPUTED AFTER EACH ITERATION CYCLE DOES NOT VARM 380
C INCREASE FOR FOUR SUCCESSIVE TIMES, THE SUBROUTINE STOPS VARM 390
C ROTATION. VARM 400
C VARM 410
C SUBROUTINES AND FUNCTION SUBPROGRAMS REQUIRED VARM 420
C NONE VARM 430
C VARM 440
C METHOD VARM 450
C KAISER'S VARIMAX ROTATION AS DESCRIBED IN 'COMPUTER PROGRAM VARM 460
C FOR VARIMAX ROTATION IN FACTOR ANALYSIS' BY THE SAME AUTHOR,VARM 470
C EDUCATIONAL AND PSYCHOLOGICAL MEASUREMENT, VOL XIX, NO. 3, VARM 480
C 1959. VARM 490
C VARM 500
C ..................................................................VARM 510
C VARM 520
SUBROUTINE VARMX (M,K,A,NC,TV,H,F,D,IER) VARM 530
DIMENSION A(1),TV(1),H(1),F(1),D(1) VARM 540
C VARM 550
C ...............................................................VARM 560
C VARM 570
C IF A DOUBLE PRECISION VERSION OF THIS ROUTINE IS DESIRED, THE VARM 580
C C IN COLUMN 1 SHOULD BE REMOVED FROM THE DOUBLE PRECISION VARM 590
C VARM 600
C DOUBLE PRECISION A,TV,H,F,D,TVLT,CONS,AA,BB,CC,DD,U,T,B,COS4T, VARM 610
C 1 SIN4T,TAN4T,SINP,COSP,CTN4T,COS2T,SIN2T,COST,SINTVARM 620
C VARM 630
C THE C MUST ALSO BE REMOVED FROM DOUBLE PRECISION STATEMENTS VARM 640
C APPEARING IN OTHER ROUTINES USED IN CONJUNCTION WITH THIS VARM 650
C ROUTINE. VARM 660
C VARM 670
C THE DOUBLE PRECISION VERSION OF THIS SUBROUTINE MUST ALSO VARM 680
C CONTAIN DOUBLE PRECISION FORTRAN FUNCTIONS. SQRT IN STATEMENTSVARM 690
C 115, 290, 330, 350, AND 355 MUST BE CHANGED TO DSQRT. ABS IN VARM 700
C STATEMENTS 280, 320, AND 375 MUST BE CHANGED TO DABS. VARM 710
C VARM 720
C ...............................................................VARM 730
C VARM 740
C INITIALIZATION VARM 750
C VARM 760
IER=0 VARM 761
EPS=0.00116 VARM 770
TVLT=0.0 VARM 780
LL=K-1 VARM 790
NV=1 VARM 800
NC=0 VARM 810
FN=M VARM 820
FFN=FN*FN VARM 830
CONS=0.7071066 VARM 840
C VARM 850
C CALCULATE ORIGINAL COMMUNALITIES VARM 860
C VARM 870
DO 110 I=1,M VARM 880
H(I)=0.0 VARM 890
DO 110 J=1,K VARM 900
L=M*(J-1)+I VARM 910
110 H(I)=H(I)+A(L)*A(L) VARM 920
C VARM 930
C CALCULATE NORMALIZED FACTOR MATRIX VARM 940
C VARM 950
DO 120 I=1,M VARM 960
115 H(I)= SQRT(H(I)) VARM 970
DO 120 J=1,K VARM 980
L=M*(J-1)+I VARM 990
120 A(L)=A(L)/H(I) VARM1000
GO TO 132 VARM1010
C VARM1020
C CALCULATE VARIANCE FOR FACTOR MATRIX VARM1030
C VARM1040
130 NV=NV+1 VARM1050
TVLT=TV(NV-1) VARM1060
132 TV(NV)=0.0 VARM1070
DO 150 J=1,K VARM1080
AA=0.0 VARM1090
BB=0.0 VARM1100
LB=M*(J-1) VARM1110
DO 140 I=1,M VARM1120
L=LB+I VARM1130
CC=A(L)*A(L) VARM1140
AA=AA+CC VARM1150
140 BB=BB+CC*CC VARM1160
150 TV(NV)=TV(NV)+(FN*BB-AA*AA)/FFN VARM1170
IF(NV-51)160,155,155 VARM1180
155 IER=1 VARM1181
GO TO 430 VARM1182
C VARM1190
C PERFORM CONVERGENCE TEST VARM1200
C VARM1210
160 IF((TV(NV)-TVLT)-(1.E-7)) 170, 170, 190 VARM1220
170 NC=NC+1 VARM1230
IF(NC-3) 190, 190, 430 VARM1240
C VARM1250
C ROTATION OF TWO FACTORS CONTINUES UP TO VARM1260
C THE STATEMENT 120. VARM1270
C VARM1280
190 DO 420 J=1,LL VARM1290
L1=M*(J-1) VARM1300
II=J+1 VARM1310
C VARM1320
C CALCULATE NUM AND DEN VARM1330
C VARM1340
DO 420 K1=II,K VARM1350
L2=M*(K1-1) VARM1360
AA=0.0 VARM1370
BB=0.0 VARM1380
CC=0.0 VARM1390
DD=0.0 VARM1400
DO 230 I=1,M VARM1410
L3=L1+I VARM1420
L4=L2+I VARM1430
U=(A(L3)+A(L4))*(A(L3)-A(L4)) VARM1440
T=A(L3)*A(L4) VARM1450
T=T+T VARM1460
CC=CC+(U+T)*(U-T) VARM1470
DD=DD+2.0*U*T VARM1480
AA=AA+U VARM1490
230 BB=BB+T VARM1500
T=DD-2.0*AA*BB/FN VARM1510
B=CC-(AA*AA-BB*BB)/FN VARM1520
C VARM1530
C COMPARISON OF NUM AND DEN VARM1540
C VARM1550
IF(T-B) 280, 240, 320 VARM1560
240 IF((T+B)-EPS) 420, 250, 250 VARM1570
C VARM1580
C NUM + DEN IS GREATER THAN OR EQUAL TO THE VARM1590
C TOLERANCE FACTOR VARM1600
C VARM1610
250 COS4T=CONS VARM1620
SIN4T=CONS VARM1630
GO TO 350 VARM1640
C VARM1650
C NUM IS LESS THAN DEN VARM1660
C VARM1670
280 TAN4T= ABS(T)/ ABS(B) VARM1680
IF(TAN4T-EPS) 300, 290, 290 VARM1690
290 COS4T=1.0/ SQRT(1.0+TAN4T*TAN4T) VARM1700
SIN4T=TAN4T*COS4T VARM1710
GO TO 350 VARM1720
300 IF(B) 310, 420, 420 VARM1730
310 SINP=CONS VARM1740
COSP=CONS VARM1750
GO TO 400 VARM1760
C VARM1770
C NUM IS GREATER THAN DEN VARM1780
C VARM1790
320 CTN4T= ABS(T/B) VARM1800
IF(CTN4T-EPS) 340, 330, 330 VARM1810
330 SIN4T=1.0/ SQRT(1.0+CTN4T*CTN4T) VARM1820
COS4T=CTN4T*SIN4T VARM1830
GO TO 350 VARM1840
340 COS4T=0.0 VARM1850
SIN4T=1.0 VARM1860
C VARM1870
C DETERMINE COS THETA AND SIN THETA VARM1880
C VARM1890
350 COS2T= SQRT((1.0+COS4T)/2.0) VARM1900
SIN2T=SIN4T/(2.0*COS2T) VARM1910
355 COST= SQRT((1.0+COS2T)/2.0) VARM1920
SINT=SIN2T/(2.0*COST) VARM1930
C VARM1940
C DETERMINE COS PHI AND SIN PHI VARM1950
C VARM1960
IF(B) 370, 370, 360 VARM1970
360 COSP=COST VARM1980
SINP=SINT VARM1990
GO TO 380 VARM2000
370 COSP=CONS*COST+CONS*SINT VARM2010
375 SINP= ABS(CONS*COST-CONS*SINT) VARM2020
380 IF(T) 390, 390, 400 VARM2030
390 SINP=-SINP VARM2040
C VARM2050
C PERFORM ROTATION VARM2060
C VARM2070
400 DO 410 I=1,M VARM2080
L3=L1+I VARM2090
L4=L2+I VARM2100
AA=A(L3)*COSP+A(L4)*SINP VARM2110
A(L4)=-A(L3)*SINP+A(L4)*COSP VARM2120
410 A(L3)=AA VARM2130
420 CONTINUE VARM2140
GO TO 130 VARM2150
C VARM2160
C DENORMALIZE VARIMAX LOADINGS VARM2170
C VARM2180
430 DO 440 I=1,M VARM2190
DO 440 J=1,K VARM2200
L=M*(J-1)+I VARM2210
440 A(L)=A(L)*H(I) VARM2220
C VARM2230
C CHECK ON COMMUNALITIES VARM2240
C VARM2250
NC=NV-1 VARM2260
DO 450 I=1,M VARM2270
450 H(I)=H(I)*H(I) VARM2280
DO 470 I=1,M VARM2290
F(I)=0.0 VARM2300
DO 460 J=1,K VARM2310
L=M*(J-1)+I VARM2320
460 F(I)=F(I)+A(L)*A(L) VARM2330
470 D(I)=H(I)-F(I) VARM2340
RETURN VARM2350
END VARM2360