c c........|.........|.........|.........|.........|.........|.........|.. c c Eispack subroutines. Main routine is CG, e-vals and e-vectors of c a complex general matrix. See Eispack documentation. c c........|.........|.........|.........|.........|.........|.........|.. c SUBROUTINE CDIV(AR,AI,BR,BI,CR,CI) DOUBLE PRECISION AR,AI,BR,BI,CR,CI C C COMPLEX DIVISION, (CR,CI) = (AR,AI)/(BR,BI) C DOUBLE PRECISION S,ARS,AIS,BRS,BIS S = DABS(BR) + DABS(BI) ARS = AR/S AIS = AI/S BRS = BR/S BIS = BI/S S = BRS**2 + BIS**2 CR = (ARS*BRS + AIS*BIS)/S CI = (AIS*BRS - ARS*BIS)/S RETURN END SUBROUTINE CSROOT(XR,XI,YR,YI) DOUBLE PRECISION XR,XI,YR,YI C C (YR,YI) = COMPLEX DSQRT(XR,XI) C BRANCH CHOSEN SO THAT YR .GE. 0.0 AND SIGN(YI) .EQ. SIGN(XI) C DOUBLE PRECISION S,TR,TI,PYTHAG TR = XR TI = XI S = DSQRT(0.5D0*(PYTHAG(TR,TI) + DABS(TR))) IF (TR .GE. 0.0D0) YR = S IF (TI .LT. 0.0D0) S = -S IF (TR .LE. 0.0D0) YI = S IF (TR .LT. 0.0D0) YR = 0.5D0*(TI/YI) IF (TR .GT. 0.0D0) YI = 0.5D0*(TI/YR) RETURN END DOUBLE PRECISION FUNCTION EPSLON (X) DOUBLE PRECISION X C C ESTIMATE UNIT ROUNDOFF IN QUANTITIES OF SIZE X. C DOUBLE PRECISION A,B,C,EPS C C THIS PROGRAM SHOULD FUNCTION PROPERLY ON ALL SYSTEMS C SATISFYING THE FOLLOWING TWO ASSUMPTIONS, C 1. THE BASE USED IN REPRESENTING FLOATING POINT C NUMBERS IS NOT A POWER OF THREE. C 2. THE QUANTITY A IN STATEMENT 10 IS REPRESENTED TO C THE ACCURACY USED IN FLOATING POINT VARIABLES C THAT ARE STORED IN MEMORY. C THE STATEMENT NUMBER 10 AND THE GO TO 10 ARE INTENDED TO C FORCE OPTIMIZING COMPILERS TO GENERATE CODE SATISFYING C ASSUMPTION 2. C UNDER THESE ASSUMPTIONS, IT SHOULD BE TRUE THAT, C A IS NOT EXACTLY EQUAL TO FOUR-THIRDS, C B HAS A ZERO FOR ITS LAST BIT OR DIGIT, C C IS NOT EXACTLY EQUAL TO ONE, C EPS MEASURES THE SEPARATION OF 1.0 FROM C THE NEXT LARGER FLOATING POINT NUMBER. C THE DEVELOPERS OF EISPACK WOULD APPRECIATE BEING INFORMED C ABOUT ANY SYSTEMS WHERE THESE ASSUMPTIONS DO NOT HOLD. C C THIS VERSION DATED 4/6/83. C A = 4.0D0/3.0D0 10 B = A - 1.0D0 C = B + B + B EPS = DABS(C-1.0D0) IF (EPS .EQ. 0.0D0) GO TO 10 EPSLON = EPS*DABS(X) RETURN END DOUBLE PRECISION FUNCTION PYTHAG(A,B) DOUBLE PRECISION A,B C C FINDS DSQRT(A**2+B**2) WITHOUT OVERFLOW OR DESTRUCTIVE UNDERFLOW C DOUBLE PRECISION P,R,S,T,U P = DMAX1(DABS(A),DABS(B)) IF (P .EQ. 0.0D0) GO TO 20 R = (DMIN1(DABS(A),DABS(B))/P)**2 10 CONTINUE T = 4.0D0 + R IF (T .EQ. 4.0D0) GO TO 20 S = R/T U = 1.0D0 + 2.0D0*S P = U*P R = (S/U)**2 * R GO TO 10 20 PYTHAG = P RETURN END SUBROUTINE CBABK2(NM,N,LOW,IGH,SCALE,M,ZR,ZI) C INTEGER I,J,K,M,N,II,NM,IGH,LOW DOUBLE PRECISION SCALE(N),ZR(NM,M),ZI(NM,M) DOUBLE PRECISION S C C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE C CBABK2, WHICH IS A COMPLEX VERSION OF BALBAK, C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971). C C THIS SUBROUTINE FORMS THE EIGENVECTORS OF A COMPLEX GENERAL C MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING C BALANCED MATRIX DETERMINED BY CBAL. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX. C C LOW AND IGH ARE INTEGERS DETERMINED BY CBAL. C C SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS C AND SCALING FACTORS USED BY CBAL. C C M IS THE NUMBER OF EIGENVECTORS TO BE BACK TRANSFORMED. C C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVECTORS TO BE C BACK TRANSFORMED IN THEIR FIRST M COLUMNS. C C ON OUTPUT C C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE TRANSFORMED EIGENVECTORS C IN THEIR FIRST M COLUMNS. C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C IF (M .EQ. 0) GO TO 200 IF (IGH .EQ. LOW) GO TO 120 C DO 110 I = LOW, IGH S = SCALE(I) C .......... LEFT HAND EIGENVECTORS ARE BACK TRANSFORMED C IF THE FOREGOING STATEMENT IS REPLACED BY C S=1.0D0/SCALE(I). .......... DO 100 J = 1, M ZR(I,J) = ZR(I,J) * S ZI(I,J) = ZI(I,J) * S 100 CONTINUE C 110 CONTINUE C .......... FOR I=LOW-1 STEP -1 UNTIL 1, C IGH+1 STEP 1 UNTIL N DO -- .......... 120 DO 140 II = 1, N I = II IF (I .GE. LOW .AND. I .LE. IGH) GO TO 140 IF (I .LT. LOW) I = LOW - II K = SCALE(I) IF (K .EQ. I) GO TO 140 C DO 130 J = 1, M S = ZR(I,J) ZR(I,J) = ZR(K,J) ZR(K,J) = S S = ZI(I,J) ZI(I,J) = ZI(K,J) ZI(K,J) = S 130 CONTINUE C 140 CONTINUE C 200 RETURN END SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE) C INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC DOUBLE PRECISION AR(NM,N),AI(NM,N),SCALE(N) DOUBLE PRECISION C,F,G,R,S,B2,RADIX LOGICAL NOCONV C C THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE C CBALANCE, WHICH IS A COMPLEX VERSION OF BALANCE, C NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971). C C THIS SUBROUTINE BALANCES A COMPLEX MATRIX AND ISOLATES C EIGENVALUES WHENEVER POSSIBLE. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX. C C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE COMPLEX MATRIX TO BE BALANCED. C C ON OUTPUT C C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE BALANCED MATRIX. C C LOW AND IGH ARE TWO INTEGERS SUCH THAT AR(I,J) AND AI(I,J) C ARE EQUAL TO ZERO IF C (1) I IS GREATER THAN J AND C (2) J=1,...,LOW-1 OR I=IGH+1,...,N. C C SCALE CONTAINS INFORMATION DETERMINING THE C PERMUTATIONS AND SCALING FACTORS USED. C C SUPPOSE THAT THE PRINCIPAL SUBMATRIX IN ROWS LOW THROUGH IGH C HAS BEEN BALANCED, THAT P(J) DENOTES THE INDEX INTERCHANGED C WITH J DURING THE PERMUTATION STEP, AND THAT THE ELEMENTS C OF THE DIAGONAL MATRIX USED ARE DENOTED BY D(I,J). THEN C SCALE(J) = P(J), FOR J = 1,...,LOW-1 C = D(J,J) J = LOW,...,IGH C = P(J) J = IGH+1,...,N. C THE ORDER IN WHICH THE INTERCHANGES ARE MADE IS N TO IGH+1, C THEN 1 TO LOW-1. C C NOTE THAT 1 IS RETURNED FOR IGH IF IGH IS ZERO FORMALLY. C C THE ALGOL PROCEDURE EXC CONTAINED IN CBALANCE APPEARS IN C CBAL IN LINE. (NOTE THAT THE ALGOL ROLES OF IDENTIFIERS C K,L HAVE BEEN REVERSED.) C C ARITHMETIC IS REAL THROUGHOUT. C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C RADIX = 16.0D0 C B2 = RADIX * RADIX K = 1 L = N GO TO 100 C .......... IN-LINE PROCEDURE FOR ROW AND C COLUMN EXCHANGE .......... 20 SCALE(M) = J IF (J .EQ. M) GO TO 50 C DO 30 I = 1, L F = AR(I,J) AR(I,J) = AR(I,M) AR(I,M) = F F = AI(I,J) AI(I,J) = AI(I,M) AI(I,M) = F 30 CONTINUE C DO 40 I = K, N F = AR(J,I) AR(J,I) = AR(M,I) AR(M,I) = F F = AI(J,I) AI(J,I) = AI(M,I) AI(M,I) = F 40 CONTINUE C 50 GO TO (80,130), IEXC C .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE C AND PUSH THEM DOWN .......... 80 IF (L .EQ. 1) GO TO 280 L = L - 1 C .......... FOR J=L STEP -1 UNTIL 1 DO -- .......... 100 DO 120 JJ = 1, L J = L + 1 - JJ C DO 110 I = 1, L IF (I .EQ. J) GO TO 110 IF (AR(J,I) .NE. 0.0D0 .OR. AI(J,I) .NE. 0.0D0) GO TO 120 110 CONTINUE C M = L IEXC = 1 GO TO 20 120 CONTINUE C GO TO 140 C .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE C AND PUSH THEM LEFT .......... 130 K = K + 1 C 140 DO 170 J = K, L C DO 150 I = K, L IF (I .EQ. J) GO TO 150 IF (AR(I,J) .NE. 0.0D0 .OR. AI(I,J) .NE. 0.0D0) GO TO 170 150 CONTINUE C M = K IEXC = 2 GO TO 20 170 CONTINUE C .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L .......... DO 180 I = K, L 180 SCALE(I) = 1.0D0 C .......... ITERATIVE LOOP FOR NORM REDUCTION .......... 190 NOCONV = .FALSE. C DO 270 I = K, L C = 0.0D0 R = 0.0D0 C DO 200 J = K, L IF (J .EQ. I) GO TO 200 C = C + DABS(AR(J,I)) + DABS(AI(J,I)) R = R + DABS(AR(I,J)) + DABS(AI(I,J)) 200 CONTINUE C .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW .......... IF (C .EQ. 0.0D0 .OR. R .EQ. 0.0D0) GO TO 270 G = R / RADIX F = 1.0D0 S = C + R 210 IF (C .GE. G) GO TO 220 F = F * RADIX C = C * B2 GO TO 210 220 G = R * RADIX 230 IF (C .LT. G) GO TO 240 F = F / RADIX C = C / B2 GO TO 230 C .......... NOW BALANCE .......... 240 IF ((C + R) / F .GE. 0.95D0 * S) GO TO 270 G = 1.0D0 / F SCALE(I) = SCALE(I) * F NOCONV = .TRUE. C DO 250 J = K, N AR(I,J) = AR(I,J) * G AI(I,J) = AI(I,J) * G 250 CONTINUE C DO 260 J = 1, L AR(J,I) = AR(J,I) * F AI(J,I) = AI(J,I) * F 260 CONTINUE C 270 CONTINUE C IF (NOCONV) GO TO 190 C 280 LOW = K IGH = L RETURN END SUBROUTINE CG(NM,N,AR,AI,WR,WI,MATZ,ZR,ZI,FV1,FV2,FV3,IERR) C INTEGER N,NM,IS1,IS2,IERR,MATZ DOUBLE PRECISION AR(NM,N),AI(NM,N),WR(N),WI(N),ZR(NM,N),ZI(NM,N), X FV1(N),FV2(N),FV3(N) C C THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF C SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK) C TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED) C OF A COMPLEX GENERAL MATRIX. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX A=(AR,AI). C C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE COMPLEX GENERAL MATRIX. C C MATZ IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF C ONLY EIGENVALUES ARE DESIRED. OTHERWISE IT IS SET TO C ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS. C C ON OUTPUT C C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVALUES. C C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVECTORS IF MATZ IS NOT ZERO. C C IERR IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN ERROR C COMPLETION CODE DESCRIBED IN THE DOCUMENTATION FOR COMQR C AND COMQR2. THE NORMAL COMPLETION CODE IS ZERO. C C FV1, FV2, AND FV3 ARE TEMPORARY STORAGE ARRAYS. C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C IF (N .LE. NM) GO TO 10 IERR = 10 * N GO TO 50 C 10 CALL CBAL(NM,N,AR,AI,IS1,IS2,FV1) CALL CORTH(NM,N,IS1,IS2,AR,AI,FV2,FV3) IF (MATZ .NE. 0) GO TO 20 C .......... FIND EIGENVALUES ONLY .......... CALL COMQR(NM,N,IS1,IS2,AR,AI,WR,WI,IERR) GO TO 50 C .......... FIND BOTH EIGENVALUES AND EIGENVECTORS .......... 20 CALL COMQR2(NM,N,IS1,IS2,FV2,FV3,AR,AI,WR,WI,ZR,ZI,IERR) IF (IERR .NE. 0) GO TO 50 CALL CBABK2(NM,N,IS1,IS2,FV1,N,ZR,ZI) 50 RETURN END SUBROUTINE COMQR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR) C INTEGER I,J,L,N,EN,LL,NM,IGH,ITN,ITS,LOW,LP1,ENM1,IERR DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N) DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2, X PYTHAG C C THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE C ALGOL PROCEDURE COMLR, NUM. MATH. 12, 369-376(1968) BY MARTIN C AND WILKINSON. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971). C THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS C (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM. C C THIS SUBROUTINE FINDS THE EIGENVALUES OF A COMPLEX C UPPER HESSENBERG MATRIX BY THE QR METHOD. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX. C C LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING C SUBROUTINE CBAL. IF CBAL HAS NOT BEEN USED, C SET LOW=1, IGH=N. C C HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX. C THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN C INFORMATION ABOUT THE UNITARY TRANSFORMATIONS USED IN C THE REDUCTION BY CORTH, IF PERFORMED. C C ON OUTPUT C C THE UPPER HESSENBERG PORTIONS OF HR AND HI HAVE BEEN C DESTROYED. THEREFORE, THEY MUST BE SAVED BEFORE C CALLING COMQR IF SUBSEQUENT CALCULATION OF C EIGENVECTORS IS TO BE PERFORMED. C C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVALUES. IF AN ERROR C EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT C FOR INDICES IERR+1,...,N. C C IERR IS SET TO C ZERO FOR NORMAL RETURN, C J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED C WHILE THE J-TH EIGENVALUE IS BEING SOUGHT. C C CALLS CDIV FOR COMPLEX DIVISION. C CALLS CSROOT FOR COMPLEX SQUARE ROOT. C CALLS PYTHAG FOR DSQRT(A*A + B*B) . C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C IERR = 0 IF (LOW .EQ. IGH) GO TO 180 C .......... CREATE REAL SUBDIAGONAL ELEMENTS .......... L = LOW + 1 C DO 170 I = L, IGH LL = MIN0(I+1,IGH) IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170 NORM = PYTHAG(HR(I,I-1),HI(I,I-1)) YR = HR(I,I-1) / NORM YI = HI(I,I-1) / NORM HR(I,I-1) = NORM HI(I,I-1) = 0.0D0 C DO 155 J = I, IGH SI = YR * HI(I,J) - YI * HR(I,J) HR(I,J) = YR * HR(I,J) + YI * HI(I,J) HI(I,J) = SI 155 CONTINUE C DO 160 J = LOW, LL SI = YR * HI(J,I) + YI * HR(J,I) HR(J,I) = YR * HR(J,I) - YI * HI(J,I) HI(J,I) = SI 160 CONTINUE C 170 CONTINUE C .......... STORE ROOTS ISOLATED BY CBAL .......... 180 DO 200 I = 1, N IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200 WR(I) = HR(I,I) WI(I) = HI(I,I) 200 CONTINUE C EN = IGH TR = 0.0D0 TI = 0.0D0 ITN = 30*N C .......... SEARCH FOR NEXT EIGENVALUE .......... 220 IF (EN .LT. LOW) GO TO 1001 ITS = 0 ENM1 = EN - 1 C .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT C FOR L=EN STEP -1 UNTIL LOW D0 -- .......... 240 DO 260 LL = LOW, EN L = EN + LOW - LL IF (L .EQ. LOW) GO TO 300 TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1)) X + DABS(HR(L,L)) + DABS(HI(L,L)) TST2 = TST1 + DABS(HR(L,L-1)) IF (TST2 .EQ. TST1) GO TO 300 260 CONTINUE C .......... FORM SHIFT .......... 300 IF (L .EQ. EN) GO TO 660 IF (ITN .EQ. 0) GO TO 1000 IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320 SR = HR(EN,EN) SI = HI(EN,EN) XR = HR(ENM1,EN) * HR(EN,ENM1) XI = HI(ENM1,EN) * HR(EN,ENM1) IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340 YR = (HR(ENM1,ENM1) - SR) / 2.0D0 YI = (HI(ENM1,ENM1) - SI) / 2.0D0 CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI) IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310 ZZR = -ZZR ZZI = -ZZI 310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI) SR = SR - XR SI = SI - XI GO TO 340 C .......... FORM EXCEPTIONAL SHIFT .......... 320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2)) SI = 0.0D0 C 340 DO 360 I = LOW, EN HR(I,I) = HR(I,I) - SR HI(I,I) = HI(I,I) - SI 360 CONTINUE C TR = TR + SR TI = TI + SI ITS = ITS + 1 ITN = ITN - 1 C .......... REDUCE TO TRIANGLE (ROWS) .......... LP1 = L + 1 C DO 500 I = LP1, EN SR = HR(I,I-1) HR(I,I-1) = 0.0D0 NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR) XR = HR(I-1,I-1) / NORM WR(I-1) = XR XI = HI(I-1,I-1) / NORM WI(I-1) = XI HR(I-1,I-1) = NORM HI(I-1,I-1) = 0.0D0 HI(I,I-1) = SR / NORM C DO 490 J = I, EN YR = HR(I-1,J) YI = HI(I-1,J) ZZR = HR(I,J) ZZI = HI(I,J) HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI 490 CONTINUE C 500 CONTINUE C SI = HI(EN,EN) IF (SI .EQ. 0.0D0) GO TO 540 NORM = PYTHAG(HR(EN,EN),SI) SR = HR(EN,EN) / NORM SI = SI / NORM HR(EN,EN) = NORM HI(EN,EN) = 0.0D0 C .......... INVERSE OPERATION (COLUMNS) .......... 540 DO 600 J = LP1, EN XR = WR(J-1) XI = WI(J-1) C DO 580 I = L, J YR = HR(I,J-1) YI = 0.0D0 ZZR = HR(I,J) ZZI = HI(I,J) IF (I .EQ. J) GO TO 560 YI = HI(I,J-1) HI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI 560 HR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR HR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR HI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI 580 CONTINUE C 600 CONTINUE C IF (SI .EQ. 0.0D0) GO TO 240 C DO 630 I = L, EN YR = HR(I,EN) YI = HI(I,EN) HR(I,EN) = SR * YR - SI * YI HI(I,EN) = SR * YI + SI * YR 630 CONTINUE C GO TO 240 C .......... A ROOT FOUND .......... 660 WR(EN) = HR(EN,EN) + TR WI(EN) = HI(EN,EN) + TI EN = ENM1 GO TO 220 C .......... SET ERROR -- ALL EIGENVALUES HAVE NOT C CONVERGED AFTER 30*N ITERATIONS .......... 1000 IERR = EN 1001 RETURN END SUBROUTINE COMQR2(NM,N,LOW,IGH,ORTR,ORTI,HR,HI,WR,WI,ZR,ZI,IERR) C INTEGER I,J,K,L,M,N,EN,II,JJ,LL,NM,NN,IGH,IP1, X ITN,ITS,LOW,LP1,ENM1,IEND,IERR DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N),ZR(NM,N),ZI(NM,N), X ORTR(IGH),ORTI(IGH) DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2, X PYTHAG C C THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE C ALGOL PROCEDURE COMLR2, NUM. MATH. 16, 181-204(1970) BY PETERS C AND WILKINSON. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971). C THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS C (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM. C C THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS C OF A COMPLEX UPPER HESSENBERG MATRIX BY THE QR C METHOD. THE EIGENVECTORS OF A COMPLEX GENERAL MATRIX C CAN ALSO BE FOUND IF CORTH HAS BEEN USED TO REDUCE C THIS GENERAL MATRIX TO HESSENBERG FORM. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX. C C LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING C SUBROUTINE CBAL. IF CBAL HAS NOT BEEN USED, C SET LOW=1, IGH=N. C C ORTR AND ORTI CONTAIN INFORMATION ABOUT THE UNITARY TRANS- C FORMATIONS USED IN THE REDUCTION BY CORTH, IF PERFORMED. C ONLY ELEMENTS LOW THROUGH IGH ARE USED. IF THE EIGENVECTORS C OF THE HESSENBERG MATRIX ARE DESIRED, SET ORTR(J) AND C ORTI(J) TO 0.0D0 FOR THESE ELEMENTS. C C HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX. C THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN FURTHER C INFORMATION ABOUT THE TRANSFORMATIONS WHICH WERE USED IN THE C REDUCTION BY CORTH, IF PERFORMED. IF THE EIGENVECTORS OF C THE HESSENBERG MATRIX ARE DESIRED, THESE ELEMENTS MAY BE C ARBITRARY. C C ON OUTPUT C C ORTR, ORTI, AND THE UPPER HESSENBERG PORTIONS OF HR AND HI C HAVE BEEN DESTROYED. C C WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVALUES. IF AN ERROR C EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT C FOR INDICES IERR+1,...,N. C C ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE EIGENVECTORS. THE EIGENVECTORS C ARE UNNORMALIZED. IF AN ERROR EXIT IS MADE, NONE OF C THE EIGENVECTORS HAS BEEN FOUND. C C IERR IS SET TO C ZERO FOR NORMAL RETURN, C J IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED C WHILE THE J-TH EIGENVALUE IS BEING SOUGHT. C C CALLS CDIV FOR COMPLEX DIVISION. C CALLS CSROOT FOR COMPLEX SQUARE ROOT. C CALLS PYTHAG FOR DSQRT(A*A + B*B) . C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C IERR = 0 C .......... INITIALIZE EIGENVECTOR MATRIX .......... DO 101 J = 1, N C DO 100 I = 1, N ZR(I,J) = 0.0D0 ZI(I,J) = 0.0D0 100 CONTINUE ZR(J,J) = 1.0D0 101 CONTINUE C .......... FORM THE MATRIX OF ACCUMULATED TRANSFORMATIONS C FROM THE INFORMATION LEFT BY CORTH .......... IEND = IGH - LOW - 1 IF (IEND) 180, 150, 105 C .......... FOR I=IGH-1 STEP -1 UNTIL LOW+1 DO -- .......... 105 DO 140 II = 1, IEND I = IGH - II IF (ORTR(I) .EQ. 0.0D0 .AND. ORTI(I) .EQ. 0.0D0) GO TO 140 IF (HR(I,I-1) .EQ. 0.0D0 .AND. HI(I,I-1) .EQ. 0.0D0) GO TO 140 C .......... NORM BELOW IS NEGATIVE OF H FORMED IN CORTH .......... NORM = HR(I,I-1) * ORTR(I) + HI(I,I-1) * ORTI(I) IP1 = I + 1 C DO 110 K = IP1, IGH ORTR(K) = HR(K,I-1) ORTI(K) = HI(K,I-1) 110 CONTINUE C DO 130 J = I, IGH SR = 0.0D0 SI = 0.0D0 C DO 115 K = I, IGH SR = SR + ORTR(K) * ZR(K,J) + ORTI(K) * ZI(K,J) SI = SI + ORTR(K) * ZI(K,J) - ORTI(K) * ZR(K,J) 115 CONTINUE C SR = SR / NORM SI = SI / NORM C DO 120 K = I, IGH ZR(K,J) = ZR(K,J) + SR * ORTR(K) - SI * ORTI(K) ZI(K,J) = ZI(K,J) + SR * ORTI(K) + SI * ORTR(K) 120 CONTINUE C 130 CONTINUE C 140 CONTINUE C .......... CREATE REAL SUBDIAGONAL ELEMENTS .......... 150 L = LOW + 1 C DO 170 I = L, IGH LL = MIN0(I+1,IGH) IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170 NORM = PYTHAG(HR(I,I-1),HI(I,I-1)) YR = HR(I,I-1) / NORM YI = HI(I,I-1) / NORM HR(I,I-1) = NORM HI(I,I-1) = 0.0D0 C DO 155 J = I, N SI = YR * HI(I,J) - YI * HR(I,J) HR(I,J) = YR * HR(I,J) + YI * HI(I,J) HI(I,J) = SI 155 CONTINUE C DO 160 J = 1, LL SI = YR * HI(J,I) + YI * HR(J,I) HR(J,I) = YR * HR(J,I) - YI * HI(J,I) HI(J,I) = SI 160 CONTINUE C DO 165 J = LOW, IGH SI = YR * ZI(J,I) + YI * ZR(J,I) ZR(J,I) = YR * ZR(J,I) - YI * ZI(J,I) ZI(J,I) = SI 165 CONTINUE C 170 CONTINUE C .......... STORE ROOTS ISOLATED BY CBAL .......... 180 DO 200 I = 1, N IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200 WR(I) = HR(I,I) WI(I) = HI(I,I) 200 CONTINUE C EN = IGH TR = 0.0D0 TI = 0.0D0 ITN = 30*N C .......... SEARCH FOR NEXT EIGENVALUE .......... 220 IF (EN .LT. LOW) GO TO 680 ITS = 0 ENM1 = EN - 1 C .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT C FOR L=EN STEP -1 UNTIL LOW DO -- .......... 240 DO 260 LL = LOW, EN L = EN + LOW - LL IF (L .EQ. LOW) GO TO 300 TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1)) X + DABS(HR(L,L)) + DABS(HI(L,L)) TST2 = TST1 + DABS(HR(L,L-1)) IF (TST2 .EQ. TST1) GO TO 300 260 CONTINUE C .......... FORM SHIFT .......... 300 IF (L .EQ. EN) GO TO 660 IF (ITN .EQ. 0) GO TO 1000 IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320 SR = HR(EN,EN) SI = HI(EN,EN) XR = HR(ENM1,EN) * HR(EN,ENM1) XI = HI(ENM1,EN) * HR(EN,ENM1) IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340 YR = (HR(ENM1,ENM1) - SR) / 2.0D0 YI = (HI(ENM1,ENM1) - SI) / 2.0D0 CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI) IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310 ZZR = -ZZR ZZI = -ZZI 310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI) SR = SR - XR SI = SI - XI GO TO 340 C .......... FORM EXCEPTIONAL SHIFT .......... 320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2)) SI = 0.0D0 C 340 DO 360 I = LOW, EN HR(I,I) = HR(I,I) - SR HI(I,I) = HI(I,I) - SI 360 CONTINUE C TR = TR + SR TI = TI + SI ITS = ITS + 1 ITN = ITN - 1 C .......... REDUCE TO TRIANGLE (ROWS) .......... LP1 = L + 1 C DO 500 I = LP1, EN SR = HR(I,I-1) HR(I,I-1) = 0.0D0 NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR) XR = HR(I-1,I-1) / NORM WR(I-1) = XR XI = HI(I-1,I-1) / NORM WI(I-1) = XI HR(I-1,I-1) = NORM HI(I-1,I-1) = 0.0D0 HI(I,I-1) = SR / NORM C DO 490 J = I, N YR = HR(I-1,J) YI = HI(I-1,J) ZZR = HR(I,J) ZZI = HI(I,J) HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI 490 CONTINUE C 500 CONTINUE C SI = HI(EN,EN) IF (SI .EQ. 0.0D0) GO TO 540 NORM = PYTHAG(HR(EN,EN),SI) SR = HR(EN,EN) / NORM SI = SI / NORM HR(EN,EN) = NORM HI(EN,EN) = 0.0D0 IF (EN .EQ. N) GO TO 540 IP1 = EN + 1 C DO 520 J = IP1, N YR = HR(EN,J) YI = HI(EN,J) HR(EN,J) = SR * YR + SI * YI HI(EN,J) = SR * YI - SI * YR 520 CONTINUE C .......... INVERSE OPERATION (COLUMNS) .......... 540 DO 600 J = LP1, EN XR = WR(J-1) XI = WI(J-1) C DO 580 I = 1, J YR = HR(I,J-1) YI = 0.0D0 ZZR = HR(I,J) ZZI = HI(I,J) IF (I .EQ. J) GO TO 560 YI = HI(I,J-1) HI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI 560 HR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR HR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR HI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI 580 CONTINUE C DO 590 I = LOW, IGH YR = ZR(I,J-1) YI = ZI(I,J-1) ZZR = ZR(I,J) ZZI = ZI(I,J) ZR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR ZI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI ZR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR ZI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI 590 CONTINUE C 600 CONTINUE C IF (SI .EQ. 0.0D0) GO TO 240 C DO 630 I = 1, EN YR = HR(I,EN) YI = HI(I,EN) HR(I,EN) = SR * YR - SI * YI HI(I,EN) = SR * YI + SI * YR 630 CONTINUE C DO 640 I = LOW, IGH YR = ZR(I,EN) YI = ZI(I,EN) ZR(I,EN) = SR * YR - SI * YI ZI(I,EN) = SR * YI + SI * YR 640 CONTINUE C GO TO 240 C .......... A ROOT FOUND .......... 660 HR(EN,EN) = HR(EN,EN) + TR WR(EN) = HR(EN,EN) HI(EN,EN) = HI(EN,EN) + TI WI(EN) = HI(EN,EN) EN = ENM1 GO TO 220 C .......... ALL ROOTS FOUND. BACKSUBSTITUTE TO FIND C VECTORS OF UPPER TRIANGULAR FORM .......... 680 NORM = 0.0D0 C DO 720 I = 1, N C DO 720 J = I, N TR = DABS(HR(I,J)) + DABS(HI(I,J)) IF (TR .GT. NORM) NORM = TR 720 CONTINUE C IF (N .EQ. 1 .OR. NORM .EQ. 0.0D0) GO TO 1001 C .......... FOR EN=N STEP -1 UNTIL 2 DO -- .......... DO 800 NN = 2, N EN = N + 2 - NN XR = WR(EN) XI = WI(EN) HR(EN,EN) = 1.0D0 HI(EN,EN) = 0.0D0 ENM1 = EN - 1 C .......... FOR I=EN-1 STEP -1 UNTIL 1 DO -- .......... DO 780 II = 1, ENM1 I = EN - II ZZR = 0.0D0 ZZI = 0.0D0 IP1 = I + 1 C DO 740 J = IP1, EN ZZR = ZZR + HR(I,J) * HR(J,EN) - HI(I,J) * HI(J,EN) ZZI = ZZI + HR(I,J) * HI(J,EN) + HI(I,J) * HR(J,EN) 740 CONTINUE C YR = XR - WR(I) YI = XI - WI(I) IF (YR .NE. 0.0D0 .OR. YI .NE. 0.0D0) GO TO 765 TST1 = NORM YR = TST1 760 YR = 0.01D0 * YR TST2 = NORM + YR IF (TST2 .GT. TST1) GO TO 760 765 CONTINUE CALL CDIV(ZZR,ZZI,YR,YI,HR(I,EN),HI(I,EN)) C .......... OVERFLOW CONTROL .......... TR = DABS(HR(I,EN)) + DABS(HI(I,EN)) IF (TR .EQ. 0.0D0) GO TO 780 TST1 = TR TST2 = TST1 + 1.0D0/TST1 IF (TST2 .GT. TST1) GO TO 780 DO 770 J = I, EN HR(J,EN) = HR(J,EN)/TR HI(J,EN) = HI(J,EN)/TR 770 CONTINUE C 780 CONTINUE C 800 CONTINUE C .......... END BACKSUBSTITUTION .......... ENM1 = N - 1 C .......... VECTORS OF ISOLATED ROOTS .......... DO 840 I = 1, ENM1 IF (I .GE. LOW .AND. I .LE. IGH) GO TO 840 IP1 = I + 1 C DO 820 J = IP1, N ZR(I,J) = HR(I,J) ZI(I,J) = HI(I,J) 820 CONTINUE C 840 CONTINUE C .......... MULTIPLY BY TRANSFORMATION MATRIX TO GIVE C VECTORS OF ORIGINAL FULL MATRIX. C FOR J=N STEP -1 UNTIL LOW+1 DO -- .......... DO 880 JJ = LOW, ENM1 J = N + LOW - JJ M = MIN0(J,IGH) C DO 880 I = LOW, IGH ZZR = 0.0D0 ZZI = 0.0D0 C DO 860 K = LOW, M ZZR = ZZR + ZR(I,K) * HR(K,J) - ZI(I,K) * HI(K,J) ZZI = ZZI + ZR(I,K) * HI(K,J) + ZI(I,K) * HR(K,J) 860 CONTINUE C ZR(I,J) = ZZR ZI(I,J) = ZZI 880 CONTINUE C GO TO 1001 C .......... SET ERROR -- ALL EIGENVALUES HAVE NOT C CONVERGED AFTER 30*N ITERATIONS .......... 1000 IERR = EN 1001 RETURN END SUBROUTINE CORTH(NM,N,LOW,IGH,AR,AI,ORTR,ORTI) C INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW DOUBLE PRECISION AR(NM,N),AI(NM,N),ORTR(IGH),ORTI(IGH) DOUBLE PRECISION F,G,H,FI,FR,SCALE,PYTHAG C C THIS SUBROUTINE IS A TRANSLATION OF A COMPLEX ANALOGUE OF C THE ALGOL PROCEDURE ORTHES, NUM. MATH. 12, 349-368(1968) C BY MARTIN AND WILKINSON. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971). C C GIVEN A COMPLEX GENERAL MATRIX, THIS SUBROUTINE C REDUCES A SUBMATRIX SITUATED IN ROWS AND COLUMNS C LOW THROUGH IGH TO UPPER HESSENBERG FORM BY C UNITARY SIMILARITY TRANSFORMATIONS. C C ON INPUT C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT. C C N IS THE ORDER OF THE MATRIX. C C LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING C SUBROUTINE CBAL. IF CBAL HAS NOT BEEN USED, C SET LOW=1, IGH=N. C C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE COMPLEX INPUT MATRIX. C C ON OUTPUT C C AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS, C RESPECTIVELY, OF THE HESSENBERG MATRIX. INFORMATION C ABOUT THE UNITARY TRANSFORMATIONS USED IN THE REDUCTION C IS STORED IN THE REMAINING TRIANGLES UNDER THE C HESSENBERG MATRIX. C C ORTR AND ORTI CONTAIN FURTHER INFORMATION ABOUT THE C TRANSFORMATIONS. ONLY ELEMENTS LOW THROUGH IGH ARE USED. C C CALLS PYTHAG FOR DSQRT(A*A + B*B) . C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW, C MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY C C THIS VERSION DATED AUGUST 1983. C C ------------------------------------------------------------------ C LA = IGH - 1 KP1 = LOW + 1 IF (LA .LT. KP1) GO TO 200 C DO 180 M = KP1, LA H = 0.0D0 ORTR(M) = 0.0D0 ORTI(M) = 0.0D0 SCALE = 0.0D0 C .......... SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) .......... DO 90 I = M, IGH 90 SCALE = SCALE + DABS(AR(I,M-1)) + DABS(AI(I,M-1)) C IF (SCALE .EQ. 0.0D0) GO TO 180 MP = M + IGH C .......... FOR I=IGH STEP -1 UNTIL M DO -- .......... DO 100 II = M, IGH I = MP - II ORTR(I) = AR(I,M-1) / SCALE ORTI(I) = AI(I,M-1) / SCALE H = H + ORTR(I) * ORTR(I) + ORTI(I) * ORTI(I) 100 CONTINUE C G = DSQRT(H) F = PYTHAG(ORTR(M),ORTI(M)) IF (F .EQ. 0.0D0) GO TO 103 H = H + F * G G = G / F ORTR(M) = (1.0D0 + G) * ORTR(M) ORTI(M) = (1.0D0 + G) * ORTI(M) GO TO 105 C 103 ORTR(M) = G AR(M,M-1) = SCALE C .......... FORM (I-(U*UT)/H) * A .......... 105 DO 130 J = M, N FR = 0.0D0 FI = 0.0D0 C .......... FOR I=IGH STEP -1 UNTIL M DO -- .......... DO 110 II = M, IGH I = MP - II FR = FR + ORTR(I) * AR(I,J) + ORTI(I) * AI(I,J) FI = FI + ORTR(I) * AI(I,J) - ORTI(I) * AR(I,J) 110 CONTINUE C FR = FR / H FI = FI / H C DO 120 I = M, IGH AR(I,J) = AR(I,J) - FR * ORTR(I) + FI * ORTI(I) AI(I,J) = AI(I,J) - FR * ORTI(I) - FI * ORTR(I) 120 CONTINUE C 130 CONTINUE C .......... FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) .......... DO 160 I = 1, IGH FR = 0.0D0 FI = 0.0D0 C .......... FOR J=IGH STEP -1 UNTIL M DO -- .......... DO 140 JJ = M, IGH J = MP - JJ FR = FR + ORTR(J) * AR(I,J) - ORTI(J) * AI(I,J) FI = FI + ORTR(J) * AI(I,J) + ORTI(J) * AR(I,J) 140 CONTINUE C FR = FR / H FI = FI / H C DO 150 J = M, IGH AR(I,J) = AR(I,J) - FR * ORTR(J) - FI * ORTI(J) AI(I,J) = AI(I,J) + FR * ORTI(J) - FI * ORTR(J) 150 CONTINUE C 160 CONTINUE C ORTR(M) = SCALE * ORTR(M) ORTI(M) = SCALE * ORTI(M) AR(M,M-1) = -G * AR(M,M-1) AI(M,M-1) = -G * AI(M,M-1) 180 CONTINUE C 200 RETURN END