*DECK COMLR2 SUBROUTINE COMLR2 (NM, N, LOW, IGH, INT, HR, HI, WR, WI, ZR, ZI, + IERR) C***BEGIN PROLOGUE COMLR2 C***PURPOSE Compute the eigenvalues and eigenvectors of a complex upper C Hessenberg matrix using the modified LR method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C2B C***TYPE COMPLEX (COMLR2-C) C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK, LR METHOD C***AUTHOR Smith, B. T., et al. C***DESCRIPTION C C This subroutine is a translation of the ALGOL procedure COMLR2, C NUM. MATH. 16, 181-204(1970) by Peters and Wilkinson. C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971). C C This subroutine finds the eigenvalues and eigenvectors C of a COMPLEX UPPER Hessenberg matrix by the modified LR C method. The eigenvectors of a COMPLEX GENERAL matrix C can also be found if COMHES 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 the two-dimensional C array parameters, HR, HI, ZR and ZI, as declared in the C calling program dimension statement. NM is an INTEGER C variable. C C N is the order of the matrix H=(HR,HI). N is an INTEGER C variable. N must be less than or equal to NM. C C LOW and IGH are two INTEGER variables determined by the C balancing subroutine CBAL. If CBAL has not been used, C set LOW=1 and IGH equal to the order of the matrix, N. C C INT contains information on the rows and columns C interchanged in the reduction by COMHES, if performed. C Only elements LOW through IGH are used. If you want the C eigenvectors of a complex general matrix, leave INT as it C came from COMHES. If the eigenvectors of the Hessenberg C matrix are desired, set INT(J)=J for these elements. INT C is a one-dimensional INTEGER array, dimensioned INT(IGH). C C HR and HI contain the real and imaginary parts, respectively, C of the complex upper Hessenberg matrix. Their lower C triangles below the subdiagonal contain the multipliers C which were used in the reduction by COMHES, if performed. C If the eigenvectors of a complex general matrix are C desired, leave these multipliers in the lower triangles. C If the eigenvectors of the Hessenberg matrix are desired, C these elements must be set to zero. HR and HI are C two-dimensional REAL arrays, dimensioned HR(NM,N) and C HI(NM,N). C C On OUTPUT C C The upper Hessenberg portions of HR and HI have been C destroyed, but the location HR(1,1) contains the norm C of the triangularized matrix. C C WR and WI contain the real and imaginary parts, respectively, C of the eigenvalues of the upper Hessenberg matrix. If an C error exit is made, the eigenvalues should be correct for C indices IERR+1, IERR+2, ..., N. WR and WI are one- C dimensional REAL arrays, dimensioned WR(N) and WI(N). C C ZR and ZI contain the real and imaginary parts, respectively, C of the eigenvectors. The eigenvectors are unnormalized. C If an error exit is made, none of the eigenvectors has been C found. ZR and ZI are two-dimensional REAL arrays, C dimensioned ZR(NM,N) and ZI(NM,N). C C IERR is an INTEGER flag set to C Zero for normal return, C J if the J-th eigenvalue has not been C determined after a total of 30*N iterations. C The eigenvalues should be correct for indices C IERR+1, IERR+2, ..., N, but no eigenvectors are C computed. C C Calls CSROOT for complex square root. C Calls CDIV for complex division. C C Questions and comments should be directed to B. S. Garbow, C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY C ------------------------------------------------------------------ C C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow, C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen- C system Routines - EISPACK Guide, Springer-Verlag, C 1976. C***ROUTINES CALLED CDIV, CSROOT C***REVISION HISTORY (YYMMDD) C 760101 DATE WRITTEN C 890531 Changed all specific intrinsics to generic. (WRB) C 890831 Modified array declarations. (WRB) C 890831 REVISION DATE from Version 3.2 C 891214 Prologue converted to Version 4.0 format. (BAB) C 920501 Reformatted the REFERENCES section. (WRB) C***END PROLOGUE COMLR2 C INTEGER I,J,K,L,M,N,EN,II,JJ,LL,MM,NM,NN,IGH,IM1,IP1 INTEGER ITN,ITS,LOW,MP1,ENM1,IEND,IERR REAL HR(NM,*),HI(NM,*),WR(*),WI(*),ZR(NM,*),ZI(NM,*) REAL SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,S1,S2 INTEGER INT(*) C C***FIRST EXECUTABLE STATEMENT COMLR2 IERR = 0 C .......... INITIALIZE EIGENVECTOR MATRIX .......... DO 100 I = 1, N C DO 100 J = 1, N ZR(I,J) = 0.0E0 ZI(I,J) = 0.0E0 IF (I .EQ. J) ZR(I,J) = 1.0E0 100 CONTINUE C .......... FORM THE MATRIX OF ACCUMULATED TRANSFORMATIONS C FROM THE INFORMATION LEFT BY COMHES .......... IEND = IGH - LOW - 1 IF (IEND .LE. 0) GO TO 180 C .......... FOR I=IGH-1 STEP -1 UNTIL LOW+1 DO -- .......... DO 160 II = 1, IEND I = IGH - II IP1 = I + 1 C DO 120 K = IP1, IGH ZR(K,I) = HR(K,I-1) ZI(K,I) = HI(K,I-1) 120 CONTINUE C J = INT(I) IF (I .EQ. J) GO TO 160 C DO 140 K = I, IGH ZR(I,K) = ZR(J,K) ZI(I,K) = ZI(J,K) ZR(J,K) = 0.0E0 ZI(J,K) = 0.0E0 140 CONTINUE C ZR(J,I) = 1.0E0 160 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 = 0.0E0 TI = 0.0E0 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 S1 = ABS(HR(L-1,L-1)) + ABS(HI(L-1,L-1)) 1 + ABS(HR(L,L)) + ABS(HI(L,L)) S2 = S1 + ABS(HR(L,L-1)) + ABS(HI(L,L-1)) IF (S2 .EQ. S1) 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) - HI(ENM1,EN) * HI(EN,ENM1) XI = HR(ENM1,EN) * HI(EN,ENM1) + HI(ENM1,EN) * HR(EN,ENM1) IF (XR .EQ. 0.0E0 .AND. XI .EQ. 0.0E0) GO TO 340 YR = (HR(ENM1,ENM1) - SR) / 2.0E0 YI = (HI(ENM1,ENM1) - SI) / 2.0E0 CALL CSROOT(YR**2-YI**2+XR,2.0E0*YR*YI+XI,ZZR,ZZI) IF (YR * ZZR + YI * ZZI .GE. 0.0E0) 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 = ABS(HR(EN,ENM1)) + ABS(HR(ENM1,EN-2)) SI = ABS(HI(EN,ENM1)) + ABS(HI(ENM1,EN-2)) 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 .......... LOOK FOR TWO CONSECUTIVE SMALL C SUB-DIAGONAL ELEMENTS .......... XR = ABS(HR(ENM1,ENM1)) + ABS(HI(ENM1,ENM1)) YR = ABS(HR(EN,ENM1)) + ABS(HI(EN,ENM1)) ZZR = ABS(HR(EN,EN)) + ABS(HI(EN,EN)) C .......... FOR M=EN-1 STEP -1 UNTIL L DO -- .......... DO 380 MM = L, ENM1 M = ENM1 + L - MM IF (M .EQ. L) GO TO 420 YI = YR YR = ABS(HR(M,M-1)) + ABS(HI(M,M-1)) XI = ZZR ZZR = XR XR = ABS(HR(M-1,M-1)) + ABS(HI(M-1,M-1)) S1 = ZZR / YI * (ZZR + XR + XI) S2 = S1 + YR IF (S2 .EQ. S1) GO TO 420 380 CONTINUE C .......... TRIANGULAR DECOMPOSITION H=L*R .......... 420 MP1 = M + 1 C DO 520 I = MP1, EN IM1 = I - 1 XR = HR(IM1,IM1) XI = HI(IM1,IM1) YR = HR(I,IM1) YI = HI(I,IM1) IF (ABS(XR) + ABS(XI) .GE. ABS(YR) + ABS(YI)) GO TO 460 C .......... INTERCHANGE ROWS OF HR AND HI .......... DO 440 J = IM1, N ZZR = HR(IM1,J) HR(IM1,J) = HR(I,J) HR(I,J) = ZZR ZZI = HI(IM1,J) HI(IM1,J) = HI(I,J) HI(I,J) = ZZI 440 CONTINUE C CALL CDIV(XR,XI,YR,YI,ZZR,ZZI) WR(I) = 1.0E0 GO TO 480 460 CALL CDIV(YR,YI,XR,XI,ZZR,ZZI) WR(I) = -1.0E0 480 HR(I,IM1) = ZZR HI(I,IM1) = ZZI C DO 500 J = I, N HR(I,J) = HR(I,J) - ZZR * HR(IM1,J) + ZZI * HI(IM1,J) HI(I,J) = HI(I,J) - ZZR * HI(IM1,J) - ZZI * HR(IM1,J) 500 CONTINUE C 520 CONTINUE C .......... COMPOSITION R*L=H .......... DO 640 J = MP1, EN XR = HR(J,J-1) XI = HI(J,J-1) HR(J,J-1) = 0.0E0 HI(J,J-1) = 0.0E0 C .......... INTERCHANGE COLUMNS OF HR, HI, ZR, AND ZI, C IF NECESSARY .......... IF (WR(J) .LE. 0.0E0) GO TO 580 C DO 540 I = 1, J ZZR = HR(I,J-1) HR(I,J-1) = HR(I,J) HR(I,J) = ZZR ZZI = HI(I,J-1) HI(I,J-1) = HI(I,J) HI(I,J) = ZZI 540 CONTINUE C DO 560 I = LOW, IGH ZZR = ZR(I,J-1) ZR(I,J-1) = ZR(I,J) ZR(I,J) = ZZR ZZI = ZI(I,J-1) ZI(I,J-1) = ZI(I,J) ZI(I,J) = ZZI 560 CONTINUE C 580 DO 600 I = 1, J HR(I,J-1) = HR(I,J-1) + XR * HR(I,J) - XI * HI(I,J) HI(I,J-1) = HI(I,J-1) + XR * HI(I,J) + XI * HR(I,J) 600 CONTINUE C .......... ACCUMULATE TRANSFORMATIONS .......... DO 620 I = LOW, IGH ZR(I,J-1) = ZR(I,J-1) + XR * ZR(I,J) - XI * ZI(I,J) ZI(I,J-1) = ZI(I,J-1) + XR * ZI(I,J) + XI * ZR(I,J) 620 CONTINUE C 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 = 0.0E0 C DO 720 I = 1, N C DO 720 J = I, N NORM = NORM + ABS(HR(I,J)) + ABS(HI(I,J)) 720 CONTINUE C HR(1,1) = NORM IF (N .EQ. 1 .OR. NORM .EQ. 0.0E0) 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) ENM1 = EN - 1 C .......... FOR I=EN-1 STEP -1 UNTIL 1 DO -- .......... DO 780 II = 1, ENM1 I = EN - II ZZR = HR(I,EN) ZZI = HI(I,EN) IF (I .EQ. ENM1) GO TO 760 IP1 = I + 1 C DO 740 J = IP1, ENM1 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 760 YR = XR - WR(I) YI = XI - WI(I) IF (YR .NE. 0.0E0 .OR. YI .NE. 0.0E0) GO TO 775 YR = NORM 770 YR = 0.5E0*YR IF (NORM + YR .GT. NORM) GO TO 770 YR = 2.0E0*YR 775 CALL CDIV(ZZR,ZZI,YR,YI,HR(I,EN),HI(I,EN)) 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 = MIN(J-1,IGH) C DO 880 I = LOW, IGH ZZR = ZR(I,J) ZZI = ZI(I,J) 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 -- NO CONVERGENCE TO AN C EIGENVALUE AFTER 30*N ITERATIONS .......... 1000 IERR = EN 1001 RETURN END