*DECK COMQR SUBROUTINE COMQR (NM, N, LOW, IGH, HR, HI, WR, WI, IERR) C***BEGIN PROLOGUE COMQR C***PURPOSE Compute the eigenvalues of complex upper Hessenberg matrix C using the QR method. C***LIBRARY SLATEC (EISPACK) C***CATEGORY D4C2B C***TYPE COMPLEX (HQR-S, COMQR-C) C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK C***AUTHOR Smith, B. T., et al. C***DESCRIPTION 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 the two-dimensional C array parameters, HR and HI, as declared in the calling C program dimension statement. NM is an INTEGER 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 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 information about C the unitary transformations used in the reduction by CORTH, C if performed. HR and HI are two-dimensional REAL arrays, C dimensioned HR(NM,N) and HI(NM,N). C C On OUTPUT C C The upper Hessenberg portions of HR and HI have been C destroyed. Therefore, they must be saved before calling C COMQR if subsequent calculation of eigenvectors is to C be performed. 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 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. C C Calls CSROOT for complex square root. C Calls PYTHAG(A,B) for sqrt(A**2 + B**2). 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, PYTHAG 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 COMQR C INTEGER I,J,L,N,EN,LL,NM,IGH,ITN,ITS,LOW,LP1,ENM1,IERR REAL HR(NM,*),HI(NM,*),WR(*),WI(*) REAL SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,S1,S2 REAL PYTHAG C C***FIRST EXECUTABLE STATEMENT COMQR IERR = 0 IF (LOW .EQ. IGH) GO TO 180 C .......... CREATE REAL SUBDIAGONAL ELEMENTS .......... L = LOW + 1 C DO 170 I = L, IGH LL = MIN(I+1,IGH) IF (HI(I,I-1) .EQ. 0.0E0) 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) = 0.0E0 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 = 0.0E0 TI = 0.0E0 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 E0 -- .......... 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)) 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) XI = 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 = 0.0E0 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) = 0.0E0 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) = 0.0E0 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. 0.0E0) GO TO 540 NORM = PYTHAG(HR(EN,EN),SI) SR = HR(EN,EN) / NORM SI = SI / NORM HR(EN,EN) = NORM HI(EN,EN) = 0.0E0 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 = 0.0E0 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. 0.0E0) 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 -- NO CONVERGENCE TO AN C EIGENVALUE AFTER 30*N ITERATIONS .......... 1000 IERR = EN 1001 RETURN END