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OpenLibm/slatec/comqr2.f
Viral B. Shah c977aa998f Add Makefile.extras to build libopenlibm-extras. 
Replace amos with slatec
2012-12-31 16:37:05 -05:00

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*DECK COMQR2
SUBROUTINE COMQR2 (NM, N, LOW, IGH, ORTR, ORTI, HR, HI, WR, WI,
+ ZR, ZI, IERR)
C***BEGIN PROLOGUE COMQR2
C***PURPOSE Compute the eigenvalues and eigenvectors of a complex upper
C Hessenberg matrix.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2B
C***TYPE COMPLEX (HQR2-S, COMQR2-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 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 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 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 0.0E0 for these elements. ORTR and ORTI are
C one-dimensional REAL arrays, dimensioned ORTR(IGH) and
C ORTI(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 information about
C the unitary transformations used in the reduction by CORTH,
C if performed. If the eigenvectors of the Hessenberg matrix
C are desired, these elements may be arbitrary. HR and HI
C are two-dimensional REAL arrays, dimensioned HR(NM,N) and
C HI(NM,N).
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, 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 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 COMQR2
C
INTEGER I,J,K,L,M,N,EN,II,JJ,LL,NM,NN,IGH,IP1
INTEGER ITN,ITS,LOW,LP1,ENM1,IEND,IERR
REAL HR(NM,*),HI(NM,*),WR(*),WI(*),ZR(NM,*),ZI(NM,*)
REAL ORTR(*),ORTI(*)
REAL SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,S1,S2
REAL PYTHAG
C
C***FIRST EXECUTABLE STATEMENT COMQR2
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 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. 0.0E0 .AND. ORTI(I) .EQ. 0.0E0) GO TO 140
IF (HR(I,I-1) .EQ. 0.0E0 .AND. HI(I,I-1) .EQ. 0.0E0) 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 = 0.0E0
SI = 0.0E0
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 = 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, 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 = 0.0E0
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))
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, 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. 0.0E0) 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
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 = 0.0E0
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
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. 0.0E0) 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 = 0.0E0
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
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
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