*DECK CINVIT
SUBROUTINE CINVIT (NM, N, AR, AI, WR, WI, SELECT, MM, M, ZR, ZI,
+ IERR, RM1, RM2, RV1, RV2)
C***BEGIN PROLOGUE CINVIT
C***PURPOSE Compute the eigenvectors of a complex upper Hessenberg
C associated with specified eigenvalues using inverse
C iteration.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2B
C***TYPE COMPLEX (INVIT-S, CINVIT-C)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure CXINVIT
C by Peters and Wilkinson.
C HANDBOOK FOR AUTO. COMP. VOL.II-LINEAR ALGEBRA, 418-439(1971).
C
C This subroutine finds those eigenvectors of A COMPLEX UPPER
C Hessenberg matrix corresponding to specified eigenvalues,
C using inverse iteration.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, AR, AI, 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 A=(AR,AI). N is an INTEGER
C variable. N must be less than or equal to NM.
C
C AR and AI contain the real and imaginary parts, respectively,
C of the complex upper Hessenberg matrix. AR and AI are
C two-dimensional REAL arrays, dimensioned AR(NM,N)
C and AI(NM,N).
C
C WR and WI contain the real and imaginary parts, respectively,
C of the eigenvalues of the matrix. The eigenvalues must be
C stored in a manner identical to that of subroutine COMLR,
C which recognizes possible splitting of the matrix. WR and
C WI are one-dimensional REAL arrays, dimensioned WR(N) and
C WI(N).
C
C SELECT specifies the eigenvectors to be found. The
C eigenvector corresponding to the J-th eigenvalue is
C specified by setting SELECT(J) to .TRUE. SELECT is a
C one-dimensional LOGICAL array, dimensioned SELECT(N).
C
C MM should be set to an upper bound for the number of
C eigenvectors to be found. MM is an INTEGER variable.
C
C On OUTPUT
C
C AR, AI, WI, and SELECT are unaltered.
C
C WR may have been altered since close eigenvalues are perturbed
C slightly in searching for independent eigenvectors.
C
C M is the number of eigenvectors actually found. M is an
C INTEGER variable.
C
C ZR and ZI contain the real and imaginary parts, respectively,
C of the eigenvectors corresponding to the flagged eigenvalues.
C The eigenvectors are normalized so that the component of
C largest magnitude is 1. Any vector which fails the
C acceptance test is set to zero. ZR and ZI are
C two-dimensional REAL arrays, dimensioned ZR(NM,MM) and
C ZI(NM,MM).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C -(2*N+1) if more than MM eigenvectors have been requested
C (the MM eigenvectors calculated to this point are
C in ZR and ZI),
C -K if the iteration corresponding to the K-th
C value fails (if this occurs more than once, K
C is the index of the last occurrence); the
C corresponding columns of ZR and ZI are set to
C zero vectors,
C -(N+K) if both error situations occur.
C
C RV1 and RV2 are one-dimensional REAL arrays used for
C temporary storage, dimensioned RV1(N) and RV2(N).
C They hold the approximate eigenvectors during the inverse
C iteration process.
C
C RM1 and RM2 are two-dimensional REAL arrays used for
C temporary storage, dimensioned RM1(N,N) and RM2(N,N).
C These arrays hold the triangularized form of the upper
C Hessenberg matrix used in the inverse iteration process.
C
C The ALGOL procedure GUESSVEC appears in CINVIT in-line.
C
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, 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 CINVIT
C
INTEGER I,J,K,M,N,S,II,MM,MP,NM,UK,IP1,ITS,KM1,IERR
REAL AR(NM,*),AI(NM,*),WR(*),WI(*),ZR(NM,*),ZI(NM,*)
REAL RM1(N,*),RM2(N,*),RV1(*),RV2(*)
REAL X,Y,EPS3,NORM,NORMV,GROWTO,ILAMBD,RLAMBD,UKROOT
REAL PYTHAG
LOGICAL SELECT(N)
C
C***FIRST EXECUTABLE STATEMENT CINVIT
IERR = 0
UK = 0
S = 1
C
DO 980 K = 1, N
IF (.NOT. SELECT(K)) GO TO 980
IF (S .GT. MM) GO TO 1000
IF (UK .GE. K) GO TO 200
C .......... CHECK FOR POSSIBLE SPLITTING ..........
DO 120 UK = K, N
IF (UK .EQ. N) GO TO 140
IF (AR(UK+1,UK) .EQ. 0.0E0 .AND. AI(UK+1,UK) .EQ. 0.0E0)
1 GO TO 140
120 CONTINUE
C .......... COMPUTE INFINITY NORM OF LEADING UK BY UK
C (HESSENBERG) MATRIX ..........
140 NORM = 0.0E0
MP = 1
C
DO 180 I = 1, UK
X = 0.0E0
C
DO 160 J = MP, UK
160 X = X + PYTHAG(AR(I,J),AI(I,J))
C
IF (X .GT. NORM) NORM = X
MP = I
180 CONTINUE
C .......... EPS3 REPLACES ZERO PIVOT IN DECOMPOSITION
C AND CLOSE ROOTS ARE MODIFIED BY EPS3 ..........
IF (NORM .EQ. 0.0E0) NORM = 1.0E0
EPS3 = NORM
190 EPS3 = 0.5E0*EPS3
IF (NORM + EPS3 .GT. NORM) GO TO 190
EPS3 = 2.0E0*EPS3
C .......... GROWTO IS THE CRITERION FOR GROWTH ..........
UKROOT = SQRT(REAL(UK))
GROWTO = 0.1E0 / UKROOT
200 RLAMBD = WR(K)
ILAMBD = WI(K)
IF (K .EQ. 1) GO TO 280
KM1 = K - 1
GO TO 240
C .......... PERTURB EIGENVALUE IF IT IS CLOSE
C TO ANY PREVIOUS EIGENVALUE ..........
220 RLAMBD = RLAMBD + EPS3
C .......... FOR I=K-1 STEP -1 UNTIL 1 DO -- ..........
240 DO 260 II = 1, KM1
I = K - II
IF (SELECT(I) .AND. ABS(WR(I)-RLAMBD) .LT. EPS3 .AND.
1 ABS(WI(I)-ILAMBD) .LT. EPS3) GO TO 220
260 CONTINUE
C
WR(K) = RLAMBD
C .......... FORM UPPER HESSENBERG (AR,AI)-(RLAMBD,ILAMBD)*I
C AND INITIAL COMPLEX VECTOR ..........
280 MP = 1
C
DO 320 I = 1, UK
C
DO 300 J = MP, UK
RM1(I,J) = AR(I,J)
RM2(I,J) = AI(I,J)
300 CONTINUE
C
RM1(I,I) = RM1(I,I) - RLAMBD
RM2(I,I) = RM2(I,I) - ILAMBD
MP = I
RV1(I) = EPS3
320 CONTINUE
C .......... TRIANGULAR DECOMPOSITION WITH INTERCHANGES,
C REPLACING ZERO PIVOTS BY EPS3 ..........
IF (UK .EQ. 1) GO TO 420
C
DO 400 I = 2, UK
MP = I - 1
IF (PYTHAG(RM1(I,MP),RM2(I,MP)) .LE.
1 PYTHAG(RM1(MP,MP),RM2(MP,MP))) GO TO 360
C
DO 340 J = MP, UK
Y = RM1(I,J)
RM1(I,J) = RM1(MP,J)
RM1(MP,J) = Y
Y = RM2(I,J)
RM2(I,J) = RM2(MP,J)
RM2(MP,J) = Y
340 CONTINUE
C
360 IF (RM1(MP,MP) .EQ. 0.0E0 .AND. RM2(MP,MP) .EQ. 0.0E0)
1 RM1(MP,MP) = EPS3
CALL CDIV(RM1(I,MP),RM2(I,MP),RM1(MP,MP),RM2(MP,MP),X,Y)
IF (X .EQ. 0.0E0 .AND. Y .EQ. 0.0E0) GO TO 400
C
DO 380 J = I, UK
RM1(I,J) = RM1(I,J) - X * RM1(MP,J) + Y * RM2(MP,J)
RM2(I,J) = RM2(I,J) - X * RM2(MP,J) - Y * RM1(MP,J)
380 CONTINUE
C
400 CONTINUE
C
420 IF (RM1(UK,UK) .EQ. 0.0E0 .AND. RM2(UK,UK) .EQ. 0.0E0)
1 RM1(UK,UK) = EPS3
ITS = 0
C .......... BACK SUBSTITUTION
C FOR I=UK STEP -1 UNTIL 1 DO -- ..........
660 DO 720 II = 1, UK
I = UK + 1 - II
X = RV1(I)
Y = 0.0E0
IF (I .EQ. UK) GO TO 700
IP1 = I + 1
C
DO 680 J = IP1, UK
X = X - RM1(I,J) * RV1(J) + RM2(I,J) * RV2(J)
Y = Y - RM1(I,J) * RV2(J) - RM2(I,J) * RV1(J)
680 CONTINUE
C
700 CALL CDIV(X,Y,RM1(I,I),RM2(I,I),RV1(I),RV2(I))
720 CONTINUE
C .......... ACCEPTANCE TEST FOR EIGENVECTOR
C AND NORMALIZATION ..........
ITS = ITS + 1
NORM = 0.0E0
NORMV = 0.0E0
C
DO 780 I = 1, UK
X = PYTHAG(RV1(I),RV2(I))
IF (NORMV .GE. X) GO TO 760
NORMV = X
J = I
760 NORM = NORM + X
780 CONTINUE
C
IF (NORM .LT. GROWTO) GO TO 840
C .......... ACCEPT VECTOR ..........
X = RV1(J)
Y = RV2(J)
C
DO 820 I = 1, UK
CALL CDIV(RV1(I),RV2(I),X,Y,ZR(I,S),ZI(I,S))
820 CONTINUE
C
IF (UK .EQ. N) GO TO 940
J = UK + 1
GO TO 900
C .......... IN-LINE PROCEDURE FOR CHOOSING
C A NEW STARTING VECTOR ..........
840 IF (ITS .GE. UK) GO TO 880
X = UKROOT
Y = EPS3 / (X + 1.0E0)
RV1(1) = EPS3
C
DO 860 I = 2, UK
860 RV1(I) = Y
C
J = UK - ITS + 1
RV1(J) = RV1(J) - EPS3 * X
GO TO 660
C .......... SET ERROR -- UNACCEPTED EIGENVECTOR ..........
880 J = 1
IERR = -K
C .......... SET REMAINING VECTOR COMPONENTS TO ZERO ..........
900 DO 920 I = J, N
ZR(I,S) = 0.0E0
ZI(I,S) = 0.0E0
920 CONTINUE
C
940 S = S + 1
980 CONTINUE
C
GO TO 1001
C .......... SET ERROR -- UNDERESTIMATE OF EIGENVECTOR
C SPACE REQUIRED ..........
1000 IF (IERR .NE. 0) IERR = IERR - N
IF (IERR .EQ. 0) IERR = -(2 * N + 1)
1001 M = S - 1
RETURN
END