*DECK INVIT
SUBROUTINE INVIT (NM, N, A, WR, WI, SELECT, MM, M, Z, IERR, RM1,
+ RV1, RV2)
C***BEGIN PROLOGUE INVIT
C***PURPOSE Compute the eigenvectors of a real upper Hessenberg
C matrix associated with specified eigenvalues by inverse
C iteration.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C2B
C***TYPE SINGLE PRECISION (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 INVIT
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 REAL 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, A and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix A. N is an INTEGER variable.
C N must be less than or equal to NM.
C
C A contains the upper Hessenberg matrix. A is a two-dimensional
C REAL array, dimensioned A(NM,N).
C
C WR and WI contain the real and imaginary parts, respectively,
C of the eigenvalues of the Hessenberg matrix. The eigenvalues
C must be stored in a manner identical to that output by
C subroutine HQR, which recognizes possible splitting of the
C matrix. WR and WI are one-dimensional REAL arrays,
C dimensioned WR(N) and 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 columns required to store the eigenvectors to be found.
C NOTE that two columns are required to store the
C eigenvector corresponding to a complex eigenvalue. One
C column is required to store the eigenvector corresponding
C to a real eigenvalue. MM is an INTEGER variable.
C
C On OUTPUT
C
C A and WI are unaltered.
C
C WR may have been altered since close eigenvalues are perturbed
C slightly in searching for independent eigenvectors.
C
C SELECT may have been altered. If the elements corresponding
C to a pair of conjugate complex eigenvalues were each
C initially set to .TRUE., the program resets the second of
C the two elements to .FALSE.
C
C M is the number of columns actually used to store the
C eigenvectors. M is an INTEGER variable.
C
C Z contains the real and imaginary parts of the eigenvectors.
C The eigenvectors are packed into the columns of Z starting
C at the first column. If the next selected eigenvalue is
C real, the next column of Z contains its eigenvector. If the
C eigenvalue is complex, the next two columns of Z contain the
C real and imaginary parts of its eigenvector, with the real
C part first. The eigenvectors are normalized so that the
C component of largest magnitude is 1. Any vector which fails
C the acceptance test is set to zero. Z is a two-dimensional
C REAL array, dimensioned Z(NM,MM).
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C -(2*N+1) if more than MM columns of Z are necessary
C to store the eigenvectors corresponding to
C the specified eigenvalues (in this case, M is
C equal to the number of columns of Z containing
C eigenvectors already computed),
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 Z are set to zero
C vectors,
C -(N+K) if both error situations occur.
C
C RM1 is a two-dimensional REAL array used for temporary storage.
C This array holds the triangularized form of the upper
C Hessenberg matrix used in the inverse iteration process.
C RM1 is dimensioned RM1(N,N).
C
C RV1 and RV2 are one-dimensional REAL arrays used for temporary
C storage. They hold the approximate eigenvectors during the
C inverse iteration process. RV1 and RV2 are dimensioned
C RV1(N) and RV2(N).
C
C The ALGOL procedure GUESSVEC appears in INVIT 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 INVIT
C
INTEGER I,J,K,L,M,N,S,II,IP,MM,MP,NM,NS,N1,UK,IP1,ITS,KM1,IERR
REAL A(NM,*),WR(*),WI(*),Z(NM,*)
REAL RM1(N,*),RV1(*),RV2(*)
REAL T,W,X,Y,EPS3
REAL NORM,NORMV,GROWTO,ILAMBD,RLAMBD,UKROOT
REAL PYTHAG
LOGICAL SELECT(N)
C
C***FIRST EXECUTABLE STATEMENT INVIT
IERR = 0
UK = 0
S = 1
C .......... IP = 0, REAL EIGENVALUE
C 1, FIRST OF CONJUGATE COMPLEX PAIR
C -1, SECOND OF CONJUGATE COMPLEX PAIR ..........
IP = 0
N1 = N - 1
C
DO 980 K = 1, N
IF (WI(K) .EQ. 0.0E0 .OR. IP .LT. 0) GO TO 100
IP = 1
IF (SELECT(K) .AND. SELECT(K+1)) SELECT(K+1) = .FALSE.
100 IF (.NOT. SELECT(K)) GO TO 960
IF (WI(K) .NE. 0.0E0) S = S + 1
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 (A(UK+1,UK) .EQ. 0.0E0) 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 + ABS(A(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 THE 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 .......... PERTURB CONJUGATE EIGENVALUE TO MATCH ..........
IP1 = K + IP
WR(IP1) = RLAMBD
C .......... FORM UPPER HESSENBERG A-RLAMBD*I (TRANSPOSED)
C AND INITIAL REAL VECTOR ..........
280 MP = 1
C
DO 320 I = 1, UK
C
DO 300 J = MP, UK
300 RM1(J,I) = A(I,J)
C
RM1(I,I) = RM1(I,I) - RLAMBD
MP = I
RV1(I) = EPS3
320 CONTINUE
C
ITS = 0
IF (ILAMBD .NE. 0.0E0) GO TO 520
C .......... REAL EIGENVALUE.
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 (ABS(RM1(MP,I)) .LE. ABS(RM1(MP,MP))) GO TO 360
C
DO 340 J = MP, UK
Y = RM1(J,I)
RM1(J,I) = RM1(J,MP)
RM1(J,MP) = Y
340 CONTINUE
C
360 IF (RM1(MP,MP) .EQ. 0.0E0) RM1(MP,MP) = EPS3
X = RM1(MP,I) / RM1(MP,MP)
IF (X .EQ. 0.0E0) GO TO 400
C
DO 380 J = I, UK
380 RM1(J,I) = RM1(J,I) - X * RM1(J,MP)
C
400 CONTINUE
C
420 IF (RM1(UK,UK) .EQ. 0.0E0) RM1(UK,UK) = EPS3
C .......... BACK SUBSTITUTION FOR REAL VECTOR
C FOR I=UK STEP -1 UNTIL 1 DO -- ..........
440 DO 500 II = 1, UK
I = UK + 1 - II
Y = RV1(I)
IF (I .EQ. UK) GO TO 480
IP1 = I + 1
C
DO 460 J = IP1, UK
460 Y = Y - RM1(J,I) * RV1(J)
C
480 RV1(I) = Y / RM1(I,I)
500 CONTINUE
C
GO TO 740
C .......... COMPLEX EIGENVALUE.
C TRIANGULAR DECOMPOSITION WITH INTERCHANGES,
C REPLACING ZERO PIVOTS BY EPS3. STORE IMAGINARY
C PARTS IN UPPER TRIANGLE STARTING AT (1,3) ..........
520 NS = N - S
Z(1,S-1) = -ILAMBD
Z(1,S) = 0.0E0
IF (N .EQ. 2) GO TO 550
RM1(1,3) = -ILAMBD
Z(1,S-1) = 0.0E0
IF (N .EQ. 3) GO TO 550
C
DO 540 I = 4, N
540 RM1(1,I) = 0.0E0
C
550 DO 640 I = 2, UK
MP = I - 1
W = RM1(MP,I)
IF (I .LT. N) T = RM1(MP,I+1)
IF (I .EQ. N) T = Z(MP,S-1)
X = RM1(MP,MP) * RM1(MP,MP) + T * T
IF (W * W .LE. X) GO TO 580
X = RM1(MP,MP) / W
Y = T / W
RM1(MP,MP) = W
IF (I .LT. N) RM1(MP,I+1) = 0.0E0
IF (I .EQ. N) Z(MP,S-1) = 0.0E0
C
DO 560 J = I, UK
W = RM1(J,I)
RM1(J,I) = RM1(J,MP) - X * W
RM1(J,MP) = W
IF (J .LT. N1) GO TO 555
L = J - NS
Z(I,L) = Z(MP,L) - Y * W
Z(MP,L) = 0.0E0
GO TO 560
555 RM1(I,J+2) = RM1(MP,J+2) - Y * W
RM1(MP,J+2) = 0.0E0
560 CONTINUE
C
RM1(I,I) = RM1(I,I) - Y * ILAMBD
IF (I .LT. N1) GO TO 570
L = I - NS
Z(MP,L) = -ILAMBD
Z(I,L) = Z(I,L) + X * ILAMBD
GO TO 640
570 RM1(MP,I+2) = -ILAMBD
RM1(I,I+2) = RM1(I,I+2) + X * ILAMBD
GO TO 640
580 IF (X .NE. 0.0E0) GO TO 600
RM1(MP,MP) = EPS3
IF (I .LT. N) RM1(MP,I+1) = 0.0E0
IF (I .EQ. N) Z(MP,S-1) = 0.0E0
T = 0.0E0
X = EPS3 * EPS3
600 W = W / X
X = RM1(MP,MP) * W
Y = -T * W
C
DO 620 J = I, UK
IF (J .LT. N1) GO TO 610
L = J - NS
T = Z(MP,L)
Z(I,L) = -X * T - Y * RM1(J,MP)
GO TO 615
610 T = RM1(MP,J+2)
RM1(I,J+2) = -X * T - Y * RM1(J,MP)
615 RM1(J,I) = RM1(J,I) - X * RM1(J,MP) + Y * T
620 CONTINUE
C
IF (I .LT. N1) GO TO 630
L = I - NS
Z(I,L) = Z(I,L) - ILAMBD
GO TO 640
630 RM1(I,I+2) = RM1(I,I+2) - ILAMBD
640 CONTINUE
C
IF (UK .LT. N1) GO TO 650
L = UK - NS
T = Z(UK,L)
GO TO 655
650 T = RM1(UK,UK+2)
655 IF (RM1(UK,UK) .EQ. 0.0E0 .AND. T .EQ. 0.0E0) RM1(UK,UK) = EPS3
C .......... BACK SUBSTITUTION FOR COMPLEX VECTOR
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
IF (J .LT. N1) GO TO 670
L = J - NS
T = Z(I,L)
GO TO 675
670 T = RM1(I,J+2)
675 X = X - RM1(J,I) * RV1(J) + T * RV2(J)
Y = Y - RM1(J,I) * RV2(J) - T * RV1(J)
680 CONTINUE
C
700 IF (I .LT. N1) GO TO 710
L = I - NS
T = Z(I,L)
GO TO 715
710 T = RM1(I,I+2)
715 CALL CDIV(X,Y,RM1(I,I),T,RV1(I),RV2(I))
720 CONTINUE
C .......... ACCEPTANCE TEST FOR REAL OR COMPLEX
C EIGENVECTOR AND NORMALIZATION ..........
740 ITS = ITS + 1
NORM = 0.0E0
NORMV = 0.0E0
C
DO 780 I = 1, UK
IF (ILAMBD .EQ. 0.0E0) X = ABS(RV1(I))
IF (ILAMBD .NE. 0.0E0) 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)
IF (ILAMBD .EQ. 0.0E0) X = 1.0E0 / X
IF (ILAMBD .NE. 0.0E0) Y = RV2(J)
C
DO 820 I = 1, UK
IF (ILAMBD .NE. 0.0E0) GO TO 800
Z(I,S) = RV1(I) * X
GO TO 820
800 CALL CDIV(RV1(I),RV2(I),X,Y,Z(I,S-1),Z(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
IF (ILAMBD .EQ. 0.0E0) GO TO 440
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
Z(I,S) = 0.0E0
IF (ILAMBD .NE. 0.0E0) Z(I,S-1) = 0.0E0
920 CONTINUE
C
940 S = S + 1
960 IF (IP .EQ. (-1)) IP = 0
IF (IP .EQ. 1) IP = -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 - ABS(IP)
RETURN
END