*DECK QZIT
SUBROUTINE QZIT (NM, N, A, B, EPS1, MATZ, Z, IERR)
C***BEGIN PROLOGUE QZIT
C***PURPOSE The second step of the QZ algorithm for generalized
C eigenproblems. Accepts an upper Hessenberg and an upper
C triangular matrix and reduces the former to
C quasi-triangular form while preserving the form of the
C latter. Usually preceded by QZHES and followed by QZVAL
C and QZVEC.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4C1B3
C***TYPE SINGLE PRECISION (QZIT-S)
C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is the second step of the QZ algorithm
C for solving generalized matrix eigenvalue problems,
C SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART,
C as modified in technical note NASA TN D-7305(1973) by WARD.
C
C This subroutine accepts a pair of REAL matrices, one of them
C in upper Hessenberg form and the other in upper triangular form.
C It reduces the Hessenberg matrix to quasi-triangular form using
C orthogonal transformations while maintaining the triangular form
C of the other matrix. It is usually preceded by QZHES and
C followed by QZVAL and, possibly, QZVEC.
C
C On Input
C
C NM must be set to the row dimension of the two-dimensional
C array parameters, A, B, and Z, as declared in the calling
C program dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrices A and B. N is an INTEGER
C variable. N must be less than or equal to NM.
C
C A contains a real upper Hessenberg matrix. A is a two-
C dimensional REAL array, dimensioned A(NM,N).
C
C B contains a real upper triangular matrix. B is a two-
C dimensional REAL array, dimensioned B(NM,N).
C
C EPS1 is a tolerance used to determine negligible elements.
C EPS1 = 0.0 (or negative) may be input, in which case an
C element will be neglected only if it is less than roundoff
C error times the norm of its matrix. If the input EPS1 is
C positive, then an element will be considered negligible
C if it is less than EPS1 times the norm of its matrix. A
C positive value of EPS1 may result in faster execution,
C but less accurate results. EPS1 is a REAL variable.
C
C MATZ should be set to .TRUE. if the right hand transformations
C are to be accumulated for later use in computing
C eigenvectors, and to .FALSE. otherwise. MATZ is a LOGICAL
C variable.
C
C Z contains, if MATZ has been set to .TRUE., the transformation
C matrix produced in the reduction by QZHES, if performed, or
C else the identity matrix. If MATZ has been set to .FALSE.,
C Z is not referenced. Z is a two-dimensional REAL array,
C dimensioned Z(NM,N).
C
C On Output
C
C A has been reduced to quasi-triangular form. The elements
C below the first subdiagonal are still zero, and no two
C consecutive subdiagonal elements are nonzero.
C
C B is still in upper triangular form, although its elements
C have been altered. The location B(N,1) is used to store
C EPS1 times the norm of B for later use by QZVAL and QZVEC.
C
C Z contains the product of the right hand transformations
C (for both steps) if MATZ has been set to .TRUE.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C J if neither A(J,J-1) nor A(J-1,J-2) has become
C zero after a total of 30*N iterations.
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 (NONE)
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 QZIT
C
INTEGER I,J,K,L,N,EN,K1,K2,LD,LL,L1,NA,NM,ISH,ITN,ITS,KM1,LM1
INTEGER ENM2,IERR,LOR1,ENORN
REAL A(NM,*),B(NM,*),Z(NM,*)
REAL R,S,T,A1,A2,A3,EP,SH,U1,U2,U3,V1,V2,V3,ANI
REAL A11,A12,A21,A22,A33,A34,A43,A44,BNI,B11
REAL B12,B22,B33,B34,B44,EPSA,EPSB,EPS1,ANORM,BNORM
LOGICAL MATZ,NOTLAS
C
C***FIRST EXECUTABLE STATEMENT QZIT
IERR = 0
C .......... COMPUTE EPSA,EPSB ..........
ANORM = 0.0E0
BNORM = 0.0E0
C
DO 30 I = 1, N
ANI = 0.0E0
IF (I .NE. 1) ANI = ABS(A(I,I-1))
BNI = 0.0E0
C
DO 20 J = I, N
ANI = ANI + ABS(A(I,J))
BNI = BNI + ABS(B(I,J))
20 CONTINUE
C
IF (ANI .GT. ANORM) ANORM = ANI
IF (BNI .GT. BNORM) BNORM = BNI
30 CONTINUE
C
IF (ANORM .EQ. 0.0E0) ANORM = 1.0E0
IF (BNORM .EQ. 0.0E0) BNORM = 1.0E0
EP = EPS1
IF (EP .GT. 0.0E0) GO TO 50
C .......... COMPUTE ROUNDOFF LEVEL IF EPS1 IS ZERO ..........
EP = 1.0E0
40 EP = EP / 2.0E0
IF (1.0E0 + EP .GT. 1.0E0) GO TO 40
50 EPSA = EP * ANORM
EPSB = EP * BNORM
C .......... REDUCE A TO QUASI-TRIANGULAR FORM, WHILE
C KEEPING B TRIANGULAR ..........
LOR1 = 1
ENORN = N
EN = N
ITN = 30*N
C .......... BEGIN QZ STEP ..........
60 IF (EN .LE. 2) GO TO 1001
IF (.NOT. MATZ) ENORN = EN
ITS = 0
NA = EN - 1
ENM2 = NA - 1
70 ISH = 2
C .......... CHECK FOR CONVERGENCE OR REDUCIBILITY.
C FOR L=EN STEP -1 UNTIL 1 DO -- ..........
DO 80 LL = 1, EN
LM1 = EN - LL
L = LM1 + 1
IF (L .EQ. 1) GO TO 95
IF (ABS(A(L,LM1)) .LE. EPSA) GO TO 90
80 CONTINUE
C
90 A(L,LM1) = 0.0E0
IF (L .LT. NA) GO TO 95
C .......... 1-BY-1 OR 2-BY-2 BLOCK ISOLATED ..........
EN = LM1
GO TO 60
C .......... CHECK FOR SMALL TOP OF B ..........
95 LD = L
100 L1 = L + 1
B11 = B(L,L)
IF (ABS(B11) .GT. EPSB) GO TO 120
B(L,L) = 0.0E0
S = ABS(A(L,L)) + ABS(A(L1,L))
U1 = A(L,L) / S
U2 = A(L1,L) / S
R = SIGN(SQRT(U1*U1+U2*U2),U1)
V1 = -(U1 + R) / R
V2 = -U2 / R
U2 = V2 / V1
C
DO 110 J = L, ENORN
T = A(L,J) + U2 * A(L1,J)
A(L,J) = A(L,J) + T * V1
A(L1,J) = A(L1,J) + T * V2
T = B(L,J) + U2 * B(L1,J)
B(L,J) = B(L,J) + T * V1
B(L1,J) = B(L1,J) + T * V2
110 CONTINUE
C
IF (L .NE. 1) A(L,LM1) = -A(L,LM1)
LM1 = L
L = L1
GO TO 90
120 A11 = A(L,L) / B11
A21 = A(L1,L) / B11
IF (ISH .EQ. 1) GO TO 140
C .......... ITERATION STRATEGY ..........
IF (ITN .EQ. 0) GO TO 1000
IF (ITS .EQ. 10) GO TO 155
C .......... DETERMINE TYPE OF SHIFT ..........
B22 = B(L1,L1)
IF (ABS(B22) .LT. EPSB) B22 = EPSB
B33 = B(NA,NA)
IF (ABS(B33) .LT. EPSB) B33 = EPSB
B44 = B(EN,EN)
IF (ABS(B44) .LT. EPSB) B44 = EPSB
A33 = A(NA,NA) / B33
A34 = A(NA,EN) / B44
A43 = A(EN,NA) / B33
A44 = A(EN,EN) / B44
B34 = B(NA,EN) / B44
T = 0.5E0 * (A43 * B34 - A33 - A44)
R = T * T + A34 * A43 - A33 * A44
IF (R .LT. 0.0E0) GO TO 150
C .......... DETERMINE SINGLE SHIFT ZEROTH COLUMN OF A ..........
ISH = 1
R = SQRT(R)
SH = -T + R
S = -T - R
IF (ABS(S-A44) .LT. ABS(SH-A44)) SH = S
C .......... LOOK FOR TWO CONSECUTIVE SMALL
C SUB-DIAGONAL ELEMENTS OF A.
C FOR L=EN-2 STEP -1 UNTIL LD DO -- ..........
DO 130 LL = LD, ENM2
L = ENM2 + LD - LL
IF (L .EQ. LD) GO TO 140
LM1 = L - 1
L1 = L + 1
T = A(L,L)
IF (ABS(B(L,L)) .GT. EPSB) T = T - SH * B(L,L)
IF (ABS(A(L,LM1)) .LE. ABS(T/A(L1,L)) * EPSA) GO TO 100
130 CONTINUE
C
140 A1 = A11 - SH
A2 = A21
IF (L .NE. LD) A(L,LM1) = -A(L,LM1)
GO TO 160
C .......... DETERMINE DOUBLE SHIFT ZEROTH COLUMN OF A ..........
150 A12 = A(L,L1) / B22
A22 = A(L1,L1) / B22
B12 = B(L,L1) / B22
A1 = ((A33 - A11) * (A44 - A11) - A34 * A43 + A43 * B34 * A11)
1 / A21 + A12 - A11 * B12
A2 = (A22 - A11) - A21 * B12 - (A33 - A11) - (A44 - A11)
1 + A43 * B34
A3 = A(L1+1,L1) / B22
GO TO 160
C .......... AD HOC SHIFT ..........
155 A1 = 0.0E0
A2 = 1.0E0
A3 = 1.1605E0
160 ITS = ITS + 1
ITN = ITN - 1
IF (.NOT. MATZ) LOR1 = LD
C .......... MAIN LOOP ..........
DO 260 K = L, NA
NOTLAS = K .NE. NA .AND. ISH .EQ. 2
K1 = K + 1
K2 = K + 2
KM1 = MAX(K-1,L)
LL = MIN(EN,K1+ISH)
IF (NOTLAS) GO TO 190
C .......... ZERO A(K+1,K-1) ..........
IF (K .EQ. L) GO TO 170
A1 = A(K,KM1)
A2 = A(K1,KM1)
170 S = ABS(A1) + ABS(A2)
IF (S .EQ. 0.0E0) GO TO 70
U1 = A1 / S
U2 = A2 / S
R = SIGN(SQRT(U1*U1+U2*U2),U1)
V1 = -(U1 + R) / R
V2 = -U2 / R
U2 = V2 / V1
C
DO 180 J = KM1, ENORN
T = A(K,J) + U2 * A(K1,J)
A(K,J) = A(K,J) + T * V1
A(K1,J) = A(K1,J) + T * V2
T = B(K,J) + U2 * B(K1,J)
B(K,J) = B(K,J) + T * V1
B(K1,J) = B(K1,J) + T * V2
180 CONTINUE
C
IF (K .NE. L) A(K1,KM1) = 0.0E0
GO TO 240
C .......... ZERO A(K+1,K-1) AND A(K+2,K-1) ..........
190 IF (K .EQ. L) GO TO 200
A1 = A(K,KM1)
A2 = A(K1,KM1)
A3 = A(K2,KM1)
200 S = ABS(A1) + ABS(A2) + ABS(A3)
IF (S .EQ. 0.0E0) GO TO 260
U1 = A1 / S
U2 = A2 / S
U3 = A3 / S
R = SIGN(SQRT(U1*U1+U2*U2+U3*U3),U1)
V1 = -(U1 + R) / R
V2 = -U2 / R
V3 = -U3 / R
U2 = V2 / V1
U3 = V3 / V1
C
DO 210 J = KM1, ENORN
T = A(K,J) + U2 * A(K1,J) + U3 * A(K2,J)
A(K,J) = A(K,J) + T * V1
A(K1,J) = A(K1,J) + T * V2
A(K2,J) = A(K2,J) + T * V3
T = B(K,J) + U2 * B(K1,J) + U3 * B(K2,J)
B(K,J) = B(K,J) + T * V1
B(K1,J) = B(K1,J) + T * V2
B(K2,J) = B(K2,J) + T * V3
210 CONTINUE
C
IF (K .EQ. L) GO TO 220
A(K1,KM1) = 0.0E0
A(K2,KM1) = 0.0E0
C .......... ZERO B(K+2,K+1) AND B(K+2,K) ..........
220 S = ABS(B(K2,K2)) + ABS(B(K2,K1)) + ABS(B(K2,K))
IF (S .EQ. 0.0E0) GO TO 240
U1 = B(K2,K2) / S
U2 = B(K2,K1) / S
U3 = B(K2,K) / S
R = SIGN(SQRT(U1*U1+U2*U2+U3*U3),U1)
V1 = -(U1 + R) / R
V2 = -U2 / R
V3 = -U3 / R
U2 = V2 / V1
U3 = V3 / V1
C
DO 230 I = LOR1, LL
T = A(I,K2) + U2 * A(I,K1) + U3 * A(I,K)
A(I,K2) = A(I,K2) + T * V1
A(I,K1) = A(I,K1) + T * V2
A(I,K) = A(I,K) + T * V3
T = B(I,K2) + U2 * B(I,K1) + U3 * B(I,K)
B(I,K2) = B(I,K2) + T * V1
B(I,K1) = B(I,K1) + T * V2
B(I,K) = B(I,K) + T * V3
230 CONTINUE
C
B(K2,K) = 0.0E0
B(K2,K1) = 0.0E0
IF (.NOT. MATZ) GO TO 240
C
DO 235 I = 1, N
T = Z(I,K2) + U2 * Z(I,K1) + U3 * Z(I,K)
Z(I,K2) = Z(I,K2) + T * V1
Z(I,K1) = Z(I,K1) + T * V2
Z(I,K) = Z(I,K) + T * V3
235 CONTINUE
C .......... ZERO B(K+1,K) ..........
240 S = ABS(B(K1,K1)) + ABS(B(K1,K))
IF (S .EQ. 0.0E0) GO TO 260
U1 = B(K1,K1) / S
U2 = B(K1,K) / S
R = SIGN(SQRT(U1*U1+U2*U2),U1)
V1 = -(U1 + R) / R
V2 = -U2 / R
U2 = V2 / V1
C
DO 250 I = LOR1, LL
T = A(I,K1) + U2 * A(I,K)
A(I,K1) = A(I,K1) + T * V1
A(I,K) = A(I,K) + T * V2
T = B(I,K1) + U2 * B(I,K)
B(I,K1) = B(I,K1) + T * V1
B(I,K) = B(I,K) + T * V2
250 CONTINUE
C
B(K1,K) = 0.0E0
IF (.NOT. MATZ) GO TO 260
C
DO 255 I = 1, N
T = Z(I,K1) + U2 * Z(I,K)
Z(I,K1) = Z(I,K1) + T * V1
Z(I,K) = Z(I,K) + T * V2
255 CONTINUE
C
260 CONTINUE
C .......... END QZ STEP ..........
GO TO 70
C .......... SET ERROR -- NEITHER BOTTOM SUBDIAGONAL ELEMENT
C HAS BECOME NEGLIGIBLE AFTER 30*N ITERATIONS ..........
1000 IERR = EN
C .......... SAVE EPSB FOR USE BY QZVAL AND QZVEC ..........
1001 IF (N .GT. 1) B(N,1) = EPSB
RETURN
END