*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