SUBROUTINE QZITW(NM,N,A,B,LOW,IGH,EPS1,MATZ,Z,IERR) C INTEGER I,J,K,L,N,EN,K1,K2,LD,LL,L1,NA,NM,ISH,ITS,KM1,LM1, X ENM2,IERR,LOR1,ENORN,LOW,IGH,LOWP1 DOUBLE PRECISION A(NM,N),B(NM,N),Z(NM,N) DOUBLE PRECISION R,S,T,A1,A2,A3,EP,SH,U1,U2,U3,V1,V2,V3,ANI, X A11,A12,A21,A22,A33,A34,A43,A44,BNI,B11,B12,B22,B33,B34, X B44,EPSA,EPSB,EPS1,ANORM,BNORM DOUBLE PRECISION DSQRT,DABS,DSIGN INTEGER MAX0,MIN0 LOGICAL MATZ,NOTLAS 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 E-7305(1973) BY WARD. C THE CODE IS A MODIFICATION OF SUBROUTINE QZIT FOUND IN EISPACK C SO AS TO ACCEPT PARTITIONED MATRICES AS PRODUCED BY THE C BALANCING ALGORITHM OF WARD IN SIAM J. SCIEN. STAT. COMPU. 2, C 141-152 (1981) 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 QZHESW AND C FOLLOWED BY QZVAL AND, POSSIBLY, QZVEC. C C ON INPUT- C C NM MUST BE SET TO THE ROW DIMENSION OF TWO-DIMENSIONAL C ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM C DIMENSION STATEMENT, C C N IS THE ORDER OF THE MATRICES, C C A CONTAINS A REAL UPPER HESSENBERG MATRIX, C C B CONTAINS A REAL UPPER TRIANGULAR MATRIX, C C LOW IS THE STARTING INDEX FOR THE SCALED SUBMATRICES OF C A AND B OBTAINED FROM THE BALANCING ALGORITHM. C IF THE MATRICES HAVE NOT BEEN BALANCED, SET LOW TO 1. C C IGH IS THE ENDING INDEX FOR THE SCALED SUBMATRICES OF C A AND B OBTAINED FROM THE BALANCING ALGORITHM. C IF THE MATRICES HAVE NOT BEEN BALANCED, SET IGH TO N. C C EPS1 IS A TOLERANCE USED TO DETERMINE NEGLIGIBLE ELEMENTS. C EPS1 = 0.0D0 (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, 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, C C Z CONTAINS, IF MATZ HAS BEEN SET TO .TRUE., THE C TRANSFORMATION MATRIX PRODUCED IN THE REDUCTION C BY QZHES, IF PERFORMED, OR ELSE THE IDENTITY MATRIX. C IF MATZ HAS BEEN SET TO .FALSE., Z IS NOT REFERENCED. 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 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 50 ITERATIONS. C C QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO R. C. WARD, C UNION CARBIDE CORP.-NUCLEAR DIVISION, P.O. BOX Y, BLDG. C 9704-1, OAK RIDGE, TENNESSEE 37830. C C ------------------------------------------------------------------ C IERR = 0 C ********** COMPUTE EPSA,EPSB ********** ANORM = 0.0D0 BNORM = 0.0D0 C DO 30 I = LOW, IGH ANI = 0.0D0 IF (I .NE. LOW) ANI = DABS(A(I,I-1)) BNI = 0.0D0 C DO 20 J = I, IGH ANI = ANI + DABS(A(I,J)) BNI = BNI + DABS(B(I,J)) 20 CONTINUE C IF (ANI .GT. ANORM) ANORM = ANI IF (BNI .GT. BNORM) BNORM = BNI 30 CONTINUE C IF (ANORM .EQ. 0.0D0) ANORM = 1.0D0 IF (BNORM .EQ. 0.0D0) BNORM = 1.0D0 EP = EPS1 IF (EP .GT. 0.0D0) GO TO 50 C ********** COMPUTE ROUNDOFF LEVEL IF EPS1 IS ZERO ********** EP = 1.0D0 40 EP = EP / 2.0 IF (1.0D0 + EP .GT. 1.0D0) 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 = IGH LOWP1 = LOW + 1 C ********** BEGIN QZ STEP ********** 60 IF (EN .LE. LOWP1) 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 = LOW, EN L = EN + LOW - LL IF (L .EQ. LOW) GO TO 95 LM1 = L - 1 IF (DABS(A(L,LM1)) .LE. EPSA) GO TO 90 80 CONTINUE C 90 A(L,LM1) = 0.0D0 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 (DABS(B11) .GT. EPSB) GO TO 120 B(L,L) = 0.0D0 S = DABS(A(L,L)) + DABS(A(L1,L)) U1 = A(L,L) / S U2 = A(L1,L) / S R = DSIGN(DSQRT(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. LOW) 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 (ITS .EQ. 50) GO TO 1000 IF (ITS .EQ. 10) GO TO 155 C ********** DETERMINE TYPE OF SHIFT ********** B22 = B(L1,L1) IF (DABS(B22) .LT. EPSB) B22 = EPSB B33 = B(NA,NA) IF (DABS(B33) .LT. EPSB) B33 = EPSB B44 = B(EN,EN) IF (DABS(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 = 0.5 * (A43 * B34 - A33 - A44) R = T * T + A34 * A43 - A33 * A44 IF (R .LT. 0.0D0) GO TO 150 C ********** DETERMINE SINGLE SHIFT ZEROTH COLUMN OF A ********** ISH = 1 R = DSQRT(R) SH = -T + R S = -T - R IF (DABS(S-A44) .LT. DABS(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 (DABS(B(L,L)) .GT. EPSB) T = T - SH * B(L,L) IF (DABS(A(L,LM1)) .LE. DABS(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) X / A21 + A12 - A11 * B12 A2 = (A22 - A11) - A21 * B12 - (A33 - A11) - (A44 - A11) X + A43 * B34 A3 = A(L1+1,L1) / B22 GO TO 160 C ********** AD HOC SHIFT ********** 155 A1 = 0.0D0 A2 = 1.0D0 A3 = 1.1605D0 160 ITS = ITS + 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 = MAX0(K-1,L) LL = MIN0(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 = DABS(A1) + DABS(A2) IF (S .EQ. 0.0D0) GO TO 70 U1 = A1 / S U2 = A2 / S R = DSIGN(DSQRT(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) = 0.0D0 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 = DABS(A1) + DABS(A2) + DABS(A3) IF (S .EQ. 0.0D0) GO TO 260 U1 = A1 / S U2 = A2 / S U3 = A3 / S R = DSIGN(DSQRT(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) = 0.0D0 A(K2,KM1) = 0.0D0 C ********** ZERO B(K+2,K+1) AND B(K+2,K) ********** 220 S = DABS(B(K2,K2)) + DABS(B(K2,K1)) + DABS(B(K2,K)) IF (S .EQ. 0.0D0) GO TO 240 U1 = B(K2,K2) / S U2 = B(K2,K1) / S U3 = B(K2,K) / S R = DSIGN(DSQRT(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) = 0.0D0 B(K2,K1) = 0.0D0 IF (.NOT. MATZ) GO TO 240 C DO 235 I = LOW, IGH 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 = DABS(B(K1,K1)) + DABS(B(K1,K)) IF (S .EQ. 0.0D0) GO TO 260 U1 = B(K1,K1) / S U2 = B(K1,K) / S R = DSIGN(DSQRT(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) = 0.0D0 IF (.NOT. MATZ) GO TO 260 C DO 255 I = LOW,IGH 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 50 ITERATIONS ********** 1000 IERR = EN C ********** SAVE EPSB FOR USE BY QZVAL AND QZVEC ********** 1001 IF (N .GT. 1) B(N,1) = EPSB RETURN C ********** LAST CARD OF QZITW ********** END