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387 lines
13 KiB
Fortran
387 lines
13 KiB
Fortran
*DECK QZIT
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SUBROUTINE QZIT (NM, N, A, B, EPS1, MATZ, Z, IERR)
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C***BEGIN PROLOGUE QZIT
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C***PURPOSE The second step of the QZ algorithm for generalized
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C eigenproblems. Accepts an upper Hessenberg and an upper
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C triangular matrix and reduces the former to
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C quasi-triangular form while preserving the form of the
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C latter. Usually preceded by QZHES and followed by QZVAL
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C and QZVEC.
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C***LIBRARY SLATEC (EISPACK)
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C***CATEGORY D4C1B3
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C***TYPE SINGLE PRECISION (QZIT-S)
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C***KEYWORDS EIGENVALUES, EIGENVECTORS, EISPACK
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C***AUTHOR Smith, B. T., et al.
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C***DESCRIPTION
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C
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C This subroutine is the second step of the QZ algorithm
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C for solving generalized matrix eigenvalue problems,
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C SIAM J. NUMER. ANAL. 10, 241-256(1973) by MOLER and STEWART,
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C as modified in technical note NASA TN D-7305(1973) by WARD.
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C
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C This subroutine accepts a pair of REAL matrices, one of them
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C in upper Hessenberg form and the other in upper triangular form.
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C It reduces the Hessenberg matrix to quasi-triangular form using
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C orthogonal transformations while maintaining the triangular form
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C of the other matrix. It is usually preceded by QZHES and
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C followed by QZVAL and, possibly, QZVEC.
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C
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C On Input
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C
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C NM must be set to the row dimension of the two-dimensional
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C array parameters, A, B, and Z, as declared in the calling
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C program dimension statement. NM is an INTEGER variable.
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C
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C N is the order of the matrices A and B. N is an INTEGER
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C variable. N must be less than or equal to NM.
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C
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C A contains a real upper Hessenberg matrix. A is a two-
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C dimensional REAL array, dimensioned A(NM,N).
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C
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C B contains a real upper triangular matrix. B is a two-
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C dimensional REAL array, dimensioned B(NM,N).
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C
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C EPS1 is a tolerance used to determine negligible elements.
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C EPS1 = 0.0 (or negative) may be input, in which case an
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C element will be neglected only if it is less than roundoff
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C error times the norm of its matrix. If the input EPS1 is
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C positive, then an element will be considered negligible
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C if it is less than EPS1 times the norm of its matrix. A
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C positive value of EPS1 may result in faster execution,
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C but less accurate results. EPS1 is a REAL variable.
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C
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C MATZ should be set to .TRUE. if the right hand transformations
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C are to be accumulated for later use in computing
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C eigenvectors, and to .FALSE. otherwise. MATZ is a LOGICAL
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C variable.
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C
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C Z contains, if MATZ has been set to .TRUE., the transformation
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C matrix produced in the reduction by QZHES, if performed, or
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C else the identity matrix. If MATZ has been set to .FALSE.,
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C Z is not referenced. Z is a two-dimensional REAL array,
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C dimensioned Z(NM,N).
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C
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C On Output
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C
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C A has been reduced to quasi-triangular form. The elements
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C below the first subdiagonal are still zero, and no two
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C consecutive subdiagonal elements are nonzero.
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C
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C B is still in upper triangular form, although its elements
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C have been altered. The location B(N,1) is used to store
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C EPS1 times the norm of B for later use by QZVAL and QZVEC.
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C
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C Z contains the product of the right hand transformations
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C (for both steps) if MATZ has been set to .TRUE.
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C
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C IERR is an INTEGER flag set to
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C Zero for normal return,
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C J if neither A(J,J-1) nor A(J-1,J-2) has become
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C zero after a total of 30*N iterations.
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C
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C Questions and comments should be directed to B. S. Garbow,
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C APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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C ------------------------------------------------------------------
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C
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C***REFERENCES B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
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C Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
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C system Routines - EISPACK Guide, Springer-Verlag,
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C 1976.
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C***ROUTINES CALLED (NONE)
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C***REVISION HISTORY (YYMMDD)
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C 760101 DATE WRITTEN
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C 890531 Changed all specific intrinsics to generic. (WRB)
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C 890831 Modified array declarations. (WRB)
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C 890831 REVISION DATE from Version 3.2
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C 891214 Prologue converted to Version 4.0 format. (BAB)
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C 920501 Reformatted the REFERENCES section. (WRB)
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C***END PROLOGUE QZIT
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C
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INTEGER I,J,K,L,N,EN,K1,K2,LD,LL,L1,NA,NM,ISH,ITN,ITS,KM1,LM1
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INTEGER ENM2,IERR,LOR1,ENORN
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REAL A(NM,*),B(NM,*),Z(NM,*)
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REAL R,S,T,A1,A2,A3,EP,SH,U1,U2,U3,V1,V2,V3,ANI
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REAL A11,A12,A21,A22,A33,A34,A43,A44,BNI,B11
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REAL B12,B22,B33,B34,B44,EPSA,EPSB,EPS1,ANORM,BNORM
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LOGICAL MATZ,NOTLAS
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C
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C***FIRST EXECUTABLE STATEMENT QZIT
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IERR = 0
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C .......... COMPUTE EPSA,EPSB ..........
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ANORM = 0.0E0
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BNORM = 0.0E0
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C
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DO 30 I = 1, N
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ANI = 0.0E0
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IF (I .NE. 1) ANI = ABS(A(I,I-1))
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BNI = 0.0E0
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C
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DO 20 J = I, N
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ANI = ANI + ABS(A(I,J))
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BNI = BNI + ABS(B(I,J))
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20 CONTINUE
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C
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IF (ANI .GT. ANORM) ANORM = ANI
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IF (BNI .GT. BNORM) BNORM = BNI
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30 CONTINUE
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C
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IF (ANORM .EQ. 0.0E0) ANORM = 1.0E0
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IF (BNORM .EQ. 0.0E0) BNORM = 1.0E0
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EP = EPS1
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IF (EP .GT. 0.0E0) GO TO 50
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C .......... COMPUTE ROUNDOFF LEVEL IF EPS1 IS ZERO ..........
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EP = 1.0E0
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40 EP = EP / 2.0E0
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IF (1.0E0 + EP .GT. 1.0E0) GO TO 40
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50 EPSA = EP * ANORM
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EPSB = EP * BNORM
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C .......... REDUCE A TO QUASI-TRIANGULAR FORM, WHILE
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C KEEPING B TRIANGULAR ..........
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LOR1 = 1
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ENORN = N
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EN = N
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ITN = 30*N
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C .......... BEGIN QZ STEP ..........
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60 IF (EN .LE. 2) GO TO 1001
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IF (.NOT. MATZ) ENORN = EN
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ITS = 0
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NA = EN - 1
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ENM2 = NA - 1
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70 ISH = 2
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C .......... CHECK FOR CONVERGENCE OR REDUCIBILITY.
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C FOR L=EN STEP -1 UNTIL 1 DO -- ..........
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DO 80 LL = 1, EN
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LM1 = EN - LL
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L = LM1 + 1
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IF (L .EQ. 1) GO TO 95
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IF (ABS(A(L,LM1)) .LE. EPSA) GO TO 90
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80 CONTINUE
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C
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90 A(L,LM1) = 0.0E0
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IF (L .LT. NA) GO TO 95
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C .......... 1-BY-1 OR 2-BY-2 BLOCK ISOLATED ..........
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EN = LM1
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GO TO 60
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C .......... CHECK FOR SMALL TOP OF B ..........
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95 LD = L
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100 L1 = L + 1
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B11 = B(L,L)
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IF (ABS(B11) .GT. EPSB) GO TO 120
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B(L,L) = 0.0E0
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S = ABS(A(L,L)) + ABS(A(L1,L))
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U1 = A(L,L) / S
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U2 = A(L1,L) / S
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R = SIGN(SQRT(U1*U1+U2*U2),U1)
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V1 = -(U1 + R) / R
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V2 = -U2 / R
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U2 = V2 / V1
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C
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DO 110 J = L, ENORN
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T = A(L,J) + U2 * A(L1,J)
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A(L,J) = A(L,J) + T * V1
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A(L1,J) = A(L1,J) + T * V2
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T = B(L,J) + U2 * B(L1,J)
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B(L,J) = B(L,J) + T * V1
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B(L1,J) = B(L1,J) + T * V2
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110 CONTINUE
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C
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IF (L .NE. 1) A(L,LM1) = -A(L,LM1)
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LM1 = L
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L = L1
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GO TO 90
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120 A11 = A(L,L) / B11
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A21 = A(L1,L) / B11
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IF (ISH .EQ. 1) GO TO 140
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C .......... ITERATION STRATEGY ..........
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IF (ITN .EQ. 0) GO TO 1000
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IF (ITS .EQ. 10) GO TO 155
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C .......... DETERMINE TYPE OF SHIFT ..........
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B22 = B(L1,L1)
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IF (ABS(B22) .LT. EPSB) B22 = EPSB
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B33 = B(NA,NA)
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IF (ABS(B33) .LT. EPSB) B33 = EPSB
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B44 = B(EN,EN)
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IF (ABS(B44) .LT. EPSB) B44 = EPSB
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A33 = A(NA,NA) / B33
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A34 = A(NA,EN) / B44
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A43 = A(EN,NA) / B33
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A44 = A(EN,EN) / B44
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B34 = B(NA,EN) / B44
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T = 0.5E0 * (A43 * B34 - A33 - A44)
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R = T * T + A34 * A43 - A33 * A44
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IF (R .LT. 0.0E0) GO TO 150
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C .......... DETERMINE SINGLE SHIFT ZEROTH COLUMN OF A ..........
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ISH = 1
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R = SQRT(R)
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SH = -T + R
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S = -T - R
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IF (ABS(S-A44) .LT. ABS(SH-A44)) SH = S
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C .......... LOOK FOR TWO CONSECUTIVE SMALL
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C SUB-DIAGONAL ELEMENTS OF A.
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C FOR L=EN-2 STEP -1 UNTIL LD DO -- ..........
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DO 130 LL = LD, ENM2
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L = ENM2 + LD - LL
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IF (L .EQ. LD) GO TO 140
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LM1 = L - 1
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L1 = L + 1
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T = A(L,L)
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IF (ABS(B(L,L)) .GT. EPSB) T = T - SH * B(L,L)
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IF (ABS(A(L,LM1)) .LE. ABS(T/A(L1,L)) * EPSA) GO TO 100
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130 CONTINUE
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C
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140 A1 = A11 - SH
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A2 = A21
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IF (L .NE. LD) A(L,LM1) = -A(L,LM1)
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GO TO 160
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C .......... DETERMINE DOUBLE SHIFT ZEROTH COLUMN OF A ..........
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150 A12 = A(L,L1) / B22
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A22 = A(L1,L1) / B22
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B12 = B(L,L1) / B22
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A1 = ((A33 - A11) * (A44 - A11) - A34 * A43 + A43 * B34 * A11)
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1 / A21 + A12 - A11 * B12
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A2 = (A22 - A11) - A21 * B12 - (A33 - A11) - (A44 - A11)
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1 + A43 * B34
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A3 = A(L1+1,L1) / B22
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GO TO 160
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C .......... AD HOC SHIFT ..........
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155 A1 = 0.0E0
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A2 = 1.0E0
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A3 = 1.1605E0
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160 ITS = ITS + 1
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ITN = ITN - 1
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IF (.NOT. MATZ) LOR1 = LD
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C .......... MAIN LOOP ..........
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DO 260 K = L, NA
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NOTLAS = K .NE. NA .AND. ISH .EQ. 2
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K1 = K + 1
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K2 = K + 2
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KM1 = MAX(K-1,L)
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LL = MIN(EN,K1+ISH)
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IF (NOTLAS) GO TO 190
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C .......... ZERO A(K+1,K-1) ..........
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IF (K .EQ. L) GO TO 170
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A1 = A(K,KM1)
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A2 = A(K1,KM1)
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170 S = ABS(A1) + ABS(A2)
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IF (S .EQ. 0.0E0) GO TO 70
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U1 = A1 / S
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U2 = A2 / S
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R = SIGN(SQRT(U1*U1+U2*U2),U1)
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V1 = -(U1 + R) / R
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V2 = -U2 / R
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U2 = V2 / V1
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C
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DO 180 J = KM1, ENORN
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T = A(K,J) + U2 * A(K1,J)
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A(K,J) = A(K,J) + T * V1
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A(K1,J) = A(K1,J) + T * V2
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T = B(K,J) + U2 * B(K1,J)
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B(K,J) = B(K,J) + T * V1
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B(K1,J) = B(K1,J) + T * V2
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180 CONTINUE
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C
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IF (K .NE. L) A(K1,KM1) = 0.0E0
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GO TO 240
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C .......... ZERO A(K+1,K-1) AND A(K+2,K-1) ..........
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190 IF (K .EQ. L) GO TO 200
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A1 = A(K,KM1)
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A2 = A(K1,KM1)
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A3 = A(K2,KM1)
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200 S = ABS(A1) + ABS(A2) + ABS(A3)
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IF (S .EQ. 0.0E0) GO TO 260
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U1 = A1 / S
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U2 = A2 / S
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U3 = A3 / S
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R = SIGN(SQRT(U1*U1+U2*U2+U3*U3),U1)
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V1 = -(U1 + R) / R
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V2 = -U2 / R
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V3 = -U3 / R
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U2 = V2 / V1
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U3 = V3 / V1
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C
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DO 210 J = KM1, ENORN
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T = A(K,J) + U2 * A(K1,J) + U3 * A(K2,J)
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A(K,J) = A(K,J) + T * V1
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A(K1,J) = A(K1,J) + T * V2
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A(K2,J) = A(K2,J) + T * V3
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T = B(K,J) + U2 * B(K1,J) + U3 * B(K2,J)
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B(K,J) = B(K,J) + T * V1
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B(K1,J) = B(K1,J) + T * V2
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B(K2,J) = B(K2,J) + T * V3
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210 CONTINUE
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C
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IF (K .EQ. L) GO TO 220
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A(K1,KM1) = 0.0E0
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A(K2,KM1) = 0.0E0
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C .......... ZERO B(K+2,K+1) AND B(K+2,K) ..........
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220 S = ABS(B(K2,K2)) + ABS(B(K2,K1)) + ABS(B(K2,K))
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IF (S .EQ. 0.0E0) GO TO 240
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U1 = B(K2,K2) / S
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U2 = B(K2,K1) / S
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U3 = B(K2,K) / S
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R = SIGN(SQRT(U1*U1+U2*U2+U3*U3),U1)
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V1 = -(U1 + R) / R
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V2 = -U2 / R
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V3 = -U3 / R
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U2 = V2 / V1
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U3 = V3 / V1
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C
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DO 230 I = LOR1, LL
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T = A(I,K2) + U2 * A(I,K1) + U3 * A(I,K)
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A(I,K2) = A(I,K2) + T * V1
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A(I,K1) = A(I,K1) + T * V2
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A(I,K) = A(I,K) + T * V3
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T = B(I,K2) + U2 * B(I,K1) + U3 * B(I,K)
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B(I,K2) = B(I,K2) + T * V1
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B(I,K1) = B(I,K1) + T * V2
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B(I,K) = B(I,K) + T * V3
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230 CONTINUE
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C
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B(K2,K) = 0.0E0
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B(K2,K1) = 0.0E0
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IF (.NOT. MATZ) GO TO 240
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C
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DO 235 I = 1, N
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T = Z(I,K2) + U2 * Z(I,K1) + U3 * Z(I,K)
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Z(I,K2) = Z(I,K2) + T * V1
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Z(I,K1) = Z(I,K1) + T * V2
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Z(I,K) = Z(I,K) + T * V3
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235 CONTINUE
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C .......... ZERO B(K+1,K) ..........
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240 S = ABS(B(K1,K1)) + ABS(B(K1,K))
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IF (S .EQ. 0.0E0) GO TO 260
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U1 = B(K1,K1) / S
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U2 = B(K1,K) / S
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R = SIGN(SQRT(U1*U1+U2*U2),U1)
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V1 = -(U1 + R) / R
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V2 = -U2 / R
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U2 = V2 / V1
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C
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DO 250 I = LOR1, LL
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T = A(I,K1) + U2 * A(I,K)
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A(I,K1) = A(I,K1) + T * V1
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A(I,K) = A(I,K) + T * V2
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T = B(I,K1) + U2 * B(I,K)
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B(I,K1) = B(I,K1) + T * V1
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B(I,K) = B(I,K) + T * V2
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250 CONTINUE
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C
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B(K1,K) = 0.0E0
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IF (.NOT. MATZ) GO TO 260
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C
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DO 255 I = 1, N
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T = Z(I,K1) + U2 * Z(I,K)
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Z(I,K1) = Z(I,K1) + T * V1
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Z(I,K) = Z(I,K) + T * V2
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255 CONTINUE
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C
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260 CONTINUE
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C .......... END QZ STEP ..........
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GO TO 70
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C .......... SET ERROR -- NEITHER BOTTOM SUBDIAGONAL ELEMENT
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C HAS BECOME NEGLIGIBLE AFTER 30*N ITERATIONS ..........
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1000 IERR = EN
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C .......... SAVE EPSB FOR USE BY QZVAL AND QZVEC ..........
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1001 IF (N .GT. 1) B(N,1) = EPSB
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RETURN
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END
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