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eigen.f

      SUBROUTINE BALANC(NM,N,A,LOW,IGH,SCALE)
C
      INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
      DOUBLE PRECISION A(NM,N),SCALE(N)
      DOUBLE PRECISION C,F,G,R,S,B2,RADIX
      LOGICAL NOCONV
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE BALANCE,
C     NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C     THIS SUBROUTINE BALANCES A REAL MATRIX AND ISOLATES
C     EIGENVALUES WHENEVER POSSIBLE.
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 MATRIX.
C
C        A CONTAINS THE INPUT MATRIX TO BE BALANCED.
C
C     ON OUTPUT
C
C        A CONTAINS THE BALANCED MATRIX.
C
C        LOW AND IGH ARE TWO INTEGERS SUCH THAT A(I,J)
C          IS EQUAL TO ZERO IF
C           (1) I IS GREATER THAN J AND
C           (2) J=1,...,LOW-1 OR I=IGH+1,...,N.
C
C        SCALE CONTAINS INFORMATION DETERMINING THE
C           PERMUTATIONS AND SCALING FACTORS USED.
C
C     SUPPOSE THAT THE PRINCIPAL SUBMATRIX IN ROWS LOW THROUGH IGH
C     HAS BEEN BALANCED, THAT P(J) DENOTES THE INDEX INTERCHANGED
C     WITH J DURING THE PERMUTATION STEP, AND THAT THE ELEMENTS
C     OF THE DIAGONAL MATRIX USED ARE DENOTED BY D(I,J).  THEN
C        SCALE(J) = P(J),    FOR J = 1,...,LOW-1
C                 = D(J,J),      J = LOW,...,IGH
C                 = P(J)         J = IGH+1,...,N.
C     THE ORDER IN WHICH THE INTERCHANGES ARE MADE IS N TO IGH+1,
C     THEN 1 TO LOW-1.
C
C     NOTE THAT 1 IS RETURNED FOR IGH IF IGH IS ZERO FORMALLY.
C
C     THE ALGOL PROCEDURE EXC CONTAINED IN BALANCE APPEARS IN
C     BALANC  IN LINE.  (NOTE THAT THE ALGOL ROLES OF IDENTIFIERS
C     K,L HAVE BEEN REVERSED.)
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      RADIX = 16.0D0
C
      B2 = RADIX * RADIX
      K = 1
      L = N
      GO TO 100
C     .......... IN-LINE PROCEDURE FOR ROW AND
C                COLUMN EXCHANGE ..........
   20 SCALE(M) = J
      IF (J .EQ. M) GO TO 50
C
      DO 30 I = 1, L
         F = A(I,J)
         A(I,J) = A(I,M)
         A(I,M) = F
   30 CONTINUE
C
      DO 40 I = K, N
         F = A(J,I)
         A(J,I) = A(M,I)
         A(M,I) = F
   40 CONTINUE
C
   50 GO TO (80,130), IEXC
C     .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE
C                AND PUSH THEM DOWN ..........
   80 IF (L .EQ. 1) GO TO 280
      L = L - 1
C     .......... FOR J=L STEP -1 UNTIL 1 DO -- ..........
  100 DO 120 JJ = 1, L
         J = L + 1 - JJ
C
         DO 110 I = 1, L
            IF (I .EQ. J) GO TO 110
            IF (A(J,I) .NE. 0.0D0) GO TO 120
  110    CONTINUE
C
         M = L
         IEXC = 1
         GO TO 20
  120 CONTINUE
C
      GO TO 140
C     .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE
C                AND PUSH THEM LEFT ..........
  130 K = K + 1
C
  140 DO 170 J = K, L
C
         DO 150 I = K, L
            IF (I .EQ. J) GO TO 150
            IF (A(I,J) .NE. 0.0D0) GO TO 170
  150    CONTINUE
C
         M = K
         IEXC = 2
         GO TO 20
  170 CONTINUE
C     .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L ..........
      DO 180 I = K, L
  180 SCALE(I) = 1.0D0
C     .......... ITERATIVE LOOP FOR NORM REDUCTION ..........
  190 NOCONV = .FALSE.
C
      DO 270 I = K, L
         C = 0.0D0
         R = 0.0D0
C
         DO 200 J = K, L
            IF (J .EQ. I) GO TO 200
            C = C + DABS(A(J,I))
            R = R + DABS(A(I,J))
  200    CONTINUE
C     .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW ..........
         IF (C .EQ. 0.0D0 .OR. R .EQ. 0.0D0) GO TO 270
         G = R / RADIX
         F = 1.0D0
         S = C + R
  210    IF (C .GE. G) GO TO 220
         F = F * RADIX
         C = C * B2
         GO TO 210
  220    G = R * RADIX
  230    IF (C .LT. G) GO TO 240
         F = F / RADIX
         C = C / B2
         GO TO 230
C     .......... NOW BALANCE ..........
  240    IF ((C + R) / F .GE. 0.95D0 * S) GO TO 270
         G = 1.0D0 / F
         SCALE(I) = SCALE(I) * F
         NOCONV = .TRUE.
C
         DO 250 J = K, N
  250    A(I,J) = A(I,J) * G
C
         DO 260 J = 1, L
  260    A(J,I) = A(J,I) * F
C
  270 CONTINUE
C
      IF (NOCONV) GO TO 190
C
  280 LOW = K
      IGH = L
      RETURN
      END
      SUBROUTINE BALBAK(NM,N,LOW,IGH,SCALE,M,Z)
C
      INTEGER I,J,K,M,N,II,NM,IGH,LOW
      DOUBLE PRECISION SCALE(N),Z(NM,M)
      DOUBLE PRECISION S
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE BALBAK,
C     NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A REAL GENERAL
C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C     BALANCED MATRIX DETERMINED BY  BALANC.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY  BALANC.
C
C        SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS
C          AND SCALING FACTORS USED BY  BALANC.
C
C        M IS THE NUMBER OF COLUMNS OF Z TO BE BACK TRANSFORMED.
C
C        Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGEN-
C          VECTORS TO BE BACK TRANSFORMED IN ITS FIRST M COLUMNS.
C
C     ON OUTPUT
C
C        Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE
C          TRANSFORMED EIGENVECTORS IN ITS FIRST M COLUMNS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (M .EQ. 0) GO TO 200
      IF (IGH .EQ. LOW) GO TO 120
C
      DO 110 I = LOW, IGH
         S = SCALE(I)
C     .......... LEFT HAND EIGENVECTORS ARE BACK TRANSFORMED
C                IF THE FOREGOING STATEMENT IS REPLACED BY
C                S=1.0D0/SCALE(I). ..........
         DO 100 J = 1, M
  100    Z(I,J) = Z(I,J) * S
C
  110 CONTINUE
C     ......... FOR I=LOW-1 STEP -1 UNTIL 1,
C               IGH+1 STEP 1 UNTIL N DO -- ..........
  120 DO 140 II = 1, N
         I = II
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 140
         IF (I .LT. LOW) I = LOW - II
         K = SCALE(I)
         IF (K .EQ. I) GO TO 140
C
         DO 130 J = 1, M
            S = Z(I,J)
            Z(I,J) = Z(K,J)
            Z(K,J) = S
  130    CONTINUE
C
  140 CONTINUE
C
  200 RETURN
      END
      SUBROUTINE CBABK2(NM,N,LOW,IGH,SCALE,M,ZR,ZI)
C
      INTEGER I,J,K,M,N,II,NM,IGH,LOW
      DOUBLE PRECISION SCALE(N),ZR(NM,M),ZI(NM,M)
      DOUBLE PRECISION S
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE
C     CBABK2, WHICH IS A COMPLEX VERSION OF BALBAK,
C     NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A COMPLEX GENERAL
C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C     BALANCED MATRIX DETERMINED BY  CBAL.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY  CBAL.
C
C        SCALE CONTAINS INFORMATION DETERMINING THE PERMUTATIONS
C          AND SCALING FACTORS USED BY  CBAL.
C
C        M IS THE NUMBER OF EIGENVECTORS TO BE BACK TRANSFORMED.
C
C        ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVECTORS TO BE
C          BACK TRANSFORMED IN THEIR FIRST M COLUMNS.
C
C     ON OUTPUT
C
C        ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE TRANSFORMED EIGENVECTORS
C          IN THEIR FIRST M COLUMNS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (M .EQ. 0) GO TO 200
      IF (IGH .EQ. LOW) GO TO 120
C
      DO 110 I = LOW, IGH
         S = SCALE(I)
C     .......... LEFT HAND EIGENVECTORS ARE BACK TRANSFORMED
C                IF THE FOREGOING STATEMENT IS REPLACED BY
C                S=1.0D0/SCALE(I). ..........
         DO 100 J = 1, M
            ZR(I,J) = ZR(I,J) * S
            ZI(I,J) = ZI(I,J) * S
  100    CONTINUE
C
  110 CONTINUE
C     .......... FOR I=LOW-1 STEP -1 UNTIL 1,
C                IGH+1 STEP 1 UNTIL N DO -- ..........
  120 DO 140 II = 1, N
         I = II
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 140
         IF (I .LT. LOW) I = LOW - II
         K = SCALE(I)
         IF (K .EQ. I) GO TO 140
C
         DO 130 J = 1, M
            S = ZR(I,J)
            ZR(I,J) = ZR(K,J)
            ZR(K,J) = S
            S = ZI(I,J)
            ZI(I,J) = ZI(K,J)
            ZI(K,J) = S
  130    CONTINUE
C
  140 CONTINUE
C
  200 RETURN
      END
      SUBROUTINE CBAL(NM,N,AR,AI,LOW,IGH,SCALE)
C
      INTEGER I,J,K,L,M,N,JJ,NM,IGH,LOW,IEXC
      DOUBLE PRECISION AR(NM,N),AI(NM,N),SCALE(N)
      DOUBLE PRECISION C,F,G,R,S,B2,RADIX
      LOGICAL NOCONV
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE
C     CBALANCE, WHICH IS A COMPLEX VERSION OF BALANCE,
C     NUM. MATH. 13, 293-304(1969) BY PARLETT AND REINSCH.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 315-326(1971).
C
C     THIS SUBROUTINE BALANCES A COMPLEX MATRIX AND ISOLATES
C     EIGENVALUES WHENEVER POSSIBLE.
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 MATRIX.
C
C        AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE COMPLEX MATRIX TO BE BALANCED.
C
C     ON OUTPUT
C
C        AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE BALANCED MATRIX.
C
C        LOW AND IGH ARE TWO INTEGERS SUCH THAT AR(I,J) AND AI(I,J)
C          ARE EQUAL TO ZERO IF
C           (1) I IS GREATER THAN J AND
C           (2) J=1,...,LOW-1 OR I=IGH+1,...,N.
C
C        SCALE CONTAINS INFORMATION DETERMINING THE
C           PERMUTATIONS AND SCALING FACTORS USED.
C
C     SUPPOSE THAT THE PRINCIPAL SUBMATRIX IN ROWS LOW THROUGH IGH
C     HAS BEEN BALANCED, THAT P(J) DENOTES THE INDEX INTERCHANGED
C     WITH J DURING THE PERMUTATION STEP, AND THAT THE ELEMENTS
C     OF THE DIAGONAL MATRIX USED ARE DENOTED BY D(I,J).  THEN
C        SCALE(J) = P(J),    FOR J = 1,...,LOW-1
C                 = D(J,J)       J = LOW,...,IGH
C                 = P(J)         J = IGH+1,...,N.
C     THE ORDER IN WHICH THE INTERCHANGES ARE MADE IS N TO IGH+1,
C     THEN 1 TO LOW-1.
C
C     NOTE THAT 1 IS RETURNED FOR IGH IF IGH IS ZERO FORMALLY.
C
C     THE ALGOL PROCEDURE EXC CONTAINED IN CBALANCE APPEARS IN
C     CBAL  IN LINE.  (NOTE THAT THE ALGOL ROLES OF IDENTIFIERS
C     K,L HAVE BEEN REVERSED.)
C
C     ARITHMETIC IS REAL THROUGHOUT.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      RADIX = 16.0D0
C
      B2 = RADIX * RADIX
      K = 1
      L = N
      GO TO 100
C     .......... IN-LINE PROCEDURE FOR ROW AND
C                COLUMN EXCHANGE ..........
   20 SCALE(M) = J
      IF (J .EQ. M) GO TO 50
C
      DO 30 I = 1, L
         F = AR(I,J)
         AR(I,J) = AR(I,M)
         AR(I,M) = F
         F = AI(I,J)
         AI(I,J) = AI(I,M)
         AI(I,M) = F
   30 CONTINUE
C
      DO 40 I = K, N
         F = AR(J,I)
         AR(J,I) = AR(M,I)
         AR(M,I) = F
         F = AI(J,I)
         AI(J,I) = AI(M,I)
         AI(M,I) = F
   40 CONTINUE
C
   50 GO TO (80,130), IEXC
C     .......... SEARCH FOR ROWS ISOLATING AN EIGENVALUE
C                AND PUSH THEM DOWN ..........
   80 IF (L .EQ. 1) GO TO 280
      L = L - 1
C     .......... FOR J=L STEP -1 UNTIL 1 DO -- ..........
  100 DO 120 JJ = 1, L
         J = L + 1 - JJ
C
         DO 110 I = 1, L
            IF (I .EQ. J) GO TO 110
            IF (AR(J,I) .NE. 0.0D0 .OR. AI(J,I) .NE. 0.0D0) GO TO 120
  110    CONTINUE
C
         M = L
         IEXC = 1
         GO TO 20
  120 CONTINUE
C
      GO TO 140
C     .......... SEARCH FOR COLUMNS ISOLATING AN EIGENVALUE
C                AND PUSH THEM LEFT ..........
  130 K = K + 1
C
  140 DO 170 J = K, L
C
         DO 150 I = K, L
            IF (I .EQ. J) GO TO 150
            IF (AR(I,J) .NE. 0.0D0 .OR. AI(I,J) .NE. 0.0D0) GO TO 170
  150    CONTINUE
C
         M = K
         IEXC = 2
         GO TO 20
  170 CONTINUE
C     .......... NOW BALANCE THE SUBMATRIX IN ROWS K TO L ..........
      DO 180 I = K, L
  180 SCALE(I) = 1.0D0
C     .......... ITERATIVE LOOP FOR NORM REDUCTION ..........
  190 NOCONV = .FALSE.
C
      DO 270 I = K, L
         C = 0.0D0
         R = 0.0D0
C
         DO 200 J = K, L
            IF (J .EQ. I) GO TO 200
            C = C + DABS(AR(J,I)) + DABS(AI(J,I))
            R = R + DABS(AR(I,J)) + DABS(AI(I,J))
  200    CONTINUE
C     .......... GUARD AGAINST ZERO C OR R DUE TO UNDERFLOW ..........
         IF (C .EQ. 0.0D0 .OR. R .EQ. 0.0D0) GO TO 270
         G = R / RADIX
         F = 1.0D0
         S = C + R
  210    IF (C .GE. G) GO TO 220
         F = F * RADIX
         C = C * B2
         GO TO 210
  220    G = R * RADIX
  230    IF (C .LT. G) GO TO 240
         F = F / RADIX
         C = C / B2
         GO TO 230
C     .......... NOW BALANCE ..........
  240    IF ((C + R) / F .GE. 0.95D0 * S) GO TO 270
         G = 1.0D0 / F
         SCALE(I) = SCALE(I) * F
         NOCONV = .TRUE.
C
         DO 250 J = K, N
            AR(I,J) = AR(I,J) * G
            AI(I,J) = AI(I,J) * G
  250    CONTINUE
C
         DO 260 J = 1, L
            AR(J,I) = AR(J,I) * F
            AI(J,I) = AI(J,I) * F
  260    CONTINUE
C
  270 CONTINUE
C
      IF (NOCONV) GO TO 190
C
  280 LOW = K
      IGH = L
      RETURN
      END
      SUBROUTINE CDIV(AR,AI,BR,BI,CR,CI)
      DOUBLE PRECISION AR,AI,BR,BI,CR,CI
C
C     COMPLEX DIVISION, (CR,CI) = (AR,AI)/(BR,BI)
C
      DOUBLE PRECISION S,ARS,AIS,BRS,BIS
      S = DABS(BR) + DABS(BI)
      ARS = AR/S
      AIS = AI/S
      BRS = BR/S
      BIS = BI/S
      S = BRS**2 + BIS**2
      CR = (ARS*BRS + AIS*BIS)/S
      CI = (AIS*BRS - ARS*BIS)/S
      RETURN
      END
      SUBROUTINE COMQR(NM,N,LOW,IGH,HR,HI,WR,WI,IERR)
C
      INTEGER I,J,L,N,EN,LL,NM,IGH,ITN,ITS,LOW,LP1,ENM1,IERR
      DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N)
      DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2,
     X       PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE
C     ALGOL PROCEDURE  COMLR, NUM. MATH. 12, 369-376(1968) BY MARTIN
C     AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 396-403(1971).
C     THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS
C     (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM.
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES OF A COMPLEX
C     UPPER HESSENBERG MATRIX BY THE QR METHOD.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  CBAL.  IF  CBAL  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX.
C          THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN
C          INFORMATION ABOUT THE UNITARY TRANSFORMATIONS USED IN
C          THE REDUCTION BY  CORTH, IF PERFORMED.
C
C     ON OUTPUT
C
C        THE UPPER HESSENBERG PORTIONS OF HR AND HI HAVE BEEN
C          DESTROYED.  THEREFORE, THEY MUST BE SAVED BEFORE
C          CALLING  COMQR  IF SUBSEQUENT CALCULATION OF
C          EIGENVECTORS IS TO BE PERFORMED.
C
C        WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVALUES.  IF AN ERROR
C          EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C          FOR INDICES IERR+1,...,N.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C                     WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C     CALLS CDIV FOR COMPLEX DIVISION.
C     CALLS CSROOT FOR COMPLEX SQUARE ROOT.
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of L to keep g77 -Wall happy
c
      L = 0
C
      IERR = 0
      IF (LOW .EQ. IGH) GO TO 180
C     .......... CREATE REAL SUBDIAGONAL ELEMENTS ..........
      L = LOW + 1
C
      DO 170 I = L, IGH
         LL = MIN0(I+1,IGH)
         IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170
         NORM = PYTHAG(HR(I,I-1),HI(I,I-1))
         YR = HR(I,I-1) / NORM
         YI = HI(I,I-1) / NORM
         HR(I,I-1) = NORM
         HI(I,I-1) = 0.0D0
C
         DO 155 J = I, IGH
            SI = YR * HI(I,J) - YI * HR(I,J)
            HR(I,J) = YR * HR(I,J) + YI * HI(I,J)
            HI(I,J) = SI
  155    CONTINUE
C
         DO 160 J = LOW, LL
            SI = YR * HI(J,I) + YI * HR(J,I)
            HR(J,I) = YR * HR(J,I) - YI * HI(J,I)
            HI(J,I) = SI
  160    CONTINUE
C
  170 CONTINUE
C     .......... STORE ROOTS ISOLATED BY CBAL ..........
  180 DO 200 I = 1, N
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200
         WR(I) = HR(I,I)
         WI(I) = HI(I,I)
  200 CONTINUE
C
      EN = IGH
      TR = 0.0D0
      TI = 0.0D0
      ITN = 30*N
C     .......... SEARCH FOR NEXT EIGENVALUE ..........
  220 IF (EN .LT. LOW) GO TO 1001
      ITS = 0
      ENM1 = EN - 1
C     .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C                FOR L=EN STEP -1 UNTIL LOW D0 -- ..........
  240 DO 260 LL = LOW, EN
         L = EN + LOW - LL
         IF (L .EQ. LOW) GO TO 300
         TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1))
     X            + DABS(HR(L,L)) + DABS(HI(L,L))
         TST2 = TST1 + DABS(HR(L,L-1))
         IF (TST2 .EQ. TST1) GO TO 300
  260 CONTINUE
C     .......... FORM SHIFT ..........
  300 IF (L .EQ. EN) GO TO 660
      IF (ITN .EQ. 0) GO TO 1000
      IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320
      SR = HR(EN,EN)
      SI = HI(EN,EN)
      XR = HR(ENM1,EN) * HR(EN,ENM1)
      XI = HI(ENM1,EN) * HR(EN,ENM1)
      IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340
      YR = (HR(ENM1,ENM1) - SR) / 2.0D0
      YI = (HI(ENM1,ENM1) - SI) / 2.0D0
      CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI)
      IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310
      ZZR = -ZZR
      ZZI = -ZZI
  310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI)
      SR = SR - XR
      SI = SI - XI
      GO TO 340
C     .......... FORM EXCEPTIONAL SHIFT ..........
  320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2))
      SI = 0.0D0
C
  340 DO 360 I = LOW, EN
         HR(I,I) = HR(I,I) - SR
         HI(I,I) = HI(I,I) - SI
  360 CONTINUE
C
      TR = TR + SR
      TI = TI + SI
      ITS = ITS + 1
      ITN = ITN - 1
C     .......... REDUCE TO TRIANGLE (ROWS) ..........
      LP1 = L + 1
C
      DO 500 I = LP1, EN
         SR = HR(I,I-1)
         HR(I,I-1) = 0.0D0
         NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR)
         XR = HR(I-1,I-1) / NORM
         WR(I-1) = XR
         XI = HI(I-1,I-1) / NORM
         WI(I-1) = XI
         HR(I-1,I-1) = NORM
         HI(I-1,I-1) = 0.0D0
         HI(I,I-1) = SR / NORM
C
         DO 490 J = I, EN
            YR = HR(I-1,J)
            YI = HI(I-1,J)
            ZZR = HR(I,J)
            ZZI = HI(I,J)
            HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR
            HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI
            HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR
            HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI
  490    CONTINUE
C
  500 CONTINUE
C
      SI = HI(EN,EN)
      IF (SI .EQ. 0.0D0) GO TO 540
      NORM = PYTHAG(HR(EN,EN),SI)
      SR = HR(EN,EN) / NORM
      SI = SI / NORM
      HR(EN,EN) = NORM
      HI(EN,EN) = 0.0D0
C     .......... INVERSE OPERATION (COLUMNS) ..........
  540 DO 600 J = LP1, EN
         XR = WR(J-1)
         XI = WI(J-1)
C
         DO 580 I = L, J
            YR = HR(I,J-1)
            YI = 0.0D0
            ZZR = HR(I,J)
            ZZI = HI(I,J)
            IF (I .EQ. J) GO TO 560
            YI = HI(I,J-1)
            HI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI
  560       HR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR
            HR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR
            HI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI
  580    CONTINUE
C
  600 CONTINUE
C
      IF (SI .EQ. 0.0D0) GO TO 240
C
      DO 630 I = L, EN
         YR = HR(I,EN)
         YI = HI(I,EN)
         HR(I,EN) = SR * YR - SI * YI
         HI(I,EN) = SR * YI + SI * YR
  630 CONTINUE
C
      GO TO 240
C     .......... A ROOT FOUND ..........
  660 WR(EN) = HR(EN,EN) + TR
      WI(EN) = HI(EN,EN) + TI
      EN = ENM1
      GO TO 220
C     .......... SET ERROR -- ALL EIGENVALUES HAVE NOT
C                CONVERGED AFTER 30*N ITERATIONS ..........
 1000 IERR = EN
 1001 RETURN
      END
      SUBROUTINE COMQR2(NM,N,LOW,IGH,ORTR,ORTI,HR,HI,WR,WI,ZR,ZI,IERR)
C
      INTEGER I,J,K,L,M,N,EN,II,JJ,LL,NM,NN,IGH,IP1,
     X        ITN,ITS,LOW,LP1,ENM1,IEND,IERR
      DOUBLE PRECISION HR(NM,N),HI(NM,N),WR(N),WI(N),ZR(NM,N),ZI(NM,N),
     X       ORTR(IGH),ORTI(IGH)
      DOUBLE PRECISION SI,SR,TI,TR,XI,XR,YI,YR,ZZI,ZZR,NORM,TST1,TST2,
     X       PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF A UNITARY ANALOGUE OF THE
C     ALGOL PROCEDURE  COMLR2, NUM. MATH. 16, 181-204(1970) BY PETERS
C     AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C     THE UNITARY ANALOGUE SUBSTITUTES THE QR ALGORITHM OF FRANCIS
C     (COMP. JOUR. 4, 332-345(1962)) FOR THE LR ALGORITHM.
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C     OF A COMPLEX UPPER HESSENBERG MATRIX BY THE QR
C     METHOD.  THE EIGENVECTORS OF A COMPLEX GENERAL MATRIX
C     CAN ALSO BE FOUND IF  CORTH  HAS BEEN USED TO REDUCE
C     THIS GENERAL MATRIX TO HESSENBERG FORM.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  CBAL.  IF  CBAL  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        ORTR AND ORTI CONTAIN INFORMATION ABOUT THE UNITARY TRANS-
C          FORMATIONS USED IN THE REDUCTION BY  CORTH, IF PERFORMED.
C          ONLY ELEMENTS LOW THROUGH IGH ARE USED.  IF THE EIGENVECTORS
C          OF THE HESSENBERG MATRIX ARE DESIRED, SET ORTR(J) AND
C          ORTI(J) TO 0.0D0 FOR THESE ELEMENTS.
C
C        HR AND HI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE COMPLEX UPPER HESSENBERG MATRIX.
C          THEIR LOWER TRIANGLES BELOW THE SUBDIAGONAL CONTAIN FURTHER
C          INFORMATION ABOUT THE TRANSFORMATIONS WHICH WERE USED IN THE
C          REDUCTION BY  CORTH, IF PERFORMED.  IF THE EIGENVECTORS OF
C          THE HESSENBERG MATRIX ARE DESIRED, THESE ELEMENTS MAY BE
C          ARBITRARY.
C
C     ON OUTPUT
C
C        ORTR, ORTI, AND THE UPPER HESSENBERG PORTIONS OF HR AND HI
C          HAVE BEEN DESTROYED.
C
C        WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVALUES.  IF AN ERROR
C          EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C          FOR INDICES IERR+1,...,N.
C
C        ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVECTORS.  THE EIGENVECTORS
C          ARE UNNORMALIZED.  IF AN ERROR EXIT IS MADE, NONE OF
C          THE EIGENVECTORS HAS BEEN FOUND.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C                     WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C     CALLS CDIV FOR COMPLEX DIVISION.
C     CALLS CSROOT FOR COMPLEX SQUARE ROOT.
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of L to keep g77 -Wall happy
c
      L = 0
C
      IERR = 0
C     .......... INITIALIZE EIGENVECTOR MATRIX ..........
      DO 101 J = 1, N
C
         DO 100 I = 1, N
            ZR(I,J) = 0.0D0
            ZI(I,J) = 0.0D0
  100    CONTINUE
         ZR(J,J) = 1.0D0
  101 CONTINUE
C     .......... FORM THE MATRIX OF ACCUMULATED TRANSFORMATIONS
C                FROM THE INFORMATION LEFT BY CORTH ..........
      IEND = IGH - LOW - 1
C      IF (IEND) 180, 150, 105
      IF (IEND .GT. 0) GOTO 180
      IF (IEND .EQ. 0) GOTO 150
C     .......... FOR I=IGH-1 STEP -1 UNTIL LOW+1 DO -- ..........
      DO 140 II = 1, IEND
         I = IGH - II
         IF (ORTR(I) .EQ. 0.0D0 .AND. ORTI(I) .EQ. 0.0D0) GO TO 140
         IF (HR(I,I-1) .EQ. 0.0D0 .AND. HI(I,I-1) .EQ. 0.0D0) GO TO 140
C     .......... NORM BELOW IS NEGATIVE OF H FORMED IN CORTH ..........
         NORM = HR(I,I-1) * ORTR(I) + HI(I,I-1) * ORTI(I)
         IP1 = I + 1
C
         DO 110 K = IP1, IGH
            ORTR(K) = HR(K,I-1)
            ORTI(K) = HI(K,I-1)
  110    CONTINUE
C
         DO 130 J = I, IGH
            SR = 0.0D0
            SI = 0.0D0
C
            DO 115 K = I, IGH
               SR = SR + ORTR(K) * ZR(K,J) + ORTI(K) * ZI(K,J)
               SI = SI + ORTR(K) * ZI(K,J) - ORTI(K) * ZR(K,J)
  115       CONTINUE
C
            SR = SR / NORM
            SI = SI / NORM
C
            DO 120 K = I, IGH
               ZR(K,J) = ZR(K,J) + SR * ORTR(K) - SI * ORTI(K)
               ZI(K,J) = ZI(K,J) + SR * ORTI(K) + SI * ORTR(K)
  120       CONTINUE
C
  130    CONTINUE
C
  140 CONTINUE
C     .......... CREATE REAL SUBDIAGONAL ELEMENTS ..........
  150 L = LOW + 1
C
      DO 170 I = L, IGH
         LL = MIN0(I+1,IGH)
         IF (HI(I,I-1) .EQ. 0.0D0) GO TO 170
         NORM = PYTHAG(HR(I,I-1),HI(I,I-1))
         YR = HR(I,I-1) / NORM
         YI = HI(I,I-1) / NORM
         HR(I,I-1) = NORM
         HI(I,I-1) = 0.0D0
C
         DO 155 J = I, N
            SI = YR * HI(I,J) - YI * HR(I,J)
            HR(I,J) = YR * HR(I,J) + YI * HI(I,J)
            HI(I,J) = SI
  155    CONTINUE
C
         DO 160 J = 1, LL
            SI = YR * HI(J,I) + YI * HR(J,I)
            HR(J,I) = YR * HR(J,I) - YI * HI(J,I)
            HI(J,I) = SI
  160    CONTINUE
C
         DO 165 J = LOW, IGH
            SI = YR * ZI(J,I) + YI * ZR(J,I)
            ZR(J,I) = YR * ZR(J,I) - YI * ZI(J,I)
            ZI(J,I) = SI
  165    CONTINUE
C
  170 CONTINUE
C     .......... STORE ROOTS ISOLATED BY CBAL ..........
  180 DO 200 I = 1, N
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 200
         WR(I) = HR(I,I)
         WI(I) = HI(I,I)
  200 CONTINUE
C
      EN = IGH
      TR = 0.0D0
      TI = 0.0D0
      ITN = 30*N
C     .......... SEARCH FOR NEXT EIGENVALUE ..........
  220 IF (EN .LT. LOW) GO TO 680
      ITS = 0
      ENM1 = EN - 1
C     .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C                FOR L=EN STEP -1 UNTIL LOW DO -- ..........
  240 DO 260 LL = LOW, EN
         L = EN + LOW - LL
         IF (L .EQ. LOW) GO TO 300
         TST1 = DABS(HR(L-1,L-1)) + DABS(HI(L-1,L-1))
     X            + DABS(HR(L,L)) + DABS(HI(L,L))
         TST2 = TST1 + DABS(HR(L,L-1))
         IF (TST2 .EQ. TST1) GO TO 300
  260 CONTINUE
C     .......... FORM SHIFT ..........
  300 IF (L .EQ. EN) GO TO 660
      IF (ITN .EQ. 0) GO TO 1000
      IF (ITS .EQ. 10 .OR. ITS .EQ. 20) GO TO 320
      SR = HR(EN,EN)
      SI = HI(EN,EN)
      XR = HR(ENM1,EN) * HR(EN,ENM1)
      XI = HI(ENM1,EN) * HR(EN,ENM1)
      IF (XR .EQ. 0.0D0 .AND. XI .EQ. 0.0D0) GO TO 340
      YR = (HR(ENM1,ENM1) - SR) / 2.0D0
      YI = (HI(ENM1,ENM1) - SI) / 2.0D0
      CALL CSROOT(YR**2-YI**2+XR,2.0D0*YR*YI+XI,ZZR,ZZI)
      IF (YR * ZZR + YI * ZZI .GE. 0.0D0) GO TO 310
      ZZR = -ZZR
      ZZI = -ZZI
  310 CALL CDIV(XR,XI,YR+ZZR,YI+ZZI,XR,XI)
      SR = SR - XR
      SI = SI - XI
      GO TO 340
C     .......... FORM EXCEPTIONAL SHIFT ..........
  320 SR = DABS(HR(EN,ENM1)) + DABS(HR(ENM1,EN-2))
      SI = 0.0D0
C
  340 DO 360 I = LOW, EN
         HR(I,I) = HR(I,I) - SR
         HI(I,I) = HI(I,I) - SI
  360 CONTINUE
C
      TR = TR + SR
      TI = TI + SI
      ITS = ITS + 1
      ITN = ITN - 1
C     .......... REDUCE TO TRIANGLE (ROWS) ..........
      LP1 = L + 1
C
      DO 500 I = LP1, EN
         SR = HR(I,I-1)
         HR(I,I-1) = 0.0D0
         NORM = PYTHAG(PYTHAG(HR(I-1,I-1),HI(I-1,I-1)),SR)
         XR = HR(I-1,I-1) / NORM
         WR(I-1) = XR
         XI = HI(I-1,I-1) / NORM
         WI(I-1) = XI
         HR(I-1,I-1) = NORM
         HI(I-1,I-1) = 0.0D0
         HI(I,I-1) = SR / NORM
C
         DO 490 J = I, N
            YR = HR(I-1,J)
            YI = HI(I-1,J)
            ZZR = HR(I,J)
            ZZI = HI(I,J)
            HR(I-1,J) = XR * YR + XI * YI + HI(I,I-1) * ZZR
            HI(I-1,J) = XR * YI - XI * YR + HI(I,I-1) * ZZI
            HR(I,J) = XR * ZZR - XI * ZZI - HI(I,I-1) * YR
            HI(I,J) = XR * ZZI + XI * ZZR - HI(I,I-1) * YI
  490    CONTINUE
C
  500 CONTINUE
C
      SI = HI(EN,EN)
      IF (SI .EQ. 0.0D0) GO TO 540
      NORM = PYTHAG(HR(EN,EN),SI)
      SR = HR(EN,EN) / NORM
      SI = SI / NORM
      HR(EN,EN) = NORM
      HI(EN,EN) = 0.0D0
      IF (EN .EQ. N) GO TO 540
      IP1 = EN + 1
C
      DO 520 J = IP1, N
         YR = HR(EN,J)
         YI = HI(EN,J)
         HR(EN,J) = SR * YR + SI * YI
         HI(EN,J) = SR * YI - SI * YR
  520 CONTINUE
C     .......... INVERSE OPERATION (COLUMNS) ..........
  540 DO 600 J = LP1, EN
         XR = WR(J-1)
         XI = WI(J-1)
C
         DO 580 I = 1, J
            YR = HR(I,J-1)
            YI = 0.0D0
            ZZR = HR(I,J)
            ZZI = HI(I,J)
            IF (I .EQ. J) GO TO 560
            YI = HI(I,J-1)
            HI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI
  560       HR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR
            HR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR
            HI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI
  580    CONTINUE
C
         DO 590 I = LOW, IGH
            YR = ZR(I,J-1)
            YI = ZI(I,J-1)
            ZZR = ZR(I,J)
            ZZI = ZI(I,J)
            ZR(I,J-1) = XR * YR - XI * YI + HI(J,J-1) * ZZR
            ZI(I,J-1) = XR * YI + XI * YR + HI(J,J-1) * ZZI
            ZR(I,J) = XR * ZZR + XI * ZZI - HI(J,J-1) * YR
            ZI(I,J) = XR * ZZI - XI * ZZR - HI(J,J-1) * YI
  590    CONTINUE
C
  600 CONTINUE
C
      IF (SI .EQ. 0.0D0) GO TO 240
C
      DO 630 I = 1, EN
         YR = HR(I,EN)
         YI = HI(I,EN)
         HR(I,EN) = SR * YR - SI * YI
         HI(I,EN) = SR * YI + SI * YR
  630 CONTINUE
C
      DO 640 I = LOW, IGH
         YR = ZR(I,EN)
         YI = ZI(I,EN)
         ZR(I,EN) = SR * YR - SI * YI
         ZI(I,EN) = SR * YI + SI * YR
  640 CONTINUE
C
      GO TO 240
C     .......... A ROOT FOUND ..........
  660 HR(EN,EN) = HR(EN,EN) + TR
      WR(EN) = HR(EN,EN)
      HI(EN,EN) = HI(EN,EN) + TI
      WI(EN) = HI(EN,EN)
      EN = ENM1
      GO TO 220
C     .......... ALL ROOTS FOUND.  BACKSUBSTITUTE TO FIND
C                VECTORS OF UPPER TRIANGULAR FORM ..........
  680 NORM = 0.0D0
C
      DO 720 I = 1, N
C
         DO 720 J = I, N
            TR = DABS(HR(I,J)) + DABS(HI(I,J))
            IF (TR .GT. NORM) NORM = TR
  720 CONTINUE
C
      IF (N .EQ. 1 .OR. NORM .EQ. 0.0D0) GO TO 1001
C     .......... FOR EN=N STEP -1 UNTIL 2 DO -- ..........
      DO 800 NN = 2, N
         EN = N + 2 - NN
         XR = WR(EN)
         XI = WI(EN)
         HR(EN,EN) = 1.0D0
         HI(EN,EN) = 0.0D0
         ENM1 = EN - 1
C     .......... FOR I=EN-1 STEP -1 UNTIL 1 DO -- ..........
         DO 780 II = 1, ENM1
            I = EN - II
            ZZR = 0.0D0
            ZZI = 0.0D0
            IP1 = I + 1
C
            DO 740 J = IP1, EN
               ZZR = ZZR + HR(I,J) * HR(J,EN) - HI(I,J) * HI(J,EN)
               ZZI = ZZI + HR(I,J) * HI(J,EN) + HI(I,J) * HR(J,EN)
  740       CONTINUE
C
            YR = XR - WR(I)
            YI = XI - WI(I)
            IF (YR .NE. 0.0D0 .OR. YI .NE. 0.0D0) GO TO 765
               TST1 = NORM
               YR = TST1
  760          YR = 0.01D0 * YR
               TST2 = NORM + YR
               IF (TST2 .GT. TST1) GO TO 760
  765       CONTINUE
            CALL CDIV(ZZR,ZZI,YR,YI,HR(I,EN),HI(I,EN))
C     .......... OVERFLOW CONTROL ..........
            TR = DABS(HR(I,EN)) + DABS(HI(I,EN))
            IF (TR .EQ. 0.0D0) GO TO 780
            TST1 = TR
            TST2 = TST1 + 1.0D0/TST1
            IF (TST2 .GT. TST1) GO TO 780
            DO 770 J = I, EN
               HR(J,EN) = HR(J,EN)/TR
               HI(J,EN) = HI(J,EN)/TR
  770       CONTINUE
C
  780    CONTINUE
C
  800 CONTINUE
C     .......... END BACKSUBSTITUTION ..........
      ENM1 = N - 1
C     .......... VECTORS OF ISOLATED ROOTS ..........
      DO  840 I = 1, ENM1
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 840
         IP1 = I + 1
C
         DO 820 J = IP1, N
            ZR(I,J) = HR(I,J)
            ZI(I,J) = HI(I,J)
  820    CONTINUE
C
  840 CONTINUE
C     .......... MULTIPLY BY TRANSFORMATION MATRIX TO GIVE
C                VECTORS OF ORIGINAL FULL MATRIX.
C                FOR J=N STEP -1 UNTIL LOW+1 DO -- ..........
      DO 880 JJ = LOW, ENM1
         J = N + LOW - JJ
         M = MIN0(J,IGH)
C
         DO 880 I = LOW, IGH
            ZZR = 0.0D0
            ZZI = 0.0D0
C
            DO 860 K = LOW, M
               ZZR = ZZR + ZR(I,K) * HR(K,J) - ZI(I,K) * HI(K,J)
               ZZI = ZZI + ZR(I,K) * HI(K,J) + ZI(I,K) * HR(K,J)
  860       CONTINUE
C
            ZR(I,J) = ZZR
            ZI(I,J) = ZZI
  880 CONTINUE
C
      GO TO 1001
C     .......... SET ERROR -- ALL EIGENVALUES HAVE NOT
C                CONVERGED AFTER 30*N ITERATIONS ..........
 1000 IERR = EN
 1001 RETURN
      END
      SUBROUTINE CORTH(NM,N,LOW,IGH,AR,AI,ORTR,ORTI)
C
      INTEGER I,J,M,N,II,JJ,LA,MP,NM,IGH,KP1,LOW
      DOUBLE PRECISION AR(NM,N),AI(NM,N),ORTR(IGH),ORTI(IGH)
      DOUBLE PRECISION F,G,H,FI,FR,SCALE,PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF A COMPLEX ANALOGUE OF
C     THE ALGOL PROCEDURE ORTHES, NUM. MATH. 12, 349-368(1968)
C     BY MARTIN AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
C
C     GIVEN A COMPLEX GENERAL MATRIX, THIS SUBROUTINE
C     REDUCES A SUBMATRIX SITUATED IN ROWS AND COLUMNS
C     LOW THROUGH IGH TO UPPER HESSENBERG FORM BY
C     UNITARY SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  CBAL.  IF  CBAL  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE COMPLEX INPUT MATRIX.
C
C     ON OUTPUT
C
C        AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE HESSENBERG MATRIX.  INFORMATION
C          ABOUT THE UNITARY TRANSFORMATIONS USED IN THE REDUCTION
C          IS STORED IN THE REMAINING TRIANGLES UNDER THE
C          HESSENBERG MATRIX.
C
C        ORTR AND ORTI CONTAIN FURTHER INFORMATION ABOUT THE
C          TRANSFORMATIONS.  ONLY ELEMENTS LOW THROUGH IGH ARE USED.
C
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      LA = IGH - 1
      KP1 = LOW + 1
      IF (LA .LT. KP1) GO TO 200
C
      DO 180 M = KP1, LA
         H = 0.0D0
         ORTR(M) = 0.0D0
         ORTI(M) = 0.0D0
         SCALE = 0.0D0
C     .......... SCALE COLUMN (ALGOL TOL THEN NOT NEEDED) ..........
         DO 90 I = M, IGH
   90    SCALE = SCALE + DABS(AR(I,M-1)) + DABS(AI(I,M-1))
C
         IF (SCALE .EQ. 0.0D0) GO TO 180
         MP = M + IGH
C     .......... FOR I=IGH STEP -1 UNTIL M DO -- ..........
         DO 100 II = M, IGH
            I = MP - II
            ORTR(I) = AR(I,M-1) / SCALE
            ORTI(I) = AI(I,M-1) / SCALE
            H = H + ORTR(I) * ORTR(I) + ORTI(I) * ORTI(I)
  100    CONTINUE
C
         G = DSQRT(H)
         F = PYTHAG(ORTR(M),ORTI(M))
         IF (F .EQ. 0.0D0) GO TO 103
         H = H + F * G
         G = G / F
         ORTR(M) = (1.0D0 + G) * ORTR(M)
         ORTI(M) = (1.0D0 + G) * ORTI(M)
         GO TO 105
C
  103    ORTR(M) = G
         AR(M,M-1) = SCALE
C     .......... FORM (I-(U*UT)/H) * A ..........
  105    DO 130 J = M, N
            FR = 0.0D0
            FI = 0.0D0
C     .......... FOR I=IGH STEP -1 UNTIL M DO -- ..........
            DO 110 II = M, IGH
               I = MP - II
               FR = FR + ORTR(I) * AR(I,J) + ORTI(I) * AI(I,J)
               FI = FI + ORTR(I) * AI(I,J) - ORTI(I) * AR(I,J)
  110       CONTINUE
C
            FR = FR / H
            FI = FI / H
C
            DO 120 I = M, IGH
               AR(I,J) = AR(I,J) - FR * ORTR(I) + FI * ORTI(I)
               AI(I,J) = AI(I,J) - FR * ORTI(I) - FI * ORTR(I)
  120       CONTINUE
C
  130    CONTINUE
C     .......... FORM (I-(U*UT)/H)*A*(I-(U*UT)/H) ..........
         DO 160 I = 1, IGH
            FR = 0.0D0
            FI = 0.0D0
C     .......... FOR J=IGH STEP -1 UNTIL M DO -- ..........
            DO 140 JJ = M, IGH
               J = MP - JJ
               FR = FR + ORTR(J) * AR(I,J) - ORTI(J) * AI(I,J)
               FI = FI + ORTR(J) * AI(I,J) + ORTI(J) * AR(I,J)
  140       CONTINUE
C
            FR = FR / H
            FI = FI / H
C
            DO 150 J = M, IGH
               AR(I,J) = AR(I,J) - FR * ORTR(J) - FI * ORTI(J)
               AI(I,J) = AI(I,J) + FR * ORTI(J) - FI * ORTR(J)
  150       CONTINUE
C
  160    CONTINUE
C
         ORTR(M) = SCALE * ORTR(M)
         ORTI(M) = SCALE * ORTI(M)
         AR(M,M-1) = -G * AR(M,M-1)
         AI(M,M-1) = -G * AI(M,M-1)
  180 CONTINUE
C
  200 RETURN
      END
      SUBROUTINE CSROOT(XR,XI,YR,YI)
      DOUBLE PRECISION XR,XI,YR,YI
C
C     (YR,YI) = COMPLEX DSQRT(XR,XI) 
C     BRANCH CHOSEN SO THAT YR .GE. 0.0 AND SIGN(YI) .EQ. SIGN(XI)
C
      DOUBLE PRECISION S,TR,TI,PYTHAG
      TR = XR
      TI = XI
      S = DSQRT(0.5D0*(PYTHAG(TR,TI) + DABS(TR)))
      IF (TR .GE. 0.0D0) YR = S
      IF (TI .LT. 0.0D0) S = -S
      IF (TR .LE. 0.0D0) YI = S
      IF (TR .LT. 0.0D0) YR = 0.5D0*(TI/YI)
      IF (TR .GT. 0.0D0) YI = 0.5D0*(TI/YR)
      RETURN
      END
      SUBROUTINE ELMHES(NM,N,LOW,IGH,A,INT)
C
      INTEGER I,J,M,N,LA,NM,IGH,KP1,LOW,MM1,MP1
      DOUBLE PRECISION A(NM,N)
      DOUBLE PRECISION X,Y
      INTEGER INT(IGH)
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE ELMHES,
C     NUM. MATH. 12, 349-368(1968) BY MARTIN AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 339-358(1971).
C
C     GIVEN A REAL GENERAL MATRIX, THIS SUBROUTINE
C     REDUCES A SUBMATRIX SITUATED IN ROWS AND COLUMNS
C     LOW THROUGH IGH TO UPPER HESSENBERG FORM BY
C     STABILIZED ELEMENTARY SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  BALANC.  IF  BALANC  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        A CONTAINS THE INPUT MATRIX.
C
C     ON OUTPUT
C
C        A CONTAINS THE HESSENBERG MATRIX.  THE MULTIPLIERS
C          WHICH WERE USED IN THE REDUCTION ARE STORED IN THE
C          REMAINING TRIANGLE UNDER THE HESSENBERG MATRIX.
C
C        INT CONTAINS INFORMATION ON THE ROWS AND COLUMNS
C          INTERCHANGED IN THE REDUCTION.
C          ONLY ELEMENTS LOW THROUGH IGH ARE USED.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      LA = IGH - 1
      KP1 = LOW + 1
      IF (LA .LT. KP1) GO TO 200
C
      DO 180 M = KP1, LA
         MM1 = M - 1
         X = 0.0D0
         I = M
C
         DO 100 J = M, IGH
            IF (DABS(A(J,MM1)) .LE. DABS(X)) GO TO 100
            X = A(J,MM1)
            I = J
  100    CONTINUE
C
         INT(M) = I
         IF (I .EQ. M) GO TO 130
C     .......... INTERCHANGE ROWS AND COLUMNS OF A ..........
         DO 110 J = MM1, N
            Y = A(I,J)
            A(I,J) = A(M,J)
            A(M,J) = Y
  110    CONTINUE
C
         DO 120 J = 1, IGH
            Y = A(J,I)
            A(J,I) = A(J,M)
            A(J,M) = Y
  120    CONTINUE
C     .......... END INTERCHANGE ..........
  130    IF (X .EQ. 0.0D0) GO TO 180
         MP1 = M + 1
C
         DO 160 I = MP1, IGH
            Y = A(I,MM1)
            IF (Y .EQ. 0.0D0) GO TO 160
            Y = Y / X
            A(I,MM1) = Y
C
            DO 140 J = M, N
  140       A(I,J) = A(I,J) - Y * A(M,J)
C
            DO 150 J = 1, IGH
  150       A(J,M) = A(J,M) + Y * A(J,I)
C
  160    CONTINUE
C
  180 CONTINUE
C
  200 RETURN
      END
      SUBROUTINE ELTRAN(NM,N,LOW,IGH,A,INT,Z)
C
      INTEGER I,J,N,KL,MM,MP,NM,IGH,LOW,MP1
      DOUBLE PRECISION A(NM,IGH),Z(NM,N)
      INTEGER INT(IGH)
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE ELMTRANS,
C     NUM. MATH. 16, 181-204(1970) BY PETERS AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C
C     THIS SUBROUTINE ACCUMULATES THE STABILIZED ELEMENTARY
C     SIMILARITY TRANSFORMATIONS USED IN THE REDUCTION OF A
C     REAL GENERAL MATRIX TO UPPER HESSENBERG FORM BY  ELMHES.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  BALANC.  IF  BALANC  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        A CONTAINS THE MULTIPLIERS WHICH WERE USED IN THE
C          REDUCTION BY  ELMHES  IN ITS LOWER TRIANGLE
C          BELOW THE SUBDIAGONAL.
C
C        INT CONTAINS INFORMATION ON THE ROWS AND COLUMNS
C          INTERCHANGED IN THE REDUCTION BY  ELMHES.
C          ONLY ELEMENTS LOW THROUGH IGH ARE USED.
C
C     ON OUTPUT
C
C        Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED IN THE
C          REDUCTION BY  ELMHES.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
C     .......... INITIALIZE Z TO IDENTITY MATRIX ..........
      DO 80 J = 1, N
C
         DO 60 I = 1, N
   60    Z(I,J) = 0.0D0
C
         Z(J,J) = 1.0D0
   80 CONTINUE
C
      KL = IGH - LOW - 1
      IF (KL .LT. 1) GO TO 200
C     .......... FOR MP=IGH-1 STEP -1 UNTIL LOW+1 DO -- ..........
      DO 140 MM = 1, KL
         MP = IGH - MM
         MP1 = MP + 1
C
         DO 100 I = MP1, IGH
  100    Z(I,MP) = A(I,MP-1)
C
         I = INT(MP)
         IF (I .EQ. MP) GO TO 140
C
         DO 130 J = MP, IGH
            Z(MP,J) = Z(I,J)
            Z(I,J) = 0.0D0
  130    CONTINUE
C
         Z(I,MP) = 1.0D0
  140 CONTINUE
C
  200 RETURN
      END
      DOUBLE PRECISION FUNCTION EPSLON (X)
      DOUBLE PRECISION X
C
C     ESTIMATE UNIT ROUNDOFF IN QUANTITIES OF SIZE X.
C
      DOUBLE PRECISION A,B,C,EPS
C
C     THIS PROGRAM SHOULD FUNCTION PROPERLY ON ALL SYSTEMS
C     SATISFYING THE FOLLOWING TWO ASSUMPTIONS,
C        1.  THE BASE USED IN REPRESENTING FLOATING POINT
C            NUMBERS IS NOT A POWER OF THREE.
C        2.  THE QUANTITY  A  IN STATEMENT 10 IS REPRESENTED TO 
C            THE ACCURACY USED IN FLOATING POINT VARIABLES
C            THAT ARE STORED IN MEMORY.
C     THE STATEMENT NUMBER 10 AND THE GO TO 10 ARE INTENDED TO
C     FORCE OPTIMIZING COMPILERS TO GENERATE CODE SATISFYING 
C     ASSUMPTION 2.
C     UNDER THESE ASSUMPTIONS, IT SHOULD BE TRUE THAT,
C            A  IS NOT EXACTLY EQUAL TO FOUR-THIRDS,
C            B  HAS A ZERO FOR ITS LAST BIT OR DIGIT,
C            C  IS NOT EXACTLY EQUAL TO ONE,
C            EPS  MEASURES THE SEPARATION OF 1.0 FROM
C                 THE NEXT LARGER FLOATING POINT NUMBER.
C     THE DEVELOPERS OF EISPACK WOULD APPRECIATE BEING INFORMED
C     ABOUT ANY SYSTEMS WHERE THESE ASSUMPTIONS DO NOT HOLD.
C
C     THIS VERSION DATED 4/6/83.
C
      A = 4.0D0/3.0D0
   10 B = A - 1.0D0
      C = B + B + B
      EPS = DABS(C-1.0D0)
      IF (EPS .EQ. 0.0D0) GO TO 10
      EPSLON = EPS*DABS(X)
      RETURN
      END
      SUBROUTINE HQR(NM,N,LOW,IGH,H,WR,WI,IERR)
C
      INTEGER I,J,K,L,M,N,EN,LL,MM,NA,NM,IGH,ITN,ITS,LOW,MP2,ENM2,IERR
      DOUBLE PRECISION H(NM,N),WR(N),WI(N)
      DOUBLE PRECISION P,Q,R,S,T,W,X,Y,ZZ,NORM,TST1,TST2
      LOGICAL NOTLAS
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE HQR,
C     NUM. MATH. 14, 219-231(1970) BY MARTIN, PETERS, AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 359-371(1971).
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES OF A REAL
C     UPPER HESSENBERG MATRIX BY THE QR METHOD.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  BALANC.  IF  BALANC  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        H CONTAINS THE UPPER HESSENBERG MATRIX.  INFORMATION ABOUT
C          THE TRANSFORMATIONS USED IN THE REDUCTION TO HESSENBERG
C          FORM BY  ELMHES  OR  ORTHES, IF PERFORMED, IS STORED
C          IN THE REMAINING TRIANGLE UNDER THE HESSENBERG MATRIX.
C
C     ON OUTPUT
C
C        H HAS BEEN DESTROYED.  THEREFORE, IT MUST BE SAVED
C          BEFORE CALLING  HQR  IF SUBSEQUENT CALCULATION AND
C          BACK TRANSFORMATION OF EIGENVECTORS IS TO BE PERFORMED.
C
C        WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVALUES.  THE EIGENVALUES
C          ARE UNORDERED EXCEPT THAT COMPLEX CONJUGATE PAIRS
C          OF VALUES APPEAR CONSECUTIVELY WITH THE EIGENVALUE
C          HAVING THE POSITIVE IMAGINARY PART FIRST.  IF AN
C          ERROR EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C          FOR INDICES IERR+1,...,N.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C                     WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of L M P Q R to keep g77 -Wall happy
c
      L = 0
      M = 0
      P = 0.0D0
      Q = 0.0D0
      R = 0.0D0
C
      IERR = 0
      NORM = 0.0D0
      K = 1
C     .......... STORE ROOTS ISOLATED BY BALANC
C                AND COMPUTE MATRIX NORM ..........
      DO 50 I = 1, N
C
         DO 40 J = K, N
   40    NORM = NORM + DABS(H(I,J))
C
         K = I
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 50
         WR(I) = H(I,I)
         WI(I) = 0.0D0
   50 CONTINUE
C
      EN = IGH
      T = 0.0D0
      ITN = 30*N
C     .......... SEARCH FOR NEXT EIGENVALUES ..........
   60 IF (EN .LT. LOW) GO TO 1001
      ITS = 0
      NA = EN - 1
      ENM2 = NA - 1
C     .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C                FOR L=EN STEP -1 UNTIL LOW DO -- ..........
   70 DO 80 LL = LOW, EN
         L = EN + LOW - LL
         IF (L .EQ. LOW) GO TO 100
         S = DABS(H(L-1,L-1)) + DABS(H(L,L))
         IF (S .EQ. 0.0D0) S = NORM
         TST1 = S
         TST2 = TST1 + DABS(H(L,L-1))
         IF (TST2 .EQ. TST1) GO TO 100
   80 CONTINUE
C     .......... FORM SHIFT ..........
  100 X = H(EN,EN)
      IF (L .EQ. EN) GO TO 270
      Y = H(NA,NA)
      W = H(EN,NA) * H(NA,EN)
      IF (L .EQ. NA) GO TO 280
      IF (ITN .EQ. 0) GO TO 1000
      IF (ITS .NE. 10 .AND. ITS .NE. 20) GO TO 130
C     .......... FORM EXCEPTIONAL SHIFT ..........
      T = T + X
C
      DO 120 I = LOW, EN
  120 H(I,I) = H(I,I) - X
C
      S = DABS(H(EN,NA)) + DABS(H(NA,ENM2))
      X = 0.75D0 * S
      Y = X
      W = -0.4375D0 * S * S
  130 ITS = ITS + 1
      ITN = ITN - 1
C     .......... LOOK FOR TWO CONSECUTIVE SMALL
C                SUB-DIAGONAL ELEMENTS.
C                FOR M=EN-2 STEP -1 UNTIL L DO -- ..........
      DO 140 MM = L, ENM2
         M = ENM2 + L - MM
         ZZ = H(M,M)
         R = X - ZZ
         S = Y - ZZ
         P = (R * S - W) / H(M+1,M) + H(M,M+1)
         Q = H(M+1,M+1) - ZZ - R - S
         R = H(M+2,M+1)
         S = DABS(P) + DABS(Q) + DABS(R)
         P = P / S
         Q = Q / S
         R = R / S
         IF (M .EQ. L) GO TO 150
         TST1 = DABS(P)*(DABS(H(M-1,M-1)) + DABS(ZZ) + DABS(H(M+1,M+1)))
         TST2 = TST1 + DABS(H(M,M-1))*(DABS(Q) + DABS(R))
         IF (TST2 .EQ. TST1) GO TO 150
  140 CONTINUE
C
  150 MP2 = M + 2
C
      DO 160 I = MP2, EN
         H(I,I-2) = 0.0D0
         IF (I .EQ. MP2) GO TO 160
         H(I,I-3) = 0.0D0
  160 CONTINUE
C     .......... DOUBLE QR STEP INVOLVING ROWS L TO EN AND
C                COLUMNS M TO EN ..........
      DO 260 K = M, NA
         NOTLAS = K .NE. NA
         IF (K .EQ. M) GO TO 170
         P = H(K,K-1)
         Q = H(K+1,K-1)
         R = 0.0D0
         IF (NOTLAS) R = H(K+2,K-1)
         X = DABS(P) + DABS(Q) + DABS(R)
         IF (X .EQ. 0.0D0) GO TO 260
         P = P / X
         Q = Q / X
         R = R / X
  170    S = DSIGN(DSQRT(P*P+Q*Q+R*R),P)
         IF (K .EQ. M) GO TO 180
         H(K,K-1) = -S * X
         GO TO 190
  180    IF (L .NE. M) H(K,K-1) = -H(K,K-1)
  190    P = P + S
         X = P / S
         Y = Q / S
         ZZ = R / S
         Q = Q / P
         R = R / P
         IF (NOTLAS) GO TO 225
C     .......... ROW MODIFICATION ..........
         DO 200 J = K, N
            P = H(K,J) + Q * H(K+1,J)
            H(K,J) = H(K,J) - P * X
            H(K+1,J) = H(K+1,J) - P * Y
  200    CONTINUE
C
         J = MIN0(EN,K+3)
C     .......... COLUMN MODIFICATION ..........
         DO 210 I = 1, J
            P = X * H(I,K) + Y * H(I,K+1)
            H(I,K) = H(I,K) - P
            H(I,K+1) = H(I,K+1) - P * Q
  210    CONTINUE
         GO TO 255
  225    CONTINUE
C     .......... ROW MODIFICATION ..........
         DO 230 J = K, N
            P = H(K,J) + Q * H(K+1,J) + R * H(K+2,J)
            H(K,J) = H(K,J) - P * X
            H(K+1,J) = H(K+1,J) - P * Y
            H(K+2,J) = H(K+2,J) - P * ZZ
  230    CONTINUE
C
         J = MIN0(EN,K+3)
C     .......... COLUMN MODIFICATION ..........
         DO 240 I = 1, J
            P = X * H(I,K) + Y * H(I,K+1) + ZZ * H(I,K+2)
            H(I,K) = H(I,K) - P
            H(I,K+1) = H(I,K+1) - P * Q
            H(I,K+2) = H(I,K+2) - P * R
  240    CONTINUE
  255    CONTINUE
C
  260 CONTINUE
C
      GO TO 70
C     .......... ONE ROOT FOUND ..........
  270 WR(EN) = X + T
      WI(EN) = 0.0D0
      EN = NA
      GO TO 60
C     .......... TWO ROOTS FOUND ..........
  280 P = (Y - X) / 2.0D0
      Q = P * P + W
      ZZ = DSQRT(DABS(Q))
      X = X + T
      IF (Q .LT. 0.0D0) GO TO 320
C     .......... REAL PAIR ..........
      ZZ = P + DSIGN(ZZ,P)
      WR(NA) = X + ZZ
      WR(EN) = WR(NA)
      IF (ZZ .NE. 0.0D0) WR(EN) = X - W / ZZ
      WI(NA) = 0.0D0
      WI(EN) = 0.0D0
      GO TO 330
C     .......... COMPLEX PAIR ..........
  320 WR(NA) = X + P
      WR(EN) = X + P
      WI(NA) = ZZ
      WI(EN) = -ZZ
  330 EN = ENM2
      GO TO 60
C     .......... SET ERROR -- ALL EIGENVALUES HAVE NOT
C                CONVERGED AFTER 30*N ITERATIONS ..........
 1000 IERR = EN
 1001 RETURN
      END
      SUBROUTINE HQR2(NM,N,LOW,IGH,H,WR,WI,Z,IERR)
C
      INTEGER I,J,K,L,M,N,EN,II,JJ,LL,MM,NA,NM,NN,
     X        IGH,ITN,ITS,LOW,MP2,ENM2,IERR
      DOUBLE PRECISION H(NM,N),WR(N),WI(N),Z(NM,N)
      DOUBLE PRECISION P,Q,R,S,T,W,X,Y,RA,SA,VI,VR,ZZ,NORM,TST1,TST2
      LOGICAL NOTLAS
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE HQR2,
C     NUM. MATH. 16, 181-204(1970) BY PETERS AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 372-395(1971).
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C     OF A REAL UPPER HESSENBERG MATRIX BY THE QR METHOD.  THE
C     EIGENVECTORS OF A REAL GENERAL MATRIX CAN ALSO BE FOUND
C     IF  ELMHES  AND  ELTRAN  OR  ORTHES  AND  ORTRAN  HAVE
C     BEEN USED TO REDUCE THIS GENERAL MATRIX TO HESSENBERG FORM
C     AND TO ACCUMULATE THE SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        LOW AND IGH ARE INTEGERS DETERMINED BY THE BALANCING
C          SUBROUTINE  BALANC.  IF  BALANC  HAS NOT BEEN USED,
C          SET LOW=1, IGH=N.
C
C        H CONTAINS THE UPPER HESSENBERG MATRIX.
C
C        Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED BY  ELTRAN
C          AFTER THE REDUCTION BY  ELMHES, OR BY  ORTRAN  AFTER THE
C          REDUCTION BY  ORTHES, IF PERFORMED.  IF THE EIGENVECTORS
C          OF THE HESSENBERG MATRIX ARE DESIRED, Z MUST CONTAIN THE
C          IDENTITY MATRIX.
C
C     ON OUTPUT
C
C        H HAS BEEN DESTROYED.
C
C        WR AND WI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE EIGENVALUES.  THE EIGENVALUES
C          ARE UNORDERED EXCEPT THAT COMPLEX CONJUGATE PAIRS
C          OF VALUES APPEAR CONSECUTIVELY WITH THE EIGENVALUE
C          HAVING THE POSITIVE IMAGINARY PART FIRST.  IF AN
C          ERROR EXIT IS MADE, THE EIGENVALUES SHOULD BE CORRECT
C          FOR INDICES IERR+1,...,N.
C
C        Z CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGENVECTORS.
C          IF THE I-TH EIGENVALUE IS REAL, THE I-TH COLUMN OF Z
C          CONTAINS ITS EIGENVECTOR.  IF THE I-TH EIGENVALUE IS COMPLEX
C          WITH POSITIVE IMAGINARY PART, THE I-TH AND (I+1)-TH
C          COLUMNS OF Z CONTAIN THE REAL AND IMAGINARY PARTS OF ITS
C          EIGENVECTOR.  THE EIGENVECTORS ARE UNNORMALIZED.  IF AN
C          ERROR EXIT IS MADE, NONE OF THE EIGENVECTORS HAS BEEN FOUND.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE LIMIT OF 30*N ITERATIONS IS EXHAUSTED
C                     WHILE THE J-TH EIGENVALUE IS BEING SOUGHT.
C
C     CALLS CDIV FOR COMPLEX DIVISION.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of L M P R S to keep g77 -Wall happy
c
      L = 0
      M = 0
      P = 0.0D0
      R = 0.0D0
      S = 0.0D0
C
      IERR = 0
      NORM = 0.0D0
      K = 1
C     .......... STORE ROOTS ISOLATED BY BALANC
C                AND COMPUTE MATRIX NORM ..........
      DO 50 I = 1, N
C
         DO 40 J = K, N
   40    NORM = NORM + DABS(H(I,J))
C
         K = I
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 50
         WR(I) = H(I,I)
         WI(I) = 0.0D0
   50 CONTINUE
C
      EN = IGH
      T = 0.0D0
      ITN = 30*N
C     .......... SEARCH FOR NEXT EIGENVALUES ..........
   60 IF (EN .LT. LOW) GO TO 340
      ITS = 0
      NA = EN - 1
      ENM2 = NA - 1
C     .......... LOOK FOR SINGLE SMALL SUB-DIAGONAL ELEMENT
C                FOR L=EN STEP -1 UNTIL LOW DO -- ..........
   70 DO 80 LL = LOW, EN
         L = EN + LOW - LL
         IF (L .EQ. LOW) GO TO 100
         S = DABS(H(L-1,L-1)) + DABS(H(L,L))
         IF (S .EQ. 0.0D0) S = NORM
         TST1 = S
         TST2 = TST1 + DABS(H(L,L-1))
         IF (TST2 .EQ. TST1) GO TO 100
   80 CONTINUE
C     .......... FORM SHIFT ..........
  100 X = H(EN,EN)
      IF (L .EQ. EN) GO TO 270
      Y = H(NA,NA)
      W = H(EN,NA) * H(NA,EN)
      IF (L .EQ. NA) GO TO 280
      IF (ITN .EQ. 0) GO TO 1000
      IF (ITS .NE. 10 .AND. ITS .NE. 20) GO TO 130
C     .......... FORM EXCEPTIONAL SHIFT ..........
      T = T + X
C
      DO 120 I = LOW, EN
  120 H(I,I) = H(I,I) - X
C
      S = DABS(H(EN,NA)) + DABS(H(NA,ENM2))
      X = 0.75D0 * S
      Y = X
      W = -0.4375D0 * S * S
  130 ITS = ITS + 1
      ITN = ITN - 1
C     .......... LOOK FOR TWO CONSECUTIVE SMALL
C                SUB-DIAGONAL ELEMENTS.
C                FOR M=EN-2 STEP -1 UNTIL L DO -- ..........
      DO 140 MM = L, ENM2
         M = ENM2 + L - MM
         ZZ = H(M,M)
         R = X - ZZ
         S = Y - ZZ
         P = (R * S - W) / H(M+1,M) + H(M,M+1)
         Q = H(M+1,M+1) - ZZ - R - S
         R = H(M+2,M+1)
         S = DABS(P) + DABS(Q) + DABS(R)
         P = P / S
         Q = Q / S
         R = R / S
         IF (M .EQ. L) GO TO 150
         TST1 = DABS(P)*(DABS(H(M-1,M-1)) + DABS(ZZ) + DABS(H(M+1,M+1)))
         TST2 = TST1 + DABS(H(M,M-1))*(DABS(Q) + DABS(R))
         IF (TST2 .EQ. TST1) GO TO 150
  140 CONTINUE
C
  150 MP2 = M + 2
C
      DO 160 I = MP2, EN
         H(I,I-2) = 0.0D0
         IF (I .EQ. MP2) GO TO 160
         H(I,I-3) = 0.0D0
  160 CONTINUE
C     .......... DOUBLE QR STEP INVOLVING ROWS L TO EN AND
C                COLUMNS M TO EN ..........
      DO 260 K = M, NA
         NOTLAS = K .NE. NA
         IF (K .EQ. M) GO TO 170
         P = H(K,K-1)
         Q = H(K+1,K-1)
         R = 0.0D0
         IF (NOTLAS) R = H(K+2,K-1)
         X = DABS(P) + DABS(Q) + DABS(R)
         IF (X .EQ. 0.0D0) GO TO 260
         P = P / X
         Q = Q / X
         R = R / X
  170    S = DSIGN(DSQRT(P*P+Q*Q+R*R),P)
         IF (K .EQ. M) GO TO 180
         H(K,K-1) = -S * X
         GO TO 190
  180    IF (L .NE. M) H(K,K-1) = -H(K,K-1)
  190    P = P + S
         X = P / S
         Y = Q / S
         ZZ = R / S
         Q = Q / P
         R = R / P
         IF (NOTLAS) GO TO 225
C     .......... ROW MODIFICATION ..........
         DO 200 J = K, N
            P = H(K,J) + Q * H(K+1,J)
            H(K,J) = H(K,J) - P * X
            H(K+1,J) = H(K+1,J) - P * Y
  200    CONTINUE
C
         J = MIN0(EN,K+3)
C     .......... COLUMN MODIFICATION ..........
         DO 210 I = 1, J
            P = X * H(I,K) + Y * H(I,K+1)
            H(I,K) = H(I,K) - P
            H(I,K+1) = H(I,K+1) - P * Q
  210    CONTINUE
C     .......... ACCUMULATE TRANSFORMATIONS ..........
         DO 220 I = LOW, IGH
            P = X * Z(I,K) + Y * Z(I,K+1)
            Z(I,K) = Z(I,K) - P
            Z(I,K+1) = Z(I,K+1) - P * Q
  220    CONTINUE
         GO TO 255
  225    CONTINUE
C     .......... ROW MODIFICATION ..........
         DO 230 J = K, N
            P = H(K,J) + Q * H(K+1,J) + R * H(K+2,J)
            H(K,J) = H(K,J) - P * X
            H(K+1,J) = H(K+1,J) - P * Y
            H(K+2,J) = H(K+2,J) - P * ZZ
  230    CONTINUE
C
         J = MIN0(EN,K+3)
C     .......... COLUMN MODIFICATION ..........
         DO 240 I = 1, J
            P = X * H(I,K) + Y * H(I,K+1) + ZZ * H(I,K+2)
            H(I,K) = H(I,K) - P
            H(I,K+1) = H(I,K+1) - P * Q
            H(I,K+2) = H(I,K+2) - P * R
  240    CONTINUE
C     .......... ACCUMULATE TRANSFORMATIONS ..........
         DO 250 I = LOW, IGH
            P = X * Z(I,K) + Y * Z(I,K+1) + ZZ * Z(I,K+2)
            Z(I,K) = Z(I,K) - P
            Z(I,K+1) = Z(I,K+1) - P * Q
            Z(I,K+2) = Z(I,K+2) - P * R
  250    CONTINUE
  255    CONTINUE
C
  260 CONTINUE
C
      GO TO 70
C     .......... ONE ROOT FOUND ..........
  270 H(EN,EN) = X + T
      WR(EN) = H(EN,EN)
      WI(EN) = 0.0D0
      EN = NA
      GO TO 60
C     .......... TWO ROOTS FOUND ..........
  280 P = (Y - X) / 2.0D0
      Q = P * P + W
      ZZ = DSQRT(DABS(Q))
      H(EN,EN) = X + T
      X = H(EN,EN)
      H(NA,NA) = Y + T
      IF (Q .LT. 0.0D0) GO TO 320
C     .......... REAL PAIR ..........
      ZZ = P + DSIGN(ZZ,P)
      WR(NA) = X + ZZ
      WR(EN) = WR(NA)
      IF (ZZ .NE. 0.0D0) WR(EN) = X - W / ZZ
      WI(NA) = 0.0D0
      WI(EN) = 0.0D0
      X = H(EN,NA)
      S = DABS(X) + DABS(ZZ)
      P = X / S
      Q = ZZ / S
      R = DSQRT(P*P+Q*Q)
      P = P / R
      Q = Q / R
C     .......... ROW MODIFICATION ..........
      DO 290 J = NA, N
         ZZ = H(NA,J)
         H(NA,J) = Q * ZZ + P * H(EN,J)
         H(EN,J) = Q * H(EN,J) - P * ZZ
  290 CONTINUE
C     .......... COLUMN MODIFICATION ..........
      DO 300 I = 1, EN
         ZZ = H(I,NA)
         H(I,NA) = Q * ZZ + P * H(I,EN)
         H(I,EN) = Q * H(I,EN) - P * ZZ
  300 CONTINUE
C     .......... ACCUMULATE TRANSFORMATIONS ..........
      DO 310 I = LOW, IGH
         ZZ = Z(I,NA)
         Z(I,NA) = Q * ZZ + P * Z(I,EN)
         Z(I,EN) = Q * Z(I,EN) - P * ZZ
  310 CONTINUE
C
      GO TO 330
C     .......... COMPLEX PAIR ..........
  320 WR(NA) = X + P
      WR(EN) = X + P
      WI(NA) = ZZ
      WI(EN) = -ZZ
  330 EN = ENM2
      GO TO 60
C     .......... ALL ROOTS FOUND.  BACKSUBSTITUTE TO FIND
C                VECTORS OF UPPER TRIANGULAR FORM ..........
  340 IF (NORM .EQ. 0.0D0) GO TO 1001
C     .......... FOR EN=N STEP -1 UNTIL 1 DO -- ..........
      DO 800 NN = 1, N
         EN = N + 1 - NN
         P = WR(EN)
         Q = WI(EN)
         NA = EN - 1
C         IF (Q) 710, 600, 800
         IF (Q .GT. 0.0D0) GOTO 710
         IF (Q .LT. 0.0D0) GOTO 800
C     .......... REAL VECTOR ..........
         M = EN
         H(EN,EN) = 1.0D0
         IF (NA .EQ. 0) GO TO 800
C     .......... FOR I=EN-1 STEP -1 UNTIL 1 DO -- ..........
         DO 700 II = 1, NA
            I = EN - II
            W = H(I,I) - P
            R = 0.0D0
C
            DO 610 J = M, EN
  610       R = R + H(I,J) * H(J,EN)
C
            IF (WI(I) .GE. 0.0D0) GO TO 630
            ZZ = W
            S = R
            GO TO 700
  630       M = I
            IF (WI(I) .NE. 0.0D0) GO TO 640
            T = W
            IF (T .NE. 0.0D0) GO TO 635
               TST1 = NORM
               T = TST1
  632          T = 0.01D0 * T
               TST2 = NORM + T
               IF (TST2 .GT. TST1) GO TO 632
  635       H(I,EN) = -R / T
            GO TO 680
C     .......... SOLVE REAL EQUATIONS ..........
  640       X = H(I,I+1)
            Y = H(I+1,I)
            Q = (WR(I) - P) * (WR(I) - P) + WI(I) * WI(I)
            T = (X * S - ZZ * R) / Q
            H(I,EN) = T
            IF (DABS(X) .LE. DABS(ZZ)) GO TO 650
            H(I+1,EN) = (-R - W * T) / X
            GO TO 680
  650       H(I+1,EN) = (-S - Y * T) / ZZ
C
C     .......... OVERFLOW CONTROL ..........
  680       T = DABS(H(I,EN))
            IF (T .EQ. 0.0D0) GO TO 700
            TST1 = T
            TST2 = TST1 + 1.0D0/TST1
            IF (TST2 .GT. TST1) GO TO 700
            DO 690 J = I, EN
               H(J,EN) = H(J,EN)/T
  690       CONTINUE
C
  700    CONTINUE
C     .......... END REAL VECTOR ..........
         GO TO 800
C     .......... COMPLEX VECTOR ..........
  710    M = NA
C        R correction
         IF (NA .EQ. 0) GO TO 800
C     .......... LAST VECTOR COMPONENT CHOSEN IMAGINARY SO THAT
C                EIGENVECTOR MATRIX IS TRIANGULAR ..........
         IF (DABS(H(EN,NA)) .LE. DABS(H(NA,EN))) GO TO 720
         H(NA,NA) = Q / H(EN,NA)
         H(NA,EN) = -(H(EN,EN) - P) / H(EN,NA)
         GO TO 730
  720    CALL CDIV(0.0D0,-H(NA,EN),H(NA,NA)-P,Q,H(NA,NA),H(NA,EN))
  730    H(EN,NA) = 0.0D0
         H(EN,EN) = 1.0D0
         ENM2 = NA - 1
         IF (ENM2 .EQ. 0) GO TO 800
C     .......... FOR I=EN-2 STEP -1 UNTIL 1 DO -- ..........
         DO 795 II = 1, ENM2
            I = NA - II
            W = H(I,I) - P
            RA = 0.0D0
            SA = 0.0D0
C
            DO 760 J = M, EN
               RA = RA + H(I,J) * H(J,NA)
               SA = SA + H(I,J) * H(J,EN)
  760       CONTINUE
C
            IF (WI(I) .GE. 0.0D0) GO TO 770
            ZZ = W
            R = RA
            S = SA
            GO TO 795
  770       M = I
            IF (WI(I) .NE. 0.0D0) GO TO 780
            CALL CDIV(-RA,-SA,W,Q,H(I,NA),H(I,EN))
            GO TO 790
C     .......... SOLVE COMPLEX EQUATIONS ..........
  780       X = H(I,I+1)
            Y = H(I+1,I)
            VR = (WR(I) - P) * (WR(I) - P) + WI(I) * WI(I) - Q * Q
            VI = (WR(I) - P) * 2.0D0 * Q
            IF (VR .NE. 0.0D0 .OR. VI .NE. 0.0D0) GO TO 784
               TST1 = NORM * (DABS(W) + DABS(Q) + DABS(X)
     X                      + DABS(Y) + DABS(ZZ))
               VR = TST1
  783          VR = 0.01D0 * VR
               TST2 = TST1 + VR
               IF (TST2 .GT. TST1) GO TO 783
  784       CALL CDIV(X*R-ZZ*RA+Q*SA,X*S-ZZ*SA-Q*RA,VR,VI,
     X                H(I,NA),H(I,EN))
            IF (DABS(X) .LE. DABS(ZZ) + DABS(Q)) GO TO 785
            H(I+1,NA) = (-RA - W * H(I,NA) + Q * H(I,EN)) / X
            H(I+1,EN) = (-SA - W * H(I,EN) - Q * H(I,NA)) / X
            GO TO 790
  785       CALL CDIV(-R-Y*H(I,NA),-S-Y*H(I,EN),ZZ,Q,
     X                H(I+1,NA),H(I+1,EN))
C
C     .......... OVERFLOW CONTROL ..........
  790       T = DMAX1(DABS(H(I,NA)), DABS(H(I,EN)))
            IF (T .EQ. 0.0D0) GO TO 795
            TST1 = T
            TST2 = TST1 + 1.0D0/TST1
            IF (TST2 .GT. TST1) GO TO 795
            DO 792 J = I, EN
               H(J,NA) = H(J,NA)/T
               H(J,EN) = H(J,EN)/T
  792       CONTINUE
C
  795    CONTINUE
C     .......... END COMPLEX VECTOR ..........
  800 CONTINUE
C     .......... END BACK SUBSTITUTION.
C                VECTORS OF ISOLATED ROOTS ..........
      DO 840 I = 1, N
         IF (I .GE. LOW .AND. I .LE. IGH) GO TO 840
C
         DO 820 J = I, N
  820    Z(I,J) = H(I,J)
C
  840 CONTINUE
C     .......... MULTIPLY BY TRANSFORMATION MATRIX TO GIVE
C                VECTORS OF ORIGINAL FULL MATRIX.
C                FOR J=N STEP -1 UNTIL LOW DO -- ..........
      DO 880 JJ = LOW, N
         J = N + LOW - JJ
         M = MIN0(J,IGH)
C
         DO 880 I = LOW, IGH
            ZZ = 0.0D0
C
            DO 860 K = LOW, M
  860       ZZ = ZZ + Z(I,K) * H(K,J)
C
            Z(I,J) = ZZ
  880 CONTINUE
C
      GO TO 1001
C     .......... SET ERROR -- ALL EIGENVALUES HAVE NOT
C                CONVERGED AFTER 30*N ITERATIONS ..........
 1000 IERR = EN
 1001 RETURN
      END
      SUBROUTINE HTRIBK(NM,N,AR,AI,TAU,M,ZR,ZI)
C
      INTEGER I,J,K,L,M,N,NM
      DOUBLE PRECISION AR(NM,N),AI(NM,N),TAU(2,N),ZR(NM,M),ZI(NM,M)
      DOUBLE PRECISION H,S,SI
C
C     THIS SUBROUTINE IS A TRANSLATION OF A COMPLEX ANALOGUE OF
C     THE ALGOL PROCEDURE TRBAK1, NUM. MATH. 11, 181-195(1968)
C     BY MARTIN, REINSCH, AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C     THIS SUBROUTINE FORMS THE EIGENVECTORS OF A COMPLEX HERMITIAN
C     MATRIX BY BACK TRANSFORMING THOSE OF THE CORRESPONDING
C     REAL SYMMETRIC TRIDIAGONAL MATRIX DETERMINED BY  HTRIDI.
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 MATRIX.
C
C        AR AND AI CONTAIN INFORMATION ABOUT THE UNITARY TRANS-
C          FORMATIONS USED IN THE REDUCTION BY  HTRIDI  IN THEIR
C          FULL LOWER TRIANGLES EXCEPT FOR THE DIAGONAL OF AR.
C
C        TAU CONTAINS FURTHER INFORMATION ABOUT THE TRANSFORMATIONS.
C
C        M IS THE NUMBER OF EIGENVECTORS TO BE BACK TRANSFORMED.
C
C        ZR CONTAINS THE EIGENVECTORS TO BE BACK TRANSFORMED
C          IN ITS FIRST M COLUMNS.
C
C     ON OUTPUT
C
C        ZR AND ZI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE TRANSFORMED EIGENVECTORS
C          IN THEIR FIRST M COLUMNS.
C
C     NOTE THAT THE LAST COMPONENT OF EACH RETURNED VECTOR
C     IS REAL AND THAT VECTOR EUCLIDEAN NORMS ARE PRESERVED.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (M .EQ. 0) GO TO 200
C     .......... TRANSFORM THE EIGENVECTORS OF THE REAL SYMMETRIC
C                TRIDIAGONAL MATRIX TO THOSE OF THE HERMITIAN
C                TRIDIAGONAL MATRIX. ..........
      DO 50 K = 1, N
C
         DO 50 J = 1, M
            ZI(K,J) = -ZR(K,J) * TAU(2,K)
            ZR(K,J) = ZR(K,J) * TAU(1,K)
   50 CONTINUE
C
      IF (N .EQ. 1) GO TO 200
C     .......... RECOVER AND APPLY THE HOUSEHOLDER MATRICES ..........
      DO 140 I = 2, N
         L = I - 1
         H = AI(I,I)
         IF (H .EQ. 0.0D0) GO TO 140
C
         DO 130 J = 1, M
            S = 0.0D0
            SI = 0.0D0
C
            DO 110 K = 1, L
               S = S + AR(I,K) * ZR(K,J) - AI(I,K) * ZI(K,J)
               SI = SI + AR(I,K) * ZI(K,J) + AI(I,K) * ZR(K,J)
  110       CONTINUE
C     .......... DOUBLE DIVISIONS AVOID POSSIBLE UNDERFLOW ..........
            S = (S / H) / H
            SI = (SI / H) / H
C
            DO 120 K = 1, L
               ZR(K,J) = ZR(K,J) - S * AR(I,K) - SI * AI(I,K)
               ZI(K,J) = ZI(K,J) - SI * AR(I,K) + S * AI(I,K)
  120       CONTINUE
C
  130    CONTINUE
C
  140 CONTINUE
C
  200 RETURN
      END
      SUBROUTINE HTRIDI(NM,N,AR,AI,D,E,E2,TAU)
C
      INTEGER I,J,K,L,N,II,NM,JP1
      DOUBLE PRECISION AR(NM,N),AI(NM,N),D(N),E(N),E2(N),TAU(2,N)
      DOUBLE PRECISION F,G,H,FI,GI,HH,SI,SCALE,PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF A COMPLEX ANALOGUE OF
C     THE ALGOL PROCEDURE TRED1, NUM. MATH. 11, 181-195(1968)
C     BY MARTIN, REINSCH, AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C     THIS SUBROUTINE REDUCES A COMPLEX HERMITIAN MATRIX
C     TO A REAL SYMMETRIC TRIDIAGONAL MATRIX USING
C     UNITARY SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        AR AND AI CONTAIN THE REAL AND IMAGINARY PARTS,
C          RESPECTIVELY, OF THE COMPLEX HERMITIAN INPUT MATRIX.
C          ONLY THE LOWER TRIANGLE OF THE MATRIX NEED BE SUPPLIED.
C
C     ON OUTPUT
C
C        AR AND AI CONTAIN INFORMATION ABOUT THE UNITARY TRANS-
C          FORMATIONS USED IN THE REDUCTION IN THEIR FULL LOWER
C          TRIANGLES.  THEIR STRICT UPPER TRIANGLES AND THE
C          DIAGONAL OF AR ARE UNALTERED.
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE THE TRIDIAGONAL MATRIX.
C
C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
C          MATRIX IN ITS LAST N-1 POSITIONS.  E(1) IS SET TO ZERO.
C
C        E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
C          E2 MAY COINCIDE WITH E IF THE SQUARES ARE NOT NEEDED.
C
C        TAU CONTAINS FURTHER INFORMATION ABOUT THE TRANSFORMATIONS.
C
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      TAU(1,N) = 1.0D0
      TAU(2,N) = 0.0D0
C
      DO 100 I = 1, N
  100 D(I) = AR(I,I)
C     .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
      DO 300 II = 1, N
         I = N + 1 - II
         L = I - 1
         H = 0.0D0
         SCALE = 0.0D0
         IF (L .LT. 1) GO TO 130
C     .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
         DO 120 K = 1, L
  120    SCALE = SCALE + DABS(AR(I,K)) + DABS(AI(I,K))
C
         IF (SCALE .NE. 0.0D0) GO TO 140
         TAU(1,L) = 1.0D0
         TAU(2,L) = 0.0D0
  130    E(I) = 0.0D0
         E2(I) = 0.0D0
         GO TO 290
C
  140    DO 150 K = 1, L
            AR(I,K) = AR(I,K) / SCALE
            AI(I,K) = AI(I,K) / SCALE
            H = H + AR(I,K) * AR(I,K) + AI(I,K) * AI(I,K)
  150    CONTINUE
C
         E2(I) = SCALE * SCALE * H
         G = DSQRT(H)
         E(I) = SCALE * G
         F = PYTHAG(AR(I,L),AI(I,L))
C     .......... FORM NEXT DIAGONAL ELEMENT OF MATRIX T ..........
         IF (F .EQ. 0.0D0) GO TO 160
         TAU(1,L) = (AI(I,L) * TAU(2,I) - AR(I,L) * TAU(1,I)) / F
         SI = (AR(I,L) * TAU(2,I) + AI(I,L) * TAU(1,I)) / F
         H = H + F * G
         G = 1.0D0 + G / F
         AR(I,L) = G * AR(I,L)
         AI(I,L) = G * AI(I,L)
         IF (L .EQ. 1) GO TO 270
         GO TO 170
  160    TAU(1,L) = -TAU(1,I)
         SI = TAU(2,I)
         AR(I,L) = G
  170    F = 0.0D0
C
         DO 240 J = 1, L
            G = 0.0D0
            GI = 0.0D0
C     .......... FORM ELEMENT OF A*U ..........
            DO 180 K = 1, J
               G = G + AR(J,K) * AR(I,K) + AI(J,K) * AI(I,K)
               GI = GI - AR(J,K) * AI(I,K) + AI(J,K) * AR(I,K)
  180       CONTINUE
C
            JP1 = J + 1
            IF (L .LT. JP1) GO TO 220
C
            DO 200 K = JP1, L
               G = G + AR(K,J) * AR(I,K) - AI(K,J) * AI(I,K)
               GI = GI - AR(K,J) * AI(I,K) - AI(K,J) * AR(I,K)
  200       CONTINUE
C     .......... FORM ELEMENT OF P ..........
  220       E(J) = G / H
            TAU(2,J) = GI / H
            F = F + E(J) * AR(I,J) - TAU(2,J) * AI(I,J)
  240    CONTINUE
C
         HH = F / (H + H)
C     .......... FORM REDUCED A ..........
         DO 260 J = 1, L
            F = AR(I,J)
            G = E(J) - HH * F
            E(J) = G
            FI = -AI(I,J)
            GI = TAU(2,J) - HH * FI
            TAU(2,J) = -GI
C
            DO 260 K = 1, J
               AR(J,K) = AR(J,K) - F * E(K) - G * AR(I,K)
     X                           + FI * TAU(2,K) + GI * AI(I,K)
               AI(J,K) = AI(J,K) - F * TAU(2,K) - G * AI(I,K)
     X                           - FI * E(K) - GI * AR(I,K)
  260    CONTINUE
C
  270    DO 280 K = 1, L
            AR(I,K) = SCALE * AR(I,K)
            AI(I,K) = SCALE * AI(I,K)
  280    CONTINUE
C
         TAU(2,L) = -SI
  290    HH = D(I)
         D(I) = AR(I,I)
         AR(I,I) = HH
         AI(I,I) = SCALE * DSQRT(H)
  300 CONTINUE
C
      RETURN
      END
      DOUBLE PRECISION FUNCTION PYTHAG(A,B)
      DOUBLE PRECISION A,B
C
C     FINDS DSQRT(A**2+B**2) WITHOUT OVERFLOW OR DESTRUCTIVE UNDERFLOW
C
      DOUBLE PRECISION P,R,S,T,U
      P = DMAX1(DABS(A),DABS(B))
      IF (P .EQ. 0.0D0) GO TO 20
      R = (DMIN1(DABS(A),DABS(B))/P)**2
   10 CONTINUE
         T = 4.0D0 + R
         IF (T .EQ. 4.0D0) GO TO 20
         S = R/T
         U = 1.0D0 + 2.0D0*S
         P = U*P
         R = (S/U)**2 * R
      GO TO 10
   20 PYTHAG = P
      RETURN
      END
      SUBROUTINE TQL1(N,D,E,IERR)
C
      INTEGER I,J,L,M,N,II,L1,L2,MML,IERR
      DOUBLE PRECISION D(N),E(N)
      DOUBLE PRECISION C,C2,C3,DL1,EL1,F,G,H,P,R,S,S2,TST1,TST2,PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TQL1,
C     NUM. MATH. 11, 293-306(1968) BY BOWDLER, MARTIN, REINSCH, AND
C     WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES OF A SYMMETRIC
C     TRIDIAGONAL MATRIX BY THE QL METHOD.
C
C     ON INPUT
C
C        N IS THE ORDER OF THE MATRIX.
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX.
C
C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE INPUT MATRIX
C          IN ITS LAST N-1 POSITIONS.  E(1) IS ARBITRARY.
C
C      ON OUTPUT
C
C        D CONTAINS THE EIGENVALUES IN ASCENDING ORDER.  IF AN
C          ERROR EXIT IS MADE, THE EIGENVALUES ARE CORRECT AND
C          ORDERED FOR INDICES 1,2,...IERR-1, BUT MAY NOT BE
C          THE SMALLEST EIGENVALUES.
C
C        E HAS BEEN DESTROYED.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE J-TH EIGENVALUE HAS NOT BEEN
C                     DETERMINED AFTER 30 ITERATIONS.
C
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of C3 and S2 to keep g77 -Wall happy
c
      C3 = 0.0D0
      S2 = 0.0D0
C
      IERR = 0
      IF (N .EQ. 1) GO TO 1001
C
      DO 100 I = 2, N
  100 E(I-1) = E(I)
C
      F = 0.0D0
      TST1 = 0.0D0
      E(N) = 0.0D0
C
      DO 290 L = 1, N
         J = 0
         H = DABS(D(L)) + DABS(E(L))
         IF (TST1 .LT. H) TST1 = H
C     .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT ..........
         DO 110 M = L, N
            TST2 = TST1 + DABS(E(M))
            IF (TST2 .EQ. TST1) GO TO 120
C     .......... E(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
C                THROUGH THE BOTTOM OF THE LOOP ..........
  110    CONTINUE
C
  120    IF (M .EQ. L) GO TO 210
  130    IF (J .EQ. 30) GO TO 1000
         J = J + 1
C     .......... FORM SHIFT ..........
         L1 = L + 1
         L2 = L1 + 1
         G = D(L)
         P = (D(L1) - G) / (2.0D0 * E(L))
         R = PYTHAG(P,1.0D0)
         D(L) = E(L) / (P + DSIGN(R,P))
         D(L1) = E(L) * (P + DSIGN(R,P))
         DL1 = D(L1)
         H = G - D(L)
         IF (L2 .GT. N) GO TO 145
C
         DO 140 I = L2, N
  140    D(I) = D(I) - H
C
  145    F = F + H
C     .......... QL TRANSFORMATION ..........
         P = D(M)
         C = 1.0D0
         C2 = C
         EL1 = E(L1)
         S = 0.0D0
         MML = M - L
C     .......... FOR I=M-1 STEP -1 UNTIL L DO -- ..........
         DO 200 II = 1, MML
            C3 = C2
            C2 = C
            S2 = S
            I = M - II
            G = C * E(I)
            H = C * P
            R = PYTHAG(P,E(I))
            E(I+1) = S * R
            S = E(I) / R
            C = P / R
            P = C * D(I) - S * G
            D(I+1) = H + S * (C * G + S * D(I))
  200    CONTINUE
C
         P = -S * S2 * C3 * EL1 * E(L) / DL1
         E(L) = S * P
         D(L) = C * P
         TST2 = TST1 + DABS(E(L))
         IF (TST2 .GT. TST1) GO TO 130
  210    P = D(L) + F
C     .......... ORDER EIGENVALUES ..........
         IF (L .EQ. 1) GO TO 250
C     .......... FOR I=L STEP -1 UNTIL 2 DO -- ..........
         DO 230 II = 2, L
            I = L + 2 - II
            IF (P .GE. D(I-1)) GO TO 270
            D(I) = D(I-1)
  230    CONTINUE
C
  250    I = 1
  270    D(I) = P
  290 CONTINUE
C
      GO TO 1001
C     .......... SET ERROR -- NO CONVERGENCE TO AN
C                EIGENVALUE AFTER 30 ITERATIONS ..........
 1000 IERR = L
 1001 RETURN
      END
      SUBROUTINE TQL2(NM,N,D,E,Z,IERR)
C
      INTEGER I,J,K,L,M,N,II,L1,L2,NM,MML,IERR
      DOUBLE PRECISION D(N),E(N),Z(NM,N)
      DOUBLE PRECISION C,C2,C3,DL1,EL1,F,G,H,P,R,S,S2,TST1,TST2,PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TQL2,
C     NUM. MATH. 11, 293-306(1968) BY BOWDLER, MARTIN, REINSCH, AND
C     WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 227-240(1971).
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES AND EIGENVECTORS
C     OF A SYMMETRIC TRIDIAGONAL MATRIX BY THE QL METHOD.
C     THE EIGENVECTORS OF A FULL SYMMETRIC MATRIX CAN ALSO
C     BE FOUND IF  TRED2  HAS BEEN USED TO REDUCE THIS
C     FULL MATRIX TO TRIDIAGONAL FORM.
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 MATRIX.
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX.
C
C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE INPUT MATRIX
C          IN ITS LAST N-1 POSITIONS.  E(1) IS ARBITRARY.
C
C        Z CONTAINS THE TRANSFORMATION MATRIX PRODUCED IN THE
C          REDUCTION BY  TRED2, IF PERFORMED.  IF THE EIGENVECTORS
C          OF THE TRIDIAGONAL MATRIX ARE DESIRED, Z MUST CONTAIN
C          THE IDENTITY MATRIX.
C
C      ON OUTPUT
C
C        D CONTAINS THE EIGENVALUES IN ASCENDING ORDER.  IF AN
C          ERROR EXIT IS MADE, THE EIGENVALUES ARE CORRECT BUT
C          UNORDERED FOR INDICES 1,2,...,IERR-1.
C
C        E HAS BEEN DESTROYED.
C
C        Z CONTAINS ORTHONORMAL EIGENVECTORS OF THE SYMMETRIC
C          TRIDIAGONAL (OR FULL) MATRIX.  IF AN ERROR EXIT IS MADE,
C          Z CONTAINS THE EIGENVECTORS ASSOCIATED WITH THE STORED
C          EIGENVALUES.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE J-TH EIGENVALUE HAS NOT BEEN
C                     DETERMINED AFTER 30 ITERATIONS.
C
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of C3 and S2 to keep g77 -Wall happy
c
      C3 = 0.0D0
      S2 = 0.0D0
C
      IERR = 0
      IF (N .EQ. 1) GO TO 1001
C
      DO 100 I = 2, N
  100 E(I-1) = E(I)
C
      F = 0.0D0
      TST1 = 0.0D0
      E(N) = 0.0D0
C
      DO 240 L = 1, N
         J = 0
         H = DABS(D(L)) + DABS(E(L))
         IF (TST1 .LT. H) TST1 = H
C     .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT ..........
         DO 110 M = L, N
            TST2 = TST1 + DABS(E(M))
            IF (TST2 .EQ. TST1) GO TO 120
C     .......... E(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
C                THROUGH THE BOTTOM OF THE LOOP ..........
  110    CONTINUE
C
  120    IF (M .EQ. L) GO TO 220
  130    IF (J .EQ. 30) GO TO 1000
         J = J + 1
C     .......... FORM SHIFT ..........
         L1 = L + 1
         L2 = L1 + 1
         G = D(L)
         P = (D(L1) - G) / (2.0D0 * E(L))
         R = PYTHAG(P,1.0D0)
         D(L) = E(L) / (P + DSIGN(R,P))
         D(L1) = E(L) * (P + DSIGN(R,P))
         DL1 = D(L1)
         H = G - D(L)
         IF (L2 .GT. N) GO TO 145
C
         DO 140 I = L2, N
  140    D(I) = D(I) - H
C
  145    F = F + H
C     .......... QL TRANSFORMATION ..........
         P = D(M)
         C = 1.0D0
         C2 = C
         EL1 = E(L1)
         S = 0.0D0
         MML = M - L
C     .......... FOR I=M-1 STEP -1 UNTIL L DO -- ..........
         DO 200 II = 1, MML
            C3 = C2
            C2 = C
            S2 = S
            I = M - II
            G = C * E(I)
            H = C * P
            R = PYTHAG(P,E(I))
            E(I+1) = S * R
            S = E(I) / R
            C = P / R
            P = C * D(I) - S * G
            D(I+1) = H + S * (C * G + S * D(I))
C     .......... FORM VECTOR ..........
            DO 180 K = 1, N
               H = Z(K,I+1)
               Z(K,I+1) = S * Z(K,I) + C * H
               Z(K,I) = C * Z(K,I) - S * H
  180       CONTINUE
C
  200    CONTINUE
C
         P = -S * S2 * C3 * EL1 * E(L) / DL1
         E(L) = S * P
         D(L) = C * P
         TST2 = TST1 + DABS(E(L))
         IF (TST2 .GT. TST1) GO TO 130
  220    D(L) = D(L) + F
  240 CONTINUE
C     .......... ORDER EIGENVALUES AND EIGENVECTORS ..........
      DO 300 II = 2, N
         I = II - 1
         K = I
         P = D(I)
C
         DO 260 J = II, N
            IF (D(J) .GE. P) GO TO 260
            K = J
            P = D(J)
  260    CONTINUE
C
         IF (K .EQ. I) GO TO 300
         D(K) = D(I)
         D(I) = P
C
         DO 280 J = 1, N
            P = Z(J,I)
            Z(J,I) = Z(J,K)
            Z(J,K) = P
  280    CONTINUE
C
  300 CONTINUE
C
      GO TO 1001
C     .......... SET ERROR -- NO CONVERGENCE TO AN
C                EIGENVALUE AFTER 30 ITERATIONS ..........
 1000 IERR = L
 1001 RETURN
      END
      SUBROUTINE TQLRAT(N,D,E2,IERR)
C
      INTEGER I,J,L,M,N,II,L1,MML,IERR
      DOUBLE PRECISION D(N),E2(N)
      DOUBLE PRECISION B,C,F,G,H,P,R,S,T,EPSLON,PYTHAG
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TQLRAT,
C     ALGORITHM 464, COMM. ACM 16, 689(1973) BY REINSCH.
C
C     THIS SUBROUTINE FINDS THE EIGENVALUES OF A SYMMETRIC
C     TRIDIAGONAL MATRIX BY THE RATIONAL QL METHOD.
C
C     ON INPUT
C
C        N IS THE ORDER OF THE MATRIX.
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE INPUT MATRIX.
C
C        E2 CONTAINS THE SQUARES OF THE SUBDIAGONAL ELEMENTS OF THE
C          INPUT MATRIX IN ITS LAST N-1 POSITIONS.  E2(1) IS ARBITRARY.
C
C      ON OUTPUT
C
C        D CONTAINS THE EIGENVALUES IN ASCENDING ORDER.  IF AN
C          ERROR EXIT IS MADE, THE EIGENVALUES ARE CORRECT AND
C          ORDERED FOR INDICES 1,2,...IERR-1, BUT MAY NOT BE
C          THE SMALLEST EIGENVALUES.
C
C        E2 HAS BEEN DESTROYED.
C
C        IERR IS SET TO
C          ZERO       FOR NORMAL RETURN,
C          J          IF THE J-TH EIGENVALUE HAS NOT BEEN
C                     DETERMINED AFTER 30 ITERATIONS.
C
C     CALLS PYTHAG FOR  DSQRT(A*A + B*B) .
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
c
c     unnecessary initialization of B and C to keep g77 -Wall happy
c
      B = 0.0D0
      C = 0.0D0
C
      IERR = 0
      IF (N .EQ. 1) GO TO 1001
C
      DO 100 I = 2, N
  100 E2(I-1) = E2(I)
C
      F = 0.0D0
      T = 0.0D0
      E2(N) = 0.0D0
C
      DO 290 L = 1, N
         J = 0
         H = DABS(D(L)) + DSQRT(E2(L))
         IF (T .GT. H) GO TO 105
         T = H
         B = EPSLON(T)
         C = B * B
C     .......... LOOK FOR SMALL SQUARED SUB-DIAGONAL ELEMENT ..........
  105    DO 110 M = L, N
            IF (E2(M) .LE. C) GO TO 120
C     .......... E2(N) IS ALWAYS ZERO, SO THERE IS NO EXIT
C                THROUGH THE BOTTOM OF THE LOOP ..........
  110    CONTINUE
C
  120    IF (M .EQ. L) GO TO 210
  130    IF (J .EQ. 30) GO TO 1000
         J = J + 1
C     .......... FORM SHIFT ..........
         L1 = L + 1
         S = DSQRT(E2(L))
         G = D(L)
         P = (D(L1) - G) / (2.0D0 * S)
         R = PYTHAG(P,1.0D0)
         D(L) = S / (P + DSIGN(R,P))
         H = G - D(L)
C
         DO 140 I = L1, N
  140    D(I) = D(I) - H
C
         F = F + H
C     .......... RATIONAL QL TRANSFORMATION ..........
         G = D(M)
         IF (G .EQ. 0.0D0) G = B
         H = G
         S = 0.0D0
         MML = M - L
C     .......... FOR I=M-1 STEP -1 UNTIL L DO -- ..........
         DO 200 II = 1, MML
            I = M - II
            P = G * H
            R = P + E2(I)
            E2(I+1) = S * R
            S = E2(I) / R
            D(I+1) = H + S * (H + D(I))
            G = D(I) - E2(I) / G
            IF (G .EQ. 0.0D0) G = B
            H = G * P / R
  200    CONTINUE
C
         E2(L) = S * G
         D(L) = H
C     .......... GUARD AGAINST UNDERFLOW IN CONVERGENCE TEST ..........
         IF (H .EQ. 0.0D0) GO TO 210
         IF (DABS(E2(L)) .LE. DABS(C/H)) GO TO 210
         E2(L) = H * E2(L)
         IF (E2(L) .NE. 0.0D0) GO TO 130
  210    P = D(L) + F
C     .......... ORDER EIGENVALUES ..........
         IF (L .EQ. 1) GO TO 250
C     .......... FOR I=L STEP -1 UNTIL 2 DO -- ..........
         DO 230 II = 2, L
            I = L + 2 - II
            IF (P .GE. D(I-1)) GO TO 270
            D(I) = D(I-1)
  230    CONTINUE
C
  250    I = 1
  270    D(I) = P
  290 CONTINUE
C
      GO TO 1001
C     .......... SET ERROR -- NO CONVERGENCE TO AN
C                EIGENVALUE AFTER 30 ITERATIONS ..........
 1000 IERR = L
 1001 RETURN
      END
      SUBROUTINE TRED1(NM,N,A,D,E,E2)
C
      INTEGER I,J,K,L,N,II,NM,JP1
      DOUBLE PRECISION A(NM,N),D(N),E(N),E2(N)
      DOUBLE PRECISION F,G,H,SCALE
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED1,
C     NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C     THIS SUBROUTINE REDUCES A REAL SYMMETRIC MATRIX
C     TO A SYMMETRIC TRIDIAGONAL MATRIX USING
C     ORTHOGONAL SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        A CONTAINS THE REAL SYMMETRIC INPUT MATRIX.  ONLY THE
C          LOWER TRIANGLE OF THE MATRIX NEED BE SUPPLIED.
C
C     ON OUTPUT
C
C        A CONTAINS INFORMATION ABOUT THE ORTHOGONAL TRANS-
C          FORMATIONS USED IN THE REDUCTION IN ITS STRICT LOWER
C          TRIANGLE.  THE FULL UPPER TRIANGLE OF A IS UNALTERED.
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE TRIDIAGONAL MATRIX.
C
C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
C          MATRIX IN ITS LAST N-1 POSITIONS.  E(1) IS SET TO ZERO.
C
C        E2 CONTAINS THE SQUARES OF THE CORRESPONDING ELEMENTS OF E.
C          E2 MAY COINCIDE WITH E IF THE SQUARES ARE NOT NEEDED.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      DO 100 I = 1, N
         D(I) = A(N,I)
         A(N,I) = A(I,I)
  100 CONTINUE
C     .......... FOR I=N STEP -1 UNTIL 1 DO -- ..........
      DO 300 II = 1, N
         I = N + 1 - II
         L = I - 1
         H = 0.0D0
         SCALE = 0.0D0
         IF (L .LT. 1) GO TO 130
C     .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
         DO 120 K = 1, L
  120    SCALE = SCALE + DABS(D(K))
C
         IF (SCALE .NE. 0.0D0) GO TO 140
C
         DO 125 J = 1, L
            D(J) = A(L,J)
            A(L,J) = A(I,J)
            A(I,J) = 0.0D0
  125    CONTINUE
C
  130    E(I) = 0.0D0
         E2(I) = 0.0D0
         GO TO 300
C
  140    DO 150 K = 1, L
            D(K) = D(K) / SCALE
            H = H + D(K) * D(K)
  150    CONTINUE
C
         E2(I) = SCALE * SCALE * H
         F = D(L)
         G = -DSIGN(DSQRT(H),F)
         E(I) = SCALE * G
         H = H - F * G
         D(L) = F - G
         IF (L .EQ. 1) GO TO 285
C     .......... FORM A*U ..........
         DO 170 J = 1, L
  170    E(J) = 0.0D0
C
         DO 240 J = 1, L
            F = D(J)
            G = E(J) + A(J,J) * F
            JP1 = J + 1
            IF (L .LT. JP1) GO TO 220
C
            DO 200 K = JP1, L
               G = G + A(K,J) * D(K)
               E(K) = E(K) + A(K,J) * F
  200       CONTINUE
C
  220       E(J) = G
  240    CONTINUE
C     .......... FORM P ..........
         F = 0.0D0
C
         DO 245 J = 1, L
            E(J) = E(J) / H
            F = F + E(J) * D(J)
  245    CONTINUE
C
         H = F / (H + H)
C     .......... FORM Q ..........
         DO 250 J = 1, L
  250    E(J) = E(J) - H * D(J)
C     .......... FORM REDUCED A ..........
         DO 280 J = 1, L
            F = D(J)
            G = E(J)
C
            DO 260 K = J, L
  260       A(K,J) = A(K,J) - F * E(K) - G * D(K)
C
  280    CONTINUE
C
  285    DO 290 J = 1, L
            F = D(J)
            D(J) = A(L,J)
            A(L,J) = A(I,J)
            A(I,J) = F * SCALE
  290    CONTINUE
C
  300 CONTINUE
C
      RETURN
      END
      SUBROUTINE TRED2(NM,N,A,D,E,Z)
C
      INTEGER I,J,K,L,N,II,NM,JP1
      DOUBLE PRECISION A(NM,N),D(N),E(N),Z(NM,N)
      DOUBLE PRECISION F,G,H,HH,SCALE
C
C     THIS SUBROUTINE IS A TRANSLATION OF THE ALGOL PROCEDURE TRED2,
C     NUM. MATH. 11, 181-195(1968) BY MARTIN, REINSCH, AND WILKINSON.
C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 212-226(1971).
C
C     THIS SUBROUTINE REDUCES A REAL SYMMETRIC MATRIX TO A
C     SYMMETRIC TRIDIAGONAL MATRIX USING AND ACCUMULATING
C     ORTHOGONAL SIMILARITY TRANSFORMATIONS.
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 MATRIX.
C
C        A CONTAINS THE REAL SYMMETRIC INPUT MATRIX.  ONLY THE
C          LOWER TRIANGLE OF THE MATRIX NEED BE SUPPLIED.
C
C     ON OUTPUT
C
C        D CONTAINS THE DIAGONAL ELEMENTS OF THE TRIDIAGONAL MATRIX.
C
C        E CONTAINS THE SUBDIAGONAL ELEMENTS OF THE TRIDIAGONAL
C          MATRIX IN ITS LAST N-1 POSITIONS.  E(1) IS SET TO ZERO.
C
C        Z CONTAINS THE ORTHOGONAL TRANSFORMATION MATRIX
C          PRODUCED IN THE REDUCTION.
C
C        A AND Z MAY COINCIDE.  IF DISTINCT, A IS UNALTERED.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      DO 100 I = 1, N
C
         DO 80 J = I, N
   80    Z(J,I) = A(J,I)
C
         D(I) = A(N,I)
  100 CONTINUE
C
      IF (N .EQ. 1) GO TO 510
C     .......... FOR I=N STEP -1 UNTIL 2 DO -- ..........
      DO 300 II = 2, N
         I = N + 2 - II
         L = I - 1
         H = 0.0D0
         SCALE = 0.0D0
         IF (L .LT. 2) GO TO 130
C     .......... SCALE ROW (ALGOL TOL THEN NOT NEEDED) ..........
         DO 120 K = 1, L
  120    SCALE = SCALE + DABS(D(K))
C
         IF (SCALE .NE. 0.0D0) GO TO 140
  130    E(I) = D(L)
C
         DO 135 J = 1, L
            D(J) = Z(L,J)
            Z(I,J) = 0.0D0
            Z(J,I) = 0.0D0
  135    CONTINUE
C
         GO TO 290
C
  140    DO 150 K = 1, L
            D(K) = D(K) / SCALE
            H = H + D(K) * D(K)
  150    CONTINUE
C
         F = D(L)
         G = -DSIGN(DSQRT(H),F)
         E(I) = SCALE * G
         H = H - F * G
         D(L) = F - G
C     .......... FORM A*U ..........
         DO 170 J = 1, L
  170    E(J) = 0.0D0
C
         DO 240 J = 1, L
            F = D(J)
            Z(J,I) = F
            G = E(J) + Z(J,J) * F
            JP1 = J + 1
            IF (L .LT. JP1) GO TO 220
C
            DO 200 K = JP1, L
               G = G + Z(K,J) * D(K)
               E(K) = E(K) + Z(K,J) * F
  200       CONTINUE
C
  220       E(J) = G
  240    CONTINUE
C     .......... FORM P ..........
         F = 0.0D0
C
         DO 245 J = 1, L
            E(J) = E(J) / H
            F = F + E(J) * D(J)
  245    CONTINUE
C
         HH = F / (H + H)
C     .......... FORM Q ..........
         DO 250 J = 1, L
  250    E(J) = E(J) - HH * D(J)
C     .......... FORM REDUCED A ..........
         DO 280 J = 1, L
            F = D(J)
            G = E(J)
C
            DO 260 K = J, L
  260       Z(K,J) = Z(K,J) - F * E(K) - G * D(K)
C
            D(J) = Z(L,J)
            Z(I,J) = 0.0D0
  280    CONTINUE
C
  290    D(I) = H
  300 CONTINUE
C     .......... ACCUMULATION OF TRANSFORMATION MATRICES ..........
      DO 500 I = 2, N
         L = I - 1
         Z(N,L) = Z(L,L)
         Z(L,L) = 1.0D0
         H = D(I)
         IF (H .EQ. 0.0D0) GO TO 380
C
         DO 330 K = 1, L
  330    D(K) = Z(K,I) / H
C
         DO 360 J = 1, L
            G = 0.0D0
C
            DO 340 K = 1, L
  340       G = G + Z(K,I) * Z(K,J)
C
            DO 360 K = 1, L
               Z(K,J) = Z(K,J) - G * D(K)
  360    CONTINUE
C
  380    DO 400 K = 1, L
  400    Z(K,I) = 0.0D0
C
  500 CONTINUE
C
  510 DO 520 I = 1, N
         D(I) = Z(N,I)
         Z(N,I) = 0.0D0
  520 CONTINUE
C
      Z(N,N) = 1.0D0
      E(1) = 0.0D0
      RETURN
      END
      SUBROUTINE RG(NM,N,A,WR,WI,MATZ,Z,IV1,FV1,IERR)
C
      INTEGER N,NM,IS1,IS2,IERR,MATZ
      DOUBLE PRECISION A(NM,N),WR(N),WI(N),Z(NM,N),FV1(N)
      INTEGER IV1(N)
C
C     THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF
C     SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK)
C     TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED)
C     OF A REAL GENERAL MATRIX.
C
C     ON INPUT
C
C        NM  MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL
C        ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C        DIMENSION STATEMENT.
C
C        N  IS THE ORDER OF THE MATRIX  A.
C
C        A  CONTAINS THE REAL GENERAL MATRIX.
C
C        MATZ  IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF
C        ONLY EIGENVALUES ARE DESIRED.  OTHERWISE IT IS SET TO
C        ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS.
C
C     ON OUTPUT
C
C        WR  AND  WI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE EIGENVALUES.  COMPLEX CONJUGATE
C        PAIRS OF EIGENVALUES APPEAR CONSECUTIVELY WITH THE
C        EIGENVALUE HAVING THE POSITIVE IMAGINARY PART FIRST.
C
C        Z  CONTAINS THE REAL AND IMAGINARY PARTS OF THE EIGENVECTORS
C        IF MATZ IS NOT ZERO.  IF THE J-TH EIGENVALUE IS REAL, THE
C        J-TH COLUMN OF  Z  CONTAINS ITS EIGENVECTOR.  IF THE J-TH
C        EIGENVALUE IS COMPLEX WITH POSITIVE IMAGINARY PART, THE
C        J-TH AND (J+1)-TH COLUMNS OF  Z  CONTAIN THE REAL AND
C        IMAGINARY PARTS OF ITS EIGENVECTOR.  THE CONJUGATE OF THIS
C        VECTOR IS THE EIGENVECTOR FOR THE CONJUGATE EIGENVALUE.
C
C        IERR  IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN ERROR
C           COMPLETION CODE DESCRIBED IN THE DOCUMENTATION FOR HQR
C           AND HQR2.  THE NORMAL COMPLETION CODE IS ZERO.
C
C        IV1  AND  FV1  ARE TEMPORARY STORAGE ARRAYS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (N .LE. NM) GO TO 10
      IERR = 10 * N
      GO TO 50
C
   10 CALL  BALANC(NM,N,A,IS1,IS2,FV1)
      CALL  ELMHES(NM,N,IS1,IS2,A,IV1)
      IF (MATZ .NE. 0) GO TO 20
C     .......... FIND EIGENVALUES ONLY ..........
      CALL  HQR(NM,N,IS1,IS2,A,WR,WI,IERR)
      GO TO 50
C     .......... FIND BOTH EIGENVALUES AND EIGENVECTORS ..........
   20 CALL  ELTRAN(NM,N,IS1,IS2,A,IV1,Z)
      CALL  HQR2(NM,N,IS1,IS2,A,WR,WI,Z,IERR)
      IF (IERR .NE. 0) GO TO 50
      CALL  BALBAK(NM,N,IS1,IS2,FV1,N,Z)
   50 RETURN
      END
      SUBROUTINE RS(NM,N,A,W,MATZ,Z,FV1,FV2,IERR)
C
      INTEGER N,NM,IERR,MATZ
      DOUBLE PRECISION A(NM,N),W(N),Z(NM,N),FV1(N),FV2(N)
C
C     THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF
C     SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK)
C     TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED)
C     OF A REAL SYMMETRIC MATRIX.
C
C     ON INPUT
C
C        NM  MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL
C        ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C        DIMENSION STATEMENT.
C
C        N  IS THE ORDER OF THE MATRIX  A.
C
C        A  CONTAINS THE REAL SYMMETRIC MATRIX.
C
C        MATZ  IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF
C        ONLY EIGENVALUES ARE DESIRED.  OTHERWISE IT IS SET TO
C        ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS.
C
C     ON OUTPUT
C
C        W  CONTAINS THE EIGENVALUES IN ASCENDING ORDER.
C
C        Z  CONTAINS THE EIGENVECTORS IF MATZ IS NOT ZERO.
C
C        IERR  IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN ERROR
C           COMPLETION CODE DESCRIBED IN THE DOCUMENTATION FOR TQLRAT
C           AND TQL2.  THE NORMAL COMPLETION CODE IS ZERO.
C
C        FV1  AND  FV2  ARE TEMPORARY STORAGE ARRAYS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (N .LE. NM) GO TO 10
      IERR = 10 * N
      GO TO 50
C
   10 IF (MATZ .NE. 0) GO TO 20
C     .......... FIND EIGENVALUES ONLY ..........
      CALL  TRED1(NM,N,A,W,FV1,FV2)
      CALL  TQLRAT(N,W,FV2,IERR)
      GO TO 50
C     .......... FIND BOTH EIGENVALUES AND EIGENVECTORS ..........
   20 CALL  TRED2(NM,N,A,W,FV1,Z)
      CALL  TQL2(NM,N,W,FV1,Z,IERR)
   50 RETURN
      END
      SUBROUTINE CG(NM,N,AR,AI,WR,WI,MATZ,ZR,ZI,FV1,FV2,FV3,IERR)
C
      INTEGER N,NM,IS1,IS2,IERR,MATZ
      DOUBLE PRECISION AR(NM,N),AI(NM,N),WR(N),WI(N),ZR(NM,N),ZI(NM,N),
     X       FV1(N),FV2(N),FV3(N)
C
C     THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF
C     SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK)
C     TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED)
C     OF A COMPLEX GENERAL MATRIX.
C
C     ON INPUT
C
C        NM  MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL
C        ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C        DIMENSION STATEMENT.
C
C        N  IS THE ORDER OF THE MATRIX  A=(AR,AI).
C
C        AR  AND  AI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE COMPLEX GENERAL MATRIX.
C
C        MATZ  IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF
C        ONLY EIGENVALUES ARE DESIRED.  OTHERWISE IT IS SET TO
C        ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS.
C
C     ON OUTPUT
C
C        WR  AND  WI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE EIGENVALUES.
C
C        ZR  AND  ZI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE EIGENVECTORS IF MATZ IS NOT ZERO.
C
C        IERR  IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN ERROR
C           COMPLETION CODE DESCRIBED IN THE DOCUMENTATION FOR COMQR
C           AND COMQR2.  THE NORMAL COMPLETION CODE IS ZERO.
C
C        FV1, FV2, AND  FV3  ARE TEMPORARY STORAGE ARRAYS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (N .LE. NM) GO TO 10
      IERR = 10 * N
      GO TO 50
C
   10 CALL  CBAL(NM,N,AR,AI,IS1,IS2,FV1)
      CALL  CORTH(NM,N,IS1,IS2,AR,AI,FV2,FV3)
      IF (MATZ .NE. 0) GO TO 20
C     .......... FIND EIGENVALUES ONLY ..........
      CALL  COMQR(NM,N,IS1,IS2,AR,AI,WR,WI,IERR)
      GO TO 50
C     .......... FIND BOTH EIGENVALUES AND EIGENVECTORS ..........
   20 CALL  COMQR2(NM,N,IS1,IS2,FV2,FV3,AR,AI,WR,WI,ZR,ZI,IERR)
      IF (IERR .NE. 0) GO TO 50
      CALL  CBABK2(NM,N,IS1,IS2,FV1,N,ZR,ZI)
   50 RETURN
      END
      SUBROUTINE CH(NM,N,AR,AI,W,MATZ,ZR,ZI,FV1,FV2,FM1,IERR)
C
      INTEGER I,J,N,NM,IERR,MATZ
      DOUBLE PRECISION AR(NM,N),AI(NM,N),W(N),ZR(NM,N),ZI(NM,N),
     X       FV1(N),FV2(N),FM1(2,N)
C
C     THIS SUBROUTINE CALLS THE RECOMMENDED SEQUENCE OF
C     SUBROUTINES FROM THE EIGENSYSTEM SUBROUTINE PACKAGE (EISPACK)
C     TO FIND THE EIGENVALUES AND EIGENVECTORS (IF DESIRED)
C     OF A COMPLEX HERMITIAN MATRIX.
C
C     ON INPUT
C
C        NM  MUST BE SET TO THE ROW DIMENSION OF THE TWO-DIMENSIONAL
C        ARRAY PARAMETERS AS DECLARED IN THE CALLING PROGRAM
C        DIMENSION STATEMENT.
C
C        N  IS THE ORDER OF THE MATRIX  A=(AR,AI).
C
C        AR  AND  AI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE COMPLEX HERMITIAN MATRIX.
C
C        MATZ  IS AN INTEGER VARIABLE SET EQUAL TO ZERO IF
C        ONLY EIGENVALUES ARE DESIRED.  OTHERWISE IT IS SET TO
C        ANY NON-ZERO INTEGER FOR BOTH EIGENVALUES AND EIGENVECTORS.
C
C     ON OUTPUT
C
C        W  CONTAINS THE EIGENVALUES IN ASCENDING ORDER.
C
C        ZR  AND  ZI  CONTAIN THE REAL AND IMAGINARY PARTS,
C        RESPECTIVELY, OF THE EIGENVECTORS IF MATZ IS NOT ZERO.
C
C        IERR  IS AN INTEGER OUTPUT VARIABLE SET EQUAL TO AN ERROR
C           COMPLETION CODE DESCRIBED IN THE DOCUMENTATION FOR TQLRAT
C           AND TQL2.  THE NORMAL COMPLETION CODE IS ZERO.
C
C        FV1, FV2, AND  FM1  ARE TEMPORARY STORAGE ARRAYS.
C
C     QUESTIONS AND COMMENTS SHOULD BE DIRECTED TO BURTON S. GARBOW,
C     MATHEMATICS AND COMPUTER SCIENCE DIV, ARGONNE NATIONAL LABORATORY
C
C     THIS VERSION DATED AUGUST 1983.
C
C     ------------------------------------------------------------------
C
      IF (N .LE. NM) GO TO 10
      IERR = 10 * N
      GO TO 50
C
   10 CALL  HTRIDI(NM,N,AR,AI,W,FV1,FV2,FM1)
      IF (MATZ .NE. 0) GO TO 20
C     .......... FIND EIGENVALUES ONLY ..........
      CALL  TQLRAT(N,W,FV2,IERR)
      GO TO 50
C     .......... FIND BOTH EIGENVALUES AND EIGENVECTORS ..........
   20 DO 40 I = 1, N
C
         DO 30 J = 1, N
            ZR(J,I) = 0.0D0
   30    CONTINUE
C
         ZR(I,I) = 1.0D0
   40 CONTINUE
C
      CALL  TQL2(NM,N,W,FV1,ZR,IERR)
      IF (IERR .NE. 0) GO TO 50
      CALL  HTRIBK(NM,N,AR,AI,FM1,N,ZR,ZI)
   50 RETURN
      END

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