relibc_openlibm/slatec/tql2.f

*DECK TQL2
      SUBROUTINE TQL2 (NM, N, D, E, Z, IERR)
C***BEGIN PROLOGUE  TQL2
C***PURPOSE  Compute the eigenvalues and eigenvectors of symmetric
C            tridiagonal matrix.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D4A5, D4C2A
C***TYPE      SINGLE PRECISION (TQL2-S)
C***KEYWORDS  EIGENVALUES, EIGENVECTORS, EISPACK
C***AUTHOR  Smith, B. T., et al.
C***DESCRIPTION
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 the two-dimensional
C          array parameter, Z, as declared in the calling program
C          dimension statement.  NM is an INTEGER variable.
C
C        N is the order of the matrix.  N is an INTEGER variable.
C          N must be less than or equal to NM.
C
C        D contains the diagonal elements of the symmetric tridiagonal
C          matrix.  D is a one-dimensional REAL array, dimensioned D(N).
C
C        E contains the subdiagonal elements of the symmetric
C          tridiagonal matrix in its last N-1 positions.  E(1) is
C          arbitrary.  E is a one-dimensional REAL array, dimensioned
C          E(N).
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.  Z is a two-dimensional REAL array,
C          dimensioned Z(NM,N).
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 an INTEGER flag 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(A,B) for sqrt(A**2 + B**2).
C
C     Questions and comments should be directed to B. S. Garbow,
C     APPLIED MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
C     ------------------------------------------------------------------
C
C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
C                 system Routines - EISPACK Guide, Springer-Verlag,
C                 1976.
C***ROUTINES CALLED  PYTHAG
C***REVISION HISTORY  (YYMMDD)
C   760101  DATE WRITTEN
C   890831  Modified array declarations.  (WRB)
C   890831  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  TQL2
C
      INTEGER I,J,K,L,M,N,II,L1,L2,NM,MML,IERR
      REAL D(*),E(*),Z(NM,*)
      REAL B,C,C2,C3,DL1,EL1,F,G,H,P,R,S,S2
      REAL PYTHAG
C
C***FIRST EXECUTABLE STATEMENT  TQL2
      IERR = 0
      IF (N .EQ. 1) GO TO 1001
C
      DO 100 I = 2, N
  100 E(I-1) = E(I)
C
      F = 0.0E0
      B = 0.0E0
      E(N) = 0.0E0
C
      DO 240 L = 1, N
         J = 0
         H = ABS(D(L)) + ABS(E(L))
         IF (B .LT. H) B = H
C     .......... LOOK FOR SMALL SUB-DIAGONAL ELEMENT ..........
         DO 110 M = L, N
            IF (B + ABS(E(M)) .EQ. B) 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) / (2.0E0 * E(L))
         R = PYTHAG(P,1.0E0)
         D(L) = E(L) / (P + SIGN(R,P))
         D(L1) = E(L) * (P + SIGN(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 = 1.0E0
         C2 = C
         EL1 = E(L1)
         S = 0.0E0
         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
            IF (ABS(P) .LT. ABS(E(I))) GO TO 150
            C = E(I) / P
            R = SQRT(C*C+1.0E0)
            E(I+1) = S * P * R
            S = C / R
            C = 1.0E0 / R
            GO TO 160
  150       C = P / E(I)
            R = SQRT(C*C+1.0E0)
            E(I+1) = S * E(I) * R
            S = 1.0E0 / R
            C = C * S
  160       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
         IF (B + ABS(E(L)) .GT. B) 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