# bandr.f

          SUBROUTINE BANDR (NM, N, MB, A, D, E, E2, MATZ, Z)
    C***BEGIN PROLOGUE  BANDR
    C***PURPOSE  Reduce a real symmetric band matrix to symmetric
    C            tridiagonal matrix and, optionally, accumulate
    C            orthogonal similarity transformations.
    C***LIBRARY   SLATEC (EISPACK)
    C***CATEGORY  D4C1B1
    C***TYPE      SINGLE PRECISION (BANDR-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 BANDRD,
    C     NUM. MATH. 12, 231-241(1968) by Schwarz.
    C     HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 273-283(1971).
    C
    C     This subroutine reduces a REAL SYMMETRIC BAND matrix
    C     to a symmetric tridiagonal matrix using and optionally
    C     accumulating orthogonal similarity transformations.
    C
    C     On INPUT
    C
    C        NM must be set to the row dimension of the two-dimensional
    C          array parameters, A and Z, as declared in the calling
    C          program dimension statement.  NM is an INTEGER variable.
    C
    C        N is the order of the matrix A.  N is an INTEGER variable.
    C          N must be less than or equal to NM.
    C
    C        MB is the (half) band width of the matrix, defined as the
    C          number of adjacent diagonals, including the principal
    C          diagonal, required to specify the non-zero portion of the
    C          lower triangle of the matrix.  MB is less than or equal
    C          to N.  MB is an INTEGER variable.
    C
    C        A contains the lower triangle of the real symmetric band
    C          matrix.  Its lowest subdiagonal is stored in the last
    C          N+1-MB  positions of the first column, its next subdiagonal
    C          in the last  N+2-MB  positions of the second column, further
    C          subdiagonals similarly, and finally its principal diagonal
    C          in the  N  positions of the last column.  Contents of storage
    C          locations not part of the matrix are arbitrary.  A is a
    C          two-dimensional REAL array, dimensioned A(NM,MB).
    C
    C        MATZ should be set to .TRUE. if the transformation matrix is
    C          to be accumulated, and to .FALSE. otherwise.  MATZ is a
    C          LOGICAL variable.
    C
    C     On OUTPUT
    C
    C        A has been destroyed, except for its last two columns which
    C          contain a copy of the tridiagonal matrix.
    C
    C        D contains the diagonal elements of the tridiagonal matrix.
    C          D is a one-dimensional REAL array, dimensioned D(N).
    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          E is a one-dimensional REAL array, dimensioned E(N).
    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          E2 is a one-dimensional REAL array, dimensioned E2(N).
    C
    C        Z contains the orthogonal transformation matrix produced in
    C          the reduction if MATZ has been set to .TRUE.  Otherwise, Z
    C          is not referenced.  Z is a two-dimensional REAL array,
    C          dimensioned Z(NM,N).
    C
    C     Questions and comments should be directed to B. S. Garbow,
    C     Applied Mathematics Division, ARGONNE NATIONAL LABORATORY
    C     ------------------------------------------------------------------
    C
    C***REFERENCES  B. T. Smith, J. M. Boyle, J. J. Dongarra, B. S. Garbow,
    C                 Y. Ikebe, V. C. Klema and C. B. Moler, Matrix Eigen-
    C                 system Routines - EISPACK Guide, Springer-Verlag,
    C                 1976.
    C***ROUTINES CALLED  (NONE)
    C***REVISION HISTORY  (YYMMDD)
    C   760101  DATE WRITTEN
    C   890531  Changed all specific intrinsics to generic.  (WRB)
    C   890831  Modified array declarations.  (WRB)
    C   890831  REVISION DATE from Version 3.2
    C   891214  Prologue converted to Version 4.0 format.  (BAB)
    C   920501  Reformatted the REFERENCES section.  (WRB)
    C***END PROLOGUE  BANDR