DSBTRD (3lapack)

reduce a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal similarity transformation

SYNOPSIS

  SUBROUTINE DSBTRD( VECT, UPLO, N, KD, AB, LDAB, D, E, Q, LDQ, WORK, INFO )

      CHARACTER      UPLO, VECT

      INTEGER        INFO, KD, LDAB, LDQ, N

      DOUBLE         PRECISION AB( LDAB, * ), D( * ), E( * ), Q( LDQ, * ),
                     WORK( * )

PURPOSE

  DSBTRD reduces a real symmetric band matrix A to symmetric tridiagonal form
  T by an orthogonal similarity transformation: Q**T * A * Q = T.

ARGUMENTS

  VECT    (input) CHARACTER*1
          = 'N':  do not form Q;
          = 'V':  form Q;
          = 'U':  update a matrix X, by forming X*Q.

  UPLO    (input) CHARACTER*1
          = 'U':  Upper triangle of A is stored;
          = 'L':  Lower triangle of A is stored.

  N       (input) INTEGER
          The order of the matrix A.  N >= 0.

  KD      (input) INTEGER
          The number of superdiagonals of the matrix A if UPLO = 'U', or the
          number of subdiagonals if UPLO = 'L'.  KD >= 0.

  AB      (input/output) DOUBLE PRECISION array, dimension (LDAB,N)
          On entry, the upper or lower triangle of the symmetric band matrix
          A, stored in the first KD+1 rows of the array.  The j-th column of
          A is stored in the j-th column of the array AB as follows: if UPLO
          = 'U', AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO =
          'L', AB(1+i-j,j)    = A(i,j) for j<=i<=min(n,j+kd).  On exit, the
          diagonal elements of AB are overwritten by the diagonal elements of
          the tridiagonal matrix T; if KD > 0, the elements on the first
          superdiagonal (if UPLO = 'U') or the first subdiagonal (if UPLO =
          'L') are overwritten by the off-diagonal elements of T; the rest of
          AB is overwritten by values generated during the reduction.

  LDAB    (input) INTEGER
          The leading dimension of the array AB.  LDAB >= KD+1.

  D       (output) DOUBLE PRECISION array, dimension (N)
          The diagonal elements of the tridiagonal matrix T.

  E       (output) DOUBLE PRECISION array, dimension (N-1)
          The off-diagonal elements of the tridiagonal matrix T: E(i) =
          T(i,i+1) if UPLO = 'U'; E(i) = T(i+1,i) if UPLO = 'L'.

  Q       (input/output) DOUBLE PRECISION array, dimension (LDQ,N)
          On entry, if VECT = 'U', then Q must contain an N-by-N matrix X; if
          VECT = 'N' or 'V', then Q need not be set.

          On exit: if VECT = 'V', Q contains the N-by-N orthogonal matrix Q;
          if VECT = 'U', Q contains the product X*Q; if VECT = 'N', the array
          Q is not referenced.

  LDQ     (input) INTEGER
          The leading dimension of the array Q.  LDQ >= 1, and LDQ >= N if
          VECT = 'V' or 'U'.

  WORK    (workspace) DOUBLE PRECISION array, dimension (N)

  INFO    (output) INTEGER
          = 0:  successful exit
          < 0:  if INFO = -i, the i-th argument had an illegal value