_BQR(3F)							      _BQR(3F)

NAME    [Toc]    [Back]

     BQR, SBQR	   -  EISPACK routine.	This subroutine	finds the eigenvalue
     of	smallest (usually) magnitude of	a REAL SYMMETRIC BAND matrix using the
     QR	algorithm with shifts of origin.  Consecutive calls can	be made	to
     find further eigenvalues.

SYNOPSYS    [Toc]    [Back]

	  subroutine  bqr(nm, n, mb, a,	t, r, ierr, nv,	rv)
	  integer	   nm, n, mb, ierr, nv
	  double precision t, r
	  double precision a(nm,mb), rv(nv)

	  subroutine sbqr(nm, n, mb, a,	t, r, ierr, nv,	rv)
	  integer	   nm, n, mb, ierr, nv
	  real		   t, r
	  real		   a(nm,mb), rv(nv)

DESCRIPTION    [Toc]    [Back]

     On	INPUT

     NM	must be	set to the row dimension of two-dimensional array parameters
     as	declared in the	calling	program	dimension statement.

     N is the order of the matrix.

     MB	is the (half) band width of the	matrix,	defined	as the number of
     adjacent diagonals, including the principal diagonal, required to specify
     the non-zero portion of the lower triangle	of the matrix.

     A contains	the lower triangle of the symmetric band input matrix stored
     as	an N by	MB array.  Its lowest subdiagonal is stored in the last	N+1-MB
     positions of the first column, its	next subdiagonal in the	last N+2-MB
     positions of the second column, further subdiagonals similarly, and
     finally its principal diagonal in the N positions of the last column.
     Contents of storages not part of the matrix are arbitrary.	 On a
     subsequent	call, its output contents from the previous call should	be
     passed.

     T specifies the shift (of eigenvalues) applied to the diagonal of A in
     forming the input matrix. What is actually	determined is the eigenvalue
     of	A+TI (I	is the identity	matrix)	nearest	to T.  On a subsequent call,
     the output	value of T from	the previous call should be passed if the next
     nearest eigenvalue	is sought.

     R should be specified as zero on the first	call, and as its output	value
     from the previous call on a subsequent call.  It is used to determine
     when the last row and column of the transformed band matrix can be
     regarded as negligible.




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_BQR(3F)							      _BQR(3F)



     NV	must be	set to the dimension of	the array parameter RV as declared in
     the calling program dimension statement.  On OUTPUT

     A contains	the transformed	band matrix.  The matrix A+TI derived from the
     output parameters is similar to the input A+TI to within rounding errors.
     Its last row and column are null (if IERR is zero).

     T contains	the computed eigenvalue	of A+TI	(if IERR is zero).

     R contains	the maximum of its input value and the norm of the last	column
     of	the input matrix A.

     IERR is set to Zero       for normal return, N	     if	the eigenvalue
     has not been
	determined after 30 iterations.

     RV	is a temporary storage array of	dimension at least (2*MB**2+4*MB-3).
     The first (3*MB-2)	locations correspond to	the ALGOL array	B, the next
     (2*MB-1) locations	correspond to the ALGOL	array H, and the final
     (2*MB**2-MB) locations correspond to the MB by (2*MB-1) ALGOL array U.
     NOTE. For a subsequent call, N should be replaced by N-1, but

     MB	should not be altered even when	it exceeds the current N. Calls
     PYTHAG(A,B) for SQRT(A**2 + B**2).	 Questions and comments	should be
     directed to B. S. Garbow, Applied Mathematics Division, ARGONNE NATIONAL
     LABORATORY


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