SLATEC Routines --- BQR ---
*DECK BQR
SUBROUTINE BQR (NM, N, MB, A, T, R, IERR, NV, RV)
C***BEGIN PROLOGUE BQR
C***PURPOSE Compute some of the eigenvalues of a real symmetric
C matrix using the QR method with shifts of origin.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A6
C***TYPE SINGLE PRECISION (BQR-S)
C***KEYWORDS EIGENVALUES, EISPACK
C***AUTHOR Smith, B. T., et al.
C***DESCRIPTION
C
C This subroutine is a translation of the ALGOL procedure BQR,
C NUM. MATH. 16, 85-92(1970) by Martin, Reinsch, and Wilkinson.
C HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 266-272(1971).
C
C This subroutine finds the eigenvalue of smallest (usually)
C magnitude of a REAL SYMMETRIC BAND matrix using the
C QR algorithm with shifts of origin. Consecutive calls
C can be made to find further eigenvalues.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, A, as declared in the calling program
C 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 an INTEGER variable.
C MB must be less than or equal to N on first call.
C
C A contains the lower triangle of the symmetric band input
C matrix stored as an N by MB array. Its lowest subdiagonal
C is stored in the last N+1-MB positions of the first column,
C its next subdiagonal in the last N+2-MB positions of the
C second column, further subdiagonals similarly, and finally
C its principal diagonal in the N positions of the last column.
C Contents of storages not part of the matrix are arbitrary.
C On a subsequent call, its output contents from the previous
C call should be passed. A is a two-dimensional REAL array,
C dimensioned A(NM,MB).
C
C T specifies the shift (of eigenvalues) applied to the diagonal
C of A in forming the input matrix. What is actually determined
C is the eigenvalue of A+TI (I is the identity matrix) nearest
C to T. On a subsequent call, the output value of T from the
C previous call should be passed if the next nearest eigenvalue
C is sought. T is a REAL variable.
C
C R should be specified as zero on the first call, and as its
C output value from the previous call on a subsequent call.
C It is used to determine when the last row and column of
C the transformed band matrix can be regarded as negligible.
C R is a REAL variable.
C
C NV must be set to the dimension of the array parameter RV
C as declared in the calling program dimension statement.
C NV is an INTEGER variable.
C
C On OUTPUT
C
C A contains the transformed band matrix. The matrix A+TI
C derived from the output parameters is similar to the
C input A+TI to within rounding errors. Its last row and
C column are null (if IERR is zero).
C
C T contains the computed eigenvalue of A+TI (if IERR is zero),
C where I is the identity matrix.
C
C R contains the maximum of its input value and the norm of the
C last column of the input matrix A.
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 a total of 30 iterations.
C
C RV is a one-dimensional REAL array of dimension NV which is
C at least (2*MB**2+4*MB-3), used for temporary storage. The
C first (3*MB-2) locations correspond to the ALGOL array B,
C the next (2*MB-1) locations correspond to the ALGOL array H,
C and the final (2*MB**2-MB) locations correspond to the MB
C by (2*MB-1) ALGOL array U.
C
C NOTE. For a subsequent call, N should be replaced by N-1, but
C MB should not be altered even when it exceeds the current N.
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 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 BQR