ratqr.f
SUBROUTINE RATQR (N, EPS1, D, E, E2, M, W, IND, BD, TYPE, IDEF,
+ IERR)
C***BEGIN PROLOGUE RATQR
C***PURPOSE Compute the largest or smallest eigenvalues of a symmetric
C tridiagonal matrix using the rational QR method with Newton
C correction.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (RATQR-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 RATQR,
C NUM. MATH. 11, 264-272(1968) by REINSCH and BAUER.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 257-265(1971).
C
C This subroutine finds the algebraically smallest or largest
C eigenvalues of a SYMMETRIC TRIDIAGONAL matrix by the
C rational QR method with Newton corrections.
C
C On Input
C
C N is the order of the matrix. N is an INTEGER variable.
C
C EPS1 is a theoretical absolute error tolerance for the
C computed eigenvalues. If the input EPS1 is non-positive, or
C indeed smaller than its default value, it is reset at each
C iteration to the respective default value, namely, the
C product of the relative machine precision and the magnitude
C of the current eigenvalue iterate. The theoretical absolute
C error in the K-th eigenvalue is usually not greater than
C K times EPS1. EPS1 is a REAL variable.
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 E2 contains the squares of the corresponding elements of E in
C its last N-1 positions. E2(1) is arbitrary. E2 is a one-
C dimensional REAL array, dimensioned E2(N).
C
C M is the number of eigenvalues to be found. M is an INTEGER
C variable.
C
C IDEF should be set to 1 if the input matrix is known to be
C positive definite, to -1 if the input matrix is known to
C be negative definite, and to 0 otherwise. IDEF is an
C INTEGER variable.
C
C TYPE should be set to .TRUE. if the smallest eigenvalues are
C to be found, and to .FALSE. if the largest eigenvalues are
C to be found. TYPE is a LOGICAL variable.
C
C On Output
C
C EPS1 is unaltered unless it has been reset to its
C (last) default value.
C
C D and E are unaltered (unless W overwrites D).
C
C Elements of E2, corresponding to elements of E regarded as
C negligible, have been replaced by zero causing the matrix
C to split into a direct sum of submatrices. E2(1) is set
C to 0.0e0 if the smallest eigenvalues have been found, and
C to 2.0e0 if the largest eigenvalues have been found. E2
C is otherwise unaltered (unless overwritten by BD).
C
C W contains the M algebraically smallest eigenvalues in
C ascending order, or the M largest eigenvalues in descending
C order. If an error exit is made because of an incorrect
C specification of IDEF, no eigenvalues are found. If the
C Newton iterates for a particular eigenvalue are not monotone,
C the best estimate obtained is returned and IERR is set.
C W is a one-dimensional REAL array, dimensioned W(N). W need
C not be distinct from D.
C
C IND contains in its first M positions the submatrix indices
C associated with the corresponding eigenvalues in W --
C 1 for eigenvalues belonging to the first submatrix from
C the top, 2 for those belonging to the second submatrix, etc.
C IND is an one-dimensional INTEGER array, dimensioned IND(N).
C
C BD contains refined bounds for the theoretical errors of the
C corresponding eigenvalues in W. These bounds are usually
C within the tolerance specified by EPS1. BD is a one-
C dimensional REAL array, dimensioned BD(N). BD need not be
C distinct from E2.
C
C IERR is an INTEGER flag set to
C Zero for normal return,
C 6*N+1 if IDEF is set to 1 and TYPE to .TRUE.
C when the matrix is NOT positive definite, or
C if IDEF is set to -1 and TYPE to .FALSE.
C when the matrix is NOT negative definite,
C no eigenvalues are computed, or
C M is greater than N,
C 5*N+K if successive iterates to the K-th eigenvalue
C are NOT monotone increasing, where K refers
C to the last such occurrence.
C
C Note that subroutine TRIDIB is generally faster and more
C accurate than RATQR if the eigenvalues are clustered.
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 R1MACH
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 RATQR