_RATQR(3F) _RATQR(3F)
RATQR, SRATQR - EISPACK routine. This subroutine finds the
algebraically smallest or largest eigenvalues of a SYMMETRIC TRIDIAGONAL
matrix by the rational QR method with Newton corrections.
subroutine ratqr(n,eps1,d,e,e2,m,w,ind,bd,type,idef,ierr)
integer n, m, ind(n), idef, ierr
double precision eps1
double precision d(n), e(n), e2(n), w(n), bd(n)
logical type
subroutine sratqr(n,eps1,d,e,e2,m,w,ind,bd,type,idef,ierr)
integer n, m, ind(n), idef, ierr
real eps1
real d(n), e(n), e2(n), w(n), bd(n)
logical type
On Input
N is the order of the matrix.
EPS1 is a theoretical absolute error tolerance for the computed
eigenvalues. If the input EPS1 is non-positive, or indeed smaller than
its default value, it is reset at each iteration to the respective
default value, namely, the product of the relative machine precision and
the magnitude of the current eigenvalue iterate. The theoretical
absolute error in the K-th eigenvalue is usually not greater than K times
EPS1.
D contains the diagonal elements of the input matrix.
E contains the subdiagonal elements of the input matrix in its last N-1
positions. E(1) is arbitrary.
E2 contains the squares of the corresponding elements of E. E2(1) is
arbitrary.
M is the number of eigenvalues to be found.
IDEF should be set to 1 if the input matrix is known to be positive
definite, to -1 if the input matrix is known to be negative definite, and
to 0 otherwise.
TYPE should be set to .TRUE. if the smallest eigenvalues are to be found,
and to .FALSE. If the largest eigenvalues are to be found. On Output
EPS1 is unaltered unless it has been reset to its (last) default value.
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_RATQR(3F) _RATQR(3F)
D and E are unaltered (unless W overwrites D). ELEMENTS of E2,
corresponding to elements of E regarded as negligible, have been replaced
by zero causing the matrix to split into a direct sum of submatrices.
E2(1) is set to 0.0e0 if the smallest eigenvalues have been found, and to
2.0e0 if the largest eigenvalues have been found. E2 is otherwise
unaltered (unless overwritten by BD).
W contains the M algebraically smallest eigenvalues in ascending order,
or the M largest eigenvalues in descending order. If an error exit is
made because of an incorrect specification of IDEF, no eigenvalues are
found. If the Newton iterates for a particular eigenvalue are not
monotone, the best estimate obtained is returned and IERR is set. W may
coincide with D.
IND contains in its first M positions the submatrix indices associated
with the corresponding eigenvalues in W -- 1 for eigenvalues belonging to
the first submatrix from the top, 2 for those belonging to the second
submatrix, etc.
BD contains refined bounds for the theoretical errors of the
corresponding eigenvalues in W. These bounds are usually within the
tolerance specified by EPS1. BD may coincide with E2.
IERR is set to Zero for normal return, 6*N+1 if IDEF is set
to 1 and type to .TRUE.
when the matrix is NOT positive definite, or
if IDEF is set to -1 and type to .FALSE.
when the matrix is NOT negative definite, 5*N+K if successive
iterates to the K-th eigenvalue
are NOT monotone increasing, where K refers
to the last such occurrence. Note that subroutine TRIDIB is generally
faster and more accurate than RATQR if the eigenvalues are clustered.
Questions and comments should be directed to B. S. Garbow, APPLIED
MATHEMATICS DIVISION, ARGONNE NATIONAL LABORATORY
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