imtql2.f
SUBROUTINE IMTQL2 (NM, N, D, E, Z, IERR)
C***BEGIN PROLOGUE IMTQL2
C***PURPOSE Compute the eigenvalues and eigenvectors of a symmetric
C tridiagonal matrix using the implicit QL method.
C***LIBRARY SLATEC (EISPACK)
C***CATEGORY D4A5, D4C2A
C***TYPE SINGLE PRECISION (IMTQL2-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 IMTQL2,
C NUM. MATH. 12, 377-383(1968) by Martin and Wilkinson,
C as modified in NUM. MATH. 15, 450(1970) by Dubrulle.
C HANDBOOK FOR AUTO. COMP., VOL.II-LINEAR ALGEBRA, 241-248(1971).
C
C This subroutine finds the eigenvalues and eigenvectors
C of a SYMMETRIC TRIDIAGONAL matrix by the implicit QL method.
C The eigenvectors of a FULL SYMMETRIC matrix can also
C be found if TRED2 has been used to reduce this
C full matrix to tridiagonal form.
C
C On INPUT
C
C NM must be set to the row dimension of the two-dimensional
C array parameter, Z, as declared in the calling program
C dimension statement. NM is an INTEGER variable.
C
C N is the order of the matrix. N is an INTEGER variable.
C N must be less than or equal to NM.
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 Z contains the transformation matrix produced in the reduction
C by TRED2, if performed. This transformation matrix is
C necessary if you want to obtain the eigenvectors of the full
C symmetric matrix. If the eigenvectors of the symmetric
C tridiagonal matrix are desired, Z must contain the identity
C matrix. Z is a two-dimensional REAL array, dimensioned
C Z(NM,N).
C
C On OUTPUT
C
C D contains the eigenvalues in ascending order. If an
C error exit is made, the eigenvalues are correct but
C unordered for indices 1, 2, ..., IERR-1.
C
C E has been destroyed.
C
C Z contains orthonormal eigenvectors of the full symmetric
C or symmetric tridiagonal matrix, depending on what it
C contained on input. If an error exit is made, Z contains
C the eigenvectors associated with the stored eigenvalues.
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 30 iterations.
C The eigenvalues and eigenvectors should be correct
C for indices 1, 2, ..., IERR-1, but the eigenvalues
C are not ordered.
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 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 IMTQL2