SLATEC Routines --- MINFIT ---

*DECK MINFIT
      SUBROUTINE MINFIT (NM, M, N, A, W, IP, B, IERR, RV1)
C***BEGIN PROLOGUE  MINFIT
C***PURPOSE  Compute the singular value decomposition of a rectangular
C            matrix and solve the related linear least squares problem.
C***LIBRARY   SLATEC (EISPACK)
C***CATEGORY  D9
C***TYPE      SINGLE PRECISION (MINFIT-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 MINFIT,
C     NUM. MATH. 14, 403-420(1970) by Golub and Reinsch.
C     HANDBOOK FOR AUTO. COMP., VOL II-LINEAR ALGEBRA, 134-151(1971).
C
C     This subroutine determines, towards the solution of the linear
C                                                        T
C     system AX=B, the singular value decomposition A=USV  of a real
C                                         T
C     M by N rectangular matrix, forming U B rather than U.  Householder
C     bidiagonalization and a variant of the QR algorithm are used.
C
C     On INPUT
C
C        NM must be set to the row dimension of the two-dimensional
C          array parameters, A and B, as declared in the calling
C          program dimension statement.  Note that NM must be at least
C          as large as the maximum of M and N.  NM is an INTEGER
C          variable.
C
C        M is the number of rows of A and B.  M is an INTEGER variable.
C
C        N is the number of columns of A and the order of V.  N is an
C          INTEGER variable.
C
C        A contains the rectangular coefficient matrix of the system.
C          A is a two-dimensional REAL array, dimensioned A(NM,N).
C
C        IP is the number of columns of B.  IP can be zero.
C
C        B contains the constant column matrix of the system if IP is
C          not zero.  Otherwise, B is not referenced.  B is a two-
C          dimensional REAL array, dimensioned B(NM,IP).
C
C     On OUTPUT
C
C        A has been overwritten by the matrix V (orthogonal) of the
C          decomposition in its first N rows and columns.  If an
C          error exit is made, the columns of V corresponding to
C          indices of correct singular values should be correct.
C
C        W contains the N (non-negative) singular values of A (the
C          diagonal elements of S).  They are unordered.  If an
C          error exit is made, the singular values should be correct
C          for indices IERR+1, IERR+2, ..., N.  W is a one-dimensional
C          REAL array, dimensioned W(N).
C
C                                   T
C        B has been overwritten by U B.  If an error exit is made,
C                       T
C          the rows of U B corresponding to indices of correct singular
C          values should be correct.
C
C        IERR is an INTEGER flag set to
C          Zero       for normal return,
C          K          if the K-th singular value has not been
C                     determined after 30 iterations.
C                     The singular values should be correct for
C                     indices IERR+1, IERR+2, ..., N.
C
C        RV1 is a one-dimensional REAL array used for temporary storage,
C          dimensioned RV1(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  MINFIT