SLATEC Routines --- SSVDC ---

*DECK SSVDC
      SUBROUTINE SSVDC (X, LDX, N, P, S, E, U, LDU, V, LDV, WORK, JOB,
     +   INFO)
C***BEGIN PROLOGUE  SSVDC
C***PURPOSE  Perform the singular value decomposition of a rectangular
C            matrix.
C***LIBRARY   SLATEC (LINPACK)
C***CATEGORY  D6
C***TYPE      SINGLE PRECISION (SSVDC-S, DSVDC-D, CSVDC-C)
C***KEYWORDS  LINEAR ALGEBRA, LINPACK, MATRIX,
C             SINGULAR VALUE DECOMPOSITION
C***AUTHOR  Stewart, G. W., (U. of Maryland)
C***DESCRIPTION
C
C     SSVDC is a subroutine to reduce a real NxP matrix X by orthogonal
C     transformations U and V to diagonal form.  The elements S(I) are
C     the singular values of X.  The columns of U are the corresponding
C     left singular vectors, and the columns of V the right singular
C     vectors.
C
C     On Entry
C
C         X         REAL(LDX,P), where LDX .GE. N.
C                   X contains the matrix whose singular value
C                   decomposition is to be computed.  X is
C                   destroyed by SSVDC.
C
C         LDX       INTEGER
C                   LDX is the leading dimension of the array X.
C
C         N         INTEGER
C                   N is the number of rows of the matrix X.
C
C         P         INTEGER
C                   P is the number of columns of the matrix X.
C
C         LDU       INTEGER
C                   LDU is the leading dimension of the array U.
C                   (See below).
C
C         LDV       INTEGER
C                   LDV is the leading dimension of the array V.
C                   (See below).
C
C         WORK      REAL(N)
C                   work is a scratch array.
C
C         JOB       INTEGER
C                   JOB controls the computation of the singular
C                   vectors.  It has the decimal expansion AB
C                   with the following meaning
C
C                        A .EQ. 0  Do not compute the left singular
C                                  vectors.
C                        A .EQ. 1  Return the N left singular vectors
C                                  in U.
C                        A .GE. 2  Return the first MIN(N,P) singular
C                                  vectors in U.
C                        B .EQ. 0  Do not compute the right singular
C                                  vectors.
C                        B .EQ. 1  Return the right singular vectors
C                                  in V.
C
C     On Return
C
C         S         REAL(MM), where MM=MIN(N+1,P).
C                   The first MIN(N,P) entries of S contain the
C                   singular values of X arranged in descending
C                   order of magnitude.
C
C         E         REAL(P).
C                   E ordinarily contains zeros.  However, see the
C                   discussion of INFO for exceptions.
C
C         U         REAL(LDU,K), where LDU .GE. N.  If JOBA .EQ. 1, then
C                                   K .EQ. N.  If JOBA .GE. 2 , then
C                                   K .EQ. MIN(N,P).
C                   U contains the matrix of right singular vectors.
C                   U is not referenced if JOBA .EQ. 0.  If N .LE. P
C                   or if JOBA .EQ. 2, then U may be identified with X
C                   in the subroutine call.
C
C         V         REAL(LDV,P), where LDV .GE. P.
C                   V contains the matrix of right singular vectors.
C                   V is not referenced if JOB .EQ. 0.  If P .LE. N,
C                   then V may be identified with X in the
C                   subroutine call.
C
C         INFO      INTEGER.
C                   the singular values (and their corresponding
C                   singular vectors) S(INFO+1),S(INFO+2),...,S(M)
C                   are correct (here M=MIN(N,P)).  Thus if
C                   INFO .EQ. 0, all the singular values and their
C                   vectors are correct.  In any event, the matrix
C                   B = TRANS(U)*X*V is the bidiagonal matrix
C                   with the elements of S on its diagonal and the
C                   elements of E on its super-diagonal (TRANS(U)
C                   is the transpose of U).  Thus the singular
C                   values of X and B are the same.
C
C***REFERENCES  J. J. Dongarra, J. R. Bunch, C. B. Moler, and G. W.
C                 Stewart, LINPACK Users' Guide, SIAM, 1979.
C***ROUTINES CALLED  SAXPY, SDOT, SNRM2, SROT, SROTG, SSCAL, SSWAP
C***REVISION HISTORY  (YYMMDD)
C   790319  DATE WRITTEN
C   890531  Changed all specific intrinsics to generic.  (WRB)
C   890531  REVISION DATE from Version 3.2
C   891214  Prologue converted to Version 4.0 format.  (BAB)
C   900326  Removed duplicate information from DESCRIPTION section.
C           (WRB)
C   920501  Reformatted the REFERENCES section.  (WRB)
C***END PROLOGUE  SSVDC