SSVDC(3F)

NAME
     SSVDC   - SSVDC is a subroutine to reduce a real NxP matrix X by
     orthogonal transformations U and V to diagonal form.  The diagonal
     elements S(I) are the singular values of X.  The columns of U are the
     corresponding left singular vectors, and the columns of V the right
     singular vectors.


SYNOPSYS
      SUBROUTINE SSVDC(X,LDX,N,P,S,E,U,LDU,V,LDV,WORK,JOB,INFO)

DESCRIPTION
     On Entry

     X REAL(LDX,P), where LDX .GE. N.
        X contains the matrix whose singular value
        decomposition is to be computed.  X is
        destroyed by SSVDC.

     LDX INTEGER
        LDX is the leading dimension of the array X.

     N INTEGER
        N is the number of columns of the matrix X.

     P INTEGER
        P is the number of rows of the matrix X.

     LDU INTEGER
        LDU is the leading dimension of the array U.
        (See below).

     LDV INTEGER
        LDV is the leading dimension of the array V.
        (See below).

     WORK REAL(N)
        work is a scratch array.

     JOB INTEGER
        JOB controls the computation of the singular
        vectors.  It has the decimal expansion AB
        with the following meaning
        A .EQ. 0  Do not compute the left singular
        vectors.
        A .EQ. 1  Return the N left singular vectors
        in U.
        A .GE. 2  Return the first MIN(N,P) singular
        vectors in U.
        B .EQ. 0  Do not compute the right singular
        vectors.
        B .EQ. 1  Return the right singular vectors
        in V.  On Return

     S REAL(MM), where MM=MIN(N+1,P).
        The first MIN(N,P) entries of S contain the
        singular values of X arranged in descending
        order of magnitude.

     E REAL(P).
        E ordinarily contains zeros.  However, see the
        discussion of INFO for exceptions.

     U REAL(LDU,K), where LDU .GE. N.  If JOBA .EQ. 1, then
        K .EQ. N.  If JOBA .GE. 2 , then
        K .EQ. MIN(N,P).
        U contains the matrix of right singular vectors.
        U is not referenced if JOBA .EQ. 0.  If N .LE. P
        or if JOBA .EQ. 2, then U may be identified with X
        in the subroutine call.

     V REAL(LDV,P), where LDV .GE. P.
        V contains the matrix of right singular vectors.
        V is not referenced if JOB .EQ. 0.  If P .LE. N,
        then V may be identified with X in the
        subroutine call.

     INFO INTEGER.
        the singular values (and their corresponding
        singular vectors) S(INFO+1),S(INFO+2),...,S(M)
        are correct (here M=MIN(N,P)).  Thus if
        INFO .EQ. 0, all the singular values and their
        vectors are correct.  In any event, the matrix
        B = TRANS(U)*X*V is the bidiagonal matrix
        with the elements of S on its diagonal and the
        elements of E on its super-diagonal (TRANS(U)
        is the transpose of U).  Thus the singular
        values of X and B are the same.  LINPACK.  This version dated 03/19/79
     .  G. W. Stewart, University of Maryland, Argonne National Lab.  External
     SROT BLAS SAXPY,SDOT,SSCAL,SSWAP,SNRM2,SROTG Fortran
     ABS,AMAX1,MAX0,MIN0,MOD,SQRT