.TH "SRC/sgesvj.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
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.SH NAME
SRC/sgesvj.f
.SH SYNOPSIS
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.SS "Functions/Subroutines"

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.ti -1c
.RI "subroutine \fBsgesvj\fP (joba, jobu, jobv, m, n, a, lda, sva, mv, v, ldv, work, lwork, info)"
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.RI "\fBSGESVJ\fP "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "subroutine sgesvj (character*1 joba, character*1 jobu, character*1 jobv, integer m, integer n, real, dimension( lda, * ) a, integer lda, real, dimension( n ) sva, integer mv, real, dimension( ldv, * ) v, integer ldv, real, dimension( lwork ) work, integer lwork, integer info)"

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\fBSGESVJ\fP  
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\fBPurpose:\fP
.RS 4

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.nf
!>
!> SGESVJ computes the singular value decomposition (SVD) of a real
!> M-by-N matrix A, where M >= N\&. The SVD of A is written as
!>                                    [++]   [xx]   [x0]   [xx]
!>              A = U * SIGMA * V^t,  [++] = [xx] * [ox] * [xx]
!>                                    [++]   [xx]
!> where SIGMA is an N-by-N diagonal matrix, U is an M-by-N orthonormal
!> matrix, and V is an N-by-N orthogonal matrix\&. The diagonal elements
!> of SIGMA are the singular values of A\&. The columns of U and V are the
!> left and the right singular vectors of A, respectively\&.
!> SGESVJ can sometimes compute tiny singular values and their singular vectors much
!> more accurately than other SVD routines, see below under Further Details\&.
!> 
.fi
.PP
 
.RE
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\fBParameters\fP
.RS 4
\fIJOBA\fP 
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!>          JOBA is CHARACTER*1
!>          Specifies the structure of A\&.
!>          = 'L': The input matrix A is lower triangular;
!>          = 'U': The input matrix A is upper triangular;
!>          = 'G': The input matrix A is general M-by-N matrix, M >= N\&.
!> 
.fi
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.br
\fIJOBU\fP 
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!>          JOBU is CHARACTER*1
!>          Specifies whether to compute the left singular vectors
!>          (columns of U):
!>          = 'U': The left singular vectors corresponding to the nonzero
!>                 singular values are computed and returned in the leading
!>                 columns of A\&. See more details in the description of A\&.
!>                 The default numerical orthogonality threshold is set to
!>                 approximately TOL=CTOL*EPS, CTOL=SQRT(M), EPS=SLAMCH('E')\&.
!>          = 'C': Analogous to JOBU='U', except that user can control the
!>                 level of numerical orthogonality of the computed left
!>                 singular vectors\&. TOL can be set to TOL = CTOL*EPS, where
!>                 CTOL is given on input in the array WORK\&.
!>                 No CTOL smaller than ONE is allowed\&. CTOL greater
!>                 than 1 / EPS is meaningless\&. The option 'C'
!>                 can be used if M*EPS is satisfactory orthogonality
!>                 of the computed left singular vectors, so CTOL=M could
!>                 save few sweeps of Jacobi rotations\&.
!>                 See the descriptions of A and WORK(1)\&.
!>          = 'N': The matrix U is not computed\&. However, see the
!>                 description of A\&.
!> 
.fi
.PP
.br
\fIJOBV\fP 
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.nf
!>          JOBV is CHARACTER*1
!>          Specifies whether to compute the right singular vectors, that
!>          is, the matrix V:
!>          = 'V':  the matrix V is computed and returned in the array V
!>          = 'A':  the Jacobi rotations are applied to the MV-by-N
!>                  array V\&. In other words, the right singular vector
!>                  matrix V is not computed explicitly; instead it is
!>                  applied to an MV-by-N matrix initially stored in the
!>                  first MV rows of V\&.
!>          = 'N':  the matrix V is not computed and the array V is not
!>                  referenced
!> 
.fi
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\fIM\fP 
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!>          M is INTEGER
!>          The number of rows of the input matrix A\&. 1/SLAMCH('E') > M >= 0\&.
!> 
.fi
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\fIN\fP 
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!>          N is INTEGER
!>          The number of columns of the input matrix A\&.
!>          M >= N >= 0\&.
!> 
.fi
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\fIA\fP 
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!>          A is REAL array, dimension (LDA,N)
!>          On entry, the M-by-N matrix A\&.
!>          On exit,
!>          If JOBU = 'U' \&.OR\&. JOBU = 'C':
!>                 If INFO = 0:
!>                 RANKA orthonormal columns of U are returned in the
!>                 leading RANKA columns of the array A\&. Here RANKA <= N
!>                 is the number of computed singular values of A that are
!>                 above the underflow threshold SLAMCH('S')\&. The singular
!>                 vectors corresponding to underflowed or zero singular
!>                 values are not computed\&. The value of RANKA is returned
!>                 in the array WORK as RANKA=NINT(WORK(2))\&. Also see the
!>                 descriptions of SVA and WORK\&. The computed columns of U
!>                 are mutually numerically orthogonal up to approximately
!>                 TOL=SQRT(M)*EPS (default); or TOL=CTOL*EPS (JOBU = 'C'),
!>                 see the description of JOBU\&.
!>                 If INFO > 0,
!>                 the procedure SGESVJ did not converge in the given number
!>                 of iterations (sweeps)\&. In that case, the computed
!>                 columns of U may not be orthogonal up to TOL\&. The output
!>                 U (stored in A), SIGMA (given by the computed singular
!>                 values in SVA(1:N)) and V is still a decomposition of the
!>                 input matrix A in the sense that the residual
!>                 ||A-SCALE*U*SIGMA*V^T||_2 / ||A||_2 is small\&.
!>          If JOBU = 'N':
!>                 If INFO = 0:
!>                 Note that the left singular vectors are 'for free' in the
!>                 one-sided Jacobi SVD algorithm\&. However, if only the
!>                 singular values are needed, the level of numerical
!>                 orthogonality of U is not an issue and iterations are
!>                 stopped when the columns of the iterated matrix are
!>                 numerically orthogonal up to approximately M*EPS\&. Thus,
!>                 on exit, A contains the columns of U scaled with the
!>                 corresponding singular values\&.
!>                 If INFO > 0:
!>                 the procedure SGESVJ did not converge in the given number
!>                 of iterations (sweeps)\&.
!> 
.fi
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.br
\fILDA\fP 
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!>          LDA is INTEGER
!>          The leading dimension of the array A\&.  LDA >= max(1,M)\&.
!> 
.fi
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\fISVA\fP 
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!>          SVA is REAL array, dimension (N)
!>          On exit,
!>          If INFO = 0 :
!>          depending on the value SCALE = WORK(1), we have:
!>                 If SCALE = ONE:
!>                 SVA(1:N) contains the computed singular values of A\&.
!>                 During the computation SVA contains the Euclidean column
!>                 norms of the iterated matrices in the array A\&.
!>                 If SCALE \&.NE\&. ONE:
!>                 The singular values of A are SCALE*SVA(1:N), and this
!>                 factored representation is due to the fact that some of the
!>                 singular values of A might underflow or overflow\&.
!>
!>          If INFO > 0 :
!>          the procedure SGESVJ did not converge in the given number of
!>          iterations (sweeps) and SCALE*SVA(1:N) may not be accurate\&.
!> 
.fi
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\fIMV\fP 
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!>          MV is INTEGER
!>          If JOBV = 'A', then the product of Jacobi rotations in SGESVJ
!>          is applied to the first MV rows of V\&. See the description of JOBV\&.
!> 
.fi
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\fIV\fP 
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!>          V is REAL array, dimension (LDV,N)
!>          If JOBV = 'V', then V contains on exit the N-by-N matrix of
!>                         the right singular vectors;
!>          If JOBV = 'A', then V contains the product of the computed right
!>                         singular vector matrix and the initial matrix in
!>                         the array V\&.
!>          If JOBV = 'N', then V is not referenced\&.
!> 
.fi
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\fILDV\fP 
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!>          LDV is INTEGER
!>          The leading dimension of the array V, LDV >= 1\&.
!>          If JOBV = 'V', then LDV >= max(1,N)\&.
!>          If JOBV = 'A', then LDV >= max(1,MV) \&.
!> 
.fi
.PP
.br
\fIWORK\fP 
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!>          WORK is REAL array, dimension (MAX(1,LWORK))
!>          On entry,
!>          If JOBU = 'C' :
!>          WORK(1) = CTOL, where CTOL defines the threshold for convergence\&.
!>                    The process stops if all columns of A are mutually
!>                    orthogonal up to CTOL*EPS, EPS=SLAMCH('E')\&.
!>                    It is required that CTOL >= ONE, i\&.e\&. it is not
!>                    allowed to force the routine to obtain orthogonality
!>                    below EPSILON\&.
!>          On exit,
!>          WORK(1) = SCALE is the scaling factor such that SCALE*SVA(1:N)
!>                    are the computed singular vcalues of A\&.
!>                    (See description of SVA()\&.)
!>          WORK(2) = NINT(WORK(2)) is the number of the computed nonzero
!>                    singular values\&.
!>          WORK(3) = NINT(WORK(3)) is the number of the computed singular
!>                    values that are larger than the underflow threshold\&.
!>          WORK(4) = NINT(WORK(4)) is the number of sweeps of Jacobi
!>                    rotations needed for numerical convergence\&.
!>          WORK(5) = max_{i\&.NE\&.j} |COS(A(:,i),A(:,j))| in the last sweep\&.
!>                    This is useful information in cases when SGESVJ did
!>                    not converge, as it can be used to estimate whether
!>                    the output is still useful and for post festum analysis\&.
!>          WORK(6) = the largest absolute value over all sines of the
!>                    Jacobi rotation angles in the last sweep\&. It can be
!>                    useful for a post festum analysis\&.
!> 
.fi
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\fILWORK\fP 
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!>          LWORK is INTEGER
!>          Length of WORK\&.
!>          LWORK >= 1, if MIN(M,N) = 0, and LWORK >= MAX(6,M+N), otherwise\&.
!>
!>          If on entry LWORK = -1, then a workspace query is assumed and
!>          no computation is done; WORK(1) is set to the minial (and optimal)
!>          length of WORK\&.
!> 
.fi
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\fIINFO\fP 
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!>          INFO is INTEGER
!>          = 0:  successful exit\&.
!>          < 0:  if INFO = -i, then the i-th argument had an illegal value
!>          > 0:  SGESVJ did not converge in the maximal allowed number (30)
!>                of sweeps\&. The output may still be useful\&. See the
!>                description of WORK\&.
!> 
.fi
.PP
 
.RE
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\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

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Univ\&. of California Berkeley 

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Univ\&. of Colorado Denver 

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NAG Ltd\&. 
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\fBFurther Details:\fP
.RS 4
The orthogonal N-by-N matrix V is obtained as a product of Jacobi plane rotations\&. The rotations are implemented as fast scaled rotations of Anda and Park [1]\&. In the case of underflow of the Jacobi angle, a modified Jacobi transformation of Drmac [4] is used\&. Pivot strategy uses column interchanges of de Rijk [2]\&. The relative accuracy of the computed singular values and the accuracy of the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic [3]\&. The condition number that determines the accuracy in the full rank case is essentially min_{D=diag} kappa(A*D), where kappa(\&.) is the spectral condition number\&. The best performance of this Jacobi SVD procedure is achieved if used in an accelerated version of Drmac and Veselic [5,6], and it is the kernel routine in the SIGMA library [7]\&. Some tuning parameters (marked with [TP]) are available for the implementer\&. 
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The computational range for the nonzero singular values is the machine number interval ( UNDERFLOW , OVERFLOW )\&. In extreme cases, even denormalized singular values can be computed with the corresponding gradual loss of accurate digits\&.
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\fBContributors:\fP
.RS 4
Zlatko Drmac (Zagreb, Croatia) and Kresimir Veselic (Hagen, Germany)
.RE
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\fBReferences:\fP
.RS 4
[1] A\&. A\&. Anda and H\&. Park: Fast plane rotations with dynamic scaling\&. 
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 SIAM J\&. matrix Anal\&. Appl\&., Vol\&. 15 (1994), pp\&. 162-174\&. 
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[2] P\&. P\&. M\&. De Rijk: A one-sided Jacobi algorithm for computing the singular value decomposition on a vector computer\&. 
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 SIAM J\&. Sci\&. Stat\&. Comp\&., Vol\&. 10 (1998), pp\&. 359-371\&. 
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[3] J\&. Demmel and K\&. Veselic: Jacobi method is more accurate than QR\&. 
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[4] Z\&. Drmac: Implementation of Jacobi rotations for accurate singular value computation in floating point arithmetic\&. 
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 SIAM J\&. Sci\&. Comp\&., Vol\&. 18 (1997), pp\&. 1200-1222\&. 
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[5] Z\&. Drmac and K\&. Veselic: New fast and accurate Jacobi SVD algorithm I\&. 
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 SIAM J\&. Matrix Anal\&. Appl\&. Vol\&. 35, No\&. 2 (2008), pp\&. 1322-1342\&. 
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 LAPACK Working note 169\&. 
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[6] Z\&. Drmac and K\&. Veselic: New fast and accurate Jacobi SVD algorithm II\&. 
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 SIAM J\&. Matrix Anal\&. Appl\&. Vol\&. 35, No\&. 2 (2008), pp\&. 1343-1362\&. 
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 LAPACK Working note 170\&. 
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[7] Z\&. Drmac: SIGMA - mathematical software library for accurate SVD, PSV, QSVD, (H,K)-SVD computations\&.
.br
 Department of Mathematics, University of Zagreb, 2008\&.
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\fBBugs, Examples and Comments:\fP
.RS 4
Please report all bugs and send interesting test examples and comments to drmac@math.hr\&. Thank you\&. 
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Definition at line \fB326\fP of file \fBsgesvj\&.f\fP\&.
.SH "Author"
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Generated automatically by Doxygen for LAPACK from the source code\&.