cgelsx.f.html

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      SUBROUTINE <a name="CGELSX.1"></a><a href="cgelsx.f.html#CGELSX.1">CGELSX</a>( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
     $                   WORK, RWORK, INFO )
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  -- LAPACK driver routine (version 3.1) --
</span><span class="comment">*</span><span class="comment">     Univ. of Tennessee, Univ. of California Berkeley and NAG Ltd..
</span><span class="comment">*</span><span class="comment">     November 2006
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Scalar Arguments ..
</span>      INTEGER            INFO, LDA, LDB, M, N, NRHS, RANK
      REAL               RCOND
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Array Arguments ..
</span>      INTEGER            JPVT( * )
      REAL               RWORK( * )
      COMPLEX            A( LDA, * ), B( LDB, * ), WORK( * )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Purpose
</span><span class="comment">*</span><span class="comment">  =======
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  This routine is deprecated and has been replaced by routine <a name="CGELSY.21"></a><a href="cgelsy.f.html#CGELSY.1">CGELSY</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  <a name="CGELSX.23"></a><a href="cgelsx.f.html#CGELSX.1">CGELSX</a> computes the minimum-norm solution to a complex linear least
</span><span class="comment">*</span><span class="comment">  squares problem:
</span><span class="comment">*</span><span class="comment">      minimize || A * X - B ||
</span><span class="comment">*</span><span class="comment">  using a complete orthogonal factorization of A.  A is an M-by-N
</span><span class="comment">*</span><span class="comment">  matrix which may be rank-deficient.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Several right hand side vectors b and solution vectors x can be
</span><span class="comment">*</span><span class="comment">  handled in a single call; they are stored as the columns of the
</span><span class="comment">*</span><span class="comment">  M-by-NRHS right hand side matrix B and the N-by-NRHS solution
</span><span class="comment">*</span><span class="comment">  matrix X.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  The routine first computes a QR factorization with column pivoting:
</span><span class="comment">*</span><span class="comment">      A * P = Q * [ R11 R12 ]
</span><span class="comment">*</span><span class="comment">                  [  0  R22 ]
</span><span class="comment">*</span><span class="comment">  with R11 defined as the largest leading submatrix whose estimated
</span><span class="comment">*</span><span class="comment">  condition number is less than 1/RCOND.  The order of R11, RANK,
</span><span class="comment">*</span><span class="comment">  is the effective rank of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Then, R22 is considered to be negligible, and R12 is annihilated
</span><span class="comment">*</span><span class="comment">  by unitary transformations from the right, arriving at the
</span><span class="comment">*</span><span class="comment">  complete orthogonal factorization:
</span><span class="comment">*</span><span class="comment">     A * P = Q * [ T11 0 ] * Z
</span><span class="comment">*</span><span class="comment">                 [  0  0 ]
</span><span class="comment">*</span><span class="comment">  The minimum-norm solution is then
</span><span class="comment">*</span><span class="comment">     X = P * Z' [ inv(T11)*Q1'*B ]
</span><span class="comment">*</span><span class="comment">                [        0       ]
</span><span class="comment">*</span><span class="comment">  where Q1 consists of the first RANK columns of Q.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  Arguments
</span><span class="comment">*</span><span class="comment">  =========
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  M       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of rows of the matrix A.  M &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  N       (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of columns of the matrix A.  N &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  NRHS    (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The number of right hand sides, i.e., the number of
</span><span class="comment">*</span><span class="comment">          columns of matrices B and X. NRHS &gt;= 0.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  A       (input/output) COMPLEX array, dimension (LDA,N)
</span><span class="comment">*</span><span class="comment">          On entry, the M-by-N matrix A.
</span><span class="comment">*</span><span class="comment">          On exit, A has been overwritten by details of its
</span><span class="comment">*</span><span class="comment">          complete orthogonal factorization.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDA     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array A.  LDA &gt;= max(1,M).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  B       (input/output) COMPLEX array, dimension (LDB,NRHS)
</span><span class="comment">*</span><span class="comment">          On entry, the M-by-NRHS right hand side matrix B.
</span><span class="comment">*</span><span class="comment">          On exit, the N-by-NRHS solution matrix X.
</span><span class="comment">*</span><span class="comment">          If m &gt;= n and RANK = n, the residual sum-of-squares for
</span><span class="comment">*</span><span class="comment">          the solution in the i-th column is given by the sum of
</span><span class="comment">*</span><span class="comment">          squares of elements N+1:M in that column.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  LDB     (input) INTEGER
</span><span class="comment">*</span><span class="comment">          The leading dimension of the array B. LDB &gt;= max(1,M,N).
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  JPVT    (input/output) INTEGER array, dimension (N)
</span><span class="comment">*</span><span class="comment">          On entry, if JPVT(i) .ne. 0, the i-th column of A is an
</span><span class="comment">*</span><span class="comment">          initial column, otherwise it is a free column.  Before
</span><span class="comment">*</span><span class="comment">          the QR factorization of A, all initial columns are
</span><span class="comment">*</span><span class="comment">          permuted to the leading positions; only the remaining
</span><span class="comment">*</span><span class="comment">          free columns are moved as a result of column pivoting
</span><span class="comment">*</span><span class="comment">          during the factorization.
</span><span class="comment">*</span><span class="comment">          On exit, if JPVT(i) = k, then the i-th column of A*P
</span><span class="comment">*</span><span class="comment">          was the k-th column of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RCOND   (input) REAL
</span><span class="comment">*</span><span class="comment">          RCOND is used to determine the effective rank of A, which
</span><span class="comment">*</span><span class="comment">          is defined as the order of the largest leading triangular
</span><span class="comment">*</span><span class="comment">          submatrix R11 in the QR factorization with pivoting of A,
</span><span class="comment">*</span><span class="comment">          whose estimated condition number &lt; 1/RCOND.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RANK    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          The effective rank of A, i.e., the order of the submatrix
</span><span class="comment">*</span><span class="comment">          R11.  This is the same as the order of the submatrix T11
</span><span class="comment">*</span><span class="comment">          in the complete orthogonal factorization of A.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  WORK    (workspace) COMPLEX array, dimension
</span><span class="comment">*</span><span class="comment">                      (min(M,N) + max( N, 2*min(M,N)+NRHS )),
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  RWORK   (workspace) REAL array, dimension (2*N)
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  INFO    (output) INTEGER
</span><span class="comment">*</span><span class="comment">          = 0:  successful exit
</span><span class="comment">*</span><span class="comment">          &lt; 0:  if INFO = -i, the i-th argument had an illegal value
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">  =====================================================================
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     .. Parameters ..
</span>      INTEGER            IMAX, IMIN
      PARAMETER          ( IMAX = 1, IMIN = 2 )
      REAL               ZERO, ONE, DONE, NTDONE
      PARAMETER          ( ZERO = 0.0E+0, ONE = 1.0E+0, DONE = ZERO,
     $                   NTDONE = ONE )
      COMPLEX            CZERO, CONE
      PARAMETER          ( CZERO = ( 0.0E+0, 0.0E+0 ),
     $                   CONE = ( 1.0E+0, 0.0E+0 ) )
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Local Scalars ..
</span>      INTEGER            I, IASCL, IBSCL, ISMAX, ISMIN, J, K, MN
      REAL               ANRM, BIGNUM, BNRM, SMAX, SMAXPR, SMIN, SMINPR,
     $                   SMLNUM
      COMPLEX            C1, C2, S1, S2, T1, T2
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Subroutines ..
</span>      EXTERNAL           <a name="CGEQPF.131"></a><a href="cgeqpf.f.html#CGEQPF.1">CGEQPF</a>, <a name="CLAIC1.131"></a><a href="claic1.f.html#CLAIC1.1">CLAIC1</a>, <a name="CLASCL.131"></a><a href="clascl.f.html#CLASCL.1">CLASCL</a>, <a name="CLASET.131"></a><a href="claset.f.html#CLASET.1">CLASET</a>, <a name="CLATZM.131"></a><a href="clatzm.f.html#CLATZM.1">CLATZM</a>, CTRSM,
     $                   <a name="CTZRQF.132"></a><a href="ctzrqf.f.html#CTZRQF.1">CTZRQF</a>, <a name="CUNM2R.132"></a><a href="cunm2r.f.html#CUNM2R.1">CUNM2R</a>, <a name="SLABAD.132"></a><a href="slabad.f.html#SLABAD.1">SLABAD</a>, <a name="XERBLA.132"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. External Functions ..
</span>      REAL               <a name="CLANGE.135"></a><a href="clange.f.html#CLANGE.1">CLANGE</a>, <a name="SLAMCH.135"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
      EXTERNAL           <a name="CLANGE.136"></a><a href="clange.f.html#CLANGE.1">CLANGE</a>, <a name="SLAMCH.136"></a><a href="slamch.f.html#SLAMCH.1">SLAMCH</a>
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Intrinsic Functions ..
</span>      INTRINSIC          ABS, CONJG, MAX, MIN
<span class="comment">*</span><span class="comment">     ..
</span><span class="comment">*</span><span class="comment">     .. Executable Statements ..
</span><span class="comment">*</span><span class="comment">
</span>      MN = MIN( M, N )
      ISMIN = MN + 1
      ISMAX = 2*MN + 1
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Test the input arguments.
</span><span class="comment">*</span><span class="comment">
</span>      INFO = 0
      IF( M.LT.0 ) THEN
         INFO = -1
      ELSE IF( N.LT.0 ) THEN
         INFO = -2
      ELSE IF( NRHS.LT.0 ) THEN
         INFO = -3
      ELSE IF( LDA.LT.MAX( 1, M ) ) THEN
         INFO = -5
      ELSE IF( LDB.LT.MAX( 1, M, N ) ) THEN
         INFO = -7
      END IF
<span class="comment">*</span><span class="comment">
</span>      IF( INFO.NE.0 ) THEN
         CALL <a name="XERBLA.163"></a><a href="xerbla.f.html#XERBLA.1">XERBLA</a>( <span class="string">'<a name="CGELSX.163"></a><a href="cgelsx.f.html#CGELSX.1">CGELSX</a>'</span>, -INFO )
         RETURN
      END IF
<span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment">     Quick return if possible
</span><span class="comment">*</span><span class="comment">
</span>      IF( MIN( M, N, NRHS ).EQ.0 ) THEN
         RANK = 0
         RETURN
      END IF
12

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