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SUBROUTINE <a name="CGELSY.1"></a><a href="cgelsy.f.html#CGELSY.1">CGELSY</a>( M, N, NRHS, A, LDA, B, LDB, JPVT, RCOND, RANK,
$ WORK, LWORK, 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, LWORK, 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"> <a name="CGELSY.21"></a><a href="cgelsy.f.html#CGELSY.1">CGELSY</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"> This routine is basically identical to the original xGELSX except
</span><span class="comment">*</span><span class="comment"> three differences:
</span><span class="comment">*</span><span class="comment"> o The permutation of matrix B (the right hand side) is faster and
</span><span class="comment">*</span><span class="comment"> more simple.
</span><span class="comment">*</span><span class="comment"> o The call to the subroutine xGEQPF has been substituted by the
</span><span class="comment">*</span><span class="comment"> the call to the subroutine xGEQP3. This subroutine is a Blas-3
</span><span class="comment">*</span><span class="comment"> version of the QR factorization with column pivoting.
</span><span class="comment">*</span><span class="comment"> o Matrix B (the right hand side) is updated with Blas-3.
</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 >= 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 >= 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 >= 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 >= 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">
</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 >= 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 permuted
</span><span class="comment">*</span><span class="comment"> to the front of AP, otherwise column i is a free column.
</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 < 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/output) COMPLEX array, dimension (MAX(1,LWORK))
</span><span class="comment">*</span><span class="comment"> On exit, if INFO = 0, WORK(1) returns the optimal LWORK.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> LWORK (input) INTEGER
</span><span class="comment">*</span><span class="comment"> The dimension of the array WORK.
</span><span class="comment">*</span><span class="comment"> The unblocked strategy requires that:
</span><span class="comment">*</span><span class="comment"> LWORK >= MN + MAX( 2*MN, N+1, MN+NRHS )
</span><span class="comment">*</span><span class="comment"> where MN = min(M,N).
</span><span class="comment">*</span><span class="comment"> The block algorithm requires that:
</span><span class="comment">*</span><span class="comment"> LWORK >= MN + MAX( 2*MN, NB*(N+1), MN+MN*NB, MN+NB*NRHS )
</span><span class="comment">*</span><span class="comment"> where NB is an upper bound on the blocksize returned
</span><span class="comment">*</span><span class="comment"> by <a name="ILAENV.114"></a><a href="hfy-index.html#ILAENV">ILAENV</a> for the routines <a name="CGEQP3.114"></a><a href="cgeqp3.f.html#CGEQP3.1">CGEQP3</a>, <a name="CTZRZF.114"></a><a href="ctzrzf.f.html#CTZRZF.1">CTZRZF</a>, <a name="CTZRQF.114"></a><a href="ctzrqf.f.html#CTZRQF.1">CTZRQF</a>, <a name="CUNMQR.114"></a><a href="cunmqr.f.html#CUNMQR.1">CUNMQR</a>,
</span><span class="comment">*</span><span class="comment"> and <a name="CUNMRZ.115"></a><a href="cunmrz.f.html#CUNMRZ.1">CUNMRZ</a>.
</span><span class="comment">*</span><span class="comment">
</span><span class="comment">*</span><span class="comment"> If LWORK = -1, then a workspace query is assumed; the routine
</span><span class="comment">*</span><span class="comment"> only calculates the optimal size of the WORK array, returns
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