This information is part of the Modelica Standard Library maintained by the Modelica Association.

Lapack documentation
    Purpose
    =======

    DGELSY computes the minimum-norm solution to a real linear least
    squares problem:
        minimize || A * X - B ||
    using a complete orthogonal factorization of A.  A is an M-by-N
    matrix which may be rank-deficient.

    Several right hand side vectors b and solution vectors x can be
    handled in a single call; they are stored as the columns of the
    M-by-NRHS right hand side matrix B and the N-by-NRHS solution
    matrix X.

    The routine first computes a QR factorization with column pivoting:
        A * P = Q * [ R11 R12 ]
                    [  0  R22 ]
    with R11 defined as the largest leading submatrix whose estimated
    condition number is less than 1/RCOND.  The order of R11, RANK,
    is the effective rank of A.

    Then, R22 is considered to be negligible, and R12 is annihilated
    by orthogonal transformations from the right, arriving at the
    complete orthogonal factorization:
       A * P = Q * [ T11 0 ] * Z
                   [  0  0 ]
    The minimum-norm solution is then
       X = P * Z' [ inv(T11)*Q1'*B ]
                  [        0       ]
    where Q1 consists of the first RANK columns of Q.

    This routine is basically identical to the original xGELSX except
    three differences:
      o The call to the subroutine xGEQPF has been substituted by
        the call to the subroutine xGEQP3. This subroutine is a Blas-3
        version of the QR factorization with column pivoting.
      o Matrix B (the right hand side) is updated with Blas-3.
      o The permutation of matrix B (the right hand side) is faster and
        more simple.

    Arguments
    =========

    M       (input) INTEGER
            The number of rows of the matrix A.  M >= 0.

    N       (input) INTEGER
            The number of columns of the matrix A.  N >= 0.

    NRHS    (input) INTEGER
            The number of right hand sides, i.e., the number of
            columns of matrices B and X. NRHS >= 0.

    A       (input/output) DOUBLE PRECISION array, dimension (LDA,N)
            On entry, the M-by-N matrix A.
            On exit, A has been overwritten by details of its
            complete orthogonal factorization.

    LDA     (input) INTEGER
            The leading dimension of the array A.  LDA >= max(1,M).

    B       (input/output) DOUBLE PRECISION array, dimension (LDB,NRHS)
            On entry, the M-by-NRHS right hand side matrix B.
            On exit, the N-by-NRHS solution matrix X.

    LDB     (input) INTEGER
            The leading dimension of the array B. LDB >= max(1,M,N).

    JPVT    (input/output) INTEGER array, dimension (N)
            On entry, if JPVT(i) .ne. 0, the i-th column of A is permuted
            to the front of AP, otherwise column i is a free column.
            On exit, if JPVT(i) = k, then the i-th column of AP
            was the k-th column of A.

    RCOND   (input) DOUBLE PRECISION
            RCOND is used to determine the effective rank of A, which
            is defined as the order of the largest leading triangular
            submatrix R11 in the QR factorization with pivoting of A,
            whose estimated condition number < 1/RCOND.

    RANK    (output) INTEGER
            The effective rank of A, i.e., the order of the submatrix
            R11.  This is the same as the order of the submatrix T11
            in the complete orthogonal factorization of A.

    WORK    (workspace/output) DOUBLE PRECISION array, dimension (MAX(1,LWORK))
            On exit, if INFO = 0, WORK(1) returns the optimal LWORK.

    LWORK   (input) INTEGER
            The dimension of the array WORK.
            The unblocked strategy requires that:
               LWORK >= MAX( MN+3*N+1, 2*MN+NRHS ),
            where MN = min( M, N ).
            The block algorithm requires that:
               LWORK >= MAX( MN+2*N+NB*(N+1), 2*MN+NB*NRHS ),
            where NB is an upper bound on the blocksize returned
            by ILAENV for the routines DGEQP3, DTZRZF, STZRQF, DORMQR,
            and DORMRZ.

            If LWORK = -1, then a workspace query is assumed; the routine
            only calculates the optimal size of the WORK array, returns
            this value as the first entry of the WORK array, and no error
            message related to LWORK is issued by XERBLA.

    INFO    (output) INTEGER
            = 0: successful exit
            < 0: If INFO = -i, the i-th argument had an illegal value.

(X, info, rank) = dgelsy(A, B, rcond)

A	Type: Real[:,:]
B	Type: Real[size(A, 1),:]
rcond	Default Value: 0.0 Type: Real Description: Reciprocal condition number to estimate rank

X	Default Value: cat(1, B, zeros(max(nrow, ncol) - nrow, nrhs)) Type: Real[max(size(A, 1), size(A, 2)),size(B, 2)] Description: Solution is in first size(A,2) rows
info	Type: Integer
rank	Type: Integer Description: Effective rank of A