.Modelica_LinearSystems2.Math.Matrices.LAPACK.dgelsx

Information

Lapack documentation:

   Purpose
   =======

   DGELSX 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.

   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.
           If m >= n and RANK = n, the residual sum-of-squares for
           the solution in the i-th column is given by the sum of
           squares of elements N+1:M in that column.

   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 an
           initial column, otherwise it is a free column.  Before
           the QR factorization of A, all initial columns are
           permuted to the leading positions; only the remaining
           free columns are moved as a result of column pivoting
           during the factorization.
           On exit, if JPVT(i) = k, then the i-th column of A*P
           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) DOUBLE PRECISION array, dimension
                       (max( min(M,N)+3*N, 2*min(M,N)+NRHS )),

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

   =====================================================================  

Interface

function dgelsx
  extends Modelica.Icons.Function;
  input Real A[:, :];
  input Real B[size(A, 1), :];
  input Real rcond = 0.0 "Reciprocal condition number to estimate rank";
  output Real X[max(nrow, ncol), nrhs] = cat(1, B, zeros(max(nrow, ncol) - nrow, nrhs)) "Solution is in first size(A,2) rows";
  output Integer info;
  output Integer rank "Effective rank of A";
end dgelsx;