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sgelsx (3)
  • >> sgelsx (3) ( Solaris man: Библиотечные вызовы )
  • 
    NAME
         sgelsx - compute the minimum-norm solution to a real  linear
         least squares problem
    
    SYNOPSIS
         SUBROUTINE SGELSX( M, N, NRHS, A, LDA, B, LDB, JPVT,  RCOND,
                   RANK, WORK, INFO )
    
         INTEGER INFO, LDA, LDB, M, N, NRHS, RANK
    
         REAL RCOND
    
         INTEGER JPVT( * )
    
         REAL A( LDA, * ), B( LDB, * ), WORK( * )
    
    
    
         #include <sunperf.h>
    
         void sgelsx(int m, int n, int  nrhs,  float  *sa,  int  lda,
                   float  *sb, int ldb, int *jpivot, float rcond, int
                   *rank, int *info);
    
    PURPOSE
         SGELSX 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 annihi-
         lated 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) REAL 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) REAL 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 posi-
                   tions; only the remaining free columns  are  moved
                   as a result of column pivoting during the factori-
                   zation.  On exit, if JPVT(i) = k,  then  the  i-th
                   column of A*P was the k-th column of A.
    
         RCOND     (input) REAL
                   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 factor-
                   ization with pivoting of A, whose estimated condi-
                   tion 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  fac-
                   torization of A.
    
         WORK      (workspace) REAL 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 ille-
                   gal value
    
    
    
    


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