IRIX Base Documentation 2002 November/SGI IRIX Base Documentation 2002 November.iso/usr/share/catman/p_man/cat3/SCSL/sgelsx.z/sgelsx

Text File | 2002-10-03 | 6.0 KB | 199 lines

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     SGELSX - routine is deprecated and has been replaced by routine SGELSY

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     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( * )

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     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

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     This routine is deprecated and has been replaced by routine SGELSY.
     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 annihilated by
     orthogonal transformations from the right, arriving at the complete
     orthogonal factorization:
        A * P = Q * [ T11 0 ] * Z
                    [  0  0 ]



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     The minimum-norm solution is then
        X = P * Z' [ inv(T11)*Q1'*B ]
                   [        0       ]
     where Q1 consists of the first RANK columns of Q.


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     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 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) 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 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.



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     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 illegal value

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     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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