dlatrs(3S)

NAME
     DLATRS - solve one of the triangular systems  A *x = s*b or A'*x = s*b
     with scaling to prevent overflow

SYNOPSIS
     SUBROUTINE DLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM,
                        INFO )

         CHARACTER      DIAG, NORMIN, TRANS, UPLO

         INTEGER        INFO, LDA, N

         DOUBLE         PRECISION SCALE

         DOUBLE         PRECISION A( LDA, * ), CNORM( * ), X( * )

IMPLEMENTATION
     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.

PURPOSE
     DLATRS solves one of the triangular systems A *x = s*b or A'*x = s*b with
     scaling to prevent overflow. Here A is an upper or lower triangular
     matrix, A' denotes the transpose of A, x and b are n-element vectors, and
     s is a scaling factor, usually less than or equal to 1, chosen so that
     the components of x will be less than the overflow threshold.  If the
     unscaled problem will not cause overflow, the Level 2 BLAS routine DTRSV
     is called.  If the matrix A is singular (A(j,j) = 0 for some j), then s
     is set to 0 and a non-trivial solution to A*x = 0 is returned.


ARGUMENTS
     UPLO    (input) CHARACTER*1
             Specifies whether the matrix A is upper or lower triangular.  =
             'U':  Upper triangular
             = 'L':  Lower triangular

     TRANS   (input) CHARACTER*1
             Specifies the operation applied to A.  = 'N':  Solve A * x = s*b
             (No transpose)
             = 'T':  Solve A'* x = s*b  (Transpose)
             = 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)

     DIAG    (input) CHARACTER*1
             Specifies whether or not the matrix A is unit triangular.  = 'N':
             Non-unit triangular
             = 'U':  Unit triangular

     NORMIN  (input) CHARACTER*1
             Specifies whether CNORM has been set or not.  = 'Y':  CNORM
             contains the column norms on entry
             = 'N':  CNORM is not set on entry.  On exit, the norms will be
             computed and stored in CNORM.

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

     A       (input) DOUBLE PRECISION array, dimension (LDA,N)
             The triangular matrix A.  If UPLO = 'U', the leading n by n upper
             triangular part of the array A contains the upper triangular
             matrix, and the strictly lower triangular part of A is not
             referenced.  If UPLO = 'L', the leading n by n lower triangular
             part of the array A contains the lower triangular matrix, and the
             strictly upper triangular part of A is not referenced.  If DIAG =
             'U', the diagonal elements of A are also not referenced and are
             assumed to be 1.

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

     X       (input/output) DOUBLE PRECISION array, dimension (N)
             On entry, the right hand side b of the triangular system.  On
             exit, X is overwritten by the solution vector x.

     SCALE   (output) DOUBLE PRECISION
             The scaling factor s for the triangular system A * x = s*b  or
             A'* x = s*b.  If SCALE = 0, the matrix A is singular or badly
             scaled, and the vector x is an exact or approximate solution to
             A*x = 0.

     CNORM   (input or output) DOUBLE PRECISION array, dimension (N)

             If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains
             the norm of the off-diagonal part of the j-th column of A.  If
             TRANS = 'N', CNORM(j) must be greater than or equal to the
             infinity-norm, and if TRANS = 'T' or 'C', CNORM(j) must be
             greater than or equal to the 1-norm.

             If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns
             the 1-norm of the offdiagonal part of the j-th column of A.

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

FURTHER DETAILS
     A rough bound on x is computed; if that is less than overflow, DTRSV is
     called, otherwise, specific code is used which checks for possible
     overflow or divide-by-zero at every operation.

     A columnwise scheme is used for solving A*x = b.  The basic algorithm if
     A is lower triangular is

          x[1:n] := b[1:n]
          for j = 1, ..., n
               x(j) := x(j) / A(j,j)
               x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
          end

     Define bounds on the components of x after j iterations of the loop:
        M(j) = bound on x[1:j]
        G(j) = bound on x[j+1:n]
     Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.

     Then for iteration j+1 we have
        M(j+1) <= G(j) / | A(j+1,j+1) |
        G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
               <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )

     where CNORM(j+1) is greater than or equal to the infinity-norm of column
     j+1 of A, not counting the diagonal.  Hence

        G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
                     1<=i<=j
     and

        |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                      1<=i< j

     Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
     reciprocal of the largest M(j), j=1,..,n, is larger than
     max(underflow, 1/overflow).

     The bound on x(j) is also used to determine when a step in the columnwise
     method can be performed without fear of overflow.  If the computed bound
     is greater than a large constant, x is scaled to prevent overflow, but if
     the bound overflows, x is set to 0, x(j) to 1, and scale to 0, and a
     non-trivial solution to A*x = 0 is found.

     Similarly, a row-wise scheme is used to solve A'*x = b.  The basic
     algorithm for A upper triangular is

          for j = 1, ..., n
               x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
          end

     We simultaneously compute two bounds
          G(j) = bound on ( b(i) - A[1:i-1,i]' * x[1:i-1] ), 1<=i<=j
          M(j) = bound on x(i), 1<=i<=j

     The initial values are G(0) = 0, M(0) = max{b(i), i=1,..,n}, and we add
     the constraint G(j) >= G(j-1) and M(j) >= M(j-1) for j >= 1.  Then the
     bound on x(j) is

          M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |

               <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                         1<=i<=j

     and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater than
     max(underflow, 1/overflow).


SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

     This man page is available only online.