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slatbs (3)
  • >> slatbs (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         slatbs - solve one of the triangular systems   A *x = s*b or
         A'*x  = s*b  with scaling to prevent overflow, where A is an
         upper or lower triangular band matrix
    
    SYNOPSIS
         SUBROUTINE SLATBS( UPLO, TRANS, DIAG,  NORMIN,  N,  KD,  AB,
                   LDAB, X, SCALE, CNORM, INFO )
    
         CHARACTER DIAG, NORMIN, TRANS, UPLO
    
         INTEGER INFO, KD, LDAB, N
    
         REAL SCALE
    
         REAL AB( LDAB, * ), CNORM( * ), X( * )
    
    
    
         #include <sunperf.h>
    
         void slatbs(char uplo, char trans, char diag,  char  normin,
                   int  n,  int  kd, float *sab, int ldab, float *sx,
                   float *sscale, float *cnorm, int *info) ;
    
    PURPOSE
         SLATBS solves one of the triangular  systems  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  STBSV  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 tri-
                   angular.  = '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.
    
         KD        (input) INTEGER
                   The number of subdiagonals  or  superdiagonals  in
                   the triangular matrix A.  KD >= 0.
    
         AB        (input) REAL array, dimension (LDAB,N)
                   The upper  or  lower  triangular  band  matrix  A,
                   stored  in  the  first KD+1 rows of the array. The
                   j-th column of A is stored in the j-th  column  of
                   the   array   AB  as  follows:   if  UPLO  =  'U',
                   AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;  if
                   UPLO   =   'L',   AB(1+i-j,j)      =   A(i,j)  for
                   j<=i<=min(n,j+kd).
    
         LDAB      (input) INTEGER
                   The leading dimension of the array  AB.   LDAB  >=
                   KD+1.
    
         X         (input/output) REAL 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) REAL
                   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 vec-
                   tor x is an exact or approximate solution to A*x =
                   0.
    
         CNORM     (input or output) REAL 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 ille-
                   gal value
    
    FURTHER DETAILS
         A rough bound on x is computed; if that is less  than  over-
         flow,  STBSV  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  STBSV
         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 over-
         flow.  If the computed bound is greater than  a  large  con-
         stant,  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 STBSV if 1/M(n) and 1/G(n)  are  both
         greater than max(underflow, 1/overflow).
    
    
    
    


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