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clatbs (3)
  • >> clatbs (3) ( Solaris man: Библиотечные вызовы )
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    NAME
         clatbs - solve one of the triangular systems   A * x =  s*b,
         A**T * x = s*b, or A**H * x = s*b,
    
    SYNOPSIS
         SUBROUTINE CLATBS( 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 CNORM( * )
    
         COMPLEX AB( LDAB, * ), X( * )
    
    
    
         #include <sunperf.h>
    
         void clatbs(char uplo, char trans, char diag,  char  normin,
                   int  n,  int kd, complex *cab, int ldab, complex *
                   x, float *sscale, float *cnorm, int *info) ;
    
    PURPOSE
         CLATBS solves one of the triangular systems
    
         with scaling to prevent overflow, where A  is  an  upper  or
         lower triangular band matrix.  Here A' denotes the transpose
         of A, x and b are n-element vectors, and s is a scaling fac-
         tor,  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 CTBSV 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**T * x = s*b  (Transpose)
                   = 'C':  Solve A**H * x  =  s*b   (Conjugate  tran-
                   spose)
    
         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) COMPLEX 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) COMPLEX 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,  A**T * x = s*b,  or  A**H * x = s*b.  If
                   SCALE = 0, the  matrix  A  is  singular  or  badly
                   scaled,  and  the vector x is an exact or approxi-
                   mate 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,  CTBSV  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  CTBSV
         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
         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**T  *x  =  b
         or  A**H *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 CTBSV if 1/M(n) and 1/G(n)  are  both
         greater than max(underflow, 1/overflow).
    
    
    
    


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