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NAME

       dlatbs.f -

SYNOPSIS

   Functions/Subroutines
       subroutine dlatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
           DLATBS solves a triangular banded system of equations.

Function/Subroutine Documentation

   subroutine dlatbs (characterUPLO, characterTRANS, characterDIAG, characterNORMIN, integerN,
       integerKD, double precision, dimension( ldab, * )AB, integerLDAB, double precision,
       dimension( * )X, double precisionSCALE, double precision, dimension( * )CNORM,
       integerINFO)
       DLATBS solves a triangular banded system of equations.

       Purpose:

            DLATBS solves one of the triangular systems

               A *x = s*b  or  A**T*x = s*b

            with scaling to prevent overflow, where A is an upper or lower
            triangular band matrix.  Here A**T 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 DTBSV 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.

       Parameters:
           UPLO

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

           TRANS

                     TRANS is 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**T* x = s*b  (Conjugate transpose = Transpose)

           DIAG

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

           NORMIN

                     NORMIN is 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

                     N is INTEGER
                     The order of the matrix A.  N >= 0.

           KD

                     KD is INTEGER
                     The number of subdiagonals or superdiagonals in the
                     triangular matrix A.  KD >= 0.

           AB

                     AB is DOUBLE PRECISION 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

                     LDAB is INTEGER
                     The leading dimension of the array AB.  LDAB >= KD+1.

           X

                     X is 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

                     SCALE is DOUBLE PRECISION
                     The scaling factor s for the triangular system
                        A * x = s*b  or  A**T* 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

                     CNORM is 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

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

       Author:
           Univ. of Tennessee

           Univ. of California Berkeley

           Univ. of Colorado Denver

           NAG Ltd.

       Date:
           September 2012

       Further Details:

             A rough bound on x is computed; if that is less than overflow, DTBSV
             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 DTBSV 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**T*x = b.  The basic
             algorithm for A upper triangular is

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

             We simultaneously compute two bounds
                  G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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 DTBSV if 1/M(n) and 1/G(n) are both greater
             than max(underflow, 1/overflow).

       Definition at line 242 of file dlatbs.f.

Author

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