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## slatbs()

subroutine slatbs | ( | character | uplo,
|  | character | trans,
|  | character | diag,
|  | character | normin,
|  | integer | n,
|  | integer | kd,
|  | real, dimension( ldab, * ) | ab,
|  | integer | ldab,
|  | real, dimension( * ) | x,
|  | real | scale,
|  | real, dimension( * ) | cnorm,
|  | integer | info )

SLATBS solves a triangular banded system of equations.

Download SLATBS + dependencies [TGZ] [ZIP] [TXT]

Purpose:

      !>
      !> SLATBS 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 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.
      !>

Parameters

  [in]

  UPLO

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

  [in]

  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)
      !>

  [in]

  DIAG

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

  [in]

  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.
      !>

  [in]

  N

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

  [in]

  KD

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

  [in]

  AB

      !>          AB is 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).
      !>

  [in]

  LDAB

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

  [in,out]

  X

      !>          X is 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.
      !>

  [out]

  SCALE

      !>          SCALE is REAL
      !>          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.
      !>

  [in,out]

  CNORM

      !>          CNORM is 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.
      !>

  [out]

  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.

Further Details:

      !>
      !>  A rough bound on x is computed; if that is less than overflow, 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 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 STBSV if 1/M(n) and 1/G(n) are both greater
      !>  than max(underflow, 1/overflow).
      !>