zlatbs.f(3)

NAME
zlatbs.f -

SYNOPSIS

  Functions/Subroutines
subroutine ZLATBS (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
ZLATBS

Function/Subroutine Documentation

  subroutine ZLATBS (CHARACTERUPLO, CHARACTERTRANS, CHARACTERDIAG, CHARACTERNORMIN, INTEGERN, INTEGERKD, COMPLEX*16, dimension( ldab, * )AB, INTEGERLDAB, COMPLEX*16, dimension( * )X, DOUBLE PRECISIONSCALE, DOUBLE PRECISION, dimension( * )CNORM, INTEGERINFO)

ZLATBS

Purpose:

     ZLATBS solves one of the triangular systems
        A * x = s*b,  A**T * x = s*b,  or  A**H * 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 ZTBSV 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**H * x = s*b  (Conjugate 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 COMPLEX*16 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 COMPLEX*16 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,  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 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:
    April 2012

Further Details:

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

Definition at line 243 of file zlatbs.f'.

Author

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