SLATRS (3lapack)

solve one of the triangular systems A *x = s*b or A'*x = s*b with scaling to prevent overflow

SYNOPSIS

  SUBROUTINE SLATRS( UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE, CNORM,
                     INFO )

      CHARACTER      DIAG, NORMIN, TRANS, UPLO

      INTEGER        INFO, LDA, N

      REAL           SCALE

      REAL           A( LDA, * ), CNORM( * ), X( * )

PURPOSE

  SLATRS solves one of the triangular systems triangular matrix, A' 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 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 triangular.  = '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.

  A       (input) REAL array, dimension (LDA,N)
          The triangular matrix A.  If UPLO = 'U', the leading n by n upper
          triangular part of the array A contains the upper triangular
          matrix, and the strictly lower triangular part of A is not
          referenced.  If UPLO = 'L', the leading n by n lower triangular
          part of the array A contains the lower triangular matrix, and the
          strictly upper triangular part of A is not referenced.  If DIAG =
          'U', the diagonal elements of A are also not referenced and are
          assumed to be 1.

  LDA     (input) INTEGER
          The leading dimension of the array A.  LDA >= max (1,N).

  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 vector 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 illegal value

FURTHER DETAILS

  A rough bound on x is computed; if that is less than overflow, STRSV 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

  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'*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 STRSV if 1/M(n) and 1/G(n) are both greater than
  max(underflow, 1/overflow).