.TH DLATRS l "15 June 2000" "LAPACK version 3.0" ")"
.SH NAME
DLATRS - solve one of the triangular systems  A *x = s*b or A'*x = s*b  with scaling to prevent overflow
.SH SYNOPSIS
.TP 19
SUBROUTINE DLATRS(
UPLO, TRANS, DIAG, NORMIN, N, A, LDA, X, SCALE,
CNORM, INFO )
.TP 19
.ti +4
CHARACTER
DIAG, NORMIN, TRANS, UPLO
.TP 19
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INTEGER
INFO, LDA, N
.TP 19
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DOUBLE
PRECISION SCALE
.TP 19
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DOUBLE
PRECISION A( LDA, * ), CNORM( * ), X( * )
.SH PURPOSE
DLATRS solves one of the triangular systems A *x = s*b or A'*x = s*b with scaling to prevent overflow. Here A is an upper or lower 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
overflow, the Level 2 BLAS routine DTRSV 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.
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.SH ARGUMENTS
.TP 8
UPLO    (input) CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U':  Upper triangular
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= 'L':  Lower triangular
.TP 8
TRANS   (input) CHARACTER*1
Specifies the operation applied to A.
= 'N':  Solve A * x = s*b  (No transpose)
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= 'T':  Solve A'* x = s*b  (Transpose)
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= 'C':  Solve A'* x = s*b  (Conjugate transpose = Transpose)
.TP 8
DIAG    (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N':  Non-unit triangular
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= 'U':  Unit triangular
.TP 8
NORMIN  (input) CHARACTER*1
Specifies whether CNORM has been set or not.
= 'Y':  CNORM contains the column norms on entry
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= 'N':  CNORM is not set on entry.  On exit, the norms will
be computed and stored in CNORM.
.TP 8
N       (input) INTEGER
The order of the matrix A.  N >= 0.
.TP 8
A       (input) DOUBLE PRECISION 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.
.TP 8
LDA     (input) INTEGER
The leading dimension of the array A.  LDA >= max (1,N).
.TP 8
X       (input/output) 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.
.TP 8
SCALE   (output) DOUBLE PRECISION
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.
.TP 8
CNORM   (input or output) 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.
.TP 8
INFO    (output) INTEGER
= 0:  successful exit
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< 0:  if INFO = -k, the k-th argument had an illegal value
.SH FURTHER DETAILS
A rough bound on x is computed; if that is less than overflow, DTRSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
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A columnwise scheme is used for solving A*x = b.  The basic algorithm
if A is lower triangular is
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     x[1:n] := b[1:n]
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     for j = 1, ..., n
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          x(j) := x(j) / A(j,j)
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          x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
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     end
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Define bounds on the components of x after j iterations of the loop:
   M(j) = bound on x[1:j]
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   G(j) = bound on x[j+1:n]
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Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
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Then for iteration j+1 we have
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   M(j+1) <= G(j) / | A(j+1,j+1) |
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   G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
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          <= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
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where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal.  Hence
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   G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
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                1<=i<=j
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and
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   |x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
                                 1<=i< j
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Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTRSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
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max(underflow, 1/overflow).
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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
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     for j = 1, ..., n
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          x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
     end
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We simultaneously compute two bounds
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     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
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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
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     M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
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          <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
                    1<=i<=j
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and we can safely call DTRSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
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