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# slatrs

```
NAME
SLATRS - 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 overflow, the Level 2 BLAS routine STRSV 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.

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 tri-
angular.  = '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 variable 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 variable 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 over-
flow, 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

Since |x(j)| <= M(j), we use the Level 2 BLAS routine STRSV
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 over-
flow.  If the computed bound is greater than a large con-
stant, 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).
```