Provided by: liblapack-doc_3.3.1-1_all NAME

LAPACK-3  -  solves  one  of  the  triangular  systems   A *x = s*b or A**T *x = s*b  with
scaling to prevent overflow

SYNOPSIS

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

CHARACTER      DIAG, NORMIN, TRANS, UPLO

INTEGER        INFO, LDA, N

DOUBLE         PRECISION SCALE

DOUBLE         PRECISION A( LDA, * ), CNORM( * ), X( * )

PURPOSE

DLATRS solves one of the triangular systems
triangular matrix, 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 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.

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**T* x = s*b  (Transpose)
= 'C':  Solve A**T* 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) 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.

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

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.

SCALE   (output) DOUBLE PRECISION
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.

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.

INFO    (output) INTEGER
= 0:  successful exit
< 0:  if INFO = -k, the k-th argument had an illegal value

FURTHERDETAILS

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

LAPACK auxiliary routine (version 3.2)     April 2011                            DLATRS(3lapack)