
ZLATRS(l) ) ZLATRS(l)
NAME
ZLATRS  solve one of the triangular systems A * x = s*b, A**T * x = s*b, or A**H * x =
s*b,
SYNOPSIS
SUBROUTINE ZLATRS( 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 CNORM( * )
COMPLEX*16 A( LDA, * ), X( * )
PURPOSE
ZLATRS 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. Here A is an upper or lower triangular matrix,
A**T denotes the transpose of A, A**H denotes the conjugate 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 prob
lem will not cause overflow, the Level 2 BLAS routine ZTRSV is called. If the matrix A is
singular (A(j,j) = 0 for some j), then s is set to 0 and a nontrivial solution to A*x = 0
is returned.
ARGUMENTS
UPLO (input) CHARACTER*1
Specifies whether the matrix A is upper or lower triangular. = 'U': Upper trian
gular
= 'L': Lower triangular
TRANS (input) CHARACTER*1
Specifies the operation applied to A. = 'N': Solve A * x = s*b (No trans
pose)
= 'T': Solve A**T * x = s*b (Transpose)
= 'C': Solve A**H * x = s*b (Conjugate transpose)
DIAG (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular. = 'N': Nonunit trian
gular
= '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) COMPLEX*16 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 trian
gular part of A is not referenced. If UPLO = 'L', the leading n by n lower trian
gular 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 ele
ments 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) COMPLEX*16 array, dimension (N)
On entry, the right hand side b of the triangular system. On exit, X is overwrit
ten by the solution vector x.
SCALE (output) 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 (input or output) DOUBLE PRECISION array, dimension (N)
If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the
offdiagonal part of the jth column of A. If TRANS = 'N', CNORM(j) must be
greater than or equal to the infinitynorm, and if TRANS = 'T' or 'C', CNORM(j)
must be greater than or equal to the 1norm.
If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1norm of
the offdiagonal part of the jth column of A.
INFO (output) INTEGER
= 0: successful exit
< 0: if INFO = k, the kth argument had an illegal value
FURTHER DETAILS
A rough bound on x is computed; if that is less than overflow, ZTRSV is called, otherwise,
specific code is used which checks for possible overflow or dividebyzero at every opera
tion.
A columnwise scheme is used for solving A*x = b. The basic algorithm if A is lower trian
gular 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 infinitynorm 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 ZTRSV 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 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 nontrivial solution to A*x = 0 is found.
Similarly, a rowwise 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:j1,j]' * x[1:j1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i)  A[1:i1,i]' * x[1:i1] ), 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(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is
M(j) <= M(j1) * ( 1 + CNORM(j) ) /  A(j,j) 
<= M(0) * product ( ( 1 + CNORM(i) ) / A(i,i) )
1<=i<=j
and we can safely call ZTRSV if 1/M(n) and 1/G(n) are both greater than max(underflow,
1/overflow).
LAPACK version 3.0 15 June 2000 ZLATRS(l) 
