ZLATRS(3F) ZLATRS(3F)
ZLATRS - solve one of the triangular systems A * x = s*b, A**T * x =
s*b, or A**H * x = s*b,
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( * )
ZLATRS solves one of the triangular systems
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 problem 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 non-trivial
solution to A*x = 0 is returned.
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**H * x = s*b (Conjugate 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
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ZLATRS(3F) ZLATRS(3F)
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 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) COMPLEX*16 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, 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 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, ZTRSV 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]
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ZLATRS(3F) ZLATRS(3F)
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 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 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 or A**H *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) |
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ZLATRS(3F) ZLATRS(3F)
<= 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).
ZLATRS(3F) ZLATRS(3F)
ZLATRS - solve one of the triangular systems A * x = s*b, A**T * x =
s*b, or A**H * x = s*b,
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( * )
ZLATRS solves one of the triangular systems
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 problem 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 non-trivial
solution to A*x = 0 is returned.
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**H * x = s*b (Conjugate 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
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ZLATRS(3F) ZLATRS(3F)
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 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) COMPLEX*16 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, 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 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, ZTRSV 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]
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ZLATRS(3F) ZLATRS(3F)
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 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 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 or A**H *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) |
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ZLATRS(3F) ZLATRS(3F)
<= 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).
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