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NAME
ZLATBS - solve one of the triangular systems A * x = s*b,
A**T * x = s*b, or A**H * x = s*b,
SYNOPSIS
SUBROUTINE ZLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB,
LDAB, X, SCALE, CNORM, INFO )
CHARACTER DIAG, NORMIN, TRANS, UPLO
INTEGER INFO, KD, LDAB, N
DOUBLE PRECISION SCALE
DOUBLE PRECISION CNORM( * )
COMPLEX*16 AB( LDAB, * ), X( * )
PURPOSE
ZLATBS solves one of the triangular systems
with scaling to prevent overflow, where A is an upper or
lower triangular band matrix. Here A' denotes the transpose
of A, x and b are n-element vectors, and s is a scaling fac-
tor, 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 ZTBSV 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**H * x = s*b (Conjugate 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.
KD (input) INTEGER
The number of subdiagonals or superdiagonals in the
triangular matrix A. KD >= 0.
AB (input) COMPLEX*16 array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored
in the first KD+1 rows of the array. The j-th column
of A is stored in the j-th column of the array AB as
follows: if UPLO = 'U', AB(kd+1+i-j,j) = A(i,j) for
max(1,j-kd)<=i<=j; if UPLO = 'L', AB(1+i-j,j) =
A(i,j) for j<=i<=min(n,j+kd).
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >=
KD+1.
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 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, ZTBSV 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 ZTBSV
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**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) |
<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
and we can safely call ZTBSV if 1/M(n) and 1/G(n) are both
greater than max(underflow, 1/overflow).