.TH DLATBS 1 "November 2006" " LAPACK auxiliary routine (version 3.1) " " LAPACK auxiliary routine (version 3.1) "
.SH NAME
DLATBS - one of the triangular systems A *x = s*b or A\(aq*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular band matrix
.SH SYNOPSIS
.TP 19
SUBROUTINE DLATBS(
UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X,
SCALE, CNORM, INFO )
.TP 19
.ti +4
CHARACTER
DIAG, NORMIN, TRANS, UPLO
.TP 19
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INTEGER
INFO, KD, LDAB, N
.TP 19
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DOUBLE
PRECISION SCALE
.TP 19
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DOUBLE
PRECISION AB( LDAB, * ), CNORM( * ), X( * )
.SH PURPOSE
DLATBS solves one of the triangular systems
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 DTBSV 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.
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.SH ARGUMENTS
.TP 8
UPLO (input) CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= \(aqU\(aq: Upper triangular
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= \(aqL\(aq: Lower triangular
.TP 8
TRANS (input) CHARACTER*1
Specifies the operation applied to A.
= \(aqN\(aq: Solve A * x = s*b (No transpose)
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= \(aqT\(aq: Solve A\(aq* x = s*b (Transpose)
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= \(aqC\(aq: Solve A\(aq* x = s*b (Conjugate transpose = Transpose)
.TP 8
DIAG (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= \(aqN\(aq: Non-unit triangular
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= \(aqU\(aq: Unit triangular
.TP 8
NORMIN (input) CHARACTER*1
Specifies whether CNORM has been set or not.
= \(aqY\(aq: CNORM contains the column norms on entry
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= \(aqN\(aq: CNORM is not set on entry. On exit, the norms will
be computed and stored in CNORM.
.TP 8
N (input) INTEGER
The order of the matrix A. N >= 0.
.TP 8
KD (input) INTEGER
The number of subdiagonals or superdiagonals in the
triangular matrix A. KD >= 0.
.TP 8
AB (input) DOUBLE PRECISION 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 = \(aqU\(aq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j;
if UPLO = \(aqL\(aq, AB(1+i-j,j) = A(i,j) for j<=i<=min(n,j+kd).
.TP 8
LDAB (input) INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
.TP 8
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.
.TP 8
SCALE (output) DOUBLE PRECISION
The scaling factor s for the triangular system
A * x = s*b or A\(aq* 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.
.TP 8
CNORM (input or output) DOUBLE PRECISION array, dimension (N)
If NORMIN = \(aqY\(aq, 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 = \(aqN\(aq, CNORM(j) must be greater than or equal
to the infinity-norm, and if TRANS = \(aqT\(aq or \(aqC\(aq, CNORM(j)
must be greater than or equal to the 1-norm.
If NORMIN = \(aqN\(aq, CNORM is an output argument and CNORM(j)
returns the 1-norm of the offdiagonal part of the j-th column
of A.
.TP 8
INFO (output) INTEGER
= 0: successful exit
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< 0: if INFO = -k, the k-th argument had an illegal value
.SH FURTHER DETAILS
A rough bound on x is computed; if that is less than overflow, DTBSV
is called, otherwise, specific code is used which checks for possible
overflow or divide-by-zero at every operation.
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A columnwise scheme is used for solving A*x = b. The basic algorithm
if A is lower triangular is
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x[1:n] := b[1:n]
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for j = 1, ..., n
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x(j) := x(j) / A(j,j)
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x[j+1:n] := x[j+1:n] - x(j) * A[j+1:n,j]
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end
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Define bounds on the components of x after j iterations of the loop:
M(j) = bound on x[1:j]
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G(j) = bound on x[j+1:n]
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Initially, let M(0) = 0 and G(0) = max{x(i), i=1,...,n}.
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Then for iteration j+1 we have
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M(j+1) <= G(j) / | A(j+1,j+1) |
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G(j+1) <= G(j) + M(j+1) * | A[j+2:n,j+1] |
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<= G(j) ( 1 + CNORM(j+1) / | A(j+1,j+1) | )
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where CNORM(j+1) is greater than or equal to the infinity-norm of
column j+1 of A, not counting the diagonal. Hence
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G(j) <= G(0) product ( 1 + CNORM(i) / | A(i,i) | )
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1<=i<=j
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and
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|x(j)| <= ( G(0) / |A(j,j)| ) product ( 1 + CNORM(i) / |A(i,i)| )
1<=i< j
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Since |x(j)| <= M(j), we use the Level 2 BLAS routine DTBSV if the
reciprocal of the largest M(j), j=1,..,n, is larger than
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max(underflow, 1/overflow).
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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\(aq*x = b. The basic
algorithm for A upper triangular is
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for j = 1, ..., n
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x(j) := ( b(j) - A[1:j-1,j]\(aq * x[1:j-1] ) / A(j,j)
end
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We simultaneously compute two bounds
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G(j) = bound on ( b(i) - A[1:i-1,i]\(aq * x[1:i-1] ), 1<=i<=j
M(j) = bound on x(i), 1<=i<=j
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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
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M(j) <= M(j-1) * ( 1 + CNORM(j) ) / | A(j,j) |
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<= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
1<=i<=j
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and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
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