
slatbs.f(3) LAPACK slatbs.f(3)
NAME
slatbs.f 
SYNOPSIS
Functions/Subroutines
subroutine slatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO)
SLATBS solves a triangular banded system of equations.
Function/Subroutine Documentation
subroutine slatbs (characterUPLO, characterTRANS, characterDIAG, characterNORMIN, integerN,
integerKD, real, dimension( ldab, * )AB, integerLDAB, real, dimension( * )X, realSCALE,
real, dimension( * )CNORM, integerINFO)
SLATBS solves a triangular banded system of equations.
Purpose:
SLATBS solves one of the triangular systems
A *x = s*b or A**T*x = s*b
with scaling to prevent overflow, where A is an upper or lower
triangular band matrix. Here A**T denotes the transpose of A, x and b
are nelement 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 STBSV 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.
Parameters:
UPLO
UPLO is CHARACTER*1
Specifies whether the matrix A is upper or lower triangular.
= 'U': Upper triangular
= 'L': Lower triangular
TRANS
TRANS is 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
DIAG is CHARACTER*1
Specifies whether or not the matrix A is unit triangular.
= 'N': Nonunit triangular
= 'U': Unit triangular
NORMIN
NORMIN is 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
N is INTEGER
The order of the matrix A. N >= 0.
KD
KD is INTEGER
The number of subdiagonals or superdiagonals in the
triangular matrix A. KD >= 0.
AB
AB is REAL array, dimension (LDAB,N)
The upper or lower triangular band matrix A, stored in the
first KD+1 rows of the array. The jth column of A is stored
in the jth column of the array AB as follows:
if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j;
if UPLO = 'L', AB(1+ij,j) = A(i,j) for j<=i<=min(n,j+kd).
LDAB
LDAB is INTEGER
The leading dimension of the array AB. LDAB >= KD+1.
X
X is REAL 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
SCALE is REAL
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
CNORM is REAL 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
INFO is INTEGER
= 0: successful exit
< 0: if INFO = k, the kth argument had an illegal value
Author:
Univ. of Tennessee
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Date:
September 2012
Further Details:
A rough bound on x is computed; if that is less than overflow, STBSV
is called, otherwise, specific code is used which checks for possible
overflow or dividebyzero 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 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 STBSV 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 nontrivial solution to A*x = 0 is found.
Similarly, a rowwise 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:j1,j]**T * x[1:j1] ) / A(j,j)
end
We simultaneously compute two bounds
G(j) = bound on ( b(i)  A[1:i1,i]**T * 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 STBSV if 1/M(n) and 1/G(n) are both greater
than max(underflow, 1/overflow).
Definition at line 242 of file slatbs.f.
Author
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Version 3.4.2 Tue Sep 25 2012 slatbs.f(3) 
