# slatbs (l) - Linux Man Pages

## NAME

SLATBS - solves one of the triangular systems A *x = s*b or Aaq*x = s*b with scaling to prevent overflow, where A is an upper or lower triangular band matrix

## SYNOPSIS

SUBROUTINE SLATBS(
UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO )

CHARACTER DIAG, NORMIN, TRANS, UPLO

INTEGER INFO, KD, LDAB, N

REAL SCALE

REAL AB( LDAB, * ), CNORM( * ), X( * )

## PURPOSE

SLATBS 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 STBSV 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. = aqUaq: Upper triangular
= aqLaq: Lower triangular
TRANS (input) CHARACTER*1
Specifies the operation applied to A. = aqNaq: Solve A * x = s*b (No transpose)
= aqTaq: Solve Aaq* x = s*b (Transpose)
= aqCaq: Solve Aaq* x = s*b (Conjugate transpose = Transpose)
DIAG (input) CHARACTER*1
Specifies whether or not the matrix A is unit triangular. = aqNaq: Non-unit triangular
= aqUaq: Unit triangular
NORMIN (input) CHARACTER*1
Specifies whether CNORM has been set or not. = aqYaq: CNORM contains the column norms on entry
= aqNaq: 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) REAL 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 = aqUaq, AB(kd+1+i-j,j) = A(i,j) for max(1,j-kd)<=i<=j; if UPLO = aqLaq, 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) 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 (output) REAL
The scaling factor s for the triangular system A * x = s*b or Aaq* 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) REAL array, dimension (N)
If NORMIN = aqYaq, 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 = aqNaq, CNORM(j) must be greater than or equal to the infinity-norm, and if TRANS = aqTaq or aqCaq, CNORM(j) must be greater than or equal to the 1-norm. If NORMIN = aqNaq, 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, STBSV 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) 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 CNORM(i) | A(i,i) )

1<=i<=j
and

|x(j)| <= G(0) |A(j,j)| product 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 non-trivial solution to A*x = 0 is found. Similarly, a row-wise scheme is used to solve Aaq*x = b. The basic algorithm for A upper triangular is

for j 1, ..., n

x(j) := b(j) - A[1:j-1,j]aq x[1:j-1] A(j,j)
end
We simultaneously compute two bounds

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
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) CNORM(j) | A(j,j) |

<= M(0) product 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).