dlatbs (l) - Linux Manuals

dlatbs: solves one of the triangular systems A x = sb or Aaqx = sb with scaling to prevent overflow, where A is an upper or lower triangular band matrix

Command to display dlatbs manual in Linux: $ man l dlatbs

NAME

DLATBS - 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 DLATBS(: 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 AB( LDAB, * ), CNORM( * ), X( * )

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.

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) 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 = 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) 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.
SCALE (output) DOUBLE PRECISION: 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) DOUBLE PRECISION 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, DTBSV 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 DTBSV 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) * ( 1 + CNORM(j) ) / | A(j,j) |

    <= M(0) * product ( ( 1 + CNORM(i) ) / |A(i,i)| )
              1<=i<=j
and we can safely call DTBSV if 1/M(n) and 1/G(n) are both greater than max(underflow, 1/overflow).