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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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LAPACK 3.12.1
LAPACK: Linear Algebra PACKage
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| subroutine dlatbs | ( | character | uplo, |
| character | trans, | ||
| character | diag, | ||
| character | normin, | ||
| integer | n, | ||
| integer | kd, | ||
| double precision, dimension( ldab, * ) | ab, | ||
| integer | ldab, | ||
| double precision, dimension( * ) | x, | ||
| double precision | scale, | ||
| double precision, dimension( * ) | cnorm, | ||
| integer | info ) |
DLATBS solves a triangular banded system of equations.
Download DLATBS + dependencies [TGZ] [ZIP] [TXT]
!> !> DLATBS 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 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. !>
| [in] | UPLO | !> UPLO is CHARACTER*1 !> Specifies whether the matrix A is upper or lower triangular. !> = 'U': Upper triangular !> = 'L': Lower triangular !> |
| [in] | 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) !> |
| [in] | DIAG | !> DIAG is CHARACTER*1 !> Specifies whether or not the matrix A is unit triangular. !> = 'N': Non-unit triangular !> = 'U': Unit triangular !> |
| [in] | 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. !> |
| [in] | N | !> N is INTEGER !> The order of the matrix A. N >= 0. !> |
| [in] | KD | !> KD is INTEGER !> The number of subdiagonals or superdiagonals in the !> triangular matrix A. KD >= 0. !> |
| [in] | AB | !> AB is 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 = '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). !> |
| [in] | LDAB | !> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !> |
| [in,out] | X | !> X is 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. !> |
| [out] | SCALE | !> SCALE is DOUBLE PRECISION !> 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. !> |
| [in,out] | CNORM | !> CNORM is DOUBLE PRECISION array, dimension (N) !> !> If NORMIN = 'Y', 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 = '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 argument and CNORM(j) !> returns the 1-norm of the offdiagonal part of the j-th column !> of A. !> |
| [out] | INFO | !> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !> |
!>
!> 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 A**T*x = b. The basic
!> algorithm for A upper triangular is
!>
!> for j = 1, ..., n
!> x(j) := ( b(j) - A[1:j-1,j]**T * x[1:j-1] ) / A(j,j)
!> end
!>
!> We simultaneously compute two bounds
!> G(j) = bound on ( b(i) - A[1:i-1,i]**T * 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).
!> Definition at line 238 of file dlatbs.f.
1.11.0