table of contents
latbs(3) | Library Functions Manual | latbs(3) |
NAME¶
latbs - latbs: triangular solve with scaling
SYNOPSIS¶
Functions¶
subroutine CLATBS (uplo, trans, diag, normin, n, kd, ab,
ldab, x, scale, cnorm, info)
CLATBS solves a triangular banded system of equations. subroutine
DLATBS (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm,
info)
DLATBS solves a triangular banded system of equations. subroutine
SLATBS (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm,
info)
SLATBS solves a triangular banded system of equations. subroutine
ZLATBS (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm,
info)
ZLATBS solves a triangular banded system of equations.
Detailed Description¶
Function Documentation¶
subroutine CLATBS (character uplo, character trans, character diag, character normin, integer n, integer kd, complex, dimension( ldab, * ) ab, integer ldab, complex, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)¶
CLATBS solves a triangular banded system of equations.
Purpose:
!> !> CLATBS solves one of the triangular systems !> !> A * x = s*b, A**T * x = s*b, or A**H * 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 CTBSV 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. !>
Parameters
!> 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**H * x = s*b (Conjugate transpose) !>
DIAG
!> DIAG is CHARACTER*1 !> Specifies whether or not the matrix A is unit triangular. !> = 'N': Non-unit 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 COMPLEX 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). !>
LDAB
!> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !>
X
!> X is COMPLEX 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, A**T * x = s*b, or A**H * 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 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. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
!> !> A rough bound on x is computed; if that is less than overflow, CTBSV !> 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 CTBSV 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 or !> A**H *x = b. The basic algorithm for A upper triangular is !> !> for j = 1, ..., n !> x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) !> end !> !> We simultaneously compute two bounds !> G(j) = bound on ( b(i) - A[1:i-1,i]' * 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 CTBSV if 1/M(n) and 1/G(n) are both greater !> than max(underflow, 1/overflow). !>
Definition at line 241 of file clatbs.f.
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.
Purpose:
!> !> 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. !>
Parameters
!> 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': Non-unit 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 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). !>
LDAB
!> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !>
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. !>
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. !>
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. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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 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 240 of file dlatbs.f.
subroutine SLATBS (character uplo, character trans, character diag, character normin, integer n, integer kd, real, dimension( ldab, * ) ab, integer ldab, real, dimension( * ) x, real scale, real, dimension( * ) cnorm, integer info)¶
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 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. !>
Parameters
!> 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': Non-unit 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 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). !>
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 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. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
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) ( 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 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 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 STBSV if 1/M(n) and 1/G(n) are both greater !> than max(underflow, 1/overflow). !>
Definition at line 240 of file slatbs.f.
subroutine ZLATBS (character uplo, character trans, character diag, character normin, integer n, integer kd, complex*16, dimension( ldab, * ) ab, integer ldab, complex*16, dimension( * ) x, double precision scale, double precision, dimension( * ) cnorm, integer info)¶
ZLATBS solves a triangular banded system of equations.
Purpose:
!> !> ZLATBS solves one of the triangular systems !> !> A * x = s*b, A**T * x = s*b, or A**H * 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 ZTBSV 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. !>
Parameters
!> 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**H * x = s*b (Conjugate transpose) !>
DIAG
!> DIAG is CHARACTER*1 !> Specifies whether or not the matrix A is unit triangular. !> = 'N': Non-unit 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 COMPLEX*16 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). !>
LDAB
!> LDAB is INTEGER !> The leading dimension of the array AB. LDAB >= KD+1. !>
X
!> X is COMPLEX*16 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 DOUBLE PRECISION !> The scaling factor s for the triangular system !> A * x = s*b, A**T * x = s*b, or A**H * 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 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. !>
INFO
!> INFO is INTEGER !> = 0: successful exit !> < 0: if INFO = -k, the k-th argument had an illegal value !>
Author
Univ. of California Berkeley
Univ. of Colorado Denver
NAG Ltd.
Further Details:
!> !> A rough bound on x is computed; if that is less than overflow, ZTBSV !> 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 ZTBSV 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 or !> A**H *x = b. The basic algorithm for A upper triangular is !> !> for j = 1, ..., n !> x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j) !> end !> !> We simultaneously compute two bounds !> G(j) = bound on ( b(i) - A[1:i-1,i]' * 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 ZTBSV if 1/M(n) and 1/G(n) are both greater !> than max(underflow, 1/overflow). !>
Definition at line 241 of file zlatbs.f.
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