--- rpl/lapack/lapack/zlatbs.f 2010/12/21 13:53:52 1.7 +++ rpl/lapack/lapack/zlatbs.f 2018/05/29 06:55:25 1.19 @@ -1,10 +1,252 @@ +*> \brief \b ZLATBS solves a triangular banded system of equations. +* +* =========== DOCUMENTATION =========== +* +* Online html documentation available at +* http://www.netlib.org/lapack/explore-html/ +* +*> \htmlonly +*> Download ZLATBS + dependencies +*> +*> [TGZ] +*> +*> [ZIP] +*> +*> [TXT] +*> \endhtmlonly +* +* Definition: +* =========== +* +* SUBROUTINE ZLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, +* SCALE, CNORM, INFO ) +* +* .. Scalar Arguments .. +* CHARACTER DIAG, NORMIN, TRANS, UPLO +* INTEGER INFO, KD, LDAB, N +* DOUBLE PRECISION SCALE +* .. +* .. Array Arguments .. +* DOUBLE PRECISION CNORM( * ) +* COMPLEX*16 AB( LDAB, * ), X( * ) +* .. +* +* +*> \par Purpose: +* ============= +*> +*> \verbatim +*> +*> 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. +*> \endverbatim +* +* Arguments: +* ========== +* +*> \param[in] UPLO +*> \verbatim +*> UPLO is CHARACTER*1 +*> Specifies whether the matrix A is upper or lower triangular. +*> = 'U': Upper triangular +*> = 'L': Lower triangular +*> \endverbatim +*> +*> \param[in] TRANS +*> \verbatim +*> 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) +*> \endverbatim +*> +*> \param[in] DIAG +*> \verbatim +*> DIAG is CHARACTER*1 +*> Specifies whether or not the matrix A is unit triangular. +*> = 'N': Non-unit triangular +*> = 'U': Unit triangular +*> \endverbatim +*> +*> \param[in] NORMIN +*> \verbatim +*> 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. +*> \endverbatim +*> +*> \param[in] N +*> \verbatim +*> N is INTEGER +*> The order of the matrix A. N >= 0. +*> \endverbatim +*> +*> \param[in] KD +*> \verbatim +*> KD is INTEGER +*> The number of subdiagonals or superdiagonals in the +*> triangular matrix A. KD >= 0. +*> \endverbatim +*> +*> \param[in] AB +*> \verbatim +*> 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). +*> \endverbatim +*> +*> \param[in] LDAB +*> \verbatim +*> LDAB is INTEGER +*> The leading dimension of the array AB. LDAB >= KD+1. +*> \endverbatim +*> +*> \param[in,out] X +*> \verbatim +*> 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. +*> \endverbatim +*> +*> \param[out] SCALE +*> \verbatim +*> 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. +*> \endverbatim +*> +*> \param[in,out] CNORM +*> \verbatim +*> 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. +*> \endverbatim +*> +*> \param[out] INFO +*> \verbatim +*> INFO is INTEGER +*> = 0: successful exit +*> < 0: if INFO = -k, the k-th argument had an illegal value +*> \endverbatim +* +* Authors: +* ======== +* +*> \author Univ. of Tennessee +*> \author Univ. of California Berkeley +*> \author Univ. of Colorado Denver +*> \author NAG Ltd. +* +*> \date November 2017 +* +*> \ingroup complex16OTHERauxiliary +* +*> \par Further Details: +* ===================== +*> +*> \verbatim +*> +*> 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). +*> \endverbatim +*> +* ===================================================================== SUBROUTINE ZLATBS( UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, $ SCALE, CNORM, INFO ) * -* -- LAPACK auxiliary routine (version 3.2) -- +* -- LAPACK auxiliary routine (version 3.8.0) -- * -- LAPACK is a software package provided by Univ. of Tennessee, -- * -- Univ. of California Berkeley, Univ. of Colorado Denver and NAG Ltd..-- -* November 2006 +* November 2017 * * .. Scalar Arguments .. CHARACTER DIAG, NORMIN, TRANS, UPLO @@ -16,159 +258,6 @@ COMPLEX*16 AB( LDAB, * ), X( * ) * .. * -* 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' 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. -* -* Arguments -* ========= -* -* UPLO (input) CHARACTER*1 -* Specifies whether the matrix A is upper or lower triangular. -* = 'U': Upper triangular -* = 'L': Lower triangular -* -* TRANS (input) 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 (input) CHARACTER*1 -* Specifies whether or not the matrix A is unit triangular. -* = 'N': Non-unit triangular -* = 'U': Unit triangular -* -* NORMIN (input) 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 (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) 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 (input) INTEGER -* The leading dimension of the array AB. LDAB >= KD+1. -* -* X (input/output) 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 (output) 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 (input or output) 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 (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, 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). -* * ===================================================================== * * .. Parameters .. @@ -192,7 +281,7 @@ $ ZDOTU, ZLADIV * .. * .. External Subroutines .. - EXTERNAL DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTBSV + EXTERNAL DSCAL, XERBLA, ZAXPY, ZDSCAL, ZTBSV, DLABAD * .. * .. Intrinsic Functions .. INTRINSIC ABS, DBLE, DCMPLX, DCONJG, DIMAG, MAX, MIN