LAPACK
3.4.2
LAPACK: Linear Algebra PACKage

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Functions/Subroutines  
subroutine  clatbs (UPLO, TRANS, DIAG, NORMIN, N, KD, AB, LDAB, X, SCALE, CNORM, INFO) 
CLATBS solves a triangular banded system of equations. 
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.
Download CLATBS + dependencies [TGZ] [ZIP] [TXT]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 nelement 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 nontrivial 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**H * x = s*b (Conjugate transpose) 
[in]  DIAG  DIAG is CHARACTER*1 Specifies whether or not the matrix A is unit triangular. = 'N': Nonunit 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 COMPLEX array, dimension (LDAB,N) The upper or lower triangular band matrix A, stored in the first KD+1 rows of the array. The jth column of A is stored in the jth column of the array AB as follows: if UPLO = 'U', AB(kd+1+ij,j) = A(i,j) for max(1,jkd)<=i<=j; if UPLO = 'L', AB(1+ij,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 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. 
[out]  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. 
[in,out]  CNORM  CNORM is REAL array, dimension (N) If NORMIN = 'Y', CNORM is an input argument and CNORM(j) contains the norm of the offdiagonal part of the jth column of A. If TRANS = 'N', CNORM(j) must be greater than or equal to the infinitynorm, and if TRANS = 'T' or 'C', CNORM(j) must be greater than or equal to the 1norm. If NORMIN = 'N', CNORM is an output argument and CNORM(j) returns the 1norm of the offdiagonal part of the jth column of A. 
[out]  INFO  INFO is INTEGER = 0: successful exit < 0: if INFO = k, the kth argument had an illegal value 
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 dividebyzero 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 infinitynorm 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 nontrivial solution to A*x = 0 is found. Similarly, a rowwise 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:j1,j]' * x[1:j1] ) / A(j,j) end We simultaneously compute two bounds G(j) = bound on ( b(i)  A[1:i1,i]' * x[1:i1] ), 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(j1) and M(j) >= M(j1) for j >= 1. Then the bound on x(j) is M(j) <= M(j1) * ( 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 243 of file clatbs.f.