# dlatbs.f man page

dlatbs.f —

## Synopsis

### Functions/Subroutines

subroutine **dlatbs** (UPLO, TRANS, DIAG, NORMIN, **N**, KD, AB, LDAB, X, SCALE, CNORM, INFO)**DLATBS** solves a triangular banded system of equations.

## Function/Subroutine Documentation

### 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*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 Tennessee

Univ. of California Berkeley

Univ. of Colorado Denver

NAG Ltd.

**Date:**December 2016

**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 244 of file dlatbs.f.

## Author

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## Referenced By

The man page dlatbs(3) is an alias of dlatbs.f(3).

Sat Jun 24 2017 Version 3.7.1 LAPACK