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slatbs.f(3)				      LAPACK				      slatbs.f(3)

       slatbs.f -

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

Function/Subroutine Documentation
   subroutine slatbs (characterUPLO, characterTRANS, characterDIAG, characterNORMIN, integerN,
       integerKD, real, dimension( ldab, * )AB, integerLDAB, real, dimension( * )X, realSCALE,
       real, dimension( * )CNORM, integerINFO)
       SLATBS solves a triangular banded system of equations.


	    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.


		     UPLO is CHARACTER*1
		     Specifies whether the matrix A is upper or lower triangular.
		     = 'U':  Upper triangular
		     = 'L':  Lower triangular


		     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 is CHARACTER*1
		     Specifies whether or not the matrix A is unit triangular.
		     = 'N':  Non-unit triangular
		     = 'U':  Unit triangular


		     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 is INTEGER
		     The order of the matrix A.  N >= 0.


		     KD is INTEGER
		     The number of subdiagonals or superdiagonals in the
		     triangular matrix A.  KD >= 0.


		     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 is INTEGER
		     The leading dimension of the array AB.  LDAB >= KD+1.


		     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 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 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 is INTEGER
		     = 0:  successful exit
		     < 0:  if INFO = -k, the k-th argument had an illegal value

	   Univ. of Tennessee

	   Univ. of California Berkeley

	   Univ. of Colorado Denver

	   NAG Ltd.

	   September 2012

       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]

	     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) | )

		|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)

	     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)| )

	     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 242 of file slatbs.f.

       Generated automatically by Doxygen for LAPACK from the source code.

Version 3.4.2				 Tue Sep 25 2012			      slatbs.f(3)
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