.TH "SRC/slatbs.f" 3 "Version 3.12.0" "LAPACK" \" -*- nroff -*-
.ad l
.nh
.SH NAME
SRC/slatbs.f
.SH SYNOPSIS
.br
.PP
.SS "Functions/Subroutines"

.in +1c
.ti -1c
.RI "subroutine \fBslatbs\fP (uplo, trans, diag, normin, n, kd, ab, ldab, x, scale, cnorm, info)"
.br
.RI "\fBSLATBS\fP solves a triangular banded system of equations\&. "
.in -1c
.SH "Function/Subroutine Documentation"
.PP 
.SS "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)"

.PP
\fBSLATBS\fP solves a triangular banded system of equations\&.  
.PP
\fBPurpose:\fP
.RS 4

.PP
.nf
!>
!> 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\&.
!> 
.fi
.PP
 
.RE
.PP
\fBParameters\fP
.RS 4
\fIUPLO\fP 
.PP
.nf
!>          UPLO is CHARACTER*1
!>          Specifies whether the matrix A is upper or lower triangular\&.
!>          = 'U':  Upper triangular
!>          = 'L':  Lower triangular
!> 
.fi
.PP
.br
\fITRANS\fP 
.PP
.nf
!>          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)
!> 
.fi
.PP
.br
\fIDIAG\fP 
.PP
.nf
!>          DIAG is CHARACTER*1
!>          Specifies whether or not the matrix A is unit triangular\&.
!>          = 'N':  Non-unit triangular
!>          = 'U':  Unit triangular
!> 
.fi
.PP
.br
\fINORMIN\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIN\fP 
.PP
.nf
!>          N is INTEGER
!>          The order of the matrix A\&.  N >= 0\&.
!> 
.fi
.PP
.br
\fIKD\fP 
.PP
.nf
!>          KD is INTEGER
!>          The number of subdiagonals or superdiagonals in the
!>          triangular matrix A\&.  KD >= 0\&.
!> 
.fi
.PP
.br
\fIAB\fP 
.PP
.nf
!>          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)\&.
!> 
.fi
.PP
.br
\fILDAB\fP 
.PP
.nf
!>          LDAB is INTEGER
!>          The leading dimension of the array AB\&.  LDAB >= KD+1\&.
!> 
.fi
.PP
.br
\fIX\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fISCALE\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fICNORM\fP 
.PP
.nf
!>          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\&.
!> 
.fi
.PP
.br
\fIINFO\fP 
.PP
.nf
!>          INFO is INTEGER
!>          = 0:  successful exit
!>          < 0:  if INFO = -k, the k-th argument had an illegal value
!> 
.fi
.PP
 
.RE
.PP
\fBAuthor\fP
.RS 4
Univ\&. of Tennessee 

.PP
Univ\&. of California Berkeley 

.PP
Univ\&. of Colorado Denver 

.PP
NAG Ltd\&. 
.RE
.PP
\fBFurther Details:\fP
.RS 4

.PP
.nf
!>
!>  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)\&.
!> 
.fi
.PP
 
.RE
.PP

.PP
Definition at line \fB240\fP of file \fBslatbs\&.f\fP\&.
.SH "Author"
.PP 
Generated automatically by Doxygen for LAPACK from the source code\&.