Provided by: liblapack-doc_3.3.1-1_all bug


       LAPACK-3 - 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




           INTEGER        INFO, KD, LDAB, N

           DOUBLE         PRECISION SCALE

           DOUBLE         PRECISION AB( LDAB, * ), CNORM( * ), X( * )


       DLATBS solves one of the triangular systems
        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.


        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**T* x = s*b  (Conjugate transpose = 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) 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    (input) INTEGER
                The leading dimension of the array AB.  LDAB >= KD+1.

        X       (input/output) 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   (output) 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   (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


        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]
        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 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)
        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 DTBSV if 1/M(n) and 1/G(n) are both greater
        than max(underflow, 1/overflow).

 LAPACK auxiliary routine (version 3.2)     April 2011                            DLATBS(3lapack)