


ZLATBS(3f)    LAPACK auxiliary routine (version 3.1)   ZLATBS(3f)


NNAAMMEE
       ZLATBS  -  one of the triangular systems   A * x = s*b, A**T * x = s*b,
       or A**H * x = s*b,

SSYYNNOOPPSSIISS
       SUBROUTINE ZLATBS( UPLO, TRANS, DIAG,  NORMIN,  N,  KD,  AB,  LDAB,  X,
                          SCALE, CNORM, INFO )

           CHARACTER      DIAG, NORMIN, TRANS, UPLO

           INTEGER        INFO, KD, LDAB, N

           DOUBLE         PRECISION SCALE

           DOUBLE         PRECISION CNORM( * )

           COMPLEX*16     AB( LDAB, * ), X( * )

PPUURRPPOOSSEE
       ZLATBS solves one of the triangular systems

       with scaling to prevent overflow, where A is an upper or lower triangu-
       lar band matrix.  Here A' 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 ZTBSV 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.


AARRGGUUMMEENNTTSS
       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**H * x = s*b  (Conjugate 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.




 LAPACK auxiliary routine (Nvoevresmiboenr 32.010)6                         1





ZLATBS(3f)    LAPACK auxiliary routine (version 3.1)   ZLATBS(3f)


       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) COMPLEX*16 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) COMPLEX*16 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,
               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.

       CNORM   (input or output) DOUBLE PRECISION array, dimension (N)

               If NORMIN = 'Y', CNORM is an input argument and  CNORM(j)  con-
               tains  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

FFUURRTTHHEERR DDEETTAAIILLSS
       A rough bound on x is computed; if that is less than overflow, ZTBSV 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)



 LAPACK auxiliary routine (Nvoevresmiboenr 32.010)6                         2





ZLATBS(3f)    LAPACK auxiliary routine (version 3.1)   ZLATBS(3f)


                 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 col-
       umn 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  ZTBSV  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 column-
       wise method can be performed without fear of overflow.  If the computed
       bound  is  greater  than a large constant, x is scaled to prevent over-
       flow, 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  or A**H *x =
       b.  The basic algorithm for A upper triangular is

            for j = 1, ..., n
                 x(j) := ( b(j) - A[1:j-1,j]' * x[1:j-1] ) / A(j,j)
            end

       We simultaneously compute two bounds
            G(j) = bound on ( b(i) - A[1:i-1,i]' * 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 ZTBSV if 1/M(n) and 1/G(n) are both greater than



 LAPACK auxiliary routine (Nvoevresmiboenr 32.010)6                         3





ZLATBS(3f)    LAPACK auxiliary routine (version 3.1)   ZLATBS(3f)


       max(underflow, 1/overflow).
























































 LAPACK auxiliary routine (Nvoevresmiboenr 32.010)6                         4


