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

          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 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

          LDAB is INTEGER


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

X

          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

          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

          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

          INFO is INTEGER


          = 0:  successful exit


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

Univ. of Tennessee

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Univ. of Colorado Denver

NAG Ltd.

December 2016

  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).